Hoel, David G.; (1966)Properties of sequential tests when the conditions for standard approximate formulae are not satisfied."

I
I
INSTI111fE OF STATISTICI
SOX 5457
STATE COLLEGE STATION
RALElGH~NORTH CAROI:;INA
f'
I
I
I
I
I
I
I
PROPERTIES OF SEQUENTIAL TESTS lmEN
THE CONDITIONS FOR STANDARD APPROXIMATE
FORHULAE ARE NOT SATISFIED
by
David Gerhard Hoe1
University of North Carolina
Institute of Statistics Mimeo Series No. 480
June 1966
{'
I
I
I
I
I
I
f
I
This research was supported by the ~~
Research Office - Durham - Grant No.
DA-ARO-D-31-124-G670.
DEPARTMENT GF. STATISTICS
University of North Carolina
Chapel Hill, N. C
- IPROPER'.rJ:eS OF SEQtIElNTIAL 'J.lESiJ!~ ~
-~(", 'THE CONDITIONS FOR STANDARD
APPROXIMATE FORMtlLAE ARE NOT
SATISFIED
David Gerhard. Hoel
Institute of Statistics
Mimeo Series No. 480
Name
Date
I
I
ii
..
I
I
I
I
I
I
I
..
I
I
I
I
I
I
{'
I
ACKNOWLEDGMENTS
I am deeply indebted to Professor Norman L. Johnson, not only
for proposing the topic of this dissertation, but also for his
guidance and encouragement throughout its development.
It has been
an invaluable experience to have had the opportunity to work under
his direction.
I also wish to give very special thanks to Professors William J.
Hall and Wassily Hoeffding for both their helpful suggestions and
their creative teachings.
The computer time needed for the tables in chapters one and five
which was provided by the Mathematics Division of the Oak Ridge
National Laboratories is greatly appreciated.
Finally, I IDuld like to thank Mrs. Elsie Camp and Miss Martha
Jordan of the secretarial staff and especially Mrs. Judy Zeuner for
her cheerful typing.
I
I
iii
..
I
I
I
I
I
I
I
TABLE OF CONTENTS
CHAPTER
ACKNOWLEDGMENTS.
f
I
...
1i
INTRODUCTION. •• .
I
v
. ....
SEQUENTIAL SAMPLE SIZE TEST
"
1.1 Introduction..
' •. '
1.2 Properties of the sequential sample
size test (s.s.s~t~) . . .
Data••
....
II
{'
I
I
I
I
I
I
PAGE
III
2
....
10
. .
OPERATING CHARACTERISTIC FUNCTION.
. ..
3.1 Introduction. .
•
3.2 Consistency.
3.3 Bounds for the o.c. function.
3.4 Stochastic ordering •
3.5 Change of scale.
3.6 General bounds for the o.c. function.
..
.
1
1
SAMPLE SIZE.
• •
• • • •
2.1 Introduction.
. • • .
2.2 A conjectured approximation to the a.s.n.
2.3 Wald's bounds for e(N) . • . . • • • . . • •
2.4 Hoeffding's lower bound for e(N).
2.5 Wald type bounds for e (~ ) .
2.6 General bounds for e(Nk ) . . .
.
IV
"
13
13
15
21
24
28
33
43
43
. .
46
57
..
65
.
APPLICATIONS....
4.1 Introduction.
4.2 Bounds on e(N) for the s.s.s.t.
4.3 Bounds on the o.c. function for the s.s.s.tv
4.4 Multivariate normal • .
•.• • • .
78
87
92
92
93
99
108
I
I
.I
I
I
I
I
I
I
,.
I
I
I
I
I
I
f
I
iv
PAGE
CHAPTER
v
TOMLmSON PROCEDURES. • • . . . . . . • •
5.1 Introduction. . ' . . . . . . . . . • .
5,2 A simple multiple decision procedure.
BIBLIOGRAPHY. • • • • •
•
II
•
•
•
,
•
II
..
124
124
125
137
I
I
v
.e
I
I
I
I
I
I
I
.e
I
I
I
I
I
I
f
I
mTRODUCTION
Prior to 1943, sequential analysis as it is YJnown and used today
did not exist except for various sequential type procedures each designed with a specific problem in mind.
Sequential analysis actually
came into its own as a recognized sector of statistical inference
11,
,with the publication of a series of papers ( [32]
[36], [37] ) by Abraham Waldo
[33], [34], [35],
It was here that he proposed and de-
termined many of the properties of the now well-Imom sequential
probability ratio
~
(s.p.r.t.) for discriminating between two simple
hypotheses.
As a brief summarization of this procedure, suppose that we have
a sequence of random variables xl'
X2' ...
such that xl' .. " xm has
gj
a probability density function fa ,m (xl' •.. , xm) ., and we wish to
discriminate between the two simple hypotheses
Then if A and B are two constants such that 0 < B < 1 < A and
f.
~,m
= fa i,m
(~,
1.
••• , x ), the s.p.r t
m
requires that we continue
sampling as long as
17
The numbers in square brackets refer to the bibliography.
gj Here and in the following chapters, most of the results are
also valid when probability density function and the operation of integration are replaced by probability mass function and summation,
respectively•
I
I
vi
f
B <
(0.1)
f
< A
l,m
o,m
and stop sampling as soon as one of these inequalities is violated.
If
f'
sampling is terminated at trial m then H
o
1\
and
is accepted if
(0.1)
l
-!L!
<
f
-
B
o,m
> A. If we define
o,m
b
= log B
a
= log A
Z0
then
r
f
is accepted if
=
0
Z.J.
fl i
= log ....::z..;:;.
f
o,i
for i
zi
= Z.J. - Z.J.- 1
for i = 1,2, •••
= 1,2,
•.•
may be written as "continue"sampling as long as
m
(0.2)
b
<
L
zi < a
i=l
and stop sampling as soon as one of these inequalities is violated. If
sampling is terminated at trial m then H
o
m
L
i=l
m
zi:S b
and Hl is accepted i f L zi
i=l
~ a.
is accepted if
As a matter of notation,
we shall assume that the symbols a, b, zi' Zi refer to the s.p.r.t.
given by (0.2).
If we allow the constants
a and b
in (0.2) to become functions
of m (to be denoted as am' bm), we then shall ·refer to the procedure
-.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
_.
..
I
I
I
.e
I
I
I
I
I
I
I
.e
I
I
I
I
I
I
f
I
vii
as a generalized sequential probability ratio test
(g.s.p.r.t.~
One
of the roost interesting examples of such a procedure is given by
Anderson [1] (see also Weiss [39] ).
For any sequential test procedure, the distribution of the sample
size, N,
and the operating characteristic function, Pe[accept H ]'
o
will, for practical purPOses, completely describe the procedures behavior.
Because of the difficulties involved in obtaining the distri-
bution ofN,
the expected value of N,
commonly called the average
sarnplenumber (a.s.n.)along with the operating characteristic function
has been used to represent the performance of a sequential test.
This paragraph contains a brief ·list of some of the properties of
an s.p.r.t. all of which may be found in more detail in Waldls book
[38] •
If we define
1
-cx =
p[ accept Ho
H
0
J
=
P[ accept Ho
I Hl
J
f3
then Waldls fundamental inequalities
a < log
b > log L
l-a
hold quite generally.
Now, for
~he
case when zl' z2' ... are indepen-
dent and identically distributed, Waldhas given the following wellknown approximations
viii
-.
a.s,n.
(0.4)
P[ accept Ho
where t
J*
e
at
-1
at
o -ebt 0
e
0
is the nonzero solution of e[ e
o
tzil.
=
]
1.
Page [21J, and
later Kemp [18J, has improved these approximations for the case when
Zi is normally distributed.
More recently Tallis and Vagholkar [29J
attempted to give general improvements to (0.4).
Unfortunately,
however, there are errors present in their paper (e.g. equation (11)
on page 77).
Continuing with the assumption of independent and
identically distributed variables, Wa1d has also given the following
bounds.
~
=
s~p e[ zl
- r
I
zl
2:
~
=
inf e[ zl + r
r
I
zl
~ -r
=
t
[ zl 0
sup pee
p
TJ
Then
(b+~ ')
=
[ z1t
infpe e
p
.
°1
e
e
r
zlt o
zlt o
1
for a-b
>r >0
>0
*J
for p
-J
for p > 1.
< 1p
pr accept Ho 1+ a f1- pr accept Ho JJ
-
>r >0
for a-b
1
>
I
I
I
I
I
I
I
I
--
Define
0
I
I
I
I
I
I
I'
< a.sn <
-
I
..
I
I
I
ix
.e
I
I
I
I
I
I
I
,.
I
I
I
I
I
I
{'
I
o
e at -1
at
e
o
Be
bt
o_e
~
bt
0
-T')e
1 _ e
at
[
J0
o
<
0
0
In the four following chapters we shall be concerned with determining the properties of an s.p.r.t.when Wald's assumptions of
independent and identically distributed variables are relaxed.
Pre-
vious work on this problem has been mainly directed towards the class
of s.p.r.t. 's obtainable by the invariance method (see Hall, Wijs:man
and Ghosh [13] ).
For some members of this class of tests, termina-
tion with probability one, ( P[ N < 00]
=1
), and monotonicity of the
o.c. function have been established (for more details and references,
see pages 586 and 587 of [13] ).
More generally, Bhate [7] and
Bartholomew [6] have given conjectured approximations to the a.s.n.
function and the o.c. function respectively.
Both of these conjectures
will be discussed in chapters two and three.
In chapter one, we begin by considering a particular s.p.r.t. in
which the
tributed.
observa~ions
are neither independent nor identically dis-
The properties of this test are obtained exactly and thus
x
we are furnished with a model with which the various bounds, approximations and conjectures of the following chapters may be compared.
Chapter two is concerned with the sample size of an s.p.r.t.
The bounds given in (0.5) for the a.s.n. are extended in a natural
way to the case of zl' z2' •.• independent but not identically distributed.
With this assumption on the Zi's, bounds similar to (0.5)
are obtained for e(rF).
We also consider extending the applicability
of the lower bounds for the a.s.n of a general sequential test procedure given by Hoeffding [14 J, [15 J.
We then conclude by gi ving a
method of general applicability for obtaining upper and lower bounds
for e(wn).
In chapter three the o.c. function of a s.p.r.t. is studied.
We
begin by introducing the concept of "consistent conjugate densities"
in connection with applying Wald f S O.c. formula given in (0.4) to
cases in which the z. 's ,are not independent and identically distriJ.
buted.
Also the bounds for P[accept HoJ
given in (0.5) are extended
to cases in which the Zi's are independent but not necessarily identically distributed.
P[accept HoJ
Next we consider conditions under which
is increased by replacing each zi by a stochastically
larger variable.
Also some discussion is given concerning the effect
upon P[accept HoJ after the Zi's have undergone a scale change.
Finally we conclude by giving a general method for obtaining upper
and lower bounds for P[accept HoJ.
In chapter four we apply the results of chapters two and three
to the s'.p.r.t. given in chapter one.
We then conclude by attempting
to determineP[accept HoJ for the s~.p.r.t. in Which zl,z2""
multivariate normal distribution.
have a
I
I
..
I
I
I
II
I:
I
-,
I
I
I
I
I'
I
..
I
I
I
..
I
I
I
I
I
I
I
{'
I
I
I
I
I
I
t
I
xi
In the fifth and final chapter we briefly discuss a simple
sequential test procedure (not an s.p.r.t.) Whose properties may be
determined exactly.
I
I
..
I
I
I
I
I
I
I
{'
I
I
I
I
I
I
f
I
CHAPTER
I
SEQUENTIAL SAMPLE SIZE TEST
1.1 Introduction
In certain "life-testing" problems, observations xl' x2' ••• , xr
are made in time until a predetermined number r have been obtained
'from a total sample of n.
Assuming that the sampling is random, we
then have available the first
size n.
r
order statistics from a sample of
From this information an inference is made concerning the
distribution from which the observations have been taken.
In this
chapter we will be concerned with the case when the population distribution is known but the sample size is unknown.
Johnson [17] has
given the properties of the fixed sample size test based upon the
likelihood ratio for discriminating between two values of n.
also given the maximum likelihood estimate of
n.
He
has
In the following
section, the properties of the s.p.r.t. discriminating between two
values of n will be considered.
One reason for being concerned
with this problem other than for its own sake, is that although the
observations are neither independent nor identically distributed, the
properties of this s.p.r.t. can be obtained exactly without resorting
to quadrature.
This will then provide a model to which the various
bounds, approximations and conjectures that are to appear in the
following chapters may be applied.
2
1.2 Properties.2f
~
sequential sample
~ ~
(s.s.s.t.)
Suppose we have a random sample of size n from an absolutely
continuous distribution G,
with g denoting the probability den-
Let Xl' ••• , xn represent the n
obser:vations after they have been ordered such that x.. < ..• < X •
.L n
Suppose that we wish to choose between two possible values of n by
sity function associated with G.
means of an s.p.r.t. discriminating between
(1.1)
H·
o·
n = n0
where no < n l • Letting f(X l , ••• , xrl n) denote the joint density
of Xl' ••• , xr we have
n-k
f(Xl'···'~In)
n:
= (n-k)!
[ i;18(Xi ) ] [ 1 - G("k) ] for
k~l, ... ,
n
Define
n -n
1
[l-G(Xk) ]
0
for k=l, .•• , no
and
={
if n>n
o
B if n=no
A
and then it follows that Wald's s.p.r.t. of (1.1) is to continue
sampling if B < R(k) < A and to stop if either of these inequalities is violated, at Which time Ho is accepted if R(k)
~
B While
I
I
..
I
I
I
I
I
I
I
..
I
I
I
I
I
I
..
I
I
I
s-
I
I
I
I
I
I
I
..
I
I
I
I
I
I
t
I
3
H is accepted if R(k) ~A.
l
and x
no
If sampling proceeds to the n
+1 does not exist, then H is true.
o
o
+ 1 stage
Thus the procedure will
be logical as long as we reject H if x +1 is observed and accept
n
o
o
H if x +1 does not exist. In this respect there is considerable
o
n
o
freedom in defining R(no+l).
stage the
seq~ntial
It should also be noted that at each
procedure depends solely on the most recently
observed variable and the stage index.
For purposes of determing the properties of the s.s.s.t., we
shall restrict attention to the family of hypotheses with
n
< n < n l • Suppose we are able to obtain
-
0-
(1.2)
Pn(j) = P[R(j)
SB
and
qn(j) = P[R(j)
~A
and B < R(i) < A for i=l, ••• ,j-lln]
B < R(i) <A for i=l, ••• ,j-lln]
for j = 1, ••• , no; then the properties of the test are known, since
if n > n
o
=
if n = n
Pn(j) + qn(j)
n
n
P[N = j In]
=
I
1 -
0
i=l
o
Pn(i)
I
o
if j < n
-
0
0
qn(i)
if j =n + 1
0
i=l
if j > n
o
+ 1 .
4
..
Define
~
= G(~)
A(k)
=
~: (no - k)!
n ! (n
0
b.
J.
a
max
I
I
I
I
I
I
I
- k)!
[-lLJ
[..LJ
= max { 0, 1 -
=
i
1
Lr 0,
1
n1 -no
A(i)
1 -
}
1
n 1 -no
A(i)
}
and observe that since A(i+l) ~ A(i), we have 1 ~ b i ~ a i ~ 0,
b i +1 ~ b i
and
a i +1
~
ai •
Also let
..
where
c (u 1 ' ••. , u i )
={
1
0
if 0'::: u ::: ••• .::: u i ::: 1
otherwise 1
and notice that B (t) is a polynomial in t of degree at. most
i
Pn(j)
= ~A(j)[ l_Uj]~-no::: B
1. Now
and B <A(i)[ l_u ]n1 -nO <A
i
for i
= 1, ••• ,j-1In} =
p. {b j ::: u j .::: 1 and a i < u i < b i for i = 1, .•• , j-ll n} .
Since
~
is the
k~
I
I
order statistic of a sample of size
(0,1) uniform distribution, we have
n
fram a
I
I
I
I
I
I
..
I
I
I
5
..
I
I
I
I
n:
(n-j) :
=
=
..
I
('
I
B _ (b _ )
j l
j l
Similarly,
I
I
(n-j+l):
n-j+l
(1 - b )
j
and
I
I
I
I
I
I
n :
n :
(n-j) :
a.J- 1
n :
for b. 1 > a j
J- -
(n-j) :
J
b
=
I
n .
(n-j):
j-l (l_t)n- j B
(t) dt + n:
[(l-b. )n-j+l
J-l
.
j-l
( n-j+l ) :
a _
j l
- ( l-a )n-j+l ] B _ (b _ )
j
j l
j l
for b. 1 < a j .
J- -
Now since B.(t)
is a polynomial, the above expressions can easily be
~
used to obtain Pn (j )
and qn (j ) • Although there does not appear to
be a simple expression for B. (t) , it can be obtained recursively as
~
6
follows:
if b.J.
if b,
> a,J. +1
J.-
..
< a'+l
J.
-
and a.J. +1
< t -< b.J.
-
b.
J
J.
Bi(U,) du + (t-b i ) B (b )lf b i
i 1
2:
ai+land biS t •
ai +l
For computing purposes the following algorithm
mination of Bi(t).
for b _ S t
k l
S bk •
and B(l,1,2)
= 1.
(a)
simplifies the deterj l
Let B(k,i,j) be the coefficient of t -
in Bi(t}
Then, since Bl(t) = t-a , we have B(l,l,l)=-a
l
l
B(k,i+l,·)
then
=0
for k
-2
I
I
I,
I
I"
I
..
I
I
I
7
(b)
I_
if b
> a.+ l
J.
and
i -
= -1-
B(k,i+l,j)
I
I
I
I
I
I
I
..
I
I
I
I
I
I
{'
I
k 1
B(k,i,j-l)
j-l
k-i -1
I
=
B(k,i+l,l)
o
i+l
I
n=l
I.
.
P
P
+n,J.,p)(b.
+
_boo
-pB(J.
o
J . n l+n- 1)
00
p=1
i+l
\' 1
.
p
pB(k,J.,P) b k _l
p=l
i+l
P
+ \']; B(i ,i,p)(b.
Lp
0
l
p=l
,mere
L
o
is defined as the smallest
(c)
if b.
l
>
-
a.+ l
l
and
B(i+l,i+l,j) =
0
=i
k
i+2
B(i+l,i+l,l)
=
I
B(i,i+l,p)
j > 2
b~-
p=l
=
I
1 i+l
-
I
B(i,i,p)
b~
p=l
i+l
B(i+l,i+l,2)
such that b.> a. l'
J- l+
then
+1
for
j
B(i,i,p)
b~
1
p=l
The following comparison was made wi. th the fixed sample results
to be found in table 2 on page 62 of Johnson [17J,
The values of
A and B were varied so that the error probabilities agree with those
found by Johnson,
8
TABLE 1.1
en
r
en
(N)
0
10
10
10
15
15
20
15
15
20
5
5
5
5
5
5
10
10
10
15
r
15
20
25
20
25
15
20
25
25
25
20
.294
.185
.123
.360
.266
.394
.256
.134
.326
.238
is the fixed sample size and
cx,
3.23
2·98
2.78
3.32
3.13
3.38
6.34
5.79
6.35
9.82
~
(N)
1
3.73
3.81
3.83
3.68
3.75
3.65
7.33
7.43
7.08
1l.18
are the type I, II errors,
respectively.
The s.s.s.t. has the interesting feature that although the sampling rule depends upon the distribution G,
test do not.
section
the properties of the
Additional tables of these properties are given in
1.3. One additional feature of the test which is necessary
for future reference will be given before proceeding.
define u
o
That is, if we
= 0, then
Zk
=
log R(k) - log R(k-l)
=
= log ~-k+1 + (n1 - no)[lOg (l-~)-lOg(l-~_tJ
n -k+l
o
for k = 1, ••• , n
o
I
I
..
I
I
I
I
I
I
I
..
I
I
I
I
I
I
..
I
I
,I
..
t
I
I
I
I
I
9
If we set
1 - u.
v.~
I
I
I
I
I
I
I'
I
= 1,
for i
••• , n
,
1 - u. 1
~-
then the joint density of (V ' ••• , Vn )
l
n
n
II
i=l
n-i
vi
for
0
is
-< v.
<
~-
= 1,
1 and i
.,., n
Hence, vl ' ••• , vn are independent and the density of vi is
(n - i + 1) v~-i
~
for
0
oS vi oS
1
Since zk may be written as
I
i'
~
=
log v.
log
~
n -i+l
,
o
we have zl'
... , zno are independent and the distribution function
F i of zi is
for z. < a.
~-
for
where
Z
i
>a
~
i
Therefore, by redefining R(no+l), we have a sequential test with
zl' z2'··· independent while xl'
~,
••• are dependent.
I
I
..
11
TABLE
n1=15
n0 =10
1.5
B=.14918
a.s.n.
A=9·304
a.s .n.
n
o.c.
:13
14
15
.188
.098
.050
n
o.c.
t
I
10
11
12
.950
.568
.342
I
n0 =10
a
n
o.c.
a.s.n.
n
o.c.
a.s.n.
10
11
I
J2
·950
.837
.704
.358
.421
.307
5.126
5.8:13
6.442
6.946
7.291
7.478
16
17
18
19
20
.218
.152
.105
.073
.050
7.532
7.487
7.376
7.226
7.056
I
I
..
I
I
I
I
I
I
f
I
7.569
8.489
9.070
TABLE
13
14
15
n =20
1
TABLE
n0 =10
1.6
B =.14845
9·305
9.286
9·121
A=11.518
1.7
n 1 =25.
B=.16529
A=11.735
n
c.c.
a.s.n.
n
c.c.
a.s.n.
10
11
12
:13
14
15
16
17
.950
.901
.831
.742
.643
.541
.445
.359
3.829
4.286
4.741
5.165
5.531
5.820
6.026
6.152
18
19
20
21
22
23
24
25
.286
.225
.176
.137
.107
.083
.064
.050
6.209
6.209
6.166
6.092
5.996
5.888
5.774
5.658
TABLE
n =20
n0 =15
1
1.8
B=.:I3533
A=9·9515
n
c.c.
a.s.n.
n
c.c.
a.s .n.
15
16
17
.950
.584
.354
11.779
12.891
13.606
18
19
20
.194
.100
.050
13.894
13.855
13.618
------
TABLE
na =15
n
15
16
17
18
19
20
n =20
a
n1=2.5
a..s.n.
a.c.
8.097
·950
.852
8.990
9.819
.727
10.497
.583
.443
10.970
.324
11.230
n =25
1
n
a.c.
20
21
22
·950
.593
.361
I
I
1.9
B=.12595
n
21
22
23
24
25
TABLE 1.10
B=.12835
a..s.n.
16.210
17.461
18.274
n.
23
24
25
A=12.4578
a.c.
.229
.159
.109
.074
.050
a..s.n.
11.303
11.229
11.053
10.806
10.496
A=10.3237
a.c.
.198
.101
.050
a..s.n.
18.618
18.686
19·002
..
I
I
I
I
I
I
I
..
I
I
I
I
I
I
..
I
I
I
..
t
I
I
I
I
I
I
..
I
I
I
I
I
I
,
I
CHAPI'ER
II
SAMPLE SIZE
2-C'1.~
Ii:l.trDduction
In this chapter we will consider extending the usual bounds for
teN) of a s'equential test procedure for discriminating between two
simple hypotheses tJ problems in which the assumption of zl' z2' ...
being independent and identically distributed has been relaxed. Also
we shall look at bounds for the higher mOm:!nts of N.
In section 2.2,
there will be same discussion concerning a conjectured approximation
to
e (N) •
Before proceeding, however, we shall need three forms of
Wald I S equation which have been proved in the 11terature.
They are
given as follows:
Lemma. 2.1 Let Xl'
~,
... be a sequence of random variables,
• •• a sequence of constants and
upon
xl'~'
••• •
N
JJ
~l' ~2'
a stopping variable depending
If the conditions
(a)
I~i I =s
(b)
e IXi I <
(c)
e (Xi +l - ~i+ll Xl"'"
(d)
e
(N)
C
< eo
co
xi)=o and e(xl - ~l)
=0
< eo
N
(e) e(
II
X
j
I) <
eo
j=l
A stopping variable is defined to be a random variable (N)
wi th positiva integer values such that the event N=n depends only on
j)
~,
... , xn •
14
are satisfied for all i, then
N
I
e(
..
eo
Xi)
I
=
i=l
Ili P [ N ~ i ]
i=l
By lemma 2 of Chow, Robbins and Teicher [10], we have
Proof:
N
e(
I
(Xi - Ili ) ) = o.
Therefore,
i=l
N
N
e(
I
Xi ) =
e
i=l
Corollary 2.1.1
eo
I
<I
Ili P [N
Ili ) =
i=l
i=l
Let Xl'
~,
e Ixii
~ C
(b)
e(N) <
co
]
••• be a sequence of independent randan
variabies and N a stopping variable.
(a)
~i
Then, i f
< eo
e (L
\' Jx..
eo
)
i=l
Proof:
=L
\'
e{x.) P [ N > i ]
J.
-
i=l
The result follows from either lemma 2.1 or the proof of
theorem 1 by Johnson [1.6].
Lemma 2.2
Let Xl'
~,
••• be a sequence of independent random. varia-
bles and N a stopping variable.
(a)
(b)
(c)
e Xi = 0
2
e (x.)
< C < eo
J. e (N) < 10
I
I
I
I
I
I
I
are satisfied for all i, it follows that
N
I
I
If the following conditions,
..
I
I
I
I
I
I
..
I
I
I
..
t
I
I
I,
I
I
I'
..
I
I
I
I
I
I
,
I
15
are satisfied for all i, then
N
\' x. ) = 0
e ( LJ.
i=l
and
(I
eo
N
e
Xi )2
=
i=l
Proof:
I
e (x~
) P [ N
~i
]
i=l
This result is a special case of theorem 2 by Chow, Robbins
and Teicher [10].
2.2 A conjectured approximation to the a.s.n.
Bhate [7] has conjectured that for an s.p.r.t., e(N) may be obtained approximately as the solution of
e(Zm)
(2.1)
=
Lb
+ (1 -
L) a
for m, where L is the probability of accepting H. For the case where
o
Z is a sum of independent and identically distributed random varn
iables, (2.1) is simply Wald f s form.ul~ for approximating e(N).
In
the more general situation it appears to be a natural extension of
Wald's formula.
In some cases of interest, for example, the sequential
t
test,
the calculation of the left hand side of (2.1) is quite laborious.
So, Ray [22] has proposed a further conjecture.
He suggests that
the left hand side of (2.1) be replaced by log J, [e(t )]
m
log J, (tN)
= ~.
where
I
I
16
By way of illustration, both approximations will be applied to
From section 1.2 we have for n > n
l -
the s.s.s.t. of chapter 1.
~
n
0
>k>l
..
I
Since '1t is the
k~
order statistic of a sample of size
n
from a
[0,1] uniform distribution, we have
en [log (1-'1t) ]
(2.3)
J
=
~
n
log (l-u) ----.;-..;.-uk-l( l-u )n-k dUe
(k-l)!(n-k)~
o
Define
J
I
I
(l_X)n-k x k - l log (l-x) dx
o
and by integrating by parts we obtain for k
I(k)
=
(k-l): (n-k)
+
n : (n-k+l)
=
k-l
..
>1
I(k-l)
n-k+l
c(k) + d(k) I(k-l),
let us say.
Now since
1
I(l)
= -
~
J
(l_X)n-l log(l-x)dx =
o
we have for k
~
J
xe-
0
1
k-l
k
I(k) = c(k) +
c(i) o_TI
d(j)
1
J-i+l
J.=
I
o
I
I
1
I (k) = -
I
I
nx
dx
= 12
n
= c(l)
I
I
I
I
I
I
,
I
I
17
I
..
=
I
I
I
,I
I
I
=
(k-l): (n-k)!
n ! (n-k+l)
i=l
k
(k-l) ~ (n-k): \'
1
L n-i+l
n :
i=l
Therefore, by (2.2) and (2.3), en(~) is equal to
k
, (2.4)
log A(k) + (nl - no)
I
1
n-i+l
i=l
For Ray's approximation we replace en[~] by
I
,
'Which is equal to
I
I
I
I
I
I
,
I
Now by equating (2.4)
values of L b
and
(2.5)
+ ( 1 - L ) a
conjectured values of en(N) •
to the previous~ determined'
and solving for
k,
we arrive at the
I
I
18
.,
TABLE 2.1
n
o
=
10
n1
= 25
C¥=
en(N)
n
10
11
13
14
15
16
17
18
19
20
21
22
23
24
25
= .05
Bhate
Ray
2.41
2.69*
2.83*
2.65*
1.71*
10.17
9.43
8.01
7.20
6.72
6.36
6.09
5.86
5.65
5.46
5.29
3.83
4.29
4.74
5.17
5.53
5.82
6.03
6.15
6•.21
. 6.21
6.17
6.09
6.00
5.89
5.77
5.66
12
13
,I
2.67
3.02*
3.23*
3.13*
I
I
-2.19*
9.72 -
I,
8.05
7.06
6.55
6.26
6.04
5.82
5.63
5.46
5.30
5.15
I
I
I
Note: For those values Which are followed by an asterisk, there are
two solutions; the one which is given and another which falls between 10 and 11.
..
I
I
I
TABLE 2.2
0: =
13 = .05
n=n
n=n0
n
0
5
5
5
10
10
10
15
15
20
n
1
10
15
20
15
20
25
20
25
25
en(N)
3.76
2.64
2.11
7.57
5.13
3.83
11.78
8.10
16.21
1
Bhate
3.04
1.39
0.77
7.32
4.06
2.41
11.97
7.37
16.75
en(N)
4.84
3.86
3.21
9.12
7.06
5.66
13.62
10.50
19·00
Bhate
4.68
3.44
2.77
9·22
6.89
5.29
13.94
10.67
18.76
I
"
I
\
I
I
19
I
..
I
I
I
II
I
,
..
I
I
I
•I·
I
,
I
Table 2.1 is typical of the cases which were considered.
Fran
the values it appears that the approximation cannot be relied on for
all n.
Siskind [25]
considered both of these approximations in
cOlUlection with the two-sided sequential
t
test.
He found the
approximation to be adequate except midway between his hypothesized
values.
Bhate's conjecture consists of two approximations. The first i8
the neglect of overshoot which is usually considered reasonable.
general, however, when
large.
teN)
is small, the overshoot might be quite
From table 2.2, we observe that the approximation is good
for the larger values' of
values.
In
teN)
but poor for several of the small
ThUS, for these small values, we might be justified in ex-
plaining away part of the inaccuracy of Bhate' s approximation by his
neglect
of overshoot •
The second approximation in Bhate's conjecture consists of
setting e(N)
= m where
m is the solution of e(1n)
= e(~).
Suppose
we restrict our attention to the case when zl' z2' •.• are independent and the conditions of corollary 2.1.1 are sati sfied (this is,
in effect, the case for the s.s.s.t., since zl' •.• , z
are inde.
no
pendent). ~us, by corollary 2.1, Bh~te's second approximation
consists of finding an m such that
[N~i]=S
(2.6)
m+l
I
e.(Zi)
,
i=l
then approximating
teN)
between m and m+1.
by a value, which is found by interpolation,
Hopefully it will then be the case that
I
I
20
.,
10
I
m<
(2.7)
P [ N 2: i ]
~m+ 1
•
i=l
I
I
I
If (2.7) holds approximately, then we may conclude that the second
approximation is reasonable.
n = 10 and ex = f3 = .05,
1
I
Now in the s.s.s.t. with n=n =5,
o
we have
10
=
P [ N 2: i ]
3.76
i=l
Thus by (2.7),
m = 3.
I
I
I
Now, m = 3 implies, by (2.6), that
I
00
- 1.432
.s
e(zi) peN
2:
i]
.~
-2.679;
i=l
but
I'
10
I
e(zi) P[N
~ i]
..
= .1612 e(z6) - 3.114
i=l
where
I,
I
I
Therefore in general, (2.6) does not necessarily imply (2.7).
It can also be shown for an s.p.r.t. that
~ (Zi)
1
2:
for all
i
eH
(z.)
o
~
-< 0
and
0, but in general it is not necessarily true that e(Zi) ~
n
or e(z.) <
~
-
0
for all
i.
either increasing or decreasing in n.
ThusE e(z.)
. 1
~=
~
need not be
Hence there is no assurance
that a unique m which satisfies (2.6) exists (this is illustrated
in table 2.1 by n = 11, ••. , 14).
0
"
..
I
I,
I
I
21
I
,I
If u >e(z.) > t >
-
m
m t.=s
I
,
I
J.-
t
I
0,
I
I
e (Zi) .=s e (Zi) P [N
i=l
i=l
~ i]
.=s
eo
eo
peN
2:
i] .=s
I
e(Zi) peN > i]
i=l
 i]
i=l
10
~
m
 u > e(z.) > t.
].=s
(m+1)
i=l
and a similar expression for
"
-
J.-
Therefore we may
conclude that the second approximation in Bhate's conjecture will be
reasonable, in the case of independence, if there is little relative
variation among the values of e (zi) and they are' bounded away from
zero.
In general, however, it is clear that Bhate's conjecture may
often lead to fallacious results when applied to cases other than
I
I
I
I
I
f
I
that
Hence we have
I
I
(2.6)
eo
i=l
t
..
it then follows from
,
(2.8)
a(l-L) + (b+g ) L .=se(~) .=s (a+~) (l-L) + bL
22
I
I
where
=P
L
[ accept H ]
o
~ = sup
i
sup t [zi -r Iz. > r]
r
I
~ = i~f inf
J.
for a-b > r >
0
J. -
t [zi+r Izi S -r] for a-b > r > 0
•
r
By corollary 2.1.1 we have
.,
,
.,
I
I
,
where
for all
i.
Now combining (2.8) and (2.9) we arrive at the following extension
I
of Walds bounds given in the introduction:
(2.10)
As long as
J
teN)
2:
~
teN)
2:
t [(a+~)
teN)
S ~
teN)
S
>
0
[a(l-L) +
(b+~')LJ
(l-L) + b LJ
[ a(l-L) +
(b+~')
LJ
t [(a+~) (l-L) + b LJ
or
for u >
0
for u <
0
for J >
0
I
•
u < 0, we will have both an upper and lower
bound for teN) from (2.10).
..
I
I
I
0
for J <
I
I
However, in some instances the lower
,
I
..
I
I
I
e
I
I
,
I
23
bound may be trivial.
One could also consider using approximative
bounds by neglecting overshoot through setting
..
I
I
I
I
I
I
I'
I
in (2.10).
~
u
= s~p
e(zi)'
such that P[N ~ n]
But if there exists -an n
=0
~
and e(z.) is increasing or decreasing in
~
i, (2.9) may be improved as
follows:
By induction we can show
,
I
0
In equation (2.9), one will typically set .t = i~ e(zi) and
J
I
s = s' =
- Cj)(d. - d.)
J
~
So, if c
is increasing and d
i
m
m
1 (\' c.
m
L
is decreasing, then
i
~
)(I
i=l
>
d )
i
i=l
Therefore, if P [N ~ n ]
=0
and if e(zi) is increasing, it would
imply that
11
e(~) S
(* I
e(Zi) )
e(N)
i=l
Similarly, if e(z.) is decreasing theu
~
11
e(~)
>
(* I
i=l
e(Zi) ) teN)
I
I
24
2.4
HOeffding's lower bound for
e(N)
In two papers, Hoeffding ( [14], [15] ) has derived lower bounds
for the expected sample size of a sequential test procedure.
In each
case the assumption of independent and identically distributed observations has been made.
This assumption may be relaxed somewhat by
the use of the versions of Wald's equation given in section 2.1.
Let Xl'
~,
••. be a sequence of random variables 'With Xl' x ' •..,
2
(x) where i
X having a joint probability density function f
O,n -n
n
belongs to some .interval
n.
Let
S be a sequential test for de-
.,
I
,
I
I
t
I
I
where
PeeS
fine
o and tAJ. are disjoint, 0: >
W
accepts Bo } +
Pees
0
and
~
> o. Assume that
accepts Bl ) • 1 for
N to be the sample s1 ze of the test
e6
S.
0 and ex + ~ ~ 1.
Also define the
following:. variables
j = 0,1
- log
and
j = 0,1 •
f I, i-l (!..i.-l)
i j ,i-l (~-l)
j
= 0,1
De-
..
S·
I
I
,~
,:
II
..
I
I
25
Now Wald has demonstrated that
I·
I
,
"-
I
I
where L(9)
=i
[5 accepts H ].
o
Therefore, we may apply Hoeffding's
[14] result to obtain
I
I
I
..
I
I
Next we ass'UIle that the sequence of randan va.rl. ables Zjl' Zj2' •..
satisfies the conditions of corollary 2.1.1, or,more generally, lelll1m
2.1.
Then by defining
Since ee(Zji)
2:
,
t
= s~p
e,(zji) it follows that
j
= 0,1.
we may apply (2.11) and (2.12) to obtain
0,
c
l-c
c l-c
-log{o: (1-13)
+ (1-0:) f3
)
- 0<C<l
c eo + (l-c-) e
l
e,eN) > sup
inf
J.1..
'.E W.
~
I
j
-<
,
,
I
J.1
~
Hoeffding [14] has also shown that
~
0:
log
0:
I:'i3
+ (1-0:) log
(2.14)
!!l
1-0:
13
for
(} E W
o
I
26
,I
and
~ lO~.~
(2.15 )
(l-~)
+
e
log
¥
for'
€ W
1
o
For Hoeffding' s [15] second bound, let Xl' ~, •.• be a sequence of independent random variables and let the probability density
function of Xi
be
f i (with respect to a
cr-finite measure ~).
Con-
sidera nonrandamized test; B , such that the probability of making
a wrong decision is less than or equal to ai' when f j = f ij for each
j (i = 1,2).
by B.
for all
Also let N denote the number of observations required
Assume that eo (N) <
eo
j ) and suppose that a
notation and proof,
(the zero Subscript denotes f j = f oj
l
+a
2
<
Now following Hoeffding's
1.
define
2
j
= max
=
T
J(
I
I
a
I
I
I
I
i
~j
.,
..
= 1,2
t
I
[s~, ~~
-
log {2j)
f
1j
1
~j
+ ~2 )
j
2
f oj d
~
"
I
I
and assume that
T
2
= sup
j
Hoeffding's proof we have
2
T.
J
and
~
= sup
j
~j
are finite.
From
I
..
I
I
I
27
,e
..-
I
I
where
n
Zin
=
L
(log f oj
j=l
~ t~
)
i
= 1,2
•
f ij
t
I
Since
t
is finite we have, by corollary 2.1.1, eo(ziN)
-r;2 is finite we have by lermna. 2.2
I
eo
II
I
..
(I
N
i=l
2
i )
Y
I e(~)
eo
=
P [ N
Therefore
to[max (ZlN'~) ] = ~ eo (ZlN
I
I,
,
I
i ]' .
i=1
t
I
I
I
2:
Thus we arrive at Hoeffding' s lower bound,
(2.16)
+ Z2N)
+
= o.
Since
28
2.5
Wald type bounds for
e(~)
Part of the attractiveness of a s,.:p•.r.t. is that often e(N) is
smaller than the sample size required for an equivalent fixed sample
size procedure.
However, if Var(N) is large oompared with e(N), much
I
I
.,
I
of this attractiveness is lost making it desirable to resort to
truncated procedures.
Therefore, before proceeding wi th an s. p. r. t.
procedure, one should consider the value of e(~).
We shall now obtain bounds on e[~] by employing the same technique which Wald used for obtaining his bounds on e(N).
~.)2
= ff~~ <
~
- C<
and e(z. ~
10
where e(zi)' =
must restrict our attention to two cases, J
J
<
~.

0
for all
and u <
Now we
1.
0
10
where
By these assumptions and lemma 2.2, we have
N
e (
~i
I
::i
To begin, assume that zl' z2' ••• are independent, e(N) <
=
I
I
I
N
o-i)
"t
..
e[I
i=l
I
e[(
I,
,I
We now define:
-. . ',I
= sup
i
sup e [Zi - r
r
I
z.
£' = inf
irrf e [Zi + r
r
I
zi oS -r ]
~
i
~
=
sup
i
> r ] for a-b > r >
~-
sup e[(Zi - r)2 lzi
r
~ rJ
for a - b
for a - b
0
>r >
I
0
>r >
0
"I,
..
I
I
I
29
,-
t'=
sup
sup
r
•
J.
e[
(z.J. +r)21 z.J .<
-
-~J
for a - b
>r >0
N
I
Ln
=P
I
[
zi
<
b -I
N
= n]
Zi
~b
i=l
"
I
,I
and by Wa1d' s method we have
N
I
I
b + g'
oS
N
e[ I
Zi
i=l
I N = n,
I
I
:::: a + g •
Therefore,
(l-L )(a.-bn
since
Now
I
I
,
I
< b
i=l
I;
..
"I
,
]
and thus
~')
+ b + £' ::::
e[~I
N=nJ:::: Ln (b-a-£) + a. +g
I
I
30
N
-,
10
(b+s') e[LfJi ] + (a-b-s') LP[N=nlfJ1+"'+fJn)(1-Ln):S
i=l
n=l
I
,,'Which can be written as
(b +
N
~') e
(LfJi ) +
i=l
(a-b-~')(l-L)
I
11~
I
I
N
e[LfJi !reject Ho ] :s
i=l
;t
If.e >
0,
..
then
-<
N
e[~ I
fJiJ:s
(a+~)
u e(N) + (b-a-
~).t L e(N!
"
accept Ho )
i=l
If bounds for e(NI reject Ho ) and e(Nl accept H )
o
we may replace (2.18) by
. ,
(2.19)
(b+~) u
N·
S. e[
-
I
e(N) + (a-b-~ )
.
1
~ i~ ~i s.
J,
.
(a+,)
are not available.,
~ e(N)
(l-L)
-.
+ (b-a-,) JL.
Corresponding to (2.18) and (2.19), we have for the case u <
0
I;
Ii
1
I;
I;
Ii
"IIi
,
I
-
I
j
I
31
I
II
I
I)
I
I
I
I
..
t
1
I
(a+; ) J, e(N) + (b-a-
~
) u L e(N laccept Ho )
N
I
:s e[ ~
1·
I'
I
J
i=l
<
-
I
I
(b~~)J e(N) + (a-b-~)
u (l-L) e(NI reject H )
0
and
(a+~)Je(N)
N
(b-a-~)uL.:s e[~I
+
Ili
]
<
i=l
I
I
(b+ ~ )J e(N) + (a-b- g ) u (l-L)
•
Again by an argument analogous to that used by Wald [38] we have
+ (l-L) (a
By defining
s = inf (j~
i
,
t = sup (j~
i
l
2
+ 2 as + ~).
end combining (2.17), (2.19)
l
and (2.20) we arrive at the following inequalities for e(m2)
For J
>
0,
(2.21)
I
I
lli
and similarly for u <
e(m2)
>
0
~
2
[ s teN) - L(b + 2b S' +
~,)
I
2
- (1_L)(a +2a S +
e(~) :s ~[t
~)
+ 2(a+ S ).1 e(N) .+ 2
e(N) - L b
2
- (l-L) a
2
(b-a-~)
u L ]
+ 2(b+S').t e(N)
+ 2(a-b- s') u (l-L) ]
•
I
••I
I.
In the application of the above bounds, e(N) and
by their appropriate bounds.
place of (2.19).
L will be replaced
If possible (2.18) should be used in
It should further be mentioned that the lower
bound for e(~) will often be trivial.
However, the maximum of the
lower bound given above and the square of the best available lower
bound for e(N) can be used as a lower bound for
e(~).
AI!. an illustration, consider the s.p.r.t. for the mean of a
normal distribution with known variance.
More specifically let
,I
I·
I
I
I
..
Then under H , z. ,.. N (- -21, 1).
o
1.
a
and b
for ex
= ~ = .05
Employing Wald' s approximati on of
we have - b
has also obtained formulae for
sand
=a
,
s,
= 2.944.
For this Wald
which in this case are
computed to be l.64 and -1.0, respectively.
Then by (2.22)
For this test Baker [5] has obtained experimentally e(N)* 7.0 and
L
* .9646.
Inserting these values we find that
e(~) :s 102.79 and s.d. (N) :s 7.33 ,
t
I
I
I
I
I'
..
I
I
'I
,I_
I
I
I
I
,I
\1
33
while the corresponding Monte Carolo values Baker arrived at are
e(~)
* 70.5
and s.d. (N)
*
4.64.
In this problem the lower bound
for e(~) turns out to be trivial, but using e(N)
* 7.0 we have
e(~) .?; 49.
2.6 General bounds for e(~)
Consider any nonrandomized sequential rule such that after the
i~
observation xi' if Zi(x l , ••• xi)
€
Ci then another observation
is taken, otherwise sampling is discontinued.
Assume that N, the
sample size, is greater than or equal to one.
Let E.~ denote the
€
C.,
then
~
I
..,
I·
I
,
I
I
,
I
Now define
P'j
~
= prE'.
L ~ E.J
J ' ...
...
•
It then follows that
p[
n
U
1=1
E.
~
l
J
,
Now, by :Bonferronis inequality,
sl - s2+· •• -sk ;S P
[·5i=l E'iJ.::s Sl-S2+··
.+S.t
'
where
J,
It is also clear th'at P[E ] > P[N > n].
n -
is odd and k is even.
Now define
+
= max
bn.t
[ 0,
l-s
1
+ s
2
Sk] ,
- ••• - . s .t ]
and
n
Hence, since P [U E.] = P[ N < n], we have bounds on the distrii=l ~
bution function of N. (Bhate [8] has also considered using a
no as
an upper bound for
P[N.5 n]
when xl' x2 ' ••• are independent.)
Since e(~) =i~l [ (i+l)ffi -i ffi] P[N > i ] + 1 and
eo
it then follows that
I
10
(2.23) 1 +
[(i+l)m_im] ai k. 2:
~
i=l
where k. is even and .t. is odd.
~
~
10
e(~)
In
eo
2:
1
m
+ I [(i+l)m_i ]
i=l
parti cular,
10
1 +I a ik 2: e(N) > 1 +
b.J,
~ .
I
i
~
i=l
i=l
I
I
-,
I
I
I
I
I
I
I
..,
,I
Ii
I
I
I
..
I
I
I
--I
I
I
I
I
,I
I
35
and
~
1 +
~
L
. 1
(2i+1) a ik . 2: e(l'f)
In many cases, s. for i
~
and.t. = 1
0
~
('
I
L
(2i+l) b . .t
~
. 1
>1
will be 'quite difficultto obtain.
~
i
~
L
1 + P [Zi€ Ci ]
i=l
i
~=
we have
~
2: e(N) 2: 1 +
L {O'l- L
P [Zj,£ Cj ] }
j=l
max
i=l
(2.24)
L
~
1 +
(2i+l)P[Zi
i=l
If we let n
o
denote the smallest value of n such that
n+l
{'
I
I
I
I
I
I
1 +
~
~=
k. =
2:
1 -
L
P [Zj
I
Cj ]
~ 0,
j=l
it then follows that
and generally
~
1 +
L
i~
~
[(i+l)m_im] P[Zi ECi ]
2: e(wn) 2:
1 +
L
~l
[(i+l)Ul_im] [l-i+
So fox
36
j=l
It should be remarked fram a pratica1 point of view that in app1ications it will be necessary to determine a value of i, j let us say,
and a sufficiently small constant
K such that
J a ik .
+ K •
J.
By way of illustration, consider the normal test of means given
at the end of the previous section.
Let z ..... N (6,1) and
J.
the distribution function of a standard normal variate.
p
[z
'E C
mm
If we assume
1
= P [b
l
III
denote
Then
< Zm
< a. = III ( r.!- Q..[m ) - III (.!L
- tim ).
~
'1m
m
fm-
,= 0, then
where c = min (a, -b)
Now it can be shown (see (40]) that for x >
_ !2
III (x) - III (-x) ~ (1 _ e2
-1
) 2
0
•
I
I
--I
I
I
I
I
I
I
..
I
I
I
I
I
I
..
I
I
I
t'
37
eo
I
I
I
I
I
I
,
I
2m
(1 - e
1
) 2
>
m=n
2
1
C
10
J
2
c
--
00
(1 - -~
e
) 2 d m
J
~
n
( 1
~ e 2m ) d m >
n
d m
=
•
10
Therefore we cannot apply the upper bound (2.24) at the midpoint between the two hypotheses.
Next, assume 9
o > o.
Let
{'
I
I
I
I
I
I
I
[Zm € Cm l,->
P
-c-
00
m=n
I
I-
2
Thus
I
0
and noting symmetry, restrict attention to
n be an integer such that
1. -
9fn < o.
Then
eo
10
I
p[zm€
I
cm] <
m=n+l
~
m=n+l
(.!..
~
- ,.Jm )
=
10
I
( 1 - ~ (e
.Jm -
m=n+l
~(x) ~t
Now by (2.25),
.!.. )
,J;;
[1
and thus
10
I p[
Zm €
m=n+l
10
cmJ:s
I (~ -~ (
m=n+l
+
)
(1 - ;
~2) ~ J for x>
0
I
I
38
J
_1( • .fm
10
.:s ~
(l-(l-e 2
_ .!..)2
.fm
1
--.
)2)d m
n
-- •"m - -)
r
IOl(
J
a2
e 2
.,Fm
I
I
I
I
I
I'
I,
d m
n
J
lO
eta
mI-
--
e 2
n
Similarly it follows that
10
I
m p[zm€ em
m=n+l
=
Under
..,
2
e
J .:s
'a in
-2
dm
..
_2
(1+n (JT)
2
Ho ' • = - ~ and by applying (2.24) we obtain
8.29 > e(N)
~
I
I
3.97
111.68 > e(~) ~ 16.74
while Baker
[5] observed e(N)
= 7.0
The merit of the bounds given in
cability.
I,
and
e(~) = 70.5.
(2.23) is their general appli-
As an illustration of the case where the bounds given in
the previous sections cannot be applied, the two-sided sequential
test (see Rushton
([23], [24]) ) is considered. Suppose a random
t
I
I
I,
..
I
I
I
f'
I
I
I
I
I
I
I
t'
I
I
I
I
I
I
f
I
39
variable
x
is normally distributed with an unknown mean
2
cr.
unknown variance
I..l.
and an
vIe wish to perform an s.p.r.t. discriminating
between
Ho :
By defining
=0
I..l.
0= I~I
and
I I..l. I =
Hl :
cr
these hypotheses become
cr
H.
o·
Now the likelihood ratio for this test may be written as
where
11
is a confluent hypergeometric function (see Slater [26 ] )
and
n
n
un
=(
I
x
i
)2 /
i=l
I
i=l
x~
For the boundary values we vlill use Wald's approximation for the case
of 0: = (3
= .05;
namely - b
creasing function of u
log
n
= a = log
and Arnold
J n (an
) = log 19 and log J (b )
·
n n
19.
Now
J (u)
n
n
is an
in-
[4] has given values a n ,bn for
=-
log 19.
Thus
I
I
40
=p[ n p [ t
nt
2
2<a ]
1 + t
n
n
<
n
d
]
n
p [ t
-
P [ t2
n
=
n
n
<
<
a (n - 1)
n
]
n - an
..
=
-d ]
n
1
where
tn
=
•../0. X
_[
s
2'
an (n - 1) ]
andd-
n
n - a
•
n
Similarly,
P [ log J, (u )
nn
vThere
=[
e
b n(n
n
> -log 19 ] = p
- 1) ]
n - b
e
h
5
6
7
8
9
10
11
12
13
14
15
.126
.h04
.566
.700
.818
·927
1.028
1.122
1.212
1.297
30.085
4·976
3.946
3.552
3.353
3.242
3.178
3.143
3.126
3.121
3.125
2.3
n
e
16
17
18
19
20
21
1.378
1.456
1.531
1.603
1.673
1.740
1.805
1.869
1.930
1.990
2.049
2.106
22
23
24
25
26
27
Zn
E
On ]
=
]
I.
d
n
'n
e
3.136
3.151
3.170
3.191
3.215
3.240
3.267
3.294
3.323
3.352
3.382
3.412
28
29
30
31
32
33
34
35
36
37
38
39
2.162
2.216
2.269
2.322
2.373
2.423
2.473
2.521
2.568
2.615
2.661
2.706
n
Thus we have
p[
l + P rLnt < -en
> e nJ
n
d
n
n
n
•
TABLE
n
[t
I
I
I
I
p[
t
n
<
d ]
n
-
p[
t
n
<
d
n
n
l
-d
n J
3.442
3.473
3.503
3.534
3.565
3.596
3.627
3.658
3.688
3.719
3.750
3.780
+
I
I
..
I
I
I)
II
I'
I,
..
Ii
I
I
41
,e
I
I
I
I
I
I
I
('
I
I
I
I
I
I
f
I
'~1ere
t
n
has a noncentral
t
distribution with
freedom and noncentrality parameter
If we set
5
= 0,
n - 1
degrees of
oJn
then by using tables of the central
t
dis-
tribution we find that
30
I
i=l
, and
~[Zi
€ Ci ]
fP[z;
=
-
C.~ ] < 10.92
€
i=l
Now in Siskind's [25]
sm~)ling
and if we may assume that
10.76
experiment
N was truncated at 61
P[ Z.E C.] is a decreasing function of
~
~
i
then
Therefore if
N is truncated at
12.1 >
-
~<N)
61 we find, from (2.24),
> 7.67 ,
',I
while Siskind observed e(N)
* 9.9 .
For the case when N is not
truncated, we do not knm" of an upper bound to ." P[Z. € C.] .
~~
~
~
However, it is observed that
< .882
.865
and since
P[Z3l€ C ]
3l
= .02587,
for
i
= 11, ••• ,30,
it is probably reasonable to set
42
00
\ ' p[Z.€ C.J
~
i=32
~
We then find
~
*
1
1 -
.89
(.02587)
=
.236
·
I
I
..
I
I
I
I
I
I
I
..
I
I
I,
II
I
I
..
I
I
I
,e
I
I
I
I
I
I
I
{'
I
I
I
I
I
I
f
I
CHAPTER
III
OPERATING CHARACTERISTIC FUNCTION
3.1 Introduction
In this chapter we ''\'Jill be concerned with the operating characteristic function
(0. c. function), of an s.p.r.t. For the case
vn1en zl,z2' .•. are independent and identically distributed, Wald
developed the well-knovrn approximation along with upper and lower
bounds for the o.c. function.
As in the previous chapter we will
begin by applying these particular results to a larger class of
s.,p.r. tests.
Then in section 3.4 we consider conditions under
Which P[accept HoJ
is increased by replacing each
stochastically larger variable.
zi
by a-
Also some discussion is given con-
cerning the relationships between monotone likelihood ratios, stoIn section 3.5 we
chastic ordering and monotone o.c. functions.
briefly look at the effect upon P[accept H J after the z. 's have
o
2
undergone a scale change. Finally we conclude chapter three by
applying the method given in section 2.6
and
lower bounds for P[accept HoJ.
in order to obtain upper
However, we shall first give
a brief outline of Wald's method of approximating the o.c. function.
Let zl,z2' ••• be independent and identically distributed with
common probability density function f(z).
unique nonzero solution of
~(t)=l
Where
Define
~(t)
t
o
to be the
is the moment
I
I
44
generating function of z
(i.e. ~(t)
= !etzf(z)dz).
-.
Wald has
given
f
(a) e(z) exists and e(z)
(b)
0
the existence of 8 > 0: such that P[z > 0] > 0
and p[z < -8] > 0
(c) ~(t) exists for all real t
as sufficient conditions for the existence of a unique nonzero t
o
•
NovT if we define
=
r(z)
and assume that t
exp( zt } f(z)
o
,..,
>
0 then f .is a probability density function
o
and
IX)
=
I
+ z.~ < a, for
Pf[b < Zl+'"
i
= l, .•• ,j-l
j=l
..
eo
and zl+'"
+ Zj;S b]
=
I
Pf [ tob 
IJ[
j=lSj
where ui e
j ,..,
exp( -t b } n f(z.)
o
i=l ~
?(z.)
log [ f(Z~)
J
~4
oJ
=
exp(-t b} P [accept H ]
J•
~
0""
f
I
I
I
I
I
I
I
0
I
I
I
II
Thus
P [accept H ] > exp {-t b} P [accept H ]
o
f
o
f
0
and similarly
(3.1)
I
I,
..
I
I
I
.e
I
I
I
I
I
I
I
{'
I
I
I
II
Since we assumed a unique positive t , it then follows that
o
{'
I
1.
t
By the definition of
we also have Pf[Z = 0]
f
1
and thus by Stein's Theorem [28],
Therefore,
(3.3 )
Wald's approximation then consists of changing the inequalities of
(3.1) and (3.2) to equalities and of solving for Pf' [accept HoJ with
the use of (3.3).
This process yields
l-e
(3.4 )
e
bt o
a t
0
- e
,
at 0
and the same result is obtained for t o < o. An alternative approach
utilizingWald's fundamental identity is, briefly,
(3.5)
I
,I
f
Pf[z = 0]
=
e
[ t ZNl_;= [t ~[
ee
O
e
eO
l-N
1=
cp(t o ) _ _
1.
at o
...
or
I
I
46
3.2 Consistency
Let f(Zl' •• " zn) denote the probability density function of
zl' ,."
Z and aSsume that there exists a unique nonzero t suCh that
n
n
J
t Z
e n n f(zl' ,."
where Zn
= zl
+ ". + z,
n
Zn)
e(z)
exists and e(zn ) f
n
(b)
there exists a
J
I
I
I
I
I,
I
I
=1
As in the previous section
(a)
(c)
dz
-n
on > 0
0
such that
tz
e
n f(zl' •• "
..'I
Zn) d!n exists for all real t
are sufficient conditions for this assumption,
Also we shall call
f(Zl' .'., Zn) the conjugate density associated with f(zl' "', zn)'
I
I
where
(3,6)
,
Z
n
-.
).
I)
Let g(xl , ,."
xi) denote the density of the first i observed random
variables, Xl' ',., x.,
~
I
Ii
Since z.~ is a function of Xl' , .• , x.,
we
~
shall assume that the density f(Zl' , " zi) is determined by
g(xl , ,."
Also define g(xl , .,.,xi ) to be the corresponding
conjugate density associated with g(xl , ,", Xi) (i.e. g(xl, ••• ,xn )
Xi)'
IV
=
..
I
I
I
..
I
I
I
I
I
I
I
..
I
I
I
I
I
I
f
I
t
e
Z
n n(
xl'
... , xn ) g(xl ,
.•. , xn
». The development outlined in section
....
3.1 could just as well have been gi ven in terms of g and: g instead
of f
and f.
It has been stated in the literature, (Bartholomew [6], Siskind
[25]), that i f t n
f1holds".
=t
for all
n,
then Wald's o.c. formula, (3.4),
At first appearances this seems to be a reasonable extension
of Wald's o.c. formula.
Hm-rever, if we examine the development out-
lined in the previous section, the expression
equa't:ton (3.1).
1
Pf[ accept
H ] appears in
o
Now for PI' [accept Ho 'to be meaningful, we must have
dz
n = f(Zl' .•. , Zn- 1) for n=2,3, .••
If (3.7) holds, we shall say that {f) is a ~sistent sequence of
densities.
( g)
It should be noted that {f) is consistent if and only if
is consistent.
By
(3.6), (f) will be consistent if zl,z2'
are independent and t n = t but in the dependent case it is not obvious
that this is true. As an example where t = t for all n but {f} is
n
not consistent, consider the following:
Example 3.1
be a sequence of Bernoulli random variables with
I
I
48
..
Now suppose we perform an s.p.r,t. discriminating between
= 0,1 •
i
I
I
I
I
I
I
I
If vTe give the parameters the following values,
1
e0
=12
el
=
e=
1J.0 =
1
12
Is
1
b
1
1J.1
=b
17
IJ.
= 720
€0 =112
1
€l = 6'
€,=
I)
1
1
=2
I)
5
= 32
I)
2i
1
a =4'
it can then be shown that
I [ _P_[_~.;::.:_X-=l;...1
-'-'-'-'_X.;::.i=_x...;;i;..,I_H..;;l-.l
'X'
, 1"'"
for all
x"
P [Xl-xl'
i
i.
, " X4 =x4
...
Thus we have t
n
...
I Ho ]
=2
J: [Xl""1'
for all n,
... ,
~=Xi]
=
I
however,
Now,since the conjugate distributions of the observations Xl'x , ••• are
2
not'cOJ!18!1s.tent,the conjugate distribations of the variables, zl' 2: , •••
2
are not consistent.
Therefore, by extension of Wald' s development
..
I
I
I
I'
I
I
..
I
I
I
.e
I
I
I
I
I
I
I
given in the previous section, it would be valid to state that if
t
n
Wald's o.c. formula, (3.4),holds.
f
I
Some situations in Which one of
these implies the other are indicated by the following lemmas.
Lenuna 3.1
A necessary and sufficient condition for t n = t n=1,2,
and {f) to be consistent is
J
for
tz
e
n
fez n I zl' ... , zn- 1) dz n = 1
n = 2,3, ..• and
Proqf.
{'
I
I
I
'I
I
I
= t for all n and the conjugate densities are consistent, then
e
If t
= t for all n and {f) is consistent, then
n
tzn l
tzn
_ [(e
f(Zl' ••• , Z ) dz _ f(Zl' .• " z
=
.
n
n
n-1J
l
and thus the necessary part follows.
Suppose
J
tzn
e
f(zl' ••• , Zn)dZn = f(Zl' .,., zn_l)
and
J
tZ
e n-
1
f(Zl' ••• , zn_l) d!n_l
= 1 •
50
-.
Then
=
and hence, by induction, it follows that t n = t for all n.
Now
=0
{f} is consistent.
and thus
,.,.
Lemma. 3.2
for
n~=
Proof
o
=t l
for n = 2, 3, .•.
We have
J
=J
.
=
e
e
t
=e
t
t
Z
n n
Z
n n
f(zl'
(
f zl'
Z
n-l n-l f(
... ,
zl'
... ,
I
I
I
I
I
I
I
..
If {f} is consistent and
2, 3, .•. then t n
I
I
... ,
I
I
I
I
I
I
..
I
I
I
.e
I
I
I
I
I
I
I
f'
I
I
I
;1
I
I
f
I
51
and thus t n
= t n _l
Lemma 3.3
If t
n
•
=t
for all n and {r} is consistent, then
for all n.
Proof
By
lennna 3. 1,
and all n and by taking expectations of both sides we have
J
e
tz
n
for all n.
Lemma
3.4 If zl' z2' ••• are independent, then {f} is consistent
if and only if t
Proof
n
= t
l
for all n.
Since
N
we have that {f} is consistent if and only if
e
t nzn-l
1etnzn
t n-lzn-l
f(zl""'z n- 1) e
f(z n )dzn=e
f(zl""'z n- 1)
for all n.
Thus {r} is consistent if and only if
52
for n
-.
= 2,
constant
Therefore
3, ...
Now the left l'land side of the above equation is
the right l1alld side is not, unless t _ - t
n
n l
whi~e
(r) is consistent
i f and only if t
n
=t l
for all
= o.
n.
The foregoing lemmas could also be stated in terms of the observations and the densities
g
and
...g.
The statements and proofs are
quite similar, althOUgh in lemma 3.3 the condition that x l ,x ' •••
2
are independent under both H and
o
H:t
needs to be added.
These
lemmas will be helpful in determining whether or not we are justified
in using Wald's o.c. formula for a given problem.
Also they may prove
to be useful in constructing a sequence of densities, (f), such that
t
n
=t
for all
nand (1) is consistent.
that if under H , x.., ••. , x
o
J.
n
have g(x ,
l
It should also be remarked
... , xn ),
for all n, as
their joint probability density function, then by lemma 3.1 it follows
that t
n
I
I
=1
for all
also holds for
n and (g)
is consistent.
A similar statement
H1 .
Returning to the proposition of using Wald's o.c. formula without
regard to consistency, we should mention that the alternative approach,
(3.5),cannot be used, since we do not have a generalized form of Wald's
identity available.
Now, although the use of Wald ISO. c. formula wi th-
out consistency is not justified, it may nevertheless be a reasonable
approximation.
With this in mind, we shall give the follCMing example
illustrating the relationship between t
and p[accept Ho ] .
EX&l1ple 3.2
In example 3.1 let
H denote the values assigned to the para-
I
I
I
I
I
I
I
..
I
I
I
I
I
I
..
I
I
I
.e
I
I
I
I
I
I
I
53
meters 't'1hich yielded t n
=2
('
I
Let H represent the following
distribution of Xl' X2 ' •••
for all n.
'for all
i.
It can then be shawn that if
and
for i
f'
I
I
I
I
I
I
"
for all n.
then t
n
= 3, 4, ..•
= t for all n. It can also be readily observed that
log
..2.
log
"2
10
3
if xl
=0
if xl
=1
if (x ,x )
l 2
= (0,1)
~
if (~,~)
= (0,0)
log 2
if (Xl'~)
= (1,1)
0
log
or (1,0)
54
Z. = Z2 +
J
tZi
={
where z.
~
i=3
Suppose 't-re require that
then follows that N
x
1
= 1
xl =
x
1
0,
= 0,
xl =
0,
~
0
3
=
2
~
if x.
log 2
if x. = 1
< a < log ~ and log
~
-.
0
~
t
x
=
2
log 3
0
0
=> accept H0
= 1, y) = 1
x2 = 1, =") =
0
I
I
=> reject H0
=> accept H
o
I
I
I
I
I'
I
I
..
I"
I
I,
It can similarly be calculated that if'
and if
t =
~l
then PH ' [accept Ho
J *.834
•
From this example we can therefore draw the following two conclusions:
I
(1)
If H and H yield the same constant value of tn' it does
not necessarily follow that PH[ accept Ho ]
= PH'
[accept H }
o
I
I'
II
..
I:
I
I
I_
I
I
I
I
I
I
I
55
,
(2)
Suppose H yields t n = t for all n
for all n.
If t > t
P [ accept Ho
H
J~
,
and H yields t
n
= t
it does not necessarily follow that
PH' [ accept Ho
J·
Since Wald's o.c. formula is a monotonically increasing function of
t, example 3.2 and its conclusions should indicate two things; in the
first place, what the error involved in neglecting consistency may be,
and secondly, the errors in the approximation itself.
We now conclude this section with a conjecture of Bartholomew
[6J concerning the o.c. function.
Suppose that the joint density
of the observed variables Xl' ~" ••• xn is p(x l ' ••. " xn Ie) for all
n and that we wish to discriminate between
{'
I
I
I
I
I
I
,
I
by means of an s.p.r.t.
NOIv, the equation for determining
t
in
Wald's o. c. approximation ray then be written as
J[ p(.!nl
p(x
I
~
p(x
I e) dx~
~
=
1.
Bartholomew suggests
be expanded as a bivariate Taylor series in
eo "el
about the point
e. Then if the terms of third order and higher are neglected and the
I
I
-.
relationships
e [ dlog pl. = 0,
de
.
are used, we f'ind that the equation for
t
=
1
-
2( e
t
yields
-eo )
e1 - e0
TIlls may be recognized as the
t
f'ound in the s.p.r.t. discriminating
betw"een two means of a normal variable with known variance. Applying
th!s conjecture to the s.s.s.t. given in table 1.8 of section 1.3 we
find
value of' n
15
conjectured value of Pn[accept
~o]
exact value of Pn [accept H0 ]
16
17
18
19
20
..
.912 .810 .639 .428 .244 .123
.950 .584 .354 .194 .100 .050
These values serve to illustrate the possible dangers involved with
the indiscriminate use of Bartholomew's conjecture.
Siskind [25] has
also applied the conjecture to the two-sided sequential
t
test.
He
found the approximation satisfactory only for values of the parameter
which were closer to HI than Ho ' Siskind went one step further by
consider~ng the third order terms in the Taylor expansion.
He found
3
that if e (d log p ) is negligible compared with e(d log p )3,
d
d e
83
then t
=1
-
2(e -e )
o.
HovTever, in many problems such as the se-
el -eo
quentia1 t test, the range of'
e
I
I
I
I
I
I
I
for which this condition is satis-
I
I
I
I
I
I
..
I
I
I
I_
I
I
I
I
I
I
57
fied is difficult to determine.
3.3 Bounds for the o.c. function
B,y assigning different values to the parameters in example
it can be shown that there exist cases where
of
n but
3.1,
is not independent
n
is consistent. It should be noted that by lemma 3.4,
(f)
t
-
we will then necessarily be restricted to zl' z2' ... being dependent.
We shall assume ,that ,(f} is consistent and that there exists a unique
"
nonzero t
t>
n
r
for all n. Define u
= sup(t
},·t = inf(t
} and
n
n
n
n
suppose that
Then by following the development given in section 3. 1 we have
0•
•
I
=
Pf[accept H ]
o
Pr[bt i < Ui < at i
for
i
= 1, ••• ,j-l
and'
j=l
I
..
I
I
I
I
I
I
,
I
•
Uj
S
bt j ]
55
I J
f(Zl'" "Zj)d!j
>
j=l Sj
10
I
J
. e
-bt
j f(Zl'"
.,z .)dz.
J
-J
>
=
j=l Sj
e
-b.t
P..., [accept H0 ]
f
vthere
Thus similar to equations
(3.8)
(3.1)
Pr[accept H ]
o
and in a like manner
and
(3.2)
we
he.ve
> e-b.t P [accept H ]
f
0
58
I
I
_I
(3.9)
We further assume that
f
< 1, it follows
+ P_[accept H ]
l
f
and by noting P [accept H]
a
(3.9) that
from
If vTe change the inequalities of
and solve for
(.3.8)
(.3.10)
and
to equalities
Pf[accept H ]' we obtain
o
e aJ _ 1
aJ
bJ
e
- e
(3.11)
I
>a .
for
J
for
u <
..
In an identical fashion, it may be shown that
1 - e
(3.12)
au
bu
au
e
- e
To continue, suppose we assume that
as the probability of UN:S c
N
such that
c
n
'Where
< Un < dn is not satisfied.
r(z , ... ,z.)
be
log
1
N
J.
f(zl"'" zi)
it follows that
J
>
a
0
Since
•
and we define
is the least value of
UJ.'
I
I
I
I
I
I
n
was defined to
I
I
I
I
I
I
..
I
I
I
I_
59
and also by the definition of u and J. Jb
,
2:
J, a.
> b,tn -> ub and ua > iatn
-
Therefore,
I
II
I
I
I
I
..
I
I
I
I
I
I
{'
I
and hence
Similarly we have
Now Wald has shown quite generally
e
-a
and thus
Pf[UN 2: ubi ,ta, ub
JS e-ub
if u <
0
ifJ > o.
Also by using the general approxinBtions
e
a
* 1·'
ex
60
b
e
...A..
*l-a
we obtain from (3.13) and (3.14) the following approximate bounds
e Ja _ 1
e Ja _ eub
•
< Pf
-•
[ac~ept
HoJ
.
<
-•
ua 1
ua
JJb
e
-
e
for J
>0
(3.15 )
.
-.
1
ua
- e
Jb
ua < Pf[accept Ho
e
- e
J~
1 - e .ta
ub
.ta
e
- e
for u <
0
(~ will be used to denote an approximate inequaJ..ity as in the sense
here).
And as expected, on comparing (3.11), (3.12) and (3.15) find
.ta 1
e
Ja
ub
e
- e
for .t >
0
for u < o.
t
G)
where G is a subset
of the positive integers, it may be possible to improve the approximate bounds given in (3 .15 ) as follows:
Define
U
o
.to
and assume J
o
= sup
(t }
n€G
=
n
n
inf {t }
n
€ G
> o. Then
Pf [ accept Ho IN€ G] P [NEG
J:5 Pf[UN =s .tob I .tob, uoa,N
_a
I
I
I
I
I
I
I
..
and
If we have an upper bound for P [ N
f
I
I
€
G]
x
I
I
I
I
I
I
..
I
I
I
61
I_
I
I
I
I
I
I
I
and by using a similar argument for the lorer bound, we have
Prr~
L -< u0 b
lu0 b, toa]
Pr[uN.s t ob
ltob, uoa] +
- Pr [ N¢ a J .s Pr [ accept
Pr[N ~ a J
for t
0
HoJ
.s
> o.
It can aJ.so be shown that
..
I
I
I
I
I
I
f
I
By applying the approximations which were used to obtain (:3.15), we
arrive at
(3.16)
1 _ e
ua
0
e tbu&
0 _ 0
t
a
1<: l-~ 0:toa
[ accept H0_
":'" U
e
o
-e
.
+
Pr[NIG Jlor
u <0.
0
62
Now consider the case where zl' z2' ..• are independent.
Let
'1!i (t) be the moment generating function of zi and D' be that part of
the
t
complex plane such that '1!i (t) exists for all
1.
e
for t
n.
N
-1
[e ~ [IT
1'. (t)
~
1=1
ll
1
=
n
€
D, where D is a subset of D' for which li~l1'i(t) I~ 1 for all
Blom [9] has further shown that if either
(a) there exists a 6 >
P [ z.
J.
and an
0
> 0] >
€
€
>
0
such that
0
such that
for all i
or
(b) there exists a 6 >
0
and anE >
P [ zi < -6 ] >
€
for all i
then
lim
n ->-0
k
n p[ N > n
l=
0
for any k
-
is bounded then (3.17) is valid.
~ o.
,
Therefore if we assume
(3.17) is valid, then it follows that
IJ.'i (t)
~1
for all i
implies
1
1'i(t)
~1
for all i
implies
loS
Mallows [20] has shown that for t >
0
> e [ e t~ ]
e [
I
I
I
I
I
I
I
..
Blom has also proved that i f (a) and (b) hold and for some real t,
{ '1!i (t»)
_a
Then follow-
ing Hald's proof, as pointed out by Mallows [20], we move
tz.._
I
I
et~
] .
I
I
I
I
I
I
..
I
I
63
I
I_
I
I
I
I
I·
I
I
..
I
I
I
I
I
I
,
I
11 (t) ebt S e [etZNI
~ S bJ
~,~)
.
eat S e [ e
t~ I ~ 2: a J
where
11(t)
oCt)
=
=
[ t(Zi+r)
inf
inf e e l
o<r<a.-b
i
sup'
sup
o<r<a.-b
i
Jl i
.s -r J
[t(Zi-r )
e e l Zt2: r
J.
Similarly, for t <0
ebt S e [etZw I
~ ~ b J So' (t)
ebt
(3.20)
11' (t) eat S ere
t~ I ~ ~ a J :s eat
where
,
11 (t)
=
[ t(z.-r)
e e
J.
inf
inf
i
o<r<a.-b
I
zi
[ t(z.+r)
e e
J.
I zi
sup
o'(t) = sup
o<r<a.-b
i
Now, if we let L denote p; [accept H]
o
11(t)ebtL + eat (l-L)
~
ere
t~JS
Le
2: r
J
.s -r J
then by
•
(3.19)
bt + (l_L)o(t)e at
and
(3.20),
if t
>0
.(:3.21)
bt
at
Le
+ 11' (t)e (l_L)
~ ere t~ JS
Lo' (t)ebt + (l_L)e at
if t < o.
I
I
64
I f we assume there exists a unique real nonzero
'l.(t.)
1.
1.
=1
, then by defining u
= sup
i
t.
1.
such that
/, = inf
{t } and
i
{til we
i
_I
have
implies
'l.(u) >1
for all
i
implies
'l.(u) <1
for all
i
/'>0 . implies
'l.(/,) <1
for all
i
implies
'l.(/,) >1
for all
i .
u<o
-
u>o
/,So
Then by
(3.18)
u<o
implies
/'~o
implies
/,so
implies
B(t) ~ 1,
for
0
I
I
I
I
I
I
I
-
1.
-
1.
(3.21)
-
-
it follows that
bu
au
+ (l-L)B(u)e
1 < Le
bU + (l_L)~'(u)eau
1> Le
a
1> L~(/')eb/' + (l_L)e /,
b
a
lS LB'(/')e /' + (l_L)e /, •
-
=s
~(t) S 1
for
t ~ 0
and. Ol(t) ~ 1 ,
t So, it follows that
u>o
implies
P[accept H ]
0
-
implies
P[accept H ]
o
u<o
(3.22)
and
implies
o oS ~I(t) S 1
-
1.
u > 0
Now since
1.
-<
au
o(u)e
- 1
B( u ) e au - ebu
-<
au
1 - T\'(u)e
bu _ ~'(u)eau
e
.-
/'~o
implies
P[accept H ]
0
>
e a /, _ 1
aJ
b/,
e
- ~(/')e
-
implies
P[accept H ]
0
>
1 _ e aJ
BI(/')eb /'
/'<0
- e
a/'
..
I
I
I
I
I
I
..
I
I
I
I_
I
I
I
I
I
I
.1
..
I
I
I
I
I
I
f
I
65
As approximate bounds we may neglect overshoot by setting ,,= 8 = ,,'
= 8' = 1.
3.4 stochastic ordering
In this section we will be concerned with the proof of the
following intuitive proposition.
ly distributed and
z.
~
Suppose
zl,z2""
are independent-
is stochastically larger (see page 73 of
.
Lehma.nn [19] ) under H than under H' , then
PH[accept Ho ]
2:
PH' [accept Ho ]
From this statement it is easy to see how bounds may be determined
for
P[accept Ho ] in many problems. Simply replace zl' z2' • . . by
a sequence of stochastically larger (smaller) random variables for
Which the probability of acceptance is known or may be approximated
by the usual techniques •
Some necessary preliminaries for proving the above proposition
are as follOWS:
Lemma 3.5 Let H and G be Lebesgue measurable functions such that
(a)
H
is nonincreasing and nonnegative
(b)
G is of bounded variation,
and G( -) S G(x)
G(--) = lim G(x)
x-. .....
exists,
for all x.
Then if
eXists, it is nonnegative.
Proof:
Define
E.n
~
=
{I i-12
x
-
n
<
H(x )
i}
<n
2 .
for
i=l, ••• ,m-1
66
E: =
{X I
n
~
-.
H(x) }
and
if x
S (x)
n
'Where
n
Ei
m = n2 n + 1.
= < x i +1'
of interval.
have
xi
Now by
~
Sn(x)dG(x)
for
S (x)
is a step function and
n
x i +1
and <
i=l, ••• ,m
> denotes some type
=
I
~~1 {
Variation of G over
E~}
'Where
Yi
m
~
I
2
i=l
denotes either x.+
(i-1)[ G(Yi) - G(Yi +1 ) ]
or x i
J.
m
I
.
Now
m+l
m
(i-1)[ G(Yi ) - G(Y +1 ) ]
i
=
i=l
I
(i-1)G(y. )
J.
-I
(i-2)G(y )
i
i=2
i=l
m
-
(m-1)G(Ym+1 )
=
I[
G(yi ) - G(Ym+1 ) ]
i=2
which is nonnegative since G(Y ) ~ G(Y + )
i
m1
by dominated convergence we have
'I.
,,
I
I
I
i=l
=
I
I
,
(E~} is a disjoint cover of the real line, we
m
J
En
i
(a),
> where xi
Since
€
I
I
=
G(-).
Therefore,
..
I
I
I
I
I
I
..
I
I
I
,
I
I
I
I
I
I
67
Corollary 3.5.1
I
I
I
I
I
I
t'
I
H and G be Lebesgue measurable functions
such that
(a)
H
is nondecreasing and nonnegative
(b)
G is of bounded variation,
G(.).:s G(x)
G(.) = lim G(x) eXists, and
x-.·
for all x.
Then if
J
exists, it is nonpositive.
HdG
Proof:
Make the corresponding changes in the proof of lemma. 3.5 .
Corol1a.ry '.5.2 Let the real valued functions on the real line
F1
I
f'
Let
and F2
have the following properties:
(a)
F.J. is a distribution function
(b)
H
(c)
H(--)
Then F 1 (x)
= 1
H(.) = 0
and
~ F2 (x)
HdF l
Let
i=1,2
is nonincreasing
J
Proof:
for all x
•
implies that
J
2:
HdF2
G = F1 - F 2
and apply lemma. 3.5 •
Lemma. 3.6 Let
~,x2""
ables with xi
having distribution function F
be a sequence of independent random vari,
and
H"
"
.
respectively.
F.J. = F.J. for
Ip ,{
H,
If
F '" (x)
l
i=2,3, ••• , then
~ F (x)
l
i
for
«
and F
i
b <x <a
,
under
H
and
If)
b <
~
+... + x j
< a for j=l, ••• ,i-l and
~
+... + xi
Sb
~
+••• +
< a for
~
+••• + xi
S b}'
}
i=l H
";"110
~
I
P "{b <
i=l H '-
X
j
j=l, ... ,i-l and
68
Proof:
Define
~
{(~, ••• ,xn ) I ~ S
=
--I
b }
for
and
Let Fi
i=2, .•• ,n
if
(~"",Xn)
€
Ai
if
(~, .•• ,xn )
fl
Ai •
denote the distribution function of xi
and define F(~, .•• ,Xn ) = F2(~)" .Fn(Xn )
•
b <
'1.
+••• + Xj
<
a
for
for
i=l, ••• ,n
Then we may write
n
I1
j=l, ••• ,i-l and
~
+••• + xi S b }
i=l
where
Now
Ai
Bh
and
is a nonincreasing function of xl
since if
(~-C'~""'Xn)€ Aj
c>o then
for some
(Xl""'X ) €
n
jSi.
Further we have
Bh
=
0
~
=
1
I
I
and since
exists, it follows that
JJf-1
HndF
is a non-
I
I
I
I
I
I
~
I
I
I
I
I
I
..;
I
I
I
,
I
I
I
I
I
,
I
t'
I
I
I
I
I
I
t'
I
69
XI
increasing function of
and
Therefore by co11ary 3.5.2 we have
and since this is true for all
n, the lemma. follows.
Theorem 3.7 In an s.p.r.t. suppose that
zl,z2""
are independent
,
and
zi
has distribution function F
respective~.
and G
i
i
P ,[N
Also assume that either
II
H
under
and H
< eo] = 1 or
H
< eo] = 1. Then F . (z) > G. (z) for b-a oS z oS a-b and
P II [N
H
J.
i=1,2, •••
J.
implies that
P
I
H
~:
-
[accept H]
0
Suppose
> P II [accept H ] .
-
H
0
P, [N < eo] = 1
and let
H
j
H
for
the hy,pothesis that the distribution function of
i=1, ••• ,j-2
any
H
for
i=j-1,j, •••.
> n -. 1] <
-
€
,
o
H
and therefore
o
.
p~ [N
is F.
J.
for
H
j
.
Also
PH [accept H0 IN
.. '
.. ~
J.
and by the definition of H ,
0
= P, [N < n - 1]
z.
denote
P ,[N < eo] = 1, for
Since
> 0 there exists an no such that
€
P ,[N
and Gi
j=2,3,...
[N
PH
n
o
<n
- 1]
0
> n0 - 1] = P ,[N -> n.0 - 1].
-
H
o
< n0 -1 ]. PH,[ accept H0 I N<n0 -1] and thus we have
70
PH
~
[accept H ] - P ,[accept H]
o
0
< PH
-
H
[accept H IN> n - 1] X
0
~
PH [N
no
> n0 -
-
-
0
1] <
€
•
if for all
j
For
> PH
0
j=3, (3.23)
and let F
> p .. [accept H ]
H
0
j2:3
PH [accept H]
j-l
I
I
I
[accept H ] •
0
follows from lemma 3.6.
Next define
denote the joint distribution function of
zl, ••• ,Zj_3.
We may then write
I
P[ b< Zl +•.. + Zk < a
i=l
zl +••• + zi ;S b]
+
f· I(C j ){
xj -3
a-z j =3
for
Under Hj
for
Zj_2.
k=o, •.• ,
and Hj -1
i~l
the
.
I
for
k=l, ••• i-l and
pc b-Zj _3 < Zj_2 +••. + Zj+k_2 <
i=o
and
Zj_2 +•.• + Zj+i_2
zi' s
oS b-Zj~3IZj_3J}
dF .
have the same distribution except
Therefore in the ,above expression for
P[accept H ] ,
o
only the term within the braces, call it T, differs under H and
j
H.--1. Now T may be considered as P [accept H] for an s. p. r. t.
J-
0
in whi=h the first observation is
Zj_2
and the stopping boundaries
b-Z j _ and a-z j _ • We may assume b < Zj_3 < a since
3
3
nm.ltiplied by I(C ) . Therefore, by lemma 3.6 , T , when H
are
j
,
I
j-3
P[acceptHo ] =
--.
I
I
Therefore we may conclude that
P ,[accept H]
H
0
I
I
j
T is
is
-a
I
I
I
I
I
I
..
I
I
I
"I
,I
I
I
I
I
I
t'
I
a
I
I
I
I
71
true, is greater than or equal to
(3.23)
is satisfied for all
the case P
~[N
j.2:
T when Hj _ is true. Hence
l
3. A similar argument holds for
< .] = 1 •
H
In a g.s.p.r.t. suppose that zl' z2' ••• are in-
Corollary 3.7.1
dependent and z.1 has distribution function F
respectively.
i
and G.1 under H' and Htr
Also assume that either PH' [ N < ..]
= 1 or PHtr[N<..]=l.
Then Fi(Z) .2: Gi(z) for b i - a i _l ~ z ~ ai-bi_land i=1,2, ••. where
a o = b 0 = 0 implies that
Proof Simply make the appropriate changes in lemma 3.6, its proof
and the proof of theorem 3.7.
Before proceeding further, we should mention the connection between theorem 3.7 and a sufficient condition for the monotonicity
of the o. c. function.
Suppose xl' x2' ••• are independently distributed with density Pe(X). Lehmann [19] has shown that Pe(x),
possessing a monotone likelihood ratio in T(x), (M.L.R.), is a
sufficient condition for any s.p.r.t., testing eo against el(eO<
say), to have a monotone o.c. function.
el
Now if Pe(x) possesses a
monotone likelihood ratio in T(x), we may write:
Pe (x.)
1
zi = log
1
()
Pe xi
= h(T(Xi ) )
o
r
t'
I
where
h is a nondecreasing function of T(x) depending upon
el • Next by a slight extension
eo
and
of lemma 2 on page 74 of Lehmann[19],
72
we have that if 'f is a nondecreasing function of T(x), then
ee
'f(T(x) ) is a nondecreasing function of
Y(T)
we see that for
~
e
t
{
>
:
if h(T)
>c
if h(T)
S
e.
Hence by setting
c
e
Also by theorem 3.7
e[
Pet [ accept HoJ monotone o.c.
Next we shall give two examples which illustrate the fact that the
above implications cannot be reversed.
Example 3.3
(x has ".L.R.
<r z stochastically ordered)
Let the random variable
values xl'~ and
--I
I
I
I
I
I
I
tta
I
be summarized as
(3.24)
I
I
x
take on only the three distinct
X, with probability Pl (e), P2(e) and P3(e) respec-
I
I
I
I
I
"I11
J
I
I
•I
I
I
I
I
I
I
tively.
Define
P1(e)
=
*-*
13
1
= -31 --e
2
P (e) = ! + g 13
323
+!
3
if-
t
<13 < ~ and observe that P (e) and P (e) are decreasing
2
1
functions of 13, While P3(e) is increasing. Also define
for -
=
Now we will show for 13 >
1
e2
and 13' > 13 , that for all c
t'
I
I
I
I
I
I
f
I
Let zl' z2' z3' Where zl
z( 13 ,13 , x) may take.
1 2
~
z2
~
z3' be the three values which
Now since P (e) and P (e) are decreasing and
l
2
P3(e) is increasing, we have zl < 1, z2 < 1, z3 > 1 and also
z3
= z(e1,e2 ,
~).
(3.25) is true.
Thus for z2 ~ c
< z3 we
have determined that
Since P (e) and P (e) are decreasing functions of
1
2
13, (3.25) also holds for zl ~ c < z2 and is thus satisfied for all
c.
Next we will show that there does not exist a T(x) and an
h( 13 , 13 , T) such that for 13 > 13 , h( e , 13 , T) is a nondecreasing
1
2
function of T and
1
2
1
2
Now it can be shown that z(o, -2/3, xl)
9/14, thus T(Xl ) < T(~).
z(1/3,-2/3, ~) =
= 3/5,
z(O,-2/3, ~)
I
I
=
However, z(1/3, -2/3, ~) = 2/5,
f and therefore h(1/3,-2/3,T)
is nat nondecreasing
in T.
Example
,.4
(z
I
I
stochastically ordered <1= monotone o.c.)
Consider an s.p-r.t. with a
2a
= -b
and
wi th probability 1 -
1
e - 'Be
I
1
with probability 'S'9
-2a
where
~
e + ~;
1
e + 'Sa
with probability
e
S e S ~. Since N S 2 it is clear that
~[ accept
hence the test has a llXlnotonic o. c. function.
is a decreasing function of
is not true for
e,
Ho
However,
J=
stochastically ordered tI implies that Wald' s approximation to
Suppose
e' > e,
then
I
--I
e 1= e'.
the o.c. function is monotonic.
I
I
so
Returning to the expression (3.24), we should point out that
liZ
--I
This can be shown as follows:
Fe' (z) S F e< z)
and by lemma 3.5 and corollary
3.5.1
for t >
0
I
II
I.
I
..
I
I
I
75
,
I
I
I
I
I
I
,
t'
,I
for t < o.
If we define t ( e)
I-
0
r
zt(e) .
by J e d ~ = 1 then for t ( e')
> 0 we
have by (3.25),
1'Thich implies that tee) 2:
tee') <
0
that
tee').
tee) 2: tee').
decreasing function of
Similarly by (3.26) we have for
Therefore tee) is a monotonically
e, and since Wald's o.c. approximation is a
monotonic function of t, we have that it is also a monotonic function of
e.
In conclusion we will briefly consider the possibility of removing the condition of independence in theorem 3.7.
To begin,
suppose we look at a natural generalization of the theorem obtained
by removing independence and strengthening the order relation to
I
I
I
I
I
t
I
The following example will then illustrate why the theorems conelusion does not follow under these conditions.
Assume that
Pl
P2
=
P[zl = t, z2 = sJ
P3
P4
Example 3.5
a = -b and
if (t,s) =
if (t,s) =
if (t, s) =
if (t, s) =
(-2a, -3a)
(-2a, -3a)
(a/2, -38.)
('a/2,3a )
I
76
where Pi
~
0
and
Pi + P2 + P3'
i~l
Pi = 1.
It then follows that P[accept HoJ
Now suppose that F assigns Pl
and G assigns P2
1
= P3 = P4 = 3'
P2
= o.
= P4 = i,
P2
=
= 1)=
0
From this it follows that
F(Zl' Z2) ~ G(Zl' Z2)' however PH' [accept HoJ
=i
and PH,,[accept HoJ
= 2/3.
Although simple ordering of the joint distribution functions
is not a sufficient condition for a generalization of theorem 3.7,
there probably exist additional assumptions which would.
However,
in applications it is unlikely that PH[accept HoJ would be known
when zl' z2' .•. are not independent.
With this in mind, the
following theorem is of possible use in problems where the dependency between the Zi 's is relatively "weak".
Theorem 3.8
In an s.p.r.t. suppose that G(Zl"'"
~n) is the joint
distribution function of zl' ... , Zn for all n under H' and
that
.
Zl' Z2' ... are independent, with zi having distribution function
Fi for all i under H".
Assume that PH' [N <
Fi (z) ~ Gi (z I Zi_l) for b-a;
S
Z
S a-b,
00
J
= 1.
Then
b < Zi_l < a and all i im-
plies that
PH' [ accept HoJ
<
PH" [accept Ho
1.
And similarly F.J. (z) < JG.. (z IZi - 1) implies that PH' [accept H0J ->
PH,,[accept Ho ]'
Proof
Let Hj , j
= 2,3,
••• , denote the hypothesis that zl" •• ,zn
I
--I
I
I
I
I
I
I
tta
,I
I
I
I
<I
I
..
I
I
I
"I
77
have
G(zl' ••. , Zj_2) Fj_l(Zj_l) ."
2:: 3
P [ accept Eo
Hj
I
I
I
I
Since Zo = 0, (3.27) .holds for
to the proof of theorem 3.7,
< PH
j-l
1
[accept Ho
.
j
=3
by lemma 3.6.
Now by referring
j-3
I
.1
I
I
l
then
I
i'
As in the proof of theorem 3.7, if
tribution function for all n.
for all j
Fn(zn) as their joint dis-
p[ b < Zl + .,. + Zk < a
i=l
for j
= 1,
..
>' i-l and zl
+ ... + z.J.
<bl +J I(C.){
-
,
.
J
~-3
,I
t
I
I
i'
t
where F denotes the joint distribution function of zl' ••• , Zj_3'
Under Hj and Hj_l' zl' .•. , Zj_3 have the same joint distribution
function. Thus in the above expression for P[accept Ho J, only the
term within the braces, let us call it D, differs under H and H _ .
j
j l
I
78
D may be considered as P[accept H ] if the first observation is
o
z. 2 and the stopping boundaries are b - Zj
J-
b < Zj_3 < a.
'X
-~
and a - Z.
'X
J-~
where
Therefore, since Zj_2' Zj_1' ••. , zn given Zj_3 has
as a joint distribution function G(Zj_2
Iz j _3 )
F j _l (Zj_1) .•• F n (Zn)
under Hj and F j _2 (Zj_2) .•• Fn(zn) under Hj _ , we have, by 1eIJll1l8,
l
3.6, that D When Hj is true is less than or equal to D when Hj _1 is
true. Thus (3,27) holds for aJ.1 j 2:: 2. The sane line of argument
will also show that Fi(Z) > G.(zi Z. 1) implies PH,[accept H ] <
-
1
1-
0
-
PH"[accept Ho ]'
3.5 Change of scale
Although the results are more general, the previous section
could be thought of as being concerned with the effects upon the
o.,c. function of an s.p.r.t. after the variables zl' z2' ... have
been translated.
A natural question concerning the effects of a
change of scale then arises.
tuitive1y:
What we shall attemp to show is, in-
Suppose the distribution of each zi is
toward the aJ.ternati va
hypoth~sis.
more favorable
Then increasing the variance of
Zi while holding its mean fixed should disturb the process and
crease the chance of acceptance.
in~
It should be net ed that as the
variance of each zi becomes very small the probability of acceptance
tends to zero, while if the variance becomes very large this probability tends to one half.
Specifically, we shall assume. that Zl' Z2' •.• are independently distributed, with zi having f i (Zi - Il ) as its probability
i
density under H' and having
under HI!.
Also assume that b
c f 1(c i (Zi - Ili ) ) as its density
1
= -a
and either PH" [N < 00]
=1
or
t
--I
I
I
I
I
I
J
Ita
I
I
I
I
I
I
..
I
I
t
,
I
I
I
I
I
79
PH' [N <
t'
I
i
I,
I
I
I
,
I
= 1.
]
We then question under what circumstances is
(3.28)
As in the proof of theorem 3.7, a sufficient condition for (3,28)
is
P
[accept H
H
l>
0_: -
j
P
Hj _l
[accept H
0
l' for
j
= 3,4, ....
where Hj denotes the hyt>Othesis that zi has density cif.(c'(Zi1
1
f.l.i»
= 1,
for i
I
I
00
f.l.i ) for i > j-2.
..• , j-2 and density fi(zi -
:Before proceeding further we shall state the following preliminary results:
Definition f(x) is said to be increasing for XES if 'xeS, yES,
y
€
S and x < y implies that f(x)
unimodel
for
if there exists
x < x
and f(x)
an x
a
~
fey).
f(x) is said to be
such that
is decreasing for
f(x)
is increasing
x >x
0 - 0
Lemma 3.9 Let h, g, k be real valued functions on the real line
such that
(a)
h(t + f.l.)
(b)
get + v)
(c)
k(x)
= -h(-t
= g( -t + v)
= J hex - u)g(u)du
Then
j
for all
+ f.l.)
k(x + f.l. + v)
= - k( -x
= J hex
Proof:
k(x + f.l. + v)
J hex
+ f.l. - t)g(t + v)dt
for all t
exists for all x •
+ f.l. + v)
for all x .
+ f.l. + v - u)g(u)du
= - J h(-x
t
=
+ f.l. + t)g(-t + v)dt
=
~ - t)g(t + v)dt
- J h(-x +
=
-
J h(-x +
~ + v - u)g(u)du
80
I
=
t
- k( -x + ~ + v) .
Lenuna 3.10 Suppose the conditions of lemma 3.9 are satisfied and
get)
(b)
h(t+~) ~
Then
k(x+~+v) ~
k(x+~+v)
Proof:
o
-J
is unimodal
(a)
if t ~
0
~
if x
0
0
0
•
•
J h(x+~-t)g(t+v)dt
=
=.
J h(u+~)g(x+v-u)du =
00
h(-u+~)g(-x+v+u)du
+
J h(u+~)g(x+v-u)du
=
0
-00
00
J [g(x+v-u) -
g(x+v+U)]h(u+~)du
o
By
of' lemma 3.9 and
(b)
is decreasing for t
x ~u ~
(i)
(ii)
u
~
x
~
0
~ 0
Therefore
[g(x+v-u) -
and hence
k(x+~+v) ~ 0
v;
(a)
of lemma 3.10 we have that g(t)
and thus,
g(x+v-u) ~ g(x+v+u)
implies
implies g(x+v-u) = g(-x+v+u)
g(x+v+U)]h(u+~) ~ 0
if x
~
if x > 0
g(x+v+u) .
and u > 0
~ 0 •
Lennna 3.11 Let f, h, k be real valued fUnctions on the real line
such that
(a)
there exists a ~ such that f'(t+~)
(b)
~ ~
(c)
f( t)
(d)
h(t) ~ h(-t)
(e)
h(t)
(f)
k(x)
= f(-t+~)
0
is unimodal
for all t ~
is increasing for t
= J f(x-t)h(t)dt
0
~
0
exists for all x •
for all t
-.
I
I
I
I
I
I
I
--I
I
I
I
I
I
..
I
I
I
81
,
I
I
I
I
I
,
I
"I
k(x)
is increasing for x
~
k(-x)
Part (i) -
f -x)
=
~
for all x
~ 0
~
y
Let x
0',
0
4
~ 0
f(t+~)
is unimodal,
and define
f(x-s) ~ f(x-y-s) . Then get)
f(~- I2
-t) = f(~I +t) - f(~+ Z +t)
2 2
= f(~+ ~
-t) -
= -g(-t) • Now since f
~
is increasing for t
and thus
0
t ~~ ~
0
implies
~~t ~
0
implies f(~+ ~ -t) = f(~- ~ +t) ~ f(~- ~ -t)
Therefore get) 2:
f(~+ ~ -t) ~ f(~- ~-t)
for t 2: o.
0
Now k(x) - k(x-y)
00
4
=
.-
f g(s+~+~ -x)h(s)dS = f h(t+x-~- ~)g(t)dt = f .(-t+x-~- ~)g(-t)dt
o
00
+
f h(t+x-~- ~)g(t)dt
o
=f [
h(t+x-~- ~) - h(-t+X-~-~) ]g(t)dt •
0
we have
by
that t 2: -x+~+ ~ 2:
(d) and (e)
implies h( t+x-~-~) 2: h (-t-x+~+~) ~ h (-t+x-~-~)
"
..
0 0 .
0
Since x-~- ~ ~
I
I
(ii)
g(s+~+
I
I
t
k(x)
Proof:
I
~
(i)
-x+~+ ~ 2: t ~
implies R(t+x-~-~) 2: h(-t+X-~-~) . Therefore
0
[ h(t+x-~- ~) -
h(-t+X-~-~) ]g(t) 2:
k(x) - k(x-y) 2:
0
get) 2:
0
0
for
t >0
and thus
g(t+~)
= f(x-t)-
•
Part (ii) f(-x-t).
Let x 2:
0
and define
As in part (i), we can show that get)
for
and
t 2: o.
= -g(-t)
It also happens that k(x) - k(-x)
and
=
00
f
o
[h(t-~)
- h(-t-~)]g(t)dt
for t > o.
Therefore,
and
[h(t-~)
k(x) > k(-x)
for
- h(-t-~)]g(t)
x > o.
>0
I
I
Theorem 3. 12
= -a.
and with b
zi
is
~i ~ 0
Consider an s. p. r. t . with zl' z2' . . .
Suppose the probability density function of
fi(Zi- ~i)
for all
independent
where
i.
fi
is unimodal,
fi(t)
= fi(-t)
, and
Then the probability density function, gn '
of Zn' given that N ~ n , has the following two properties:
Proof:
(i)
g (x)
(ii)
gn(x)
> g (-x) for x > 0
n
-
n
-
is increasin~ for
The result is clearly true for
it is also true for
n=l.
a
=
f
-a
for-a < x < a
~_l(X)dx
o
then
~
Now let us suppose
Let
n~-l.
gm_l(x)
h (x)
m
x So.
elsewhere,
is the densi t~ of Zm_l' given that N ~ m, and it is
h (x) > h (-x) for x >
m
- m
creasing for x S o. Now gm(x) =
clear that
lemma 3.11 ~ satisfies
0
f
and that
h (x) is inm
fm(x-t-I-lm)hm(t)dt , and by
(i) and (ii) •
Returning to the problem of establishing
(3.28) and (3.29),
we begin by assuming that
(a)
~i ~ 0
(b)
0
(c)
fi(t)
(d)
f (t)
i
fi(t)
(e)
hold for all
= fi(-t)
i.
for all
t
is unimodal
is continuous
(so, in view of
Now define gj
given N ~ j , and let
I
I
I
I
I
I
I
~
,
I
I
:s 1
< ci
-.
(d)
bounded)
to be the density of Zj_l'
I
t
I
..
I
I
I
83
,
I
I
I
I
Pj(X)
= PH' [accept Ho IN> j
Tj(X)
= l(x S -a)
Fj(X)
= f fj(t)dt
and Zj = x]
+ l(-a < x < a)Pj(x)
x
x
-
and
Gj(X)
I N 2:
j] 2: PH
=_
f gj(t)dt •
00
00
By the definition of Hj ,
(3.30)
for
PH
[accept H
j+2'
0
j =1,2, . . .
is equivalent to
(3. 29).
j+l
[accept Ho
IN~
j]
It can also be seen that
I
I
I
"I
I
I
I
I
I
where F
* G(t)
L(C j ) =,!Tj(X)d[F/Cj(X-J.l j »
Define
* G/x)]
* Gj(X)]
- !Tj(X)d[F/X-J.lj )
=,r [F/X-J.lj ) - Fj(C/X-J.lj »] * G/X)dTj(X)
. and L(c.) >
for x
t
J
-
will then imply
0
(-a,a) , f j
(3.30).
Since T.(x)
J
is bounded and by assumption
(e)
is constant
g.
J
we have
L'(C j )
where f
=
dL(c j )
dc
* get)
Sj(X)
we have
Next define
"I
= f F(t-u)dG(u).
j
.[
=:
[(X-J.lj)fj(cj(x-J.lj
= f f(t-u)g(u)du.
»] * gj (X)dTj (x)
By defining
= (J.l j - X)fj(Cj(X - J.l j »
is bounded,
84
*
rl(u)
{
=
r 2*(u)
g~(:)
for uSo
gj(-U)
for u>o
'a
-
* - rl(u)
*
gj(u)
=
vThere
gj_1(U)
a
*
[Cj_lfj_l(Cj_l(u-~j_l»]
-
for -80S uS a
,;rgj~l(u) * [Cj_lfj_l(Cj_l(u-~j~l»]dU
-a
o
By assumption
gj(u)
*
(e),
is continuous and bounded on
(page 491 of Apostol [2]);
thUS, given
€
>
0 ,
I
[-80,80]
I
there exists a step
I
k:(t) such that
i=l, n
I r;(t)
(:3.31)
-
I
<
k:<t)
€
r (u)
=
{
1
gj(~)
gj(-u)
r 2 (u)
=
0
for u >
0
gj(u) - rl(u)
it then follows from (3.31)
that
and from
gj(u)
-
L
k i (t)
I<
-a < u < a
elsewhere
I
I
n
I r 2 (t)
where
for u S
I
I
~
I
I
•
i=l
Now if we define
I
I
I
elsewhere"
n
function
I
I
€
i=l
for
-a < t < a
elsewhere
I
I
..
I
I
I
,
85
for all
By theorem 3.12
i.
I
I'
I
I
·1
I
I
,
I
I
I
I
,I
I
,
I
we have
(a)
r (u)
is unimodal
(b)
r (u)
>
0
(c)
r (u)
2
=
0
1
2
for
-
u<o
and hence we may assume tl1at
k. (t)
~
ai >
where
o
-< a.
~
•
0
=
if
aoi
{
t E < ai' a i +1 >
otherwise
< ai' a i +1 > indicates some type of interval with
and
Define
for
i=l, •.. ,n,
we then have
(3.32)
n
S
a
J{.eo(x) I
+ "i (x)
i=l
+
E.J I
s/x-u)
I du
} dT/X)
S
-a
n
I J.e
i (X)dTj(X)
+
EM
i=o
where. M~ 2a(2a + ~j)fj(O)'
-Sj(t+~j) = Sj(-t+~j)
Since
- .eo (x+~.)
= .e0 (-x+~ j ).
J
= r (-t) we have by lemma. 3.9 that
l
r (t) is unimodal and -S • (t+~ .) > 0
1
J
J-
if
t
~ 0
-
v. >
~
0
and k.(u+v.)
~
~
rl(t)
Also
and thus by lemma 3.10
- .e (X+~j) > 0 if x > 0 and -.e (X+~j) < 0 if x < o.
o
0
Sj(t) is continuous, we may consider < ai' a i +l > = (ai'
for purposes of determining properties of .e.(x).
Let v.~ =
~
and observe that
and
= ki(-u+v.).
~
Since
a i +l )
1
-2(a.+a.+
)
~
~ 1
Hence by le:mma.s
I
I
86
3.9 and 3.10 we have
= Ai (-X+J.1j +vi)
2: 0 if x 2: 0 ..
- Ai (X+J.1j +vi)
- Ai (X+J.1j +vi )
At this point it is necessary to make a :f'urther assUfIII)tion based on
some heuristic considerations.
assUfIII)tion
(f) :
If'
M 2:
then Tj (x+M) + Tj (-x+M)
0
creasing function of
Let
M = J.1 j +v
i
i
2:
then
0 ,
J
Ai (X)dT j (x)
since
T/X)
is decreasing and
M < a , then
i
Ai (x)
2:
S
0
J
AidTj(X)
x for x 2: o.
0
for
a-Mi
is an in-
for
Mi
2:
a ,
x S a S M • Assume
i
0
=JAi(~+Mi)dT/X+Mi)
-a-M
i
= -fii(X+Mi)dTj(-X+Mi )
a+M
.
i
+
a..M.
a-M
i
J.
J
=
Ai (X+Mi )dT j (X+Mi )
o
J
Ai (X+Mi )d(Tj (X+Mi ) + Tj (-x+M )]
i
+
o
a+M
i
.J
Ai (X+Mi )d.T j
(~X+Mi)
-<
a-M.
0
J.
since
Ai (x+Mi ) So, and both T/-X+M )
i
are increasing.
Therefore
(3.32)
O.
by
Lt(c ) S
j
which implies
J
Thus
Ai (X)dTj(X) S
L(C j )
2: L(l)
and Tj(X+M ) + Tj(-x+Mi )
i
for
0
=
0
for
i=o,l".· ,n and
0
< cj S 1 ,
(3.30).
Concerning the assumptions (c), (d) and (e), possibly (c) is too
restrictive" however (e) is reasonable and (d) can easily be shown to
be necessary.
The problem is with assUfIII)tion (f), since in practiee
we will not know how closely (f) is satisfied.
However, it is con-
--I
I
I
I
I
I
I
--I
I
I
I
I
I
..
I
I
I
"I
I
I
I
I
I
I
--\1
I
I
I
I
I
.e
I
87
jectured that (f) will be approximately true (at least that the in..
tegral Ja-Mi t (x
i
o
.+ Mi ) d[Tj(X + Mi ) + T/-X + Mi ) ] is nonpositive)
if there is not a great dea.l of difference between the individual
densitites f
i
J
~tuitive
argument in support of
(f~
is
Putting a = -b, from (a), (c) and theorem 3.7 we have
as follows.
T.(o)
i=1,2, .•• An
<t.
Suppose we could apply Wald's o. c. approximation.
-
since a = -band Tj(O) ~
i
Then,
-
it is reasonable to assume that h<
.
0
where
Therefore,
h(a-x) 1
Pj(x)
eh(a_x)- -h(a+x)
e
-e
and it follows that
*
[ Tj(M +
*
X2)
J -[ Tj(M + Xl)
,
+ Tj(M - Xl]
1
. - e
hM
= 2-e
-ha
e
=
ha [eheM + X2 ) +eh(M - X2) _~heM + Xl::-h(M - Xl)
-h;
e
+ Tj(M -x2 )
ha
e_ha-e ha
e-e
hx
ha
[ cosh h
~
1
- cosh h Xl. >
0
for a
~ ~
2:: Xl 2:: o.
- e
Therefore when conditions (a) through (e) are satisfied and the reSUlts
are applied, the conclusions are assumed to be approximately correct.
3. 6 General bounds for the o. c. functions
The technique for determining bounds on the moments of the
sample size used in section 2.6 may also be applied to the o.c. function.
As in that particular section, let us consider any nonrandom-
ized sequential rule such that after the i~ observation Xi' i f Zi (xl'
88
••• ,
X
i
)€ C , then another observation is taken.
i
is discontinued.
otherwise, sampling
If sampling is terminated at the
i~
trial, we will
accept H if Zi(x, ... , x.)€ A., where Ai and C are disjoint for
i
o
1
J.
J.
all i; otherwise we will reject H.
o
For j
= 1,2,
... ,
••• , j-l
We then have
Next define
=
·i =i{i
~pi,·~'=j~~l
L {k , ...
sj - sj +
1
2
or.
+ sj
-
I,
I
I
I
I
I'
I
I
It then follows that
=
•I
--I
pi p[ E"i 1, Pi,k p[ E"i E"~ J'
=
..• define
I
I,
j
I'
By Bonferroni' s inequality
1'1
...
I'
..1\
1
where t is odd and k is even.
Since
J
]
J
J
where
p[ E~ 1, sr
a jk = min {
1 -
+
s~
-... +
s~
}
]
1
1
J~
J
and
1
Since p[ accept Ho =
P [accept Ho and N = nl it follows that
n=l
00
00
I
(3.33)
a.n k
n=l
J
where k
J
I
00
all n.
n
n
> p[ accept Ho
1~ I
bn .en
n=l
is an even integer for all nand t
n
is an odd integer for
If we set
1
•••
-)
J
~e
J
)
where Rj = 'OJ n Aj ; we find, in a similAr manner, that
00
00
(3.34) \' c
L
~l
> p[ reject H
~n -
1>
0-
\'
L
~l
d
n~
I
I,
where
~
•I
n
cn P = min ,
P n 1. '
n
".
[F
}
{
dn
= max
~.
am
tf
.1
n
n
~ n
1 - t l + t 2 - ••. - t.
,
~.
0,
si,
is defined similar to
bining
(3.33)
and
(3.34), we
Pn is even and
is odd..
Com-
conclude that
00
00
\' a
[ L
n k n,
n=l
~
1 -
\'
L
d
n=l
l.I. >
n~ .. -
p[
accept H
0
I
I
I'
J>
-
00
By
setting kn
= Pn
=
0
1 -
') c
L
n Pn
n=l
and
~
(3.35) ;
(3.36) 00
l
I:
= .en = 1, we obtain, as a special case
of
min {
I
P[Zn€, AnI}
n=l
> P[accept Hal >
n-l
00
1
~"
I
max [
0 ,
P[~~€ Rn l
I
-
P[Zit
i=l
n=l
III8X {
1 -
I
P[Zn' Rn ]
n=l
,
I
III8X [
ci I
in section 2.6:
I
P[Zit Ci J
i=l
I)
--I
I,
I
n=l
P[Zn € An] -
(3.36)
l}
a ,
n-l
As an illustration of
I,
J} ·
let us consider the two examples given
I'
II,
I
..
I
I
•I
I
I
I
I
I
I
t'
'I
I
I
I
I
I
..II
91
E.xa.mple 3.6
For the test of means from a normal distribution we
obtain from (3.36)
.895 ~ PH [accept Ho ] < .9997,
o
,
while Baker[5] has observed a value
PH [accept H ]
o
0
*
.9646.
For the sequential t-test given in section 2.6, we find, from
(3.36), that
.8666 ~ PH [accept Ho ] ~
o
.9958.
The lower value is valid, assuming that the test was terminated at or
before n=30.
The experimental value observed by Siskind[25] is
PH [accept Ho ] = ,960.
o
I
I,
II
CHAPTER IV
APPLICATIONS
4.1
Introduction
The primary purpose of this chapter is to illustrate the
methods given in chapters two and three.
We begin by applying these
results to the s.s.s.t. described in chapter one.
test,
zl"",zn were shown to be independent;
For this particular
so most of the
o
methods will be applicable.
ing an s.p.r.t. in which the
We then conclude the chapter by considerzl,z2""
are multivariate normal.
This appears to be a natural case in which there is dependency between
the
zi f s.
It should also be noted that if
variate normal,
Xl' x ' . . . are multi2
N(.I:!J 1:) , then the Wald s.p.r.t. for. discriminating
between
H:.I:!=u
,1:=1:
0-00
leads to the conclusion that
zl' z2' • .•
An add! tional purpose for designating
are also multivariate normal.
zl' z2' • . •
asrnultivariate nor-
mal is because of the interest in the following technique.
that
zl' z2' . • .
Suppose
are independent and identically distributed and that
the a.s.n. is large.
It may then be reasonable to set
I
I,
I
I
I
I
II
--I
I
I
I
I)
I
..
I
I
93
and consider the s.p.r.t. with increments
"I
I
I
I
i
3
]
J"
J
]
mate~
independent normal variables.
4.2 Bounds on e(N) for the s.s.s.t.
In this section the results of chapter two will be applied to
the s. s. s. t.
Let us begin by considering the Wald type bounds for
e(N) which were given in section 2.3.
At the end of section 1. 2 it was shown that zl" •• z
dependent~
distributed.
(4.1)
are inno
is redefined by setting
Now if R(n +1)
o
if n > n
o
= { a-b
b-a
J'
J
= no
,
we will then be assured that R(n + 1) > A if n > n and R(n + 1)
o
0
0
< B if n=no • Thus we may assume that the criterion used in the
s.s.s.t.
~
be expressed in terms of the variables
zl"",zn + 1
o
which are independent, where
zn + 1 is defined
by (4.1). This oper.
o
.
ation then permits the application of the bounds for e (N) which are
given by (2.10).
It was shown in section 1.2 that for
J
1
J
J
if n
n-i+1
where ci =
_n
n1 0
and
a
i=l, ... ,n , the distrio
n ... i + 1
= log -=1~r--""":"
i
no - i + 1
It then follows that
e[
hence
zi+ r
I
zi -< -r]
=
n - n
0
1
n-i+l
for i=l, .•• ,n ,
o
n
~'= inf
inf e[zi+ rlz.~ -r]
i a-b>r>o
1:
Similarly, for
=
n - n
o
1
n - no + 1
=
no
if n > n .
o
< r < a i we have
0
ai - r
which is decreasing in r .
sup e[zi- rlz i ~ r]
r>O
Therefore,
=
I.
for
n >n
= no
,
=
n
1
&.-1.
c
for
i
o
i=l, •.. n , we are in a position to
o
calculate the bounds.
Example 4.1
values insure that Gt = 13= .05.
are given in table 1.2.
e (z.)
n
1.
for
n=n , ..• ,n
1
0
Some of the properties of this test
The following table gives the values of
and i=l, •.. ,n + 1.
0
TABIE 4.1
n
1
2
i
3
4
5
-.307
-.439
6
-.140
-.685
-.189
-.268
-1.247
-.414
•I
I
I
I
Finally,
=
I
I
7
-.021
-.022
8
.068
-.018
.149
.003
.253
.097
9
.137
.186
.268
•420
10
.193
.255
.357
.539
I
II
--.
I
I
I
I
Ii
I
..
I
I
•I
I
I
I
I
I
I
95
5
6
-.708
'3.725
-3.208
-3.725
bounds on e(N)
3.725
·959
3.725
found in section 2.3 are:
TABIE 4.2
n
upper bound
lower bound
lower bound based
on averages
Since e(z.)
,~
5
20.3
.38
6
8
66.0
.32
7
.10
-.33
1,49
.89
is monotonic in i for
at the end of section 2.3, use
.44
10
28.1
.49
1.76
L83
9
37.1
n=5,8,9,10, we may, as stated
n +1
1
n + 1 i~l e(zi)
in place of
o
sup e(z.) , or inf e(z.)
i
i
~
(i.e.
~
I
I
I
I
variation among the e(z.).
~
..fl
3.725
.792
3.725
o
t'
I
.542
Using the correct values for P[accept H] given in table 1.2, the
values of n
,I
.125
in calculating the inequalities for these
line 3 of table
4.2)~
For each value of n there is a rather large amount of relative
From the development of the inequalities
we would expect poor results when this is indeed the case.
Hence the
preceding example appears to emphasize and illustrate the necessity
of having only a small amount of relative variation among the e(zi)
before one could expect reasonably tight bounds.
We now apply Hoeffding' s bOUlJ.ds, (2.14) and (2.15), to the same
example.
Since the bounds were developed in terms of the probability
densities of the observations, it is necessary to discuss the density
of x
n +
o
f(~,
1 instead of simply defining
z
n +
1 as a constant.
Let
0
••• ,xn In)
o
represent the joint density of ~, •.• ,xn
0
When the
true sample size is
h(t) =
n.
{
~.
get)
=
{
'".
I
I
Define
•I
if 0 < t < 1
elsewhere
-
1
0
exp(b-a)
if
oS t S
exp(a-b)
elsewhere
0
and consider the s.p.r.t. discriminating between
and
Hl : p(Xl""'Xn + 1) = f(Xl"v"Xn Inl)h(Xn + 1)
0 0 0
where p(xl'·.·'X + 1) represents the joint density of xl""'x +1"
no
no
Let the hypothesis H represent p(Xl""'Xn + 1 ) = f(x-, ••. ,x I n)
o
L
no
x hex + 1) where n is assumed to be greater than n. Then employ~
0
ing the notation of section 2.4,
Zo,n + 1 = a - b with probability one under H
o
Zl,n + 1 = 0 with probability one under H.
o
It is therefore clear that the above s.p.r.t. leads to the same decision and distribution of N as the s. s. s. t .
n ... i + 1
log n - J.. + 1
o
and both eH(zoi) and eH(Zli)
Now
n - no
n - i + 1
i=l, ••. ,n
n - ~
n - i + 1
i=l, ., .,n
are increasing in i
o
o
for
i=l, ••. ,n
o
Typically we will have
a - b > log (n - n + 1)
-
If this is not the case, we may change
= eH(z0, no ).
0
n - no
n - n + 1
o
.
g(Xn + 1) so that eH(Zo,n +1)
o
0
Therefore we may assume that eH(zoi) is increasing in
I
I
I
I
I
II
--I
I
I
I
I
I
..
I
I
ft
I
I
I
I
I
I
I
97 ,
for
..II
In light of the discussion at the end of section
o
2.3, it then follows that
e (n)
o
=
1
n + 1
+
o
n
o
I
'-J
i=l
Similarly
el(n)
.
r n-i+l
-1 log n - i + 1
n - no
n - i + 1
0
~
}l
n
o
=
~ i=l
I{
n - nl
n .. i + 1
log n- i + 1
n-i+l
1
"I
J
Therefore by equation (2.13),
(4.2)
sup
o<c<l
If we redefine
tt
I
I
I
I
I
I
i=l, .•• ,n + 1.
get)
=
f
exp(a-b)
L 0
if
0
oS t oS exp(b-a.)
elsewhere,
it follows, in a similar manner, that
(4.3)
ex + (
) log T
l-a
et ,log ~
l-a
f
l
where
n0-- n 1
no - i + 1
}l
As a special case of (4.2), we obtain from equation (2.15),
98
f3 log
(4.4)
h
+
(1-f3) log
¥
eo (n l '
As an illustration, let no
For these values a
= f3
= .05 and by (4,3) and (4.4)
en (N)
o
e
n1
= 5, n l = 10, a = 2.064 and b =-1.661.
(N)
> 1.65
> 2.64 •
-
These bounds are considerably better than those of example 4.1 and
also they are true for any sequential test procedure, not exclusively
an s.p.r.t.
To conclude this section, the general bounds given in section
2.6 are applied to teN) and e(~).
Following the notation of sections
1.2 and 2.6 we have
-
[0,1] uniform distribution.
Therefore, P[ Zi
€
C ] may be obtained
i
from tables of the incomplete beta function.
Exat!l1>le 4.2
Let no
a = f3 = .05.
--I
I
I
I
I:
I
I
Iti·
where u. is the ith order statistic of a sample of size n from a
J.
I
I
= 5, nl = 10, a = 2.064 and b = -1.661 which ensure that
Also denote Pi = P[ Zi
€
Ci ]. Then for the case of
n = no we find tha.,t
Pl = .9051, P2 = .7715, P3 = .6135, P4 = .4233, P = .2285 and
5
I
I
I
I
I'
I
..
I
I
I
,
I
I
I
I
I
I
I
t'
I
I
I
I
I
I
('
II
99
P. =
J
0
for j > 5.
Therefore by (2.24)
we obtain
3.94 ~ e(N) ~ 2.87
27.00 ~ e(~) ~'9.13,
while the correct values are
ly for n
P5
= nl
= .3617
we find Pl
and Pj
=0
teN) = 3.76
= .9910,
for j > 5.
4.94 > e(N)
P2
and e(~)
= ·9825
, P3
= 16.59.
= .9367,
Similar-
P4
= .6686,
It therefore follows that
~
4.45
25.44 > e(~) ~ 20.43,
while e(tf)
= 4.84
and e(~)
= 24.45.
From examples 4.1 and 4.2 it appears that the general method
given in section 2.6 will produce superior resultsWben there is
considerable variation among the distributions of the zi' s.
At least \
we are safe in stating that the Wald type bounds are likely to yield
poor results in this particular situation.
4.3 Bounds on the O.c. function for the s.s.s.t.
We begin this section by applying the bounds
to the s.s.s.t.
g1. ven
by (3.22)
In order to do so, we must first have zl' ""zn +1
'0
independent and hence Zn +1 -> a - b if n > n0 and Zn +1 -< b - a, if
o
0
n
= no'
If this is the case, then the equation 'In +1 (t) = 1 will not
o
have a nonzero solution.
Therefore we define the s.p.r.t.,
us say, as the same test as the s.s.s.t"
S, let
except that z.~ has the
same distribution as z for i > n , and zl' z2' ••• are independent.
no
0
It then clearly follows that
100
if
n
= no
if n >n.
o
Therefore we shall apply (3.22) to the test S and in this manner ob-
> n0 ].
tain bounds for P[ accept Ho ] in terms of P[ N
Now it can be shown that
ta
c e i
i
> - c i and
for t
1= -
t(zi + r)
e
e l z < -r
[
i _'
ci
ci+t
= 1,
i
for r >
.•• , no
t > -c. and
0,
J.
=.1,
i
t(ai-r)
.•. , no
J
[ 1 - l-e -c (a.-r)
i
J.
l-e
Suppose t > -c i for i
for t
> -ci'
and i
= 1,
n - n
=
n - n
o
o
••• , no .
n-n +1
no (i .e. t > n -n0
= 1, ... ,
1
that
TJ(t)
< r < ai
0
+ 1
+ 1 +(nl-n )t
0
for
t
>
0
)
0
then it follows
I
I
--I
I
I
I
I
I
I
--I
I
I
I
I
I
..
I
I
I
101
,
I
I
I
I
I
I
I
t'
I
I
I
I
I
I
,
II
n - n
5' (t)
o + t
for
t
< o.
tai
l-e
( 1 -ciai
l-e
)
for
It can be shown that
e
sup
r >0
[
t(z. or)
e
~ Izi ~ r
1= t+c:c.
~
~
~
t > o.
Hence
5(t)
= max
>i >
n
0-
f
tai
i
l-e
c
( 1 -c ai ) } for
1 ~t+ci
i
l-e
t
> o.
t
< o.
SiInilarly we have
T)'
(t) = min
n >i
0-
J2.. ( 1
> llt+C i
tao
_
,;;.l-....;e_~_
l-e
) }
for
-ciai
We are now in a position to apply (3. 22) to the test S.
easily shown that for i
= 1, •••
, no
if n
= no
So for simplicity, we will use these two values of
tion.
i
= 1,
n
= no'
First of all, it should be
note~
•.. , no is satisfied for both t
respectively.
It can be
Now assume n =n
n - i + 1
x = l
as an illustra-
that the condition t > c
= -1
l
n
and t
and t
=1
= -1
when n
i
= nl
and define
forand
I
I
102
We may then write
c
i
-t+c. (1~
l-etai
)
l_e -ciai
=1
_ ( x;~)
1 -
x-l
=1
(~)x
x
1
(x-l)
[
_
l'
x x
(x:r)-1
Since x > 1, we can show that the last expression is increasing in x.
Now x is decreasing in i
is decreasing in i.
so that
= nl ,
- 1
1-(:1.) n1 -no
n
for n = n •
o
We also have
for n
= Dl
for n = no
The following table gives the upper and lower bounds for P[accept HJS]
based upon (3.22),
I
I
--.
and by a similar argument
n
_
. 0
o
I
I
I
I
I.
Hence,
for n
8(1) =
~
The values of a and b are the sane as those
I
I
I
I
I
I
..
I
I
I
"I
I
I
I
I
I
I
t'
I
I
I
I
I
I
f'
I
103
given in section 1.3, so that for the s.s.s.t. we have ex = f3
= .05
in each case.
. TABLE
n
0
n
1
n
lower bound
.896
.018
.211
.902
.967
,015
.254
.895
·927
.950
.~
.145
.012
.146
5
15
20
10
n • n1
lower bound upper bound
.183
.877
10
upper bound
.~8
10
15
20
4.3
·930
.956
5
5
10
= Do
.914
.009
.163
·902
.958
.926
.020
25
·921·
.948
.011
.131
.124
25
.905
·925
.019
.125
15
25
20
15
20
.916
Finally we arrive at the bounds for P[accept H J by (4.5) and by
o
using the correct value of peN
> noJ.
It should be noted, however,
that in some cases peN > n ] is quite large.
o
Also the values in the
following table have been truncated to [o,lJ.
TABLE 4.4
n = DO
n • n1
lower bound upper bound
n
1
lower bound
upper bound
5
10
.877
1.000
.000
5
15
20
.896
·989
.976
1.000
.000
.183
,211
.000
.254
.000
.145
.001
.001
.146
.163
.000
.131
n0
5
10
15
·902
.895
.914
.916
10
20
10
25
20
·902
.961
1.000
25
·921
.961
.003
.124
25
·905
1.000
.000
.125
15
15
20
.• 967
Although the calculations will be much more complicated, the same
104
technique can be used for value s of
n
other than no and n .
1
Bounds for the o. c. function nay also be obtained by using the
methods of section
= n o,
As before, we require that Zl' z2' ... are
To ensure that R(n +1) > A if n > n and R(n +1) < :I if
o
0
0-
independent.
n
3.4.
we define
if n > n
if n
where d > a-b.
o
= no
Suppose we are able to find a distribution function
G such that
G > Fi
where
F
where
S
for i
= 1,
0
Then by theorem 3.1
denotes an s.p.r.t. with zl' z2' •.• independent and zi
having distribution function
P[accept H IS]
o
G.
Next we may either apProximate
or find an upper bound for it by the usual techniques.
The same procedure also yields a lower bound for P[accept H ].
o
Returning to the s. s. s. t., let us suppa se that n
we wish to obtain an upper bound for P[accept H ].'
o
> no and that
Now d
may be
made arbitrarily large without changing the outcome of the procedure,
so we may restrict attention to finding a
for i
Recall that
= 1,
•.. , n •
o
--I
I
I
I
I.
••. , n +1
is the distribution function of zi'
i
I
I
G such that
I
I
--I
I
I
I
I
I
..
I
I
I
105
,
I
I
I.
I
I
I
I
~
I
I
I
I
I
I
t'
I
r exp [ ci (z-ai )
l
=
]
1
for Z
for Z
. > a.J.
where c. is decreasing and a. is increasing.
J.
i
f
:s a i
J.
Thus F.(z) and Fj(z) ,
J.
j, intersect at exactly one point, other than at Fi(Z)
= Fj(Z)=l,
namely
ciai
c
i
=
f
. Also note that for j
1
2: Fi (z)
determine a
each F ..
J.
-c
j
i, d ij < min (ai,a j ).
for z
2:
d ·
l
:By conti~uing in this fashion we may
G which will be composed of, at m:>st, one segment from
As an illustration let no = 5 n = n l
shown that
i
dli
d
2i
~i
d4i
1
and
If we define
= max
i > 1
d
then F1 (z)
- c.aJ j
2
-.370
3
-.463
-.555
4
-.613
-.735
-·915
5
-·955
-1.15
-1.45
-1.98
= 10.
It may then be
106
G(z)
=
F (z)
1
F (Z)
2
F (Z)
3
F (z)
4
F (z)
5
if
z
~
-.37
~
if -.37
if -,555
z
~
~
z
•I
-,555
~
-,915
if -.915 ~ z ~ -1.98
if -1.98 ~ z •
In order to approximate P[accept H IS]
o
by Wa1d l s o.c. formula, it is
necessary to solve
which is calculated as,
+' 0672 e
+
-·915t _ 015 -1.98t
•
e
+
t + 1.4
.216e-· 37t _.1538e-· 555t
t + 1.8
+
2e .693t -. 24 e -.37t
It can be verified that the solution t
.
By taking the values of
a
t+2
0
-=
1
•
falls between -,65 and
-.64.
and b which yield ex = f3 = .05 and
applying Wa.ld I s formula, we obtain
It should be noted tbat if Wald I s approximations for
the five percent level are used, then
I
I
a
and b at
I
I
I
I
I
I
--I
I
I
I
I
I
..
I
I
I
t'
I
I
107
If we are interested in a lower 1::Pund, then the variable z
+1 will
no
cause difficulties, since it:may happen that G = Fn +1" So the best
o
course of action in this case is to replace the test by another, as
was done in the beginning of this section.
In conclusion we will illustrate the use of bounds (:3.36).
Using the notation of sections 1.2 and 3.6 we he. ve
I.
I
I
I
I
t'
I
I
I
I
I
I
t
'I
and, as was shown at the end of the previous section, this value may
be found from tables of the incomplete beta function.
% = P[Zi €
Ri ]
Denote
and by taking the same parameter values as those
given in example 4.2, we find that for n = no'
ql
= 0,
~
= 0,
~
:=
.0046, ~
= .0252,
~
= .0312
and
qj =
0
> 5.
for j
Now by using the values of P[Zi €C ]
i
2:
1
P [accept Ho
given in example 4.2, we have
J2: .939·
Similarly, 'When n = n we have
l
ql
0,
:=
and qj = 1 for j
~
= 0,
~
= .0411, Qq. = .3094,
> 5. Thus
.0859 > p[
-
acce~tH0 1-> .0175
•
~
= .6231
108
I
4.4 MUltivariate normal
In this section we will look at the problem of determining
P[accept Ho ] and e{N) for an s.p.r.t. in which zl,z2"" have a multivariate normal distribution. MOre specifically, assume that the distribution of zl,z2" .• ,zn is N[ ~ , Ln ] for each n and that these
distributions are consistent. We begin by finding the restrictions on
J:! and
L which are necessary in order to apply the bounds on
P[accept H] which were given in section
o
~. 3 •
Define
and t n as the nonzero solution of CPn (t) = 1. Then, in the terminology of section 3.2, we require that the conjugate densities be
consistent.
Now
=
,
where
J
=
exp
j ,
J L J
{ -2 -n
n-n
+
Since L is positive definite,
~
LJ
is
positive and therefore
(4.6)
t
n
ea
I
I
I
I
I
I
I
ea
,
(l, .•• ,l)
I
I
=
n
Hence cpn{t) • 1 has a unique nonzero solution if and only if i~ J.li
n
is nonzero. Thus we shall that inJ.li 1: 0 for all n. Now it is clear
that the conjugate density is N[ .-on
u + t nn-n
L J , Ln ]. Therefore a
necessary and sufficient condition for consistent conjugate densities
I
I
I
I
I
I
..
I
I
I
~
I
I·
I
I
I
I
I
it
I
I
I
I
I
I
t'
I
109
is
for n=2,3, .. ,
(4.7)
(~)1
where
denotes the vector ~-i' Condition (4.7) may also be
written as
(4.8)
n
t
\ ' CY .
n 1-, j J.
i=l
for
j=1, ... ,n-1 and n=2,3, ••••
we may determine t
n
Thus, if we are given
~
and I:,
for all n and decide if the conjugate densities
,are consistent by the use of (4.6) and (4.8).
Now supposing the con-
jugate densities are consistent, (4,8) then implies that
(4.9)
By (4.6) we have
n-1 n-1
(4.10)
t n _1
I I
n-1
-2 \ ' IJ.
CYji =
L
j=l i=l
j
j=l
and
n
(4.11)
=
-2
I
IJ.
j=l
Thus if the conjugates are consistent, we obtain by (4.9), (4.10) and
(4.11) that if IJ.n
F0
then
-2 IJ
(4.12)
tn =
n
n
j~lCYjn
,
110
which is less complicated than (4.6).
In order to apply Wald's o.c. formula, as discussed in section
3.2, we mst have
t n= t
for alln, and the consistency of the
l
IJ.n
1J.1
Suppose E is diagonal and --= --CJ'nn
CJ'11
conjugate densities.
for all n, then by (4.6) we have t
=
n
we have consistent conjugates. If t =
n
jugate densities are consistent, then E
Iln
for all n by (4.6), and
CJ'nn
t
t
1J.
1
= ---
CJ'11
l
for all n
for all n and the conl
is diagonal by (4.8), IJ.nf 0
for all n by (4.32).
Therefore a necessary and sufficient condition for
t
n
and consistent conjugates, is that E is diagonal and
for all
,
and by (4.8)
n.
=t l
for all n,
--- = --IJ.n
CJ'nn
1J.1
CJ'11
If we are not permitted to use Wald ISO. c. formula, we may,
however, apply the bounds given by (3.15) and (3.16) if consistency
> 0 for all n or t n < 0 for all n. :By
(4.6),
. a sufficient condition for t n > 0 for all n is IJ.<
n 0 for
holds and either t
n
< o.
n
Next we shall consider the, important case of
all n, and similarly for
t
= IJ. for all n.
n
If we can find conditions on E, such that the conjugates are consis-
IJ.
tent, then in light of the above remarks, the bounds (3.15) will be
applicable.
By (4.6) and
n = IJ. for all n, the necessary and sufficient
condition for consistency given by (4.8) becomes
(4.13)
IJ.
n
I
I I
n
n
n-l
CJ'ij
i=l
n
i=l k=l
CJ'ik
(n-l)
=
IO'ij
1=1
n-l n-l
I I
i=l k=l
for j=l, ... ,n-l and
n=2,3, •.•
CJ'ik
I
I
ea
I
I
I
I
I
I
I
ea
I
I
I
I
I
I
..
I
I
I
tt
I
I
I
I
I
I
I
t'
I
I
I
I
I
I
t
I
ill
We shall next show that
(4.14 )
0"in
(4.15)
(n-1)0"1n
a~l
for
n
(4.13).
-
for
0"ln
+
O"nn
i=l, •• , ,n-1
=
+
0"
n+1,n+1
(n-1)0"1,n+1
are a sufficient, and possibly, necessary condition for
Now it can be shown by induction and (4.15) that
n-1
(4.16)
)' O"li
i=l
-
for all
(n-2)CT.
ln
'_....J
n.
From (4.14) we have
n
n
n
I L
(4.17)
O"ij
=
i=l j=l
L
O"ii
n
+ 2
i=l
L
O"ij
i<j
n
2
n
L
(i-1)0"1i
=
-
(j-2)0"1j} +
i=2
+
2L
(i-1)0"1i
i=2
n
0"11
+
=
n
j-1
LL
O"li
+
j=2 i=l
L
n
iO"li
i=2
=
n
L
O"li
i=l
Also by (4.14) and (4.16),
=
(4.18)
j-1
n
.~
O"jj +
O"li
n
i=j+1
n
L
O"li
i=j+1
=
j
L...J
i=l
O"li
=
(j-1)0"1j +
L
O"li
i=l
+
=
112
Now by (4.17) and (4.18) we obtain for
j=l, ••• ,n,
ea
n
I
LL
n
O"ij
i=l
n
n
=
1,
If
O"nn = 0"11
O"ij
i=l j=l
and (4.13) is therefore satisfied.
for all n, then
(4.14) and (4.15) imply that the off diagonal elenents are equal.
Thus
the form
1 p p ...
p 1 p
;
=
p p l •••
p p p ...
is the only type which has equal variances and satisfies (4.14) and
(4.15) .
Now let us consider an alternative approach to the problem by
making use of the nethods given in sections 3.4 and 3.5.
b = ..& and P [N < 00] = 1.
Assume that
Then define
n..1 n..1
I I
=
for
n=2,3, •••
for
n =
O"ij
i=l j=l
o
and
a
=
{
I
I
=a
if ., >
0
if n., <
n
0
•
0
I
I
I
I
I.
I
I
ea
I
I
I
I
I
I
..
I
I
I
t'
I
I
I
I
I
I
I
Let H'
t
I
zl,z2' •• '
are
f
independent and the distribution of zn is N[a,
n 13n ] , 'Where
n-l
13n = CJ'nn
?'n
CJ'ni
i=l
I
-
n-l
an =
Also let H"
fJn
+ ?'
n
I
Jl cn
fJi
}
~
i=l
denote the hypothesis which specifies that
zl,z2".'
are independent and the distribution of zn is N[a,
13n J where
n
an
=
fJn +
?'n
r
l
n-l
-C n
-
\'
L
fJi
i=l
Since the distribution of z given Z 1 is
n
nn-l
it
I
I
I
I
I
I
denote the hypothesis which specifies' that
N { fJn +
?'n [ Zn_l -
I
fJi
J'
i=l
. by theorem 3.2,
(4.19)
Next let us a.ssume tha.t a nf
0,
Then we have
H" :
zn -
N[ an' rnn
aJ
and define
114
Also define
Now, if a
section
-r
=
sup ( r.)
J.
i
-r
=
inf ( r )
i
i
s
=
sup ( s.)
J.
i
s
=
inf ( si)
i
n
ea
> 0 for all n, we are able to apply the results of
3.5 to obtain
(4.20)
PH,,[accept HoJ
u
where
H"u ••
I f an
<
for all n, we may, by SytlllIetry considerations, also apply
0
the results of section
p~[accePtIS.J
3.5 to obtain
$
PH,,[accept IS.J
$
PHJ[accept 1\J
,
and by assuming termination with probability one, we have (4.20).
,
In a similar manner, if an
,
> 0 for all n, or a n <
0
for all n,
it then follows that
(4.21)
where
,
,
H' •
'J •
Z
~:
zn .... H[ an' s an J for all n
n
....
N[ a , s a J for all n
n - n
,
-
,
and
zl,z2""
independent
and
Zl' z2" .• independent.
From the remarks made earlier, we may then use Wald' so. c. formula to
obtain approximate values of P[accept HoJ
~.
under
I
I
Hi, ~ , H.e and
Thus we find approximate upper and lower bounds for
P[accept HoJ •
I
I
I
I
I
I
I
ea
I
I
I
I
I
I
..
I
I
I
tt
I
I
I
I
I
I
I
t'
I
I
I
I
I
I
t'
I
115
t
= IJ. > 0
for all i , then al > 0 and a l > 0 • So,
in order to apply the above technique, we require that an > 0 and
Suppose lJ.i
,
an >
for all n.
0
thus a
n
>
implies that
0
then a n = IJ. + rn (-a-(n-l)lJ.)
< '1 < (n-l+ ! )-1 • If r < 0
n
rn > 0
If
0
n
IJ.
a n = IJ. + r n (a-(n-l)lJ.) , and 'thus a n > 0
IrDlst imply that
0
and
,
,
then
requires that .! > n-l
fJ.
> r > (n-l- ! ) -~ This may be summarized as
n
fJ.
follo'tvs:
(4.22)
a
n
>
0
if and only if
(a)
rn >
0
implies
(b)
1
<
0
and .! > n-l implies
and
n
rn <
(n-l+.! )-1
fJ.
IJ.
rn>
(n-l+ !):l
IJ.
,
t
> 0 and
If r n > 0 , then a n = IJ. + r n (a-(n-l)lJ.)
n
--,
,
requires
that
r
<
(n-l-.!
)-1.
If r n < 0 , then 0:n =
n-l
n
IJ.,
r n (-a-(n-l)lJ.) , and therefore an > 0 • In summarizing this, we
and thus a
! <
IJ.
IJ.
+
find
that
t
(4.23)
an >
0
if and only if
(a)
r > 0 and .! < n-l implies r < (n-l- .! )-1.
n
n
IJ.
Similar results are obtainable for the case
IJ.
<0
IJ.
.
To illustrate the above procedure, we shall consider three examples of common covariance matrices; in each case let us assume that
IJ..
J.
= IJ.
for all i
Example 4.3
and a >1 p. > 0
•
Let
...
==
2
cr
...
It then can be shown that
116
Ipl
1
<
and
1
1p'
[ 2(n:2) + ~
=
7n
If we assume that
a
n
> 0 if 2(n-2)
12 -> p > 0 , then
+ E:1 > n-l + !
p
J.L
if
n
..
I~I >0
implies
2
for
n>l.
> 0 , and by (4.22),
n
Therefore, a > 0 for all
n
'1
1
o 0
(n-l) (1+
However,
if n-l >!.
J.L
1)
p
> 1+ 1 > 2 > 2- !.
P
J.L
I
an > 0 for all n if 0 < P:S
(n-l) (1+ 1) > 2-!. •
p
J.L
implies
1
2"
for
n >1
and therefore
Next let us asswne that
-
i:s
p < o.
Then 7 < 0 and 7n > 0 for n > 2. By (4.22), a n > 0
2
for n > 2 if (n-l) (1+ 1) > 2 +!. but (n-l) (1+ 1
p) <0. Therep
J.L
fore there does not exist a p < 0 such that an > 0 for all n.
,
By (4.23), if a
> 0 for all n then the following condition must
n
be satisfied for all n:
!. < n-l
implies
J.L
Since
-(n-l) (1+ 1)
(n-l) (1+
1)
P
> 2- !.
J.L
can be made arbitrarily large for a sufficiently
p-
(n-1) (1+ 1) < 2-! is true for large n. Therefore there
p
J.L
I
does not exist a p < 0 such that an > 0 for all n. The above
large
n,
:may be summarized as follows:
(4.24)
a n > 0 for all n
if and only if
o 0 for all n
if and only if
o <
p
,
Now for
p
2: 0 we have
~n = u2 (1
a
n
=
- p '1n )
J.L + 7 (-a - (n-l)J.L)
n
< (1+ !.)-l
J.L
1,.
< 2 •
-
I
I
I
I
I
I
I
I
I
ea
I
I
I
I
I
I
..
I
I
117
I
tt
,
Thus,
r
I
I
I
I,
I
I
I
it
I
I
I
I,
I
I
t'
I
~ + 7n (a - (n-1)~) .
=
an
for
p
n
> 0.
d2
=
and
2(n-2) + (n_1)p-1 -
p
~[2(n-2) + (n_1)p-1] - [a + (n-1)~]
n
> 1,
and
for
rn =
p
=0
or n
= 1.
Since the numerator and denominator of r
are linear in n, the
dr
n
sign of dn·n
is independent of n for n >1. Hence,
-r
=
=
!
max( r , r , lim r )
1
2
n~oo
n
min( r 1, r , lim r n )
2 n~oo
2
NoW
and
lim r n
n~oo
r
= ~
;
1
(2p + l)(p + 1)-1,
~
=
2
2
a)-l
,
~ (1 - p )(1 -p - p ~
(1_ p2) (l-p-p !)-1 > (1_ p2)(1_p)-1
Since
= -~ .
r
-
~
=
~ (2p+1)(p+1)-1 ~ 1, we have r 2 ~ lim r ~ r 1
n~oo n
Therefore,
_
(j'2
r
= -~
r
=
2
(1 - ~ )(1 - p - p -a)-l
~
2
(j'
~
Using a similar argument,
where
8
1
=
-s
=
max (
8
8
=
min (
8
(j'2
~'
2
sr.
~
2
, s2' lim 8 )
n
n~oo
1
, s2' lim s )
n~oo n
S
1
a -1
-)
and
2
= -(j'2
(l-P ) (l-P+p
~
Now, since a > ~ >
(2p+1)(p+1) -1.
(1_p2)(1_p+p)-1
s
1
S
(1+2p)(1+p)-1
~
0
,
A
Therefore,
lim s
n~oo
n
=
•
118
-s
=
~(1
IJ
s
=
CT (
2
-
IJ
+ 2p)(1 + p)-l
ttt
1· p2) (1 - p + P -a)-l
IJ
I
I
I
I
I
'I
I
If we define
H(t)
then
by
=
e at _ 1
at
-at
e - e
'
Wa1d's o.c. approximation,
PH' [accept H ]
....a.
PH,,[accept Ho ]
u
....a.
J,
~
where H,t'
0
are as defined in (4.20) and (4.21).
Thus we may
conclude that:
(a)
a).l
If o p 2:
0
will ensure that
I~ I
I
I
> o. Therefore we will assume
tta"
I
I
I,
I
I
I
..
I
I
I
tt
I
I
I
I
I
I
I
119
that
'Y
n
1 > p
2: o.
Now
i'n
= (n-2+~) -1
for
n >1
and hence,
> 0 for n > 1. As in example 4.3, it can be shown that
a >
n

n
0
for all n
if and only if
0 ~
r
=
~ [n-2+ -p -p [fJ(n-2+ -p ) - (a+(n-l)fJ' ]
-
t
Also,
n
2
and as before,
1].
r = max(
fJ
p < 1 .
1 ) -1
for
n >1 ,
and similarly for !..
r , r , lim r )
2 n-. CXl n
l
2
lim r = CXl , r = E:. (1_p2)(1_p_p !) -1, lim s = CXl and
n-. CXl n
2
fJ
fJ
n-. CXl n
2
E:. (1_p2)(1_p+p !)-l
There f ore we may conc 1u d e tha t if
fJ
fJ
s2
<
0 _
Now
=
p
< 1,
then
tt·
I
I
I
I
I
I
f'
I
It is known that
Ir; I > 0
o.:s p < 1.
consider
for
n
is greater than or equal to
p2) + 2plog p] > 0, we have that
t
n
n>l
However, we shall only
Now it can be shown that
Since the denominator of i'
to have a
Ip I < 1.
., >0
n-
for all
n > 1.
(n-l) [(1In order
> 0 and an > 0 for all n, it is required that for
120
c is defined to be ~ for the an
whenever n-1+c > 0 and where
case and - -a for the a • case.
tit
This is equivalent to
n
J.L
>
(4.25)
for all n > 1 whenever n-1+c > o.
c +
I
I
I
I
2
r:p
Next we define x=n-1, d=p ,
e=(1+p)p-1 and rewrite (4.25) as
hex) == x[ e(l_dx ) -1 - 1] > c +
where x 2: 1, 0 < d < 1
e
~d
~
2.
Now
h'(X) = [1_dx]-2[e(1_dx)_(1_dx)2+edx10g(dx)]
x
x
g'(d ) = e10g(~) + 2(1_d )
x
X
x
where g(dX) == e(l_dx ) - (1_d )2 + ed 10g(dx ). Hence, g' (d ) S 0
since e 2: 2. Thus g(dx ) ~ g(d) and since .~ = 1-d+d1og(d)
and
x
0, we have g(d ) ~ 1_d2+d10g(d2 ) ~ o.
I
~
Therefore h' (x) 2: 0, so that
heX) is a nondecreasing function of x. Hence (4.25) is satisfied if
(4.26)
(1+p2)[p(1_p)r1 > 'c + 2(1_p)-1
which is equivalent to
•
an >
0
p < (l+c)-l for
c > -1 and
p > (l+c)-l for
for all n if and only if 0 S p < 1.
If we define
p(1-pn-1 ) (1-p )-1 7n
1 - (n-2-k)7
n
1
then
.!L s
;. n
J:L r
;.
n
~
a
if k = -
J.L
if k
= - -aJ.L -
I
I
..
I
I
I
I
11
II
- 1
1
,
I
I
•
..I
I
,I
,I
.'I
I
,I
',1
,
;
rJ
i
fl
I
121
.
Next let us assume that n
~
for all n, we have . 1-(n-2-k) 'Yn >
corresponding to whether k
since the denominator of
=
~
~
I
Then, since an >
2.
-
0
and an >
0
0
for the appropriate range of p
1 or k
= - ~~ - 1,
Therefore,
rn is positive, we may write
'Where the denominator is positive.
We wish to show that
sn+1" s2
~ 0
and this will follow if rk(n+l) - r k (2) ~ 0 for k> o. Now
. rk(n+l) - r (2) ~ 0 if and only if [n(1..p2) _ 2p(1_pn) _ p2(1_pn)2 J
k
X[l+pk] + [n(1_p2) - 2p(1_pn) - (n..l_k)p(1_p)(1_pn)][p2_ l ] ~ 0 ,
which may be written as
(4.27)
'Where
it
I
I
I
I
I
I
,
I
Now gn
=
(l..p)(1_p2)pn[n(1_p) - p(l_pn)]
and suppose that we set
= n(l-p) - p(l_pn). Then tn~
o o t = (l_p)(l_pn+l) > 0 and
n
n
therefore g > 0 since t > t = (1_p)2. Next we wish to show that
nn- l
l' > g. It follows from the definition of l' and g
that
n- n
n
n
l' _ g = n(1_p2) (l+pn(l_p» _ (1_pn)[(1_p)2 + p3 + pn+l(1_2p)J.
t
n
Suppose
n
P:S
+ p2(1_2p)]
i, then we have
= (l-p)[n(l+p) n
2
f - gn ~ n(1_p2) - (l..pn) [(1_p)2 + p3
n
(1_pn)(1_p+p2)] • If we set t =
n
.
2
then it follows that t n+l - t n ~ 1 + 2p 3p3 + p4 > o. Therefore l' - g > (l-p)t > (l-p)t = 2p + p(1_p)2
. nl
n n~ o. The case p ~ i may be shown in a similar manner. Since we
n(l+p) - (loop )(l-p+p)
have shown that g >
n -
0
and l' > g , (4.27) follows.
n- n
Also we have
p[(l+k)f + g ] < 0 for k < -2. Therefore r - r < 0 for
n
n n 2 n > 2 and since r (2) = (1_p2)(1+kp)-1 we have s2 S Sl and
k
that
Hence
r
=
s
=
IJ
-
IJ
2
S
l-p, l ... pilp
= j:;~ .(2)(
2
a)-l •
~
We finally conclude that
(a)
if
0
S p < (1+ ~)-l , then
IJ
P[accept H ]
o
(b)
if
0
$
H{
-,(l-P-P~) (1_p2) -1
}
,
P[accept H]
0
in the above examples as follows.
'
a.s.n. is quite large, it may then be reasonable to obtain
If the
r
and r
from r n , r n +1".' instead of r l ,r2,... . Thus it is felt that
o
0
for large a.s.n. an approximation for PHil [accept Ho ] and
PH' [accept Ho ]
is
H{
~ li; r
n-+oo
tively.
If
lJi=
IJ
n
}
for all i, then
and
H { - li; Sn }
,
respec~
n-+oo
lim r n = lim s n , so we then
n-+oo
n-+oo
have an approximation for
P[accept H J.
o
Therefore, assuming large
a. s .n. we have
(a)
example 4.;:
P[accept Ho ]
=1=
I
I'
I
oS p < 1 , then
Basing our arguments on intuitive ideas, we may obtain approximations
to
.
~
I:
I-
2
a)-l
-~ (2)(
l-p l-p-p-
and
I
I
H {-,(1+P)(1+2 P)-1 }
I
I
.,
I
I
I
I
I
I
..
I
I
I
123
I_
I
I
I
I
I
I
I
(b)
example 4.4
P[accept H]
o
(c)
example 4.5
P[accept H ]
o
Then if we set
1:
o
=
*
~
2
(J" I , it follows that
This may have some intuitive appeal, remembering that p
IJ.
>0
and
> o.
Since
zl' z2' ... are not independent, our only available means
for calculating the a.s.n. is the general method given in section 2.6.
Now Zn has a normal distribution with mean
n
n
j~,i~(J"ij .
n
i~ 1\
and variance
Thus
tt'
I
I
I
I
I
I
co
Although an upper bound for n~ P[Zn € en]
nnlst be found, there should
o
not be any difficulty in applying the bounds given by (2.24).
It
should also be remarked that the general bounds of section 3.6 could
e~ual1y
be applied to
P[accept H ].
o
,
I
.1
I
I
-,
CHAPTER V
TOMLINSON PROCEDURES
5.1 Introduction
In
1957, Tomlinson [30] proposed a simple sequential procedure
for discriminating between
H:
m<1l
and
o
'Where m is the median Of the univariate distribution F
random observations are sequentially observed.
sists of choosing two points on the real line
".e <
from 'Which
Tomlinson I s plan conAI,
and
\.
such that
131 = (-00, "'.e]'
Il < '\ and defining four regions, 11"" ,:84, where
B = (AJ"Il], :B = (Il, \ ] and 134 = (\,00). Next, random observations
2
3
are sequentially drawn from F until one of the following four events
occurs:
(a)
(b)
one observation falls into B
l
two consecutive observations fall into B
2
(c)
two consecutive observations fall into
(d)
one observation falls into :1
13
3
4 .
Sampling is then terminated and
or (b) has occurred, otherwise
H
o
11.
is accepted if either. event (a)
is accepted.
Pl == F(AJ,)_
P2 = F(Il) - F(AJ,)
P3 = F(\.) - F(Il)
P4 = .~•. r- F(,\) ..
Suppose we define
I
I
I
I.
t'
I,
I
--.
I
I"
I
I
I
I
..
I
I
I
I_
I
I
I
125
Then Tomlinson has shown that
Pf accept Ho
2
(1+P2) (P3 + P4 (1+P3) )
1= 1. -
1 - P2 P3
(5.1)
a.s.n.
=
(1+P2) (1+P3)
1 - P~3
and has observed that the a.s.n. has an upper bound of three.
=~
~+
I
I
I
I
~ and if F is a normal distribution with mean e and variance
it
I
I
:1
I
I
I
,
I
"A£
-2cr and \
=
further suggests that
standard deviation of an observation.
He
2cr 'Where cr is the
If we use these values of "A ,
u
d2,
tl1en it can be shown that P[accept H ] is approximately equal to
o
.99, .95, .05, .01 for
e=~
-2cr,
~-cr,
f..1-foO", f..1+2cr respectively.
Tomlinson's procedure, along with others of an equally simple
nature, was considered by Craig [11], who pointed out a major weakness in Tomlinson's scheme.
of
e
The difficulty is, that given two values
and their corresponding required probabilities of acceptance,
there does not appear to be a straight forward method of determining
~ and \. (assuming they exist) Which will guarantee these proba-
bilities.
In practical applications, however, this problem may be
attacked by tria1-and-error methods using high speed computers,
The remaining part of this chapter is concerned wi th procedures
of Tomlinson's type, gerera1ized and applied to a :multiple decision
problem.
5.2
A simple multiple decision procedure.
Let CB =
Cll' ., . , B£ ) be a finite collection of disjoint
Borel sets on the real line, and Ge
a distribution function
126
depending on the parameter
e,
which may be a vector.
l1.,
m distinct simple hypotheses
that e =
ei
for i =1, .•. , m.
~,
... ,
Suppose we have
Hm, and let Hi spec! fy
Next we associate with each Hi a set
CB , i = 1, .... , .m, where the CB., i
i
J.
.
collections of CB, such that UBi
= 1,
= ill•.
... , m, are disjoint sub-
We now propose the following
sequential procedure.
At the nth stage a real-valued random observation x is taken
n
from Ge and if x is the kith successive observation to fall in B. for
n
J.
some i = 1, ••• , p"
then sampling is terminated, otherwise sampling
continues to stage n + 1 with the xn+l;th observation from Ge.
sampling is terminated at the nth stage with x
-
n
If
falling in :8 , then
i
the hypothesis H is accepted where CB j is the unique set to which B
i
j
belongs.
By applYing some results from recurrent event theory (see chapter
13 of Feller [12]), the mean and variance of the sample size, N, and
P [ a.ccept Hi ] may be shown to have quite simple expressions.
begin, we define Pi
= P [. X
€
:Ii] and ~
= l-Pi
To
where the distribution
of x is Ge. Next, let the recurrent event E represent the occurrence
i
of k
i
successive observations falling in the set Bi , and let E
= UE i •
Then if Ui(s) denotes the probability generating function of the
occurrence of E at trial n
i
and U(s) the probability generating
function of E at trial n, we have
=
l-s+~Pi
ki
s
ki +l
I
I
-,
I
I
I
I
I
I
t
--I
I
I
I
I
I
..
I
I
I
I_
I
I
I
,J
I,
,I
127
If f
is the probability of the first occurrence of the event E at
n
trial n, then it follows that
co
F(s)
J
I
I
I
,
I
=1
.
1
- ---...;;;,---
~
L
Ui
(s) - (.t-l)
i=l
Suppose we define
= l-F(s)
Q(s)
l-s
Then, since
= fn
we have
e(N) = Q(l)
Var(N) =
,-'
I
I
= L
i-l
I
tt
'\.
f i s~
2Q'(1)
+
Q(l) - Q2(1).
Now,
Q(s)
=
1
£
I
U (s) (l-s) -
i
i=l
and from (5.2) it follows that
£
Q(l) ={
I
i=l
£
Q' (1) = Q2(1) { 1-£
+
I
i=l
(£-1)
(l-s)
128
Therefore,
~
e(N) = {
I
i=l
.1 - P i
k
i
1
(-) -1
I
I
-.
-1
}
Pi
I
"
,
In order to obtain P[accept Hil, we define the recurrent event E to
be U E ' and
jfi'
t
n
j
= P{' Ei
.
rn
=P {
occurs £or the first time at trial n while Ef has not
occurred }
E' occurs for the first time at trial n while Ei has not
'yet occurred}
vn = P {E i occurs for the first time at trial n}
wn =
1
E I . occurs for the first time at trial n }.
Now, by letting T(s) denote the probability generating function of
tn' and by using a similar notation for the others, we have
T(s)
= V(s)
- R(s) V(s)
R(s)
= W(s)
- T(s)
and therefore
W{s)
I
I
:1
I
I
..,I
I
,
I
I
I
..
I
I
I
129
I·
I
I
Since
V(s)
I
J
it
,
j
,
=1
=1
W(s)
(U.(8)
_
)-1
~
- ( \
/,
j!i
U.(s) _ (l-2) )-1
J
i-Te arrive at
') (l-s)
I
t
t
V(s) [l-W(s) 1
1 - V(s) W(s)
=
T(s)
T(s)
u/s) -
= f~j1i....--i- - - - - -
U.(s) -
(1-8)
~
(5.4)
~ Ei
{
occurs before E I occurs}
~
k
qj P
j}i
k
j
j
(l-P
j
.
=
T(l)
j
)-1 }
If we define
1 - p.
ci =
1
( -)
Pi
I~
-"
J~
-1
k.
~
~
for i
-1
I
.
(l-s)
-1
Thus it follows that
I
,
,
(l-s) (.e-2)
then by
(5.3)
and
(5.4)
we have
= 1,
..• , .e
=
J,30
A
e(N)
= [
I
.,"
-1
L 1
Ci
i=l
I
if
-,
It should be noted that if kl = k4 = 1, k2 = ~ = 2, J i = (1,2) and
J = C3,4), we obtain T~mlinson's formulae given in (5.1). Also,
2
as with Tomlinson's procedure, ifffi is a cover for the real line,
then e(N) has an upper bound depending upon J and k
can be shown as fol-lows.
= max
i
k..
~
This
Define
,
t
,.
v(p)
I
I
I
Itt
-
then ci > v(P.)"since
c.~ is a decreasing function of k . Because
~"
i
we have assumed that CB is a cover, it follows that max Pi ~ j.
i
Now, since v(p) is an increasing function of P,
1
> v(ao)
J.x;
c.
i=l
and therefore
A
e(N)
=(
L
i=l
-1
1
c~) < (v(-))
...
A
-1
I
I...
J
I
I
=-!A-l
k
(J -1).
I
I
I
..
-I
,
I
II
I
I
I
I
t
t
it
,
•I
I
t
I
,
I
131
It should be remarked that this bo..md i2 not particularly good and
equa1ity can never be achieved because
I
Ci > v(j) •
i=l
An immediate criticism of this procedure is that decisions
will o:ften be based upon only a small portion of the available data.
It is felt that a similar procedure in which
"a total of k. ob1.
servations fall in B." replaces "k. successive observations fall in
1.
1.
B. II is preferable in many cases.
1.
Unfortunately, we would then not
be able to express simply the exact properties of the test as was
1/ .
done in (5.5)
Thus in applying our procedure, we should con-
sider attempting to keep k
small,
(3 or 4), and varying the Bi
in order to obtain the desired error levels.
As an example,
Tomlinson has shown that his procedure, while not significantly
altering the o.C. function, has a uniformly smaller a.s.n. than
a modified procedure (for use where ~ is unknown) in Which~=
and
~
=
_00
00.
Suppose Go possesses a probability density function go • Then
the type of regions described by Armitage [3] should be helpful in
determining a reasonable choice of ill.
These regions are as follows.
If we define
for
j
r i}
for i = 1, ".", m
then it seems plausible to associate an observation belonging to
11
Tsao [31] has obtained the properties of this test for
the case of m = ~ = 2. He also discusses the optimum choice of
B and )2"
1
132
R.~ with the hypothesis H..
~
Thus, as a starting point
~or
deter-
mining CB, we may require
~or
= 1,
i
I
... , m.
Finally, we shall conclude this chapter with the
~ollowing
Example 5.1
SUpPOse G
is a normal distribution with mean
e
e and
variance one and that m = 3 with
9=-1
Although there are many reasonable choices
this example, restrict ourselves to the
:1
1
1
2
~
and set CB
i
o~
CB, we shall,
~ollowing
class.
~or
De~ine
= (-(Xl, -t)
= (-t, t)
= (t,oo
= (Bi )
~or
i
J
,I
,
t
Itt
I
)
= 1,2,3.
to only those cases in 'Which k
l
I~
= ~,
we further restrict ourselves
then we may obtain equal
error probabilities by finding that value
o~
t
such that
(5.6)
I
I
I
I
"-/
and some d.
e(NI~)
t
I
illustration.
I1.:
,
I
.,
Now, by the symmetry
= e(NI~).
several choices
o~
o~
the problem we will have
The ~ollowingtable illustrates these values for
k
l
and k •
2
I
..
I
I
,.
..,
I
,
I
I
I
I
t'
t
,
I
133
TABLE 5.1
t
1
1
1
2
1
3
1
4
2
1
2
2
2
3
2
4
2
2
5
6
3
3
3
3
3
4
3
5
1
2
d
.803
1.264
1.514
1.680
.381
.750
.966
1.115
1.227
1.317
.216
.522
.720
.860
.968
.578.630
.658
.677
.619
.697
.746
1.00
l.00
1.59
2.17
2.73
2.00
3.11
4.28
.779
.804
.824
.655
5.49
6.74
8.03
3.26
.744
5·01
6.98
9.12
11.41
1·79
2.63
3.48
2.08
3.61
5.32
7.13
9·00
10.91
3.84
6.55
.801
.839
,867
In figure 5.1 i'le have plotted the values of
max( e(NI~), e(NI~)
9·81
13.38
17·15
d against
}
for the various pairs (kl , k2 ) given in
table 5.1 (a few additional pairs have aiso been given). In this
figure we have drawn a line connecting (k , k2 ) and (kl , k +1). Now,
2
1
if our criterion for selecting a procedure is to pick the one which
t
has the smallest max( e(NI~), e(NIH ) ), then figure 5.1 will be
2
quite helpful. One simply decides which value of d he desires and
I
then finds the lowest point in the figure which is on or to the right
t
,
I
of the vertical line running through his choice of d.
l34
Figure 5.2 is the same as figure 5.1 except that "a total of k
i
observations in Bi" has replaced "ki successive observations in Bi".
A different value of
hold.
was used in order to ensure that (5.6) would
t
,
I
..
I
t
I
In comparing these two figures, we observe that for the larger
values of
d
and hence larger
~,
k , the procedure represented by
2
figure 5.1 is not particularly good.
This seems to substantiate to
some extent what was said earlier.
One further point of interest is the following.
in table 5 .1 given by k
1
achieve its maximum at
= 2 and k = 2, we find. that
2
e = ..t
and
e= t
For the test
ee(N)
does, not
as in the test given by
Sobel and Wald [27].
In fact, from table 5.2, it appears as though
the maximum occurs at
e=
o.
TABLE 5.2
e
o
.1
.2
;.611
e
.6
.7
.8
;.417
;.;50
;.276
.;
.4
;.561
;.52; -
.9
;.195
1.0
;.109
.',
I
I
--I,
I"'
t
I
t
\
I
136
FIGURE 5.2
I
I
..
I
t
II
l
\5
I,
I
I
..
I
I'
I
,t
I
.40
.70
."15
d.
t
,
I
,
I
fI
I
•'II,
t
BIBLlOORAPHY
[1]
ratio test to reduce the sample size", Annals of Mathematical
Statistics, Volume 31 (1960), p.p 165-197.
[2]
[3]
,
(1950), pp. 137-144.
[4]
Washington:
[5]
I
U.S. Dept. of Commerce,
1951~
Baker, A. G., "Properties of some tests in sequential analysis",
Biometrika, Volume 37 (1950), pp. 334-346.
[6]
Bartholomew, D.J., "A sequential test for randomness of intervals",
Journal of the Royal Statistical Society, Series B, Volume 18
(1956), pp. 95-103.
I
,
Arnold, K. J.(Ed.), Tables to facilitate Sequential t-tests,
(National Bureau of Standards Applied Mathematics Series,7 )
[7]
t
Armitage, P., "Sequential analysis with nx>re than two alternative
Journal of the Royal Statistical Society, Series B, Volume 12
I
II
Mathematical Analysis, Addison-Wesley Publish-
hypotheses and its relation to discriminant function analysis",
I
1
Apostol, T. M.,
ing Company, Inc., London, 1960.
I
t'
Anderson, T. H., "A oodification of the sequential probability
Bhate, D. H., "Sequential analysis with special reference to
distributions of sample size", London University Ph.D. Thesis,1955.
[8]
Bhate, D. H., "Approximations to the distribution of sample size
for sequential tests, I.
Tests of simple hypotheses", Biometrika,
Volume 46 (1959), pp. 130-138.
138
[9J
Blom, G.,
"A generalizarion of Wald' s fundamental identity",
Annals of Mathematical Statistics, VolUllX'! 20 (1949), pp.439-444.
[lOJ
Chow, Y. S., Robbins, H. and Teicher, H., "Moments of randomly
stopped sums", Annals of Mathematical Statistics, VolUllX'! 36(1965),
pp.789-799.
[llJ
Craig, C. C., "On a class of simple sequential tests on means",
Technometrics, Volume 4 (1962), pp. 345-359.
[J2J Feller, W., An Introduction to Probability Theory and Its
Applications,
Volume I, Second Edition, John Wiley and Sons, Inc.
New York, 1958.
[13J
Hall, W. J., Wijsma.n, R.A. and Ghosh, J. K., "The relations between sufficiency and invariance with applications in sequential
analysis", Annals of Mathematical Statistics,
lblume 36 (1965),
pp. 575-614.
[14]
Hoeffding, W., "A lower bound for the average sample number of
a sequential test", Annals of Mathematical Statistics, Volume 24
(1953), pp. J27-130.
[15]
Hoeffding, W., "Lower bounds for the expected sample size and
the average risk of a sequential procedure", Annals of Mathematical Statistics,
[16]
Johnson, N. L"
Volume 31 (1960), pp. 352-368.
"A proof of Wald's theorem on cWl1Ulative surne",
Annals of Mathematical Statistics, Volume 30 (1959), pp. J2451247.
[17]
Johnson, N. L., "Estimation of Sample Size", Technometrics,
Volume 4 (1962), pp. 59-67.
[18]
Kemp, K. W., "Formulae for calculating the operating characteristic and the average sample number of some sequential tests",
I
I,.,
ea·
I
t
:1
1
t
I
t
..
I
t
(I
t
I
t
,
I
I
I
~
I
,
I
I
I
I
I
.,
I
II
139
Journal of the Royal Statistical Society, Series B, Volume 20
(1958), pp. 379-386.
[19]
Lehmann, E. L., Testing Statistical H;y;potheses, John Wiley and
Sons, Inc., New York, 1959.
[20]
Mallows,
c.
L., "Sequential discrimination", Sankhya, Volume 12
(1953), pp. 321-338.
[21]
Page, E. S., "An improvement to Wa1d I s appr ax:ima.tion for some
properties of sequential tests", Journal of the Royal Statisti-
E.!.!
[22]
Society,
Series B, Volume 16 (1954), pp. 136-139.
Ray, W. D., "Sequential analysis applied to certain experimental designs in the analysis of variance", Biometrika, Volume 43
(1956), pp. 388-403.
[23]
Rushton, S.,
"On a sequential t-test", BiolOOtrika, Volume 37
(1950), pp. 326-333.
[24]
Rushton, S., "On a two-sided sequential t-test", Bio:n:etrika,
Volume 39 (1952), pp. 302-308.
[25]
Siskind, V., "On certain suggested formulae applied to the
sequential t-test", Bio:n:etrika, Volume 51 (1964), pp.97-106.
[26]
Slater, L. J., Confluent H;y;pergeometric Functions, Cambridge
Universi ty Press,
[27]
()I. mbridge,
1960.
Sobel, M, and 'Vla1d, A., "A sequential decision procedure for
choosing one of three hypotheses concerning the unknown mean
I
of a normal distribution", Annals of Mathematical Statistics,
I
Volume 20 (1949), pp. 5~-522.
•
-
,•-
[28]
Stein,C., "A note on cumulative awns", Annals of Mathematical
Statistics, Volume 17 (1946), pp. 498..499.
140
[29J Tallis, G. M. and Vagholkar, M K., "Formulae to improve Wald's
approximation for SOme properties of sequential tests", Journal
of the Royal Statistical Society,~,Volume 27(1965), pp 74-81.
[30J Tomlinson, R. C., "A simple sequential procedure to test
whether average conditions achieve a certain standard",
Applied Statistics, Volume 6 (1957), pp. 198-207.
[31J Tsao, C. K., "A sillple sequential procedure for testing
statistical hypotheses", Annals of Mathematical Statistics,
Volume 25 (1954), pp. 687-702.
[32] Wald, A., "On cumulative sums of random variables", Annals of
Mathematical statistics, Volume 15 (1944), pp. 283-296.
[33 J Wald, A., "Sequential tests of statistical hypotheses", Annals
of Mathematical Statistics, Volume 16 (1945), pp. 117-186.
[34J Wald, A., "Sequential method of sampling for deciding between
two courses of action", Journal of the American Statistical
Association, Volume 40 (1945), pp. 277-306.
[35J Wald, A., "Some generalizations of the theory of cUllD.llative
sums of random variables ", Annals of Mathematical Statistics,
Volume 16 (1945), pp. 287-293~
[36J Wald, A., "Some improvements in setting limits for the expected
number of observations required by a sequential probability
test", Annals of Mathematical Statistics, Volume 17 (1946), pp.
466-474.
[37J Wald, A., "Differentiation under the expectation sign in the
fundamental identity of sequential analysis", Annals of Mathematical Statistics, Volume 17 (1946), pp. 493-497.
I
I
e,:
I
I·
I
,:
Il
I;
II
-.i
j
11,
-II
I
11
11
I'
I:
\,
i
I
I
I
.I
I
I
I
I
I
I
,
I
I
I
I
I
•
I
,
I
141
[38] Wald, A., Sequential Analysis, John Wiley and Sons, Inc.,
New York, 1947.
[39] Weiss, L., "Testing one simple hypothesis against another",
Annals of Mathematical Statistics, Volume 24 (1953), pp. 273281-
[40]
Williams, J. D., "An approximation to the probability
integral", Annals of Mathematical Statistics, Volume 17 (1946),
pp. 363-365.

Download Report

Hoel, David G.; (1966)Properties of sequential tests when the conditions for standard approximate formulae are not satisfied."

Paperzz.com

Your Paperzz