UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill , N. C.
SEVERAL TEST PROCEDURES (SEQUE1"TIAL AND
NON-SEQUENTIAL) FOR NORMAL MEANS
by
Michael D. Hogan
December 1963
This research was su:pporte-d by Public Health Service
Research Grant GM-10397-0l, from the Division of
General Medical Sciences, National Institutes of
Health.
Institute of Statistics
Mimeo Series No. 379
TABLE OF CONTENTS
CHAPTER
PAGE
.. .. iii
ACKNOWLEDGMEM'S
I
INTRODUCTION
.. - .. - -
1
l~
Summary
2.
Statement of Problem and Background
1
Discussion - - - .. - - - - - - - - - -
2
3. General Intersection Test Procedure .. - - .. - 10
II
DEVELOPMEW.r OF THE THREE-DECISION PROCEDURES -
15
1. The Sobel-Wald Procedure - -
15
2.
20
The Fixed SamFle Size Test
3. The Stein Two-Stage Procedure
....
25
4. The Baker-Hall Sequential Analog of
-- - ---- - The Minimum Probability Ra.tio Test
Procedure - - - - - - - - - - - - .. Stein's Two-Stage Procedure
5.
28
38
6. Hall's Modification of the M.P.R.T. for
the Case of Unknown Variance III
EMPIRICAL EVAWATIONS
BIBLIOGRAPHY
- - --
---- -
. ---
44
- -- -
52
63
iii
ACKNOWLEDGMENTS
I am deeply indebted to Professor William Jackson Hall,
not only for proposing my thesis topic and conscientiously
directing my research, but also for guiding me through the
more trying moments of my graduate studies.
He has been a
great source of personal warmth and inspiration for me.
I would also like to thank Professor Normal L. Johnson
for his many valuable comments and suggestions.
I am very
fortunate to have had the opportunity to avail myself of
his abundant talent.
Finally, I would like to thank my father, for without
his continued support and encouragement, I would never have
reached this phase of my academic career.
CHAPTER I
INTRODUCTION
1.
Summary
In this paper the problem of choosing one of three hypotheses
about the unknown mean of a normal distribution when there are specified bounds on the error probabilities is considered. Three-decision
analogs of the fixed sample size test" Wald sequential probability
ratio test, Hall minimum probability ratio test" Stein two-stage
test" Baker-Hall sequential analog of Stein's two-stage test, and
the modified minimum probability ratio test are developed by use of
an intersection technique suggested by Sobel and Wald (1949).
The
first three procedures are applicable when the variance is knownj
the latter three are of use in the case of unknown variance.
In
addition either exact or approximate formulas for the operating
characteristic functions are presented for several of the procedures.
Finally" the results of an empirical sampling investigation of the
terminal sample sizes required in three different test cases by each
of the six procedures mentioned above .are presented.
dice~te
(i)
~e·.:tesults
that, as in ,t»case of the t":'lo-deCis ion procedures"
the three-decision sequential procedures lead to a
substantial savings in the average amount of required
sampling,
in-
..
2
(ii)
the truncated sequential procedures (i.e., the M.P.R.T.
and its modification for the case of unknown variance)
are much less variable than the non-truncated or twostage procedures, and
(iii)
all of the procedures tend to be somewhat conservative
w. r •t.
2.
their error bounds •
Statement of Problem and Background Discussion.
Suppose that one is sampling from a normal distribution with un-
known mean,
a,
and that one wishes to test the composite hypotheses
,
H:
o
where
(i)
91 < e2 ~ e < e4'
3
and
(ii)
82 - 91 = 84 - 8
3
= ~,
say
..
Thus, one is dealing With a three-decision problem, i.e., one of
the three hypotheses is to be "accepted It and the remaining two "rejected". Alternatively, this problem is frequently treated as a
two-sided, two-decision problem by merging H-
and
rl
into a
single hypothesis
H to be tested aaainst H •
o
l
Suppose, further, that it is desired that the test be constructed
so as to ensure that the following bounds on the error probabilities
will be satisfied:
.
Pr ( rejecting H- Ie
}~
Y1 '
Pr ( rejecting Hole } ~ Y2 '
(1)
Pr ( rejecting H+
Ie }~
Y3 '
for
e~
for
e2 ~ e ~
for
e -> 94 ,
el ;
93 ; and
for specified Y , Y2' Y3· No requirements are placed on the del
cision probabilities for a-values between 9 1 and 8 and between
2
9 and 94; these intervals might thus be tenmed "indifference
3
regions" ..
To illustrate how such a problem could arise in a practical
situation, it might be illuminating to consider an emmple from the
field of quality controL
Suppose that a contractor who is engaged
in the production of concrete for highway construction wants to
detenmine the mean breaking point,
8,
of his product.
Suppose,
also, that past experience indicates that (i) if the mean breaking
point falls below 1600 Ibs./in2 the composition of the concrete will
be materially altered so as to render it unsuitable for nprmal highway traffic, (ii) if the mean breaking point lies between 1900 Ibs./in
2
and 2100 Ibs./in. , the concrete will be able to withstand the
stresses of a nonmal load of traffic, and (iii) if the mean breaking
2
point exceeds 2200 Ibs./in. , the concrete will contain an unnecessarily large proportion of cement, thereby raising production costs.
Under such conditions it seems likely that the contractor's quality
control department might want to test the hypotheses
2
4
with the bounds
Y1' Y2' Y;' on the error probabilities being de-
termined by various economic considerations.
In this paper we shall consider six different procedures for
testing
H-
satisfied.
vs.
H+
so that the guarantees in
(1) are
Three of these procedures are suitable for the case of
known variance
is unknown.
H vs.
o
a
2
;
the remaining three being applicable when a
2
Of the first three, one is non-sequential and two are
strictly sequential (one of which is truncated).
The latter three
are analogs of the first three procedures, except that an initial
sample is taken in each largely for the purpose of estimating the
unknown variance.
Each of these procedures is developed from a uni-
form approach (see Chapter I, Section ;), which may be described
simply as the intersection of two one-sided test procedures.
This
approach is essentially due to Sobel and Wa1d [8J * •
The Sobe1-Wa1d procedure (see Chapter II, Section 1) is a sequentia1. test procedure that is applicable to the preceding general
problem when a 2
is known.
Essentially this technique is based on
the simultaneous performance of two Wa1d S.P.R.T .. is, one of
9
2
to
and the other of
H- 1 H ' and H+
o
e;
vs.
94.
The final decision with respect
depends on the outcomes of the respective sub-
tests, which are designed so as to ensure the error bounds in
*Numbers
9 vs.
1
in square brackets refer to the bibliography.
(1).
5
It should be noted that the proposed procedures all tend to
give. a weak inequality for the bound on Pr {rejecting HIe}
o
over the interval [e , 9 ] unless the difference, 9 - 82 , is
2 3
3
quite small (see Chapter I, Section 3). Thus, in this respect,
they are rather conservative. A less conservative, three-decision
fixed-sample size test can be constructed for the case where (8 -8 2 )
3
is significantly greater than zero (see Chapter II, Section 2) by
departing from the general method of development employed throughout this paper.
In section 1 of Chapter II the Sobel-Wald technique, denoted by
I.S.P.R.T., is reviewed in some detail; and in the remaining sections
of Chapter II three-decision analogs are successively developed for
the fixed-sample size test, denoted by I.F.S.S.T., Stein's twostage test
[9J, designated by I.SoT.S.T"(m)l, the Baker-Hall sequen-
tial analog of Stein's two-stage test [2, 5], designated by
I.S.P .R aT. (m)' the minimum probability ratio test [4], denoted by
I.M.P.R.T., and Hall's modification of the minimum probability ratio
test for the case of unknown variance [5], designated by I.M"P.R.T.(m).
Thus, three-decision analogs of the following one-sided test procedures
are treated:
Sampling procedure
(j
2
unknown
one-or two-stage
F"S.S.T.
S.T.S.T. (m)
sequential
S..P.R.T.
M.P.R.T.
S.
:r .R.,;!:, .(m)
MoP.R.T·(m)
~- th1'ee-decision analog of Stein f s two-stage procedure is denoted by
I.S.11.S ..T~(m)'; sinc·e it is formed by intersecting two one.sided
BoToS.Te(m) .procedur~ and since the unknowp variance is estimated
fiom an
initial sample of size m.
6
Finally, Chapter III presents the results of an empirical sampling
investigation of the terminal sample sizes required in three different test cases by each of the six procedures under consideration.
To see when the above test procedures might be applied to a
practical problem, let's reconsider the contractor engaged in the
mixing of concrete for highway construction.
If the variance of
the concretefsbreaking point is known (perhaps the quality control
department has at its disposal a substantial quantity of pervious
inspection data), then an I.F.S.S.T., I.S.P.R.T., or I.M.P.R.T.
could be utilized to test H- vs
H vs.
o
H+.
2
On the other hand
if the variance is not known, then an I.S.T.S.T.(m)' I.S.P.R.T.(m)'
or I.M.P.R.T.{m)
•
could be employed.
The relative merits of these
test procedures are not known explicitly.
bilities have been
controll~d
Since the error proba- .
in the same way for each of the pro-
cedurea, they do not provide much basis for choosing among the
various tests.
Furthermore, although the minimum sample size needed
to ensure that the guarantees stated in (1)
will be satisfied can
readily be determined for the one- and two-stage procedures, little
is known about the average sample number functions (A.S.N.) for the
sequential and modified sequential techniques.
able to obtain a lower bound on the A. S. N.
S.P .R.T.
[Sobel and Wald were
for the three-decision
which seems to be a fairly close approximation to the
A.S.N. except over the range
(9 , 9 ).
2 3
By considering a slightly
2It should be noted that the I.M.P.R.T. and I.M.P.R.T.(m)
applicable only when Yl=V3=Y2/2
•
are
7
less general problem they were also able to derive a rather complica~ed
expression for an upper bound on the A.S.N.
that appears
to provide a reasonably close approximation when certain restrictions are satisfied. J Numerical and theoretical studie.a t;hat have
been made of the one-sided test procedures listed in the table on
page 5
seem to indicate that if one of the (two) hypotheses being
tested is true, then one can expect substantial savings on the
average by using a suitable analog of a S.P.R.T. instead of the
comparable one- or two-stage procedures.
However, even though the
average sample numbers (sizes) will not be too large with such a
procedure, actual sample numbers will in. fact be random and also
unbounded.
On the other hand the 'truncated sequential procedures,
i.e., the M.P.R.T.
and the M.P.R.T.(m)' tend to provide more
control over the variability and the maximum amount of sampling
than do the non-truncated procedures.
Further;more, their A.S.N.'s
tend to be a little flatter than those of the S.P.R.T. f S
slightly lower between the hypotheses being tested and slightly
higher when one of the hypotheses is true.
Hence, the truncated
sequential tests may be preferable to the non-truncated techniques,
especially if neither of the hypotheses under consideration is
true.
Now, the empirical investigations summarized in Chapter III
indicate that these same general conclusions can be drawn for the
various three-decision procedures being discussed in this paper.
A portion of the data contained in Chapter III is presented below.
8
Here, "s.d." refers to the standard deviations of the requiredsample sizes.
Each of' these numbers is based on 165 repetitions
of the tests, so all of the figures are reasonably precise.
(The
standard deviations of the averages range from about 0.5 to about
Procedure
Test Case 1
Correct Hypothesis
Median Sample Size
s.d.
I.F.S.S.T.
H-
44
I.S.P .R.T.
H-
-
19
15
I.M.P.R.T.
H
-
23
11
I .S.T.S.T. (m) 3
H-
4:;
19
I.S.,P .R~T. (m)
H-
24
21
I.M.P.R.T·(m)
H-
25
15
Test Case 2
Procedure
Correct HxP0thesis
I.F.S.S.T.
H
I.S.P.R.T.
H
o
Median Sample Size
s.d.
44
12
o
I.M.P .R.T.
7
I.S,T,S.T·(m)
H0
4:;
19
I.S.P .R.T, (m)
H0
H0
26
17
27
13
I .M.P.R.T. (m)
3m
= 16
9
Test Case;
Procedure
Median Sample Size
Correct HYPothesis
s.d.
I.F.S.S.T.
none
44
-
I.S.P.R.T.
none
26
22
I.M.P.R.T.
none
29
12
I.S ~T.S.T. (m)
none
4;
19
I.S.P.R.Te(m)
none
;4
;1
I .M.P .R.T. (m)
none
;1
19
It should be noted that the
the I.M.P.R.T.(m)
I~S~T.S.T.(m)1
I.S.P.R.T.(m)' and
all require the choice of an initial sample size
of m.
This choice is arbitraryl but in every case the estimation
of cr2
is based solely on this sample.
If a small value is chosen
for m, then cr2 may be poorly estimated and the rules for continuing sampling badly chosen as a consequence; on the other baUd,
if a large value is chosen for ml one may be doing more sampling
than is necessary.
Some d1ecussion of this choice of the initial
sample size for the S.T.S.T.(m) procedure appears in [7J. As a
rule of thumb l one might guess at the true value of cr2 and then
choose m to be roughly 1/4 to 1/2 the re<nlired sample size
for the corresponding I.F.S.S.T.
Finally" an alternative and commonly used approach to testing
H-
vs.
H vs.
o
H+ which may be employed in place of the
10
I.S.T.S. T.(m)" the I.S.P.R.T_(m)' or
I.M.P.R.T.(m)
is to "re_
formulate" the hypotbeses in terms of (unknown) standard deviation
units" i.e."
for specified
8i" 82" 83"
and
84'
or else to pev.mit the error
probabilities to vary with the unknown 0'2..
Then a three-decision
sequential (or non-sequential) t ..test could be used.
Neither of
these approaches may be completely satisfactory" and they will not
be considered further here.
It should be noted" however, that non..
sequential tests meeting the guarantees in
when
0'2.
(1)
are not possible
is unknown unless one makes such a reformulation of the
problem.
3. General Intersection Test Procedure.
In the preceding section it was noted that each of the procedures considered in this paper was to be developed from a uniform approach -- namely, the intersecting of two two-decision
tests.
More explicitly" this intersection technique may be
thought of as a four-step process.
First" the original variables"
by setting
Thus, the
Xl' X2.' ••• , are transformed
11
and
Next" two sub-tests T-
and T+ are constructed so that T-
is a test" whose specific form depends on the particular threedecision procedure under consideration" of e- = 0 vs.
pre-aS$:i.g:ned~stra14sth(~-"~-),, and T+
e+
e-
is a test of e+
=1, with
=0
vs.
with pre-assigned strength (a+,~+). [Because of the mono-
= 1,
tonicity of the power function for a normal mean test, it follows
e- = 0
that an a- - level test of
a-
... level test of e-
test of
e+<_
0
~ 0
vs.
+
e = 0 vs. e+ = 1
+
vs. e > 1.J
-
Then
sub-tests T-
vs.
e- _>
1.
e-
=1
is also an
Similarly" an a + - level
is also an a+ - level test of
and T+ are performed simultaneously,
and the testing procedure is continued until decision
or decision
d~ (e+
= 0)
di (e-
= il)
or decision
d~
(e-
= 0)
has been made under T- and decision
a;:,
(e+
= 1)
has been made under T+ •
Finally, an overall test T is defined as the intersection of
sub-tests
T-
+ with sampling rule as described in the preand T,
ceding paragraph and with terminal deci£f1onsD-(e ~ e ),
l
Do(e 2 ~ e ~ 8,), and D+ (e ~ 84). Since T = T-" T+, the
following combinations of sub-decisions could arise with test T:
(~n d~)
"
Expressing the various sub-decisions in their corresponding equivalent forms of inequalities involving
e,
one can see that making
12
decision
d- (e ~ e )
1
l
d- (e ~ 8 )
2
0
d- (e~ e )
0
2
with test
,,
with test
T-, and making decision
I
5 e~)
d~(e 5 e~)
d;(e
\ a;.(e ~ 84)
+ would logically lead to a terminal decision of
T,
D-(e 5 ell
Do (Q2
j
)
)
)
~ e ~ 83 ) .(
D+(e ~ e4)
for test
T.
Furthermore, it is apparent
)
that the sub-tests
T-
and T+
should be designed so as to render
d~ and decision d~ under test T
the making of both decision
impossible, since no' :reasonable conclusion can be drawn from such
a combination..
decisions for
-
-
D =~,
With this restriction on the sub..tests, the terminal
T can be expressed as
Do
= do-.f\
~ \
+
do'
+
and D
= d+l
•
Now,
Pr{D- using Tie}
= Prt d~
using T-je}
~ 1 -
13-, for 8 ~ e l ;
, } = Prtdl
r + using T+J 8 } ::: 1 - f3 +, for 8
Prtr D+ using TIe
+J }
- Prtr d+
l using T e
~ 1 - ex- - ex+, for
82 ~
~
8 ~
84; and
8 ..
3
13
Thus it is obvious that the bounds on the pre-assigned error
probabilities given in (1) will be obtained using T if ~-, ~+,
a.,
and a,+ are chosen so that
t:l<
t:l+ _< Y3'
tJ
_ Yl' tJ
and ",'"'" + ,.,,+
'"'" < Y2' i .e.,
Now, even though there is some loss of generality, we shall for
convenience take
throughout this paper.
pr{D o using TIe}
It should be noted that the inegUl11ity on
is weak if 92 < 9 , since
3
+1
pr{d+
l using T eJ
= Pr{di+
+1
using Te
= 83 } = a,+ •
Finally, analogous to the power function or operation characteristic function of a two-decision test procedure, one can obtain
exact or approximate expressions for the respective decision probabilities,
Pr{D- using TleJ, Pr(D+ using TIe}, and pr{D o using TIe},
if the SUb-test power functions for the three-decision procedures
under consideration are available, Lie.,
14
Pr{D· using Tie) = Pr{di using T-Ie) ,
Pr{D+ using TIe)
=
pr(d~ using
T+ Ie) , and
Pr{Do using TIe} = 1 - pr{di using T-Ie} - pr(d~ using T+Je) •
CHAPTER II
DEVE.t.OPMENT OF THE THREE-DECISION PROCFJ.JURES
1.
The Sohel-Wald Procedure
[SJ is a three-decision sequential procedure
The IoS~P.R.T.
which is applicable to the general problem described in Section 2
2
of Chapter I when the variance,
procedure we let
based on
T-
denote the S.P.R.T. of
X;:', X;, u.,
8+
=0
assigned strength
In the Sobel-Wald
e- = 0
B-
- and we let
and A;
e+ = 1,
vs.
= 1,
vs. 8-
with pre-assigned strength (a-, ~-)
termination boundaries
S.P.R.T. of
is known.
(j,
T+
and
denote the
+ X+, ••• , with prebased on Xl'
2
(a+, ~+) and termination boundaries
B+
and
A+, where
b
(2)
(A- )(B+) ~ 1,
B- ~ Yl'
+
(A+)(B-)
and
B+ ~ Y ,
3
= log
B+, and a +
= log
A+;
> 1; and
A- ~
2/Y2'
and A+
~ a!Y2.
Then, computing the relevant probability ratios, one finds
that the terminal decision rule for test
stop sampling and make decision
n
~ Xi ~ (b-)
i=l
2
(j
T-
is given by:
d~ (8- = 0)
/(8 1 - 82 ) + neal + 82 )/2
if
,
16
stop sampling and make decision
d~
(e-
= 1)
if
n
2
~ xi ~ (a-) a feel - e 2 ) + n(e l + e 2 )/2 1
i=l
and co:ntinue sampling if
Similarlyl the terminal decision rule for test T+
is given by:
stop sampling and make decision d+{8+ = 0) if
a
n
+ 2
.~ xi ~ (b )a /(84 - e~) + n(8~ + 84)/2 1
~=l
stop sampling and make decision
~(e+
n
+ 2
~ xi ~ (a )a /(e4-e~) + n(e~ + 84)/2
i=l
= 1)
if
1
and continue sampling if
Now, a necessary and sufficient condition for the impossibility
of making both decision d- and decision
l
at n=l, the following inequalities should
d+ with test T is that
l
hold (see Figure 1):
Acceptance number (of e- = 1) for T" < Acceptance number
( of e+ = 0) for T+ 1 and
Rejection number (of e- = 1) for T- < Rejection number
(of e+
= 0)
for T+ 1 i.e.1
17
and
These inequalities can be written as
+ ~2/~
(l/A- • B )
~
1/2[(e4+93) - (9 1+e 2 )J
e·
and
•
Now, since
the inequalities in (;) will certainly be satisfied if
/
1
(dj):
~x1 ::: (a+)
(d+):
~x1
(d-):
~x1 ::: _(b-)cr2 /b. + n{e l + 92 )/2
(0.;:);
~x. ::: _(a-)cr2/b. + n{e + ( )/2
1.
l
2
0
0
=
(b+)
(l/b.
cl/b.
+
nee:;
+ (4)/2
+ nee:; + 94)/2
FIGURE 1 •
,
,
,
and
•
19
[Obviously, for any given problem the actual shape of the graphic
representation of the Sobel-Wald test procedure will depend on the
+ a.+ ,
values of '\)-:-,
2 and b., which jointly determine the dis-
CT ,
tance between the parallel decision lines, and on the quantities
(8 1+82) and
(e3+~4)' which determine the slopes of the (pairs
of) decision lines.]
Now, reca.lling Wald's conservative bounds on the error probaS~P.R.T.
bilities of a
[10] and referring to (2), one can readily
see that
(i)
~-
-<
<
B
(ii)
~+
<
B+
(11i)
cit
<
Y2 /2
'Vl '
<
Y3 '
and
•
Thus, the bounds on the overall error probabilities for test T are
satisfied, as 'Was shown in section 3 of Chapter I.
Finally, recalling Wald's approximation formula for the operating
characteristic function of a
1 _ eah(e)
= Pr[accept1ng the null hypothesis'e} e ---:b:":h...,(-e"'\")----.ah:r"'(,..e~) ,
L(e)
(4)
S.P.R.T. (see p. 50 [10]) , namely
e
- e
where
h( e) is a non-zer2 solution of
Jooff° (x)/feo(x)]h(e) d Fe(x)
-eo
=
1
,
1
arid "elf implies that Ifexcesslt over the boundaries at the termination
of the procedure is neglected,
20
one can obtain the following apl?roxiIllations to the various decision
probabilities for test T:
Pr{D-
using Tie}
,
e
Pr{D o
and
.+
Pr{D
using TIe}
where
h-(e)
Now, at
= 0,
h-(e)
(5)
= [2e
(5)
+ e4) - 2eJ/~ •
+ e2 )/2 and at e = (e3 + 64)/2 [i.e.,
or h+(e) = 0], (4) can be replaced by
~
a/fa-b)
,
is obtained by taking the limits (as
(4) and using
2.
= [(e 3
e = (e 1
L(e)
where
- (e l + e2)]/~' and h+(e)
h(e) -> 0) in
1\
l'Hopital's rule.
The Fixed Sample S:i3e Test
For the I.F.S.S.T. procedure, which can be used to solve the
general problem under consideration in this paper when
T-
denotes a
F.S.S.T.
of
e- = Ovs. e- = 1,
2
~
is known,
based on a sample
21
T+
denotes a
F.S.S.T.
of
(X~" x~"
•.• " X;)"
+
~ ~
2
a-
e+
e+
= 0 vs.
= I"
based on a sample
and
+ z
~-
et. ~
'V2/2
~+)" where
(a+,
with pre-assigned strength
""5'
N = max,. Uo- (z
(6)
of
;
)2/62J + Ii [o-2{z + + z +)2/~2] + 1)
~
a
is used symbolically to denote tfthe largest integer in";
"[]f1
and z
~
is that number Which is exceeded by a standard normal
.
random variable with probability I.l.
[Before we consider the respective terminal decision rules for subtests
T-
and T+ , it should be noted that by defining the overall
sample size,
N, as in
(6), we are actually ensuring that
and
Hence if 'V ~ 'V3" test T will be more conservative than is neces1
sary.]
+
Now" the terminal decision rules for sub-tests T"" and T can
be stated as
= 0)
-
> - 6 c + 82
make decision diCe - = 1) if ~ < - 6 c + e2
make decision
d-Ce0
if
~
-
-
if ~ <
+
6 c + 83
,
+
6 c + e
,
and
= 0)
make decision
d+(e+
make decision
d~(e + = 1) if ~ >
0
3
1
,
22
respectively, where
•
9 ~ 9 , and since c and c+ are always non-negative
2
3
y /2 is always ~ 1/2), it is obvious that
2
Since
(i.e.,
e = 1)
Acceptance no. (of
e+
= 0)
for T+, and
e· = 1)
Rejection no. (of
e+
= 0)
for T- < Acceptance no. (of
for
for T· < Rejection no. (of
T+ , i.e.,
- ~ c· + 8 ~ 6 c+ + 9
2
3
•
Hence, it is impossible to make both decisions
d~ and d~ using
test T.
Furthermore, since
(3-
(i)
~
Yl
(3+ ~ Y3
(ii)
c?<
(iii)
1
and
y /2
as was stipulated in
[given in (l)J
,
2
(6), the bounds on the error probabilities
for test T are satisfied.
A less conservative, three-decision F.S.S.T. procedure can be
obtained by choosing c·, c+, and N so that
Pr{making decision D-19}
> 1 - Yl '
Pr{making decision D+!9}
> 1 - Y3
,
> 1 • Y2
(iii) inf. Pr{making decision Die}
0
e~~ 93
,
(i)
info
e~
(ii)
91
info
9 ~ 94
and
where the terminal decision rule is of the form
i
make decision
D-;
if
~~
D+;
if
~?:
t D0'.
- -
+ 82
b,c+ + 9;
boc
,
,
otherwise.
Thus, ~or N fixed and equality in (i) and (ii), we find, using the
monotonicity
~.n
c
=
and
e
of the probabilities, that
1
-
z
"1
•
~/6 ~
,
•
Observing that
Prtmaking decision DIe}
o
= i(~
~
(6 c+ +
e~.,/
- e)}
we find, upon taking first and second derivatives w.r.t.
Pr(making decision D Ie}
o
eo =
that
attains its maximum over the interval
2
,
at either 9
or e;.
If the nd.nimum occurs
2
8 = 8 , then equality in (iii) implies that
2
and i ts
at
e,
Illi~imum
24
A tria1-and-error solution for N in the above expression, say N-,
can be obtained by referring to a table of the standard normal distribution.
Similal~ly,
minimum over [e , e ]
2
3
if Pr{making decision D fe}
o'
at
e = e3 '
attains its
then equality in qiii) implies
that
IN
(e:;-9 1 )
cr
}
•
+
Denoting the tria1-and-error solution for N in the above by N,
we can define the overall sample size as the first integer that
exceeds max .. (N-, N+). This N is certainly less than the conservative one obtained for the I.F.S.S.T.
which was, in effect,
chosen by replacing the minimization in (iii), where the probability
to be minimized may be written as the sum of two different terms,
with the sum of the minimizations of these separate terms.
(A more
detailed discussion of the above test procedure appears in the
Appendix of [3].)
Finally, by performing some simple algebraic manipulations
one sees that the decision probabilities for test T can be written
as
Pr{D- using Tiel = Pr{Z < - z
+
a
Pr{D o using Tie}
= Pr{Z
- Pr{Z > z
a+
~ - z .., +
of-
a
IN
IN
IN
(02 - e)/cr}
(e 2 - e)/cr}
(e~ - e)/cr} ,
.
,
25
and
where
N is defined as in
(6),
and
Z is a standard normal random variable.
3. 9:he fltein Two-Stage Procedure
Steinls test [9)
is a two-stage procedure that can be applied
to the general problem introduced in Section 2 of Chapter I when
the variance, 0-2 , is unknown. Under this technique the sub-tests
T-
and T+ are constructed as follows:
Let
*+ represent
(T)
e+
the Stein two-stage test, here denoted by S.T.S.T.(m),Of
e+ = 1,
(+
Xl' X+'2, ••• , X+') and (+
X +1' X+ 2' ••• 'X+) ,
m
m
~
+
based on samples
with initial sample size of m (a fixed integer ~ 2) and preassigned strength
(7)
.x
m
t
J.l.
=
m
N
(at, ,,+), where
+
N = max. tm, [s2(t + t
m a,+
,,+
,,- :5 min.(1/2"
= 0 vs.
'Vl)'
E x./m, and
i=l ~
,,+ :5 min.
s
2
m
m
= E
i=l
,2/t}]
+ l}
;
(1/2, 'V3)' and
(Xi -
2
x ) /(m-l);
m
rt.:5 min.(1/2''V2/2 );
and
is that number which is exceeded by a Student's
variable (with m-l d.f.) With probability J.l..
t
random
26
Then sub-tests T-
(T*)-
and T+ are equivalent to tests
and
(T* )+ , respectively, except that the sample sizes N- and N+
N = max. (N-, N+) •
are replaced by
[As was noted in the discussion of the I.F.S.S.T.
in the pre-
N as above leads to a more conservative
ceding section, defining
test T than is actually required if Yl ~ Y3 ].
Now, the terminal decision rules for sub-tests
+
and T
T-
are given, respectively, by
make decision d- (S-
= 0)
if
XN >
make decision
~ (8-
= 1)
if
XN <
make decision
d+ (e+
= 0) if
make decision
ai (8+ = 1)
0
-
tJ.k + 92 '
- tJ. k
-
+ 82 '
and
0
if
~
<
~ >
6,
k+ +
t k
+
93 '
+ 9 '
3
where
k- = t
CY:
I(t
a-
+ t
f3"
), and k+
Since it is given that
that
r:.~ and
c+
8
2
:5
8
= t +I(t
a
3
a+
+ t
f3+
and since (7)
).
implies
are always non-nega.tive, it obviously follows
that
Acceptance no. (of
e+
= )
0
for T+
Rejection no. (of
e+ = 0)
e- = 1)
f
I
for T-
< Acceptance no. (of
and
e- = 1)
or T+'
,~.e.,
for T- < Rejection no. (of
27
Therefore, a situation in which both decision
d~
and decision
~ are made will never arise when test T is employed.
Finally, since
,
(i) 13- ~ Yl
(ii) 13+ ~ Y ,
and
3
(iii)
as stated in
at -<
,
Y2/2
(7), it follows from Section 3 of Chapter I that the
bounds on the pre-assigned error probabilities for test Tare
satisfied.
Now, in order to derive expressions for the various decision
probabilities under test T we shall make use of the fact that
/N~
8
-
e)
m
has a Student's.t distribution with (m-l) d.f. (see [9]).
With
this result in mind one can readily see that
Pr[D- using
Tie}
= Pr{T(m-l)
< (IN/8m) (- 6 k- + (e 2 - e)]} ,
Pr{D o using Tie} = Pr[T(m-l) ~
(IN/8m)
(- 6 k- + (e 2 - e)]}
(IN/8m)
[6 k+ + (8
and
Pr{D+ using Tie} = Pr{T{m-l) >
3-
e)]} ,
where T{m-l) is, marginally, a Student's-t random variable with
(m-l) degrees of freedom.
Now, replacing the random variable (/N/S ) by its lower bound,
m
max. {(t
ex-
+ t
~-
)/~, (t + + t
ex
~+
)/~}, we can obtain either an upper
bound or a lower bound on each of the various decision probabilities,/
depending on the value of
[- ~ k - +
(e 2
e•
For eXQ.mple, if
- e*)]
<
0 ,
then
Pr{D- using TJe*} ~ Pr{T(m-I)<;-t _ + (e 2 - e*){t _ + t _)/~};
ex
ex
~
and if
then
Pr{D- using T{e*} ~ Pr{T(m-l) < -t _ + (e 2 - e*)(t _ + t _)/~} •
ex
ex
f3
These bounds are approximate equalities if the probability of continuing sampling beyond the initial sample is close to unity.
4. The Baker-Hall Sequential Analog of Stein's Two-Stage Procedure.
The· Baker-Hall procedure [2,
5J can be thought of as a se-
quential analog of Stein's two-stage test, because it is, essentially,
a S.P .R.T. with the unknown variance replaced by an estimate based
on an initial sample.
It tends to be somewhat less restrictive than
the Stein procedure since it can be applied directly to the general
problem set forth in the Introduction rather than to the slightly
diluted version that was, in effect, considered in the preceding
section, i.e., where T was designed so that
29
and
Furthermore, the Baker-Hall technique may result in a substan·
tial savings of the total number of observations required to reach
a final decision, just as, in the case of known ~2, the I.S.P.R.T.
may result in savings over the I.F.S.S.T.
Now" in order to apply the Baker-Hall procedure the sub-tests
T- and T+ should be constructed as follmvs:
S.P.R.T. of 9-
=0
vs.
e-
= 1" based on
Let T-
denote a
X-,
m
Xl' X·m+ 2' ••• ,
m+
with pre-assigned strength (or, 13-), termination boundaries
2
2
B- and A- , and with (~.)
[know] replaced by (a·)
•
m
m
m
Similarly, let T+ denote a S.P.R.T. of e+ = 0 vs. ~ + = 1,
x:'
X:.rl" X:+ 2 ' ... , with pre-assigned strength (a+"I3+),
+
+ 2
termination boundaries B+ and A" and with (~) [knowJ replaced
m
m
by fSm+)2
\'
, 1vhere
based on
(A-) (B+) > 1"
m
m -
and
b; = log B; < - v
( ~)
b+
=
log B+
a!
m
=
log
m
- 2
(sm)
m
(y~2/V
v = m• 1
1)/2"
,
< _ v (y.2/v • 1)/2 ,
;
A!-m->
2
= sm/(el
(B-) (A+) _> 1;
m
m
2 V
v [(y,..j2r / - IJ/2;
- 92 )
t:.
2
= sm2/62 = sm2/{94
-
and
8;) 2
=
(+ 2
sm)
•
Then, computing the relevant probability ratios, one finds that
the terminal decision rules for sub-tests T-
and T+ are given by
30
stop sampling and make decision d·(e-=O) if r (s-) < b- ,
o
n
m -
m
dl-(e-=l) ifr (s-) > a- ,
n m - m
continue sampling if b- < r (s·) < a- ,
m
n m
m
stop sampling and make decision
and
stop sampling and make decision
d+(e+=O) if r (s+) < b+ ,
stop sampling and make decision
a::--1(e+=1)
o
n
m -
m
if r nm-m
(s+) > a+ ,
continue sampling if b+ < r (s+) < a+ ,
m
n m
m
respectively, where
+
n
+
r (s-) = ~ (X- .
n m
i=l i
+ 2
1/2)/(s-)
m
n > m.
,
Equivalent forms of the above terminal decision rules, which greatly
facilitate numerical computations, can be stated as follows:
stop sampling and make decision d-
o
n
E Xi ~
i=l
if
2
(b-) s /(.~) + n{e l + e2 )/2
m m
stop sampling and make decision
d~
if
2
n
E Xi ~ (a·) s /(.~) + n(e l + e2 )/2
i=l
m m
continue sampling if
and
stop sampling and make decision
n
E
i=l
,
X.
2
/~ +
m m
+
< (b)s
1. -
n(e~ +
.,/
stop sampling and make decision
n
i:l Xi
~ (a:)s;/a
+
d+ if
o
e4)/2 ,
d+ if
1
n(9 3 + 94)/2 ,
I
31
continue sampling if
Now, a necessary and sufficient condition to ensure that both
decisions
n
d- and d+ will not be made under test
l
1
T is that at
=m I
Acceptance no. (of
e- = 1)
for
T- ~ Acceptance no. (of
e+=0)
+ and
for T,
Rejection no. (of
e- = 1)
forT- < Rejection no. (of
e+=o)
+
for T I i.e",
and
Now, these inequalities can be expressed in the equivalent form
"
Thus, since
they will certainly be satisfied if
32
In order to determine whether or not test T meets the bounds
on the pre-assigned error probabilities given in (1), it is necessary
to consider in some detail a conditional SoP.R.T.,
T(s
,a), applied
m
to the general problem introduced in Section 2 of' Chapter 1.
So,
(8 , 0') let T-(s ,a) denote the eond1tional S.P.R.T.
m
m
of e- ~ 0 vs. e = 1, based on X -m , X·m+ l' X-m+c,.,,1 . . . . , with
termination boundaries ]3-, i-. Similarly, let T\s ,a) dem
m
m
+
+
note the conditional S.P.R.T. of e = 0 vs. e = 1, based on
-+
-+ X+
+
-+
Xm ' m+l' Xm+2' .... , with termination boundaries B m and Am'
where
for given
log
Bm-
=
and
•
Then" the terminal decision rules for sub-tests T-
and T+
can be expressed as
stop sampling and make decision d-
o
-
(e- = 0)
if
stop sampling and make decision dl (e- = 1) if
continue sampling if b- < r (a-) <
m
n
a-m "
and
-~'pe dec i s i on d+(e+
s t op samp ling and ~
o
stop sampling and make decision
d.i+( e+
= 0)
if r
= 1 ) if r
continue sampling if -+
b m < r n '\(+)-+
cr < a m •
n
n
(a+) < b + ,
- m
(a+)
Z>
-
a+
m
,
It can be shown by straightforward algebraic manipulation that
the necessary and sufficient condition for the impossibility of
making both decision
di
d~ using test T(Sm,rr)
and decision
is the same as that for test T.
Furthermore, since
+
+ 2
r n (&-)/(0-)
m
-+
2 + I 2
~ = s ([).:. )lrr,
m
m m
= sm2 ~
2
r n (s!)/rr
m
2 +
2
and -a m-+ = .sm
(a:m>/rr
,
,
it is obvious that T(s ,rr) has precisely the same decision at each
m
stage as does test T.
T(sm,rr)
Since the terminal decision rules for
and T are equivalent, it follows that
ES {Pr(D* using T(s ,rr)IS , rr,
m
m
m
=
e1J
ES {Pr(D* using Tis ,u, e)}
m
m
1
where D* represents some specified decision.
Now, consider the random variable, w, where
w = (" I;
)
\. 0;
if decision D* is made using test T,
otherwise
Clearly,
and
ES {E[wlsm,
m
~,
eJ} = E{wla, e}
=
Pr{D* using Tl~, e} •
34
Thus,
(10)
ES {Pr(D* using T(sm,~)ISm,~, e)J
m
;
Pr{D* using TI~, eJ
•
Now, making use of Wald's conservative bounds on the error
probabilities of a S.P.R.T., we have
Pr{d! using T!(s ,~)ls , ~, ~
o
mm
=
l}
< B:t
= exp. (~s2/rl) ,
m
m m
and
Pr{a;l using T~(s,
m ~)Ism, ~, ~ = o}
Hence, it follows from (10) that
(11) pr{d~ using T-f~, e- = 1} ~ E{exp. (b; S;/~2)J
= (1 - 2b- /v fV/2, for all
m
since
2/ 2
vSm/fr
= Xv2
,
Similarly,
(12)
=
(1
-
+/v) -\)/2
2bm
,
~2
,
35
Pr{a.:'
using T-lfj,
--1
= (1
+ 2a
e- = oJ. -< E
2
(exp.(-a-m sm
/(ln
-Iv tV 12 ,
m
and
(14)
pr(d~ using T+lfj" e+ = 0J:5 E{exp. (-a: s,;lfj2)}
= (1
+ 2a+Iv )-v/2
m
,
for all fj 2 • Thus, by substituting for ~
m
and
a±m
from (8) in
(11) .. (14), one finds that
(i) 13- ~ Yl'
(ii) 13+:5
Y,'
and
(iii) at:5 Y 2 / 2 "
which guarantees that the bounds on the error probabilities given
in
(1) areaatisfied by test T.
Now, in order to develop approxilllation formulae 4 for the various
decision probabilities for the I.S,P.R.To(m) it is necessary" once
again, to refer to Wald's approximation formula
for the operating
characteristic function of a S.P.R.T. Thus, for the conditional
S.P.R.T., T(s ,fj), one has
m
Pr{D- using T(s ,fj)ls , fj" e}
mm
(A- )h-(e) - 1
= pr{di.
using T- (sm,fj) }sm,fj"e} ~ 1 _
m
,
(A-) h-(e).(B- )h-(e)
m
m
4.The approximation formulae for the various decision probabilities
which are derived above are taken from [5J, and the reader who is
interested in a more complete discussion of the mathematics involved in their development should consult this paper.
;6
= [2e
- (e l + 82)]/60 Taking expectations with respect
te s; in (15), substituting from (9), and ignoring the subscripts
where
h-(e)
s;,
on a;, b; and
we obtain
2J
[ - _h- _s 2/
(16) Pr{D- using TI~,e} ~ E {l _ 1 - expo_-a
~~ 2 2 }
1- exp.[-(a -b )h S /~ 1
= 1 - E{[l .. expo( ..a-h"s2/~2)]
(1)
~ [exp.(_i(a-_b-)h-s2/~2J} ~'
i=l
for
+ e )/2, which does not depend on ~"
2
l
and then taking expectations term-by-term,
h-(e) > 0, Le., e > (e
Expanding the RHS of
(16)
one has
-[1 + 2(a--b-)h- /vJ-V/ 2 + [1 + 2(2a--b-)h-/vJ-\)/2
-[1 + 4(a-_b-)h-/vJ- V/ 2 + •••
(1$) Pr{D- using
TI~,e} ~
1 - (1 + 2b-h-/v)-V/2
+ [1 + 2(b--a-)h-/vJ-V / 2 - [1 + 2{2b--a-)h-/vrV/ 2
+ [1 + 4(b-_a-)h-/vJ- V/ 2 - •••
Analogously, for
h+(e)
(19)
TI~,
Pr{D+ using
= [(8;+84)
e}
~
- 2eJ/t. > 0, we obtain
(1 + 2a+h+/V) ..V/2
-[1 + 2(a+-b+)h+/vJ- V/ 2 + (1 + 2(aa+-b+)h+ /\1]-\1/2
-[1 + 4(a+-b+)h+ /vJ- V/ 2 +
~oo
I.
37
and for
h+(e) < 0, i.e.,
e
> (9 + 94)/2 ,
TI~,e} ~
1
~
(20) Pr{D+ using
3
2b+h+/~)-V/2
(1+
V2
+ [1 + 2 (b +-a+) h +/~ J- / - •••
~(9) = 0, then, as was noted in Section 1 of Chapter II,
Now, if
~/(~ ~ a,.± )
e
;::
by l'Hopital's rule.
Finally, recalling that
Pr{D using TJ~, e}
O
=1
- pr{d~
using T-I~1 e}
- pr{di using T+J~, e}
J
we have
(21a)
Pr{D using
o
TI~J9} ~
.(1 + 2a+h+ /~)-~/2
+[1 + 2(a+ (21b)
pr{D
o
TI~J
-(1 + 2a+h+ /~f~/2 +
2b~h~/~)~~/2
_ [1 + 2(b- _ a-)h-/~r·V/2
b+)h+/~J~V/2
using
(1 +
e}
+ ••• J
~
[1 +
1
~
2(a~
e < (e l
for
(1 +
-
+ 82)/2
2a-h~/~)-~/2
b-)h~/vr~/2
+[1 + 2{a+-b+)h+ /~J-~/2 - ".J for (e +8 )/2 < e
1 2
(2lc) pr{D using TJ~J e} ~ {l + 2b+h+/~)-~/2
o
- (1 +
;
< (e +e4)/2 ;
3
2a~h~/~r·~/2 ~ [1 + 2(b+.a+)h+ /vJ-~/2
+ [1 + 2(a--b-)h- /~r~/2 + .... J
for
e>
(93 + 94)/2
•
:?8
7. !l:e Minimum Prrobability Ratio Test Procedure5
The minimum probability ratio test, denoted by M.P.R.T., is
a modified version of Waldls S.P.RoT. with converging straight-line
boundaries.
The three-decision analog of the M.P.R.T., designated
by I.M.P.R.T., is only applicable to the symmetric version (i.e.,
Yl
= V:? = Y2 /2
=
y.,
say) of the general problem described in
Section 2 of Chapter I when the variance is known.
For this
special case it is, presumably, intermediate between the I.F.S.S.T.
and the I.S.P.R.T. with respect to the area at best performance.
In order to develop the I.M.P.R.T.,
and T+
should be constructed as follows:
M.P.R.T.
of
e- = 0
vs.
e = 1,
e+ = 0
e+ = 1,
VB.
=
denote a
T-
denote a
with
M.P.R.T.
+ +
based on Xl' X2 ' ••• , with pre-assigned
(a, a), where a:5 min.(y, 1/2).
Ji
Let T-
based on Xi, X;, ••• ,
pre-assigned strength (a, a); and let T+
of
the sub-tests
J1 -
Then, if
1/2 ,
so that
of
y1: = N{(e-+ - 1/2),
i
=
the decision rules for sub-tests
+
+2
N{,r-,
(0-) },
...
T
+
and T
+ 2
(~)
}
say,
can be stated as
5Anderson [1] proposed a number of test procedures for the case of
known (T2 as modifications of Wald's S..P.R.T., the purpose being to
effect a reduction in the required sample size ,at values of the unknown mean intermediate between those being tested. His methods
were approximate, being based on a Weiner process approximation to
the sequence of random variables under study. Hall [4], utU;l.zing
some methods due to Hoeffding [6], showed that one of Anderson's
procedures could be treated exactly. He called this procedure the
sequential minimum probability ratio test.
39
2
n
stop sampling if lEy:' > 2 c(o--)
i=l J. and
= 0)
make decision d~ (emake decision
-
n/4 '
n
if E y~ < 0 ,
i::::1
n
a;: (e- = 1) if i=l
E y~ > 0 ,
n
(arbitrary if E Y~ = 0), or
i=l
n
continue sampling if I ~ y~J < 2c (0--)2 - n/4
i=l
,
and
stop sampling if
and
lEy+i I ~
+ 2
2c (0-)
.. n/4
,
··i=l
+ +
make decision d (e
o
make decision
a;.
n
n
if E
i=l
n
(e+ = 1) if E
i=l
+
= 0)
(arbitrary if E y.
i=l J.
= 0),
continue sampling if
IE
n
i=l
respectively, where c
= log
+
y. <
J.
o,
+
Yi >
o,
or
y~l
< 2c
(1/2 a)"
2
(0-+)
- n/4 ,
Equivalent forms for the
above decision rules are given by
I.. En
2
xi + n(e l +e 2 )/21 ~ 2 0- c/~ .. n~/4 ,
i=l
.
and
n
make decision d~ if .E xi > n(e l + e 2 )/2: ,
J.=l
n
make decision d~ if ~ xi < n(a + 9 )/2
2
l
i=l
stop sampling if
n
(arbitrary if E xi = n(e l + e 2 )/2) ,
:1.=1
continue sampling if
and
stop sampling if
and
.
make decision
n
d~ if i:1 xi < n(8 3
+ 84)/2
,
make decision d~ if E xi > n(e + e4)/2
3
i=l
I
n
n
(arbitrary if E xi
i=l
continue sampling if
= n(e 3
+ e4)/2)
2
n
f E xi ~ n(8 +84)/21 < 20' c/t,. nt,/4
3
i=l
respectively
1
1
0
From the preceding decision rules for sub-tests
T-
and T+
it can readily be seen that y. 1 (or y+ 1) is observed if and
n+
n+
only if the inequality
1
holds,
n
n +
E Y~ (or E yi)1
i=l
i=l
2
< 2c (0' )/t, - n/4
Furthermore, it is obvious that this inequality will be
violated if
i.e., if
n >
Thus,
by
ac
(0'2 ) / t, 2
•
a.n upper bound on the total sample size for test T is given
41
Now, a necessary and sufficient condition to ensure that both
decision d- and decision d+ will not be made with test T is
l
l
that the :fb llowing inequalities always hold:
Acceptance no. (of
e+ = 0)
e- = 1)
for T- < Acceptance no. (of
for T+1 and
Rejection no. (of
e- = 1)
for T- < Rejection no. (of
e+ = 0)
for T+ •
Since the sub-tests have converging straight-line boundaries, the
above inequalities will certa inly hold for all values of n > 1
if they hold at n = 1 (see Figure 2), i.e., if
and
or equivalently, if
and
which follows in:nned.1ately from the initia.l restrictions imposed on
the
a-values in the Introd.uction.
\
,\
"
:
\
.. \.!
-'--r:
!
I!
.._I!
_-
_.. /
I
Accept H+
Accept H
0
,
::
H-
Accept
\ \
\
\ \
\
'
make d+
make
(<\):
EXi =
(d~):
EXi
(d~):
EXi
(di):
EXi
2(l(c)/6
= -2cl(c)/6
= 2rr2(c)/6
= _2rr2 (c)/6
+
n(:~9,
+ 84)lh
+
n(3e4
+ 93)/4
+
n(3e 1
+ 9
)/4
+
n(3e 2
+ 9
)/4
FIGURE 2
2
Q
1
,
and
•
R~
NowJ let
denote the acceptance region for sub-test
n
i.e.J let
{Y(n)
e;
R~}
:<--,...:;;
{N = n, and decision d~ is made} ,
where
and
N = the terminal sample size for sub-test
T~.
Then,
= zS
n Rn
= Z
f+
n dy(n)
S-
{e
[z Y~ /2( U J.
n R
n
-< zn
S-
(f+/ro) fO
n n n dY(n)
= zJ
n Rn
r,
c.
2
)
- n/6(u~) J
} fO dY(n)
n
222
[(-2C(u-) + n/4)/2(a-) ~n/8«(j-) J.
{e
}fo dY(n)
n
R
n
= z
n
JR- {e-c } fOn
dy(n)
=
n
=
(2a) (1/2)
=
Z
n
S- (20:)
R
n
a < y
-
J
fO
n dY(n)
T~J
44
since
J f~
E
n Rn
dY(n) =
pr{d~
using
T-I'~-
= o}
because of the symmetry of the test procedure.
=
1/2
,
S1m11ar1y, it can
be shown that
Pr{rejecting
H+IH+
Pr{rejecting
H
is true}
< Y
is true}
<
J
and
IH
0 0 -
Little can be said about the
2y
•
various decision probabilities
under test T other than the basie guarantees stated above, since
little is known about the operating characteristic functions for
the sub-tests
T-
and T+
other than that they are monotone.
However, this monot.onicity does imply that
and Pr{D-
using T Je}
are monotone.
pr(D+ using Tie}
Also, by the symmetry of
the formulation of the procedure T, it is readily seen that
Pr(D O using TJe}
and at least approximately monotone (for y
of
e*
= (8 2+8;)/2
small) om either side
is symmetric about the point
e*.
8. Rallis
lJ.l~f~ation
of the M.P ..RQT. for the Case of Unknown
Variance.
Hall's analog of "the M.P.R.T.
[5]
for the case of unknown va.riance
is similar to the Baker-Hall sequential analog of the S.T.S.T.(m)
procedure in that an initial sample of size m is drawn in order
to estimate
2
cr after which sampling is continued as in the case
of the M.P.R.T.
procedure.
As was true for the I.M.P.R.T., the
three-decision analog of the M.P.R.T.(m)' denoted by I.M.P.R.T.(m)'
is applicable to the problem stated in the Introduction only if
y 1 = Y3 = y 2/2
==
y, say •
In order to obtain the I.M~P.R.T.{m)' the sub-tests T- and
T+ are constructed as follows~ Let T-
e- = 0
vs.
e- = 1,
X-,
m
X-+ , X- ~, ••• , with preml
m+c;
and with (0'-)2 [known] replaced by
+.+
denote a M..P ..R.T.. of e = 0 VS .. El == 1,
based on
assigned strength (a, a )
m m
_2
+
( sm) • Similarly, let T
x:'
denote a M.P.R.T. of
X:+ l , X:+ 2 , .... , with pre-assigned streIJ.gth (am' am)'
+ 2
+ 2
and with (o') [known] replaced. by. (s)
,where
m
based on
-1/2(c )
= (1/2) e
a
m
(22)
,
m
and
.
Then, letting
YI+
+
~
==
I
- 1 2
as before and computing the rele-
vant probability ratios, one can derive the following respective
decision rules for sub-tests Tstop sampling if
I~
i=l
Yi-
l
and T+ :
> c {s-)2 - n/4;
m m
and
make decision
make decision
d~
di
n
(e- = 0) if
(e-'
= 1)
if
E
i=l
!1
i==l
y:
<
0
,
y~ >
0
I
n> m ,
J.
n
(arbitrary if .E y~
i=l
=
0), or
continue sampling if
I
.E
n
i=1
2
Yi- 1
< c (s-) - n/4 ,
m m
46
and
,
atop sampling if
and
+ +
make decision d (9
o
= 0)
(9 +
= 1)
make decision
a!
J.
n
+
(arbitrary if E Yi
i=l
continue sampling if
n +
if E Yi < 0
1
,
0
~=
of
~
n
~
LJ
i=l
+
Yi
>
0,
= 0), or
n
f E1
+
yiJ
°
~=
+ 2
< cm(s)
- n/4 •
ill
For computational purposes it is more convenient to express the
above decision rules in the following form:
and
n
make decision d- if E xi > n(e l +9 2 )/2
o
i=l
I
n
make decision d~ if .& xi < n(9 1 + 92 )/2
i=l
n
(arbitrary if E xi
i=l
= n(91+9 2 )/2)
continue sampling if
f-. ~l
E xi
and
stop sampling if
n
n
Ii=l
E xi
I
I
or
+ n(91+9 2 )/2!
2
< c (s )/6 - n6/4 ,
m m
2
- n(9,+94)/2; ~ c (s )/6 - n6/4
m m
1
47
and
n
make decision d+
o if Z Xi < n(e 3 + e4)/2
i=l
make decision ~
,
n
if .Z xi > n(e + e4)/2
3
J.=l
,
n
(arbitrary if Z xi = nee; + e4)/2) , or
i=l
n
continue sampling if
Ii=l
Z xi
- n(0~ + 94)/21 < c (s2)/~ - n~/4 •
~
m m
As 'Was observed in tne preceding section, simple algebraic
manipulations of the above decision rules readily yield an upper
bound on the overall sample size for test T, once
s
2
is known,
m
namely
Since the outcomes of the respective sub-tests depend on the
sampling rules as well as the terminal decision rules, a necessary
and sufficient condition for the impossibility of making both
d-l and d+ using test T is that of n
l
following inequalities hold:
decisions
Acceptance no. (of e
-
= 1) for T-
= m,
the
< Acceptance no. of
(9+ = 0) for T\ and
Rejection no. (of e
..
=
1)
-
for T
< Rejection no. of
(e+ = 0) for T+, i.e.,
-Cm(s;)/~ + m(;9 2+e l )/4 ~ -cm(s;)/~ + m(;e4 + e;)/4 ,
and
which, as before, must necessarily hold because of the initial
restrictions placed on the
e-values in Sectim 2 of Chapter I.
Analogous to the derivation of' the Baker-Hall analog of Stein's
two-stage test, it is necessary" in order to show that T satisfies the bounds on the pre-assigned error probabilities [see (l)J,
T(sm"a).
denote a conditional M.P.R.T.
to consider the properties of a conditional M.P.R.T.,
(sm~a) let T-(sm,a)
So" for given
of
e-
=0
vs.
e- = 1"
assigned strength
..
based on X m
- :J X-nH- l' X-m+ 2 ' " •• , with preSimilarly" let T+(s ,0") denote
(ex,:lI1 ex).
m
.
a conditional M.P.R.T. of' e+
=0
vs"
m
e+
= 1"
based on
..., with pre-assigned strength (am, am) , where
am = (1/2)
(23)
-1/2 C
e
m
and
Then" computing the relevant probability
that the decision rules for sub-tests
n
stop sampling if
IE
i=l
y~I
~atios,
T- and T+
one finds
are given by
and
make decision
d~
n
if I: y~
i=l
< 0,
make decision
_
0. .
1
n
if I: y~
1=1 J.
>
0
,
n
(arbitrary if I: y~ = 0),
i=l
or
n
continue sampling if
I I:
i=1
y~ 1
and
+ 2
stop sampling if
cm( s)
m
and
n
make decision
0.+
make decision
a+
1
0
if I:
i=l
n
if I:
. 1
J.=
n
(arbitrary if I: Yi+
i=l
continue sampling if
= 0),
n
- n/4;
n> m,
+
Yi < 0
+
y. > 0
J.
,
or
+
+ 2
J I: y. J < c (s)
1=1 J.
m m
- n/4 ,
respectively.
Now, since T(s ,cr) and T have the same decision rules,
m
it is fairly obvious that the necessary and sufficient condition
mentioned earlier to ensure that both decision
di and decision
d~ will not be made with test T also holds for T(sm'cr).
Furthermore, as was demonstrated in the development of the BakerHall procedure,
50
ES {Pr(D* using Tesm,,0') I.s m
" 0'" e)}
m
=
where
=
ES {Pr(D* using Tlsm, 0', e)}
m
1)*
Pr(D* using Tlo, e) ,
'
denotes some specified decision.
Finally, using the
same basic line of reasoning that was employed in the preceding
section to prove that the M.P.R.T. satisfied the bounds on the
error-probabilities given in (l), it can be shown that for
s2
m
fixed,
--
Pr{d- using T-(s ,O')1s , 0', ~- = 1/2} < a 1
0
m
m
m
pr{d~
using T-(s 10')1 s , 0', ~m
m
+
Pr{d+ using T+(s ,,0') Js 1 0',
0
m
m
~
pr{d~ using T+(sm,0') Ism, 0',
~
+
= -1/2} -< am ,
-
=
1/2} < a , and
m
=
-1/2} ~ am •
-
Thus, taking expectations with respect to S on both sides of the
m
preceding inequalities, one obtains
(24)
Pr{d~ using T-IO'" ~-
< (1/2) ES {e
m
-1/2 c s2/O'2
m m }
using
(26)
and
= 1/2}
using
= (1
/
+ cm/vfV 2/2
,
51
using
for all values of
2 • Now from (22) we have
1
0-
so a < y which tmplies that test T: satisfies the bounds on
mthe error probabilities stated in (l), as was demonstrated in
Section :3 of Chapter I.
Once again our knowledge about the decision probability functions is limited to the type of general comments made in the preceding section.
CHAPTER III
EMPIRICAL EVAWATIONS
In order to make some empirical evaulation of the rela.tive
merits (wor.t. terminal sample sizes) of the various procedures
developed in the preceding sections of this paper, we shall consider the following test situation:
Letting
test
H-~
e -< -0.6,
vs. H : -0.1 < e < 0.1,
0
--
vs. It: e _> 0.6 ,
and compare the average sample size of the test procedures at three
e-values, namely
(1) a
= -0.6 (i.e., H-
(2) a
=
(,) a
=
is correct) ,
0.0 (i.e., Ho is correct), and
0.2 (i.e., none of the hypotheses are correct).
A Univac 1105 was used to generate 165 random samples of 100
observations 6 each from the first 20,000 listings in the Rand
table of 100,000 standard normal deviates.
Then the terminal boun-
daries of the six test procedures were transformed so as to make
the true mean of the data correspond to the particular a-value
60ccasiOnallY, an individual trial for a particular test procedure
required more than 100 observations to reach a decision. When this
situation arose, additional observations were drawn at random.
5;
specified by the example under investigation.
Although the variance
was obviously equal to one" it 'Was assumed to be unknown for the
I.S.T.S.TO(m)" I.S.P.RoT.(m)" and I.M.P.R.T.(m)
procedures.
Since
all six of the tests vere performed on the same set of data for
each example" the results that were obtained are correlated" pre...
sUlllably positively.
Thus differences that occurred (e.g., in
variability of sample size) among the various procedures are probably more meaningful than they would have been if the trials were
independent; and similarities are less meaningful than they would
have been if the trials were independent.
The results of the empirical investigation are presented in
the tables and figures below.
Table 1 (and 2) presents some
characteristics of the terminal sample size distributions for the
six procedures.
From the calculated standard deviations it can
be shown that the coefficients of variation of the average sample
size for the various procedures range from about 2 to 5 percent"
which implies that the average (and median) sample sizes given in
the table are quite precise.
Figures;,
4, and 5 present some histo-
grams of the actual sample size distributions. Table; contains empirical data on the frequency of the various decisions in each of
the examples.
These data serve as estimates of the true decision
probabilities.
However" these estimates are not very precise as
can be seen by appealing to the binomial coefficient of variation
formula,
-5L
np
• For example, with an n of 165 and a
p
the true coefficient of variation of a frequency is about 34 percent. Much more empirical sampling would be required to get precise estimates of the decision
probabi1ities~
Some interpretations
of these tables follow.
TABLE I
Test Case 1 (e = -0.6)
Range of
sample sizes
Median
sample
A.S.N.
n(l)
n(165)
size
Procedure
IoF.S.S.T.
I.S.P.R.T.
,
7
I.M.-P .R.T.
44
4
s
9
J.1
6-
n
5
44
108
66
44
19.33
22.88
44
24.02
24.42
15.49
10.61
1.21
0.83
I.S.T.S .T.(m)16
I .S .P .R,.T•(m)16
I.M.P .RoT '(m)16
108
161
91
42.67
24.17
24.61
48.46
31.35
29.92
19.02
21.• 41
14.98
1.48
1.67
1.17
a
Test Case 2 (8
Procedure
~
n(165)
~
= 0.0)
n(83)
A.S.N.
s
n
sn
44
4
11
44
96
54
44
23,.05
24,,29
44
26.44
26.85
12.03
7.37
0.94
0.57
I.S.T~S.T~(m) 16
108
115
92
42,,67
26.36
26.75
48.46
19~O2
31.• 92
30.33
16053
13,,11
1,.48
1.29
1.02
I.F.S.S .•T,.
I .•S.P.R.T.
L.M.P .R.T.
I.S.•P.R.T'.(m) 16
I.M.P.R.T.(m) 16
7For the IoM.P ..R.T .. (0'"2 known)# N = 74 is an upper bound on the
terminal sample size.
8m = 16.
9~ = snl fi65 , the estimated standard deviation of the A.S.N., which
serves as a measure of the accuracy of the estimated A.S.N.
55
Test Case 3 (e
Procedure
n(l)
I.F.•S.S.T.•
44
I.S.P .H.T.•
4
I .M.P .R.T.
8
n(165)
44
116
64
I.S.T.SoT"(m) 16
I.S.P.R.T.(m) 16
IoM.P.R.T.(m) 16
108
166
109
= 0.2)
n(83)
44
26..17
28.83
42.67
33.50
31.36
A.S.N.
sn
s_
n
44
33.69
31.93
22.26
11.67
1.•73
0..91
48.• 46
45,.16
36.67
19.02
31.27
19.27
1.48
2.43
1.50
~
,.[)
I .........
Distribution of samp~e s;izes in Test Case ~ for the :;r:..s"T"S.T"(m)'
FIGURE 3:
I"S.P"R"T"(m)' and I"M.PeR"T"(m)
~5
I :
i
.
5th.
- -rl
I I
.L iT! , I
58
rl
I
I
-
h-, Il-lI rll·tl-'d h~ I
j j
(m=~6)"
-n
I~.rr
10 .
procedures
I!
-.-
I
I"S.T"S.T,,(m )
-
i n ' '-,
j
i-C! cr.
'.,
rl
-,
n
il
I
II
15
i
10
i-"
!;
m' I
I
5 I'-
;
I,
;
i
iii
I
:- ,
I'
r,,
1T1 ~I ~, HTf
•
,-'
I ;
1
,"
r!-il l}'i
I
i
I"S"P"R.T"(m)
~n
,-j
.,
......... '
n
46 n
.,
".
",
15:
10
~,
~
1
rh ,-I I
; I •
I 1-, ,-;-:
1-'1-
I .M.P .R.T. (m)
:lliLi
,r':,,;! I I
I-I: : 1 i ~ j-1 ! 1:-1 ~ - I i , . .
! ; ,
, ~ , ! ! ! ! I!
!-rl- _...:.:::......:-.:::L __._~--!:!::':~,
1-:, 1-
-20
e
i
1k>
60
n
80
e
e
I:"LC\
FIGURE 4:
Distribution of sample sizes in Test Case 2 for the I.S.T.S ..T.(m)'
I.S.P.RoT.(m)' and IfM.P.R.T.(m)
I
15
t
r-!-1'!
'I!
t1~'II:
I
leS .T.S.T •(mO
i
:i
i
5 L__ ,__ II
I , ,-rj i !-! II
.
.
i
~
1,,'; -II1 r~
_~'
i I r-rn UI ~i 'Ii
....I!
I
r'
i-"
i
I
i
I
,
i i-,
i-
f"Tj
!
i
-l
'I
I,
,.......,.,..-.- .
i'.
i;-.
~'~~,
n
.
I.SoPoRoTO(m)
I
I ., ~
r"
51:!!I Ii:
! i i
;;~rr-i
. i I i'l!
I
~;!
10
(m=16).
_
10 I
15
20
procedures
",-'
I
,
":=i=
.. -
'''.-=-~
I
-, L-I
-1-'--'
!
I
•
=
, ,
,
1
,--c-,
'_',
-----
n
,-r")
,
20 t
15 i-,
1__;
_
-I: :
pool : ,__ II
10 til i
5
r!
I (-,
I
j
1! I I
: '! ', I,
I i
,
1oM.P.R.T·
ll;_
~ ! i! i: Lr-III40[i i-~-i-('.1.7'_
,~
1I i
!:il~
20
e
i-:__
_
I
I I
I
0
l~",
,-I
80
.'-:
100
e
Cm )
120
,"",.'
--~-n
•
co
.
I!'\
FIGURE 5:
Distribution of sample sizes in Test Case 3 for the I.S.T.S.T·(m)'
I.S.P.R.T.(m)' and I.M.P.R.T.(m) procedures
(m=16).
15
::,,_
1_. ,
I: . . ,." '-'.
.
5~_;~
I, I L r:,-, ii' LI ,-, 'n '
:'' 1 I I
I.S.T.S.T·(m)
lOt
!
:
I'
;
i
' ~ !
'·'1·'
i
"-"",~.-
I
1
~ !
"
1 I l
, ( ,
I.
l
roO. , i I
.,
I : , . ,_ .
"r,
j
I .
1_'
'T'-'
t.
,
n
20b
15' ,
10
1.S.P.R.T"(m)
I,
III-II 1-:, .r:'-'."._
Ii.
"
' .'
1I'. .'.
i I
, 'r 'i.
·..- "
5 tI. 1'
-1'
I I I
-,-,.-.~
.
I
I
'."
. I
'
.. ,
. I,
I I l-. i , I ... i;;i, I
•
. ,
"
..
.
"
.
1 .I
_!
'.
.
•
I .
'
r
'_. !
I
I ,
" •• II f"
(_'
_"
d'"
I
._,
,~
.
CJ.
,
!
n
201
i
15-1--'1
!
I . - ,.--
10i,
"-
I _I Il •, !
I
I I r-, , i-I j,,--:
-.
5 I! . I
I - :.i !
Ii'" I.
I .M.P .R.T. (m)
I-
I
I
,
I
I
I
"._ . . ~. t.~ I
I
I
~'O
e
I
I
iT"1f·_·..
i'
~ I 'j . ;'_;..:"
I
40
I
;
,
•......,.-..
"
: . '-,,
i-:· i
t
60
1
~
iii
r-;-r-r,
1-'\
80
!
j
r-
1 Of)
e
,.w
,
I
1
....
1 ?f)
1 ha
160
n
•
59
Comments
1.) There is essentially no apparent difference between the
I.S.P.R.T.
and
t~e
I.M.P.R.T.
procedures nor between the I.S.P.R.T_(m)
and the I.M.P.R.T.(m) procedures in average or median sample size,
though they are superior in this respect to the two-stage or nonsequential procedure.
However, if we were to employ the less con-
servative non-sequential procedure proposed in Section 2 of Chapter
II, the required sample size would be reduced from
44 to ;6 which
is more in liine with the various sequential procedures, particularly
w.r.t. test case;.
(This reduction in average sample size from
44 to;6 serves as an indication of the extent to which the
I.F.S.S.T. and, indirectly, the other procedures developed in this
paper are conservative when
2.)
8
2
:f:
8;.)
The I.M.P.R.T. and the I.M"P"R.T.(m) are less variable
w.r.t. sample size
(See Table 2)
tha~
are the non-truncated sequential procedures 10 •
Similarly, the I.M.P.RQT"(m) is less variable than
the corresponding two-stage procedure except in Ce,se; when they
are approximately the same.
TABLE 2
Test Case
I.S.P .R ..T.
ratio of
sdn.'s
I .M.P.R.T.
ratio of
ranges
IoS.PoR.T.(m): LJMoP.R.To(m)
ratio of
ratio of
sdp.'s
ranges
1
2
;
10
ObViously, the greater variability of the LS.P.R..T. and I.S.P"R"T.(m)
procedures may, for an individual trial, be favorable instead of
unfavorable, as is indicated by the n(l) values given in Table 1.
60
38) The statistics given in Table 1 seem to indicate that
the only two situations in which it would be reasonable to make
use of a I.F.S.S.T. procedure are (a.) if sequential sampling is
not feasible, or (b.) if the major consideration for choosing a
test procedure is to obtain the smallest possible upper bound on
the sample size.
Furthermore, as indicated previously, the three..
decision F.S.S.T.
developed in Section2 of Chapter II would be
y
preferable to the I.F.S.S.T. in either of these situations.
4.) It is interesting to note that the mean sample size for
the I.S.[1 ..SS.(m) is only 4.5 . observations larger than the required sample size for the I.F.SQS~TQ Thus, the cost (in terms
of sample units) of not knoWing the true variance
I.S.T.S.T.(m) is relatively smallo
were found in the sequential cases.
~
2
for the
Slightly larger differences
I.F~S.S.T.
none
0
I.S.P .R.T,.
none
1
I,.M.•P.R,.T,.
none
I.S..'f ,.S ;.T • (m) none
I.SoP.R.T"(m) none
LM"P.R ..T.(m), n~ne
l~or "the
0
138
145
142
27
19
23
2
0
0
137
153
149
26
12
16
j;,3 ,8;o:'8.T.. , we'
Pr{rejec"ting H Ie = o}
o
:+
and Pr{accep"ting H
Ie
hav.e P-rJruetlae1iitig H-Ie:;;; .0.. 6.1 ~~ot0473 ,
= 0.0210, Pr(accep"ting
= 0.2} = 0.1636.
H
0
Ie = 0.2} = .8327,
62
5.) The observed error probabilities tended, on the whole, to
'be cOllsidera:bly smal1er12 than the theoretical significance levels.
This consideration suggests that slightly larger theoretical error
probabilities than are actually desired could be employed in
determining the boundary conditions for the various test procedures
(particularly in the cases of the· Baker-Hall and Sobe1-Wa1d tests,
where differences between the observed error probabilities and the
significance levels seem to be the greatest).
Although some pre-
liminary tria1-and-error investigation would be required to determine the maximum allowable
"y" levels, one would expect to ob-
tain smaller average sample sizes
~nd
still satisfy the desired
significance levels by such a technique.
6.) It should
~lso
be noted that truncation did not tend to
produce any great change in the decision-making pattern for the
sequential procedures.
For example, although under the LM.P.R.T.(m)'
the truncated counterpart of the I.S.P.R.T.(m)' decision D+ was
made 16 times in Test Case .3, as opposed to 12 times for the BakerHall test, the same 12 trials resulted in decision D+ for both
procedures.
l~he conservatism observed in Test Case 2 is largely due to the
fact that B2 ~ B.3 •
BIBLIOGRAPHY
[lJ
Anderson1 T.
w.
(1960).
"A modification of the sequential
probability ratio test to .reduce the sample size. rt Ann. Math.
~.1
[2J
.2!,
165-197.
Baker, A. G. (1950).,
'~roperties of some tests in sequential
analysis." ~iometrika" 37, 334-346.
[3J
Hall, W. J.(1954).
"Most economical multiple-decision rules."
Institute of Statistics, Mimeograph Series No. 115.
[4J
Hall" W. J. (1961).
problems. "
"Some sequential tests for s;ymnetric
Invited address at Eastern Regional Meeting of
the Institute of Mathematical Statistics, Ithaca" N. 1.,.. to
be submitted to Anno Math o
[5J
Hall, W.J. (1962),.
stage test."
[ 6J
Stat~
"Some sequential analogs of Stein f S two-
Biometrika" 49, 367-378.
Hoeffding, W. (1960).
"Lower bounds for the expected sample
size and the average risk of a sequential procedure."
Ann.
Math. Stat. 1 31" 352-368.
-
[7J
Mosbman, J. (1958).
itA method for selecting the size of the
initial sample in Stein"s two sample procedure. 1t
~., ~"
[8J
Ann. Math.
1271-1275.
Sobel" M. and vlald, A. (1949).
"A sequential decision procedure
for choosing one of three hypotheses concerning the unknown mean
of a normal distribution."
[9J
Stein" C. (1945).
Ann. Math. Stat." 20" 502-522.
itA two-stage test for a linear hypothesis
whose power function is independent of the variance." Ann.
Math. Stat., 16, 243-258.
[10] Wald, A. (1947).
and Sons.
Sequential Analysis.
New' York:
John Wiley