I
..I
I
I
I
I
I
I
••I
I
I
I
I
I
I
,.
I
SEQUENTIAL RANK TESTS FOR LOCATION
By
Pranab Kumar Sen and Malay Ghosh
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 688
June 1970
•
I
SEQUENTIAL RANK TESTS FOR LOCATION
~
I
I
I
I
I
I
I
•I
I
I
I
I
I
I
~
I
*
BY PRANAB KUMAR SEN AND MALAY GHOSH
University of North Carolina, Chapel Hill.
1. Introduction. Recent years have seen the development of sequential rank tests.
~
Some strong theoretical motivations for the use of such tests may be found in Hall,
Wijsman and Ghosh [9J. Though, still at an early stage, quite a few research papers
have come up in the area of two sample sequential rank tests ([2,3,4,13,21 J). Most
of these tests, however, are based on Lehmann alternatives, and as a result, are
not applicable for the location problem. Besides, with the exception of termination
with probability 1 (wp 1) and finite moment generating function (mgf) ( for Wilcoxon
scores), any other characteristics like the OC and ASN functions of such tests are
very little known. Recently, Professor W.J.Hall [8J has suggested a sequential
Wilcoxon test for the two sample location problem. His procedure achieves asymptotically the prescribed strength (a,S) and possesses the Wald [1 9 J optimality for
logistic shift. On the contrary, in the specification of the alternative hypothesis,
it involves the use of a functional of the unknown distribution and thereby demands
its knowledge
this, however, does not appear to be very realistic.
Compared to the two sample location problem, relatively little attention has
been paid to the so called one sample problem. Efforts in this direction made by
Weed [20J ( see also [5J) are subject to the same criticisms as in the two sample
problem dealing with Lehmann alternatives.
In the present paper, we develop a general class of sequential rank tests for
the one and two sample location problems. In the classical two sample location
problem, once observations are taken in pairs at each stage of experimentation and
we work with their differences ( which are distributed symmetrically about the
difference of the two locations) , the problem reduces to the corresponding one
sample case. Hence, we shall only deal specifically with the one sample problem
while the above remark permits us to handle equally the two sample case.
* Work
supported by the National Institutes of Health, Grant GM- 12868.
I
~
I
I
I
I
I
I
Ie
I,
2
In section 2, we start with an asymptotically equivalent form of the Wald [19J
SPRT based on the maximum likelihood estimator (MLE) of location. Since this procedure demands the knowledge of the underlying distribution, in section 3, we consider
an alternative procedure based on the sample means which along with some mild regularity conditions requires a strongly consistent estimator of the population variance.
Both these procedures are vulnerable to gross errors or outliers, and, in addition,
the second one may be quite inefficient for distributions with'heavy tails'. In
section 4, we work with robust estimators of location based on rank statistics, and
after using the asymptotic linearity of rank statistics, we ultimately express our
'stopping rule' in terms of some well known rank statistics. It is shown that the
proposed procedure terminates wp 1 when the underlying score functions are square
integrable. Section 5 deals with the OC and ASN of the proposed tests, while section
6 is devoted to the study of the allied ARE (asymptotic relative efficiency) results.
The last section includes, by way of remarks, a comparison of the proposed procedure
with some alternative ones suggested by Albert [lJ and others; it appears that the
others may achieve the asymptotic optimality of the fixed sample size procedure but
not of the SPRT.
I
I
I
I
I
I
,-
I
~~equivalentform of the Wald SPRT. Let { x ,X ""
l
2
ad in£} be
a sequence of iidrv ( independent and identically distributed random variables )
with a df ( distribution function) F (x)
8
= F(x
-8), 8 (s 8) is an unknown parameter.
We want to test
(2.1)
H : 8
O
=
°
against
HI: 8
We assume that (i) F(x) + F(-x)
~,where ~
is small.
1 for all real x i.e., F is symmetric about 0, (ii)
F is absolutely continuous wrt Lebesgue measure admitting a density f, (iii) f is
strongly unimodal
i.e., -logf(x) is a convex function of x, and (iv) hex)
=
-f'(x)/f(x) is absolutely continuous wrt Lebesgue measure. This implies that (a)
hex) is t in x , and
(2.2)
I(f)
00
(b)
2
002
= £oo(f'/f)dF = £ooh
(x)dF(x)
00
= J
_00
h' (x)dF(x) <
00
•
I
~
I
I
I
I
I
I
--I
I
I
I
I
I
I
,-
I
3
Finally, we assume that h'(x) is uniformly continuous in x , so that
limo+ o {sup
(2.3)
x
I h'(x
I }=
+ 0) - h'(x)
0.
These assumptions are all met by a broad class of df's including the normal, logistic,
double exponential, Cauchy
and many other df's.
Consider now the Wald [19J SPRT of strength ( a, S), and define
A
(2.4)
m
and two numbers A and B such that
° <B <1 <A <
~)
00. Then, the stopping variable N(
is defined to be the first positive integer for which the inequality
log B
(2.5)
A
<
m
AN(~) ~
is vitiated. In the event
<
log A
HI is accepted. From the results of Wald [19J it follows that
<
(1- S)/ a.
B
we have
~
S/(l-a), A
~
a
and
P {Type II error }
~ +
~
S.
0 ) equivalent form of SPRT. Suppose
e is the true value of the location parameter, and write e =
¢ E I
=
B > S/(l -a) and A
, the excess over the boundaries can be neglected, and
Next we propose an asymptotically ( as
(2.7)
{x:
Ix
I ~K,
for the probability computed under
e =
N(~)
~ +
as
(2.9)
Since,
Z.(~)
l
= loge
p¢{ ~-2cl (E)
f(X.l
~)/f(X.)J,
l
<
N(~)
< ~-2 C (E) }
2
i _>1, and Zn*(~) =
P¢[ N(~) >n J ~ P¢[ 10gB < Z~(~) < logA J
(Z*(~) - E Z*(~) )
(10gB - E Z*(~))
_ _ _ _..lp:-...:..n_~_ <
n
¢ n
= P¢ [( n V¢(Zl(~))) ~
k
( n V¢(Zl(~)))2
Z*(~)
n
finite positive
Cl(E), C (E), such that
2
lim~ +0 [ suP¢ E I
Proof. Let
stands
0, where P¢
¢~.
Lemma 2.1. Under (2.2) and (2.3), for every E > 0, there exist
(2.8)
where we assume that
¢~,
where K( 1< K <00) is fixed.}
In the following lemma we give an asymptotic order of
constants
log A,
(1- S)/ a, and
~
P {Type I error }
(2.6)
~
For small
AN(~) ~
log B, HO is accepted, while if
J
n
> 1 -
~. lZ,(~),
l=
l
( logA <
( n
E.
Then,
E¢Z~(~)
k
V¢(Zl(~)))2
)
]
.
involves a sum of iidrv's with a non-zero and finite variance, by the
k
classical central limit theorem, one gets that for large n,(Z~(~) -E¢Z~(~))/(nV¢(Zl(~»)2
I
..I
I
I
I
I
J
4
is
~
N(O,l). Hence, it suffices to show that for n
(2.10)
10g(AB
-1
rr;::::
1:
)/(nVep(Zl(6)))2
<V27TE
~
as
C
2
(E)/6
6+
2
(+
00
as 6 + 0 ),
O.
6h(x- 612) + O( 6 2 ),
Now, when 6 is small and (2.3) holds, 10g{f(x- 6)/£(x)}=
uniformly in real x. So, for ep E I, 6 small, under (2.2) and (2.3),
(2.11)
Vep(Zl(6))
~ 6 2I(f) ~ nVep(Zl (6»
~ C 2 (E)I(f) •
1:
and this can be made smaller than (27T) 2E by proper choice of
C
2
-1'
1:
)/(C (E)I(f»2,
2
(E). Again, on
Thus, the lhs of (2.10) is approximately equal to ( for small 6 ) 10g(AB
writing 2d = mine -10gB, 10gA ), we have for small 6 ,
Pep{N(6) <n }= Pep {Z:(6) ~ (logB,logA), for some m: 1< m< n }
(2.12)
IZ*(6) 1>2d} < P { max IZ*-(6) __ E Z*(6) I> d },
< P {max
- ep 1 <m 2.n m'
.
- ep 1 <m <n m
ep m
as for all m 2. n
~cl (E)/6 2 , IEepZ:(6) I.:. m 6 2 (lep-!-.21 I(f) + 0(1)) 2. c 1 (E) Jep -!-.2II(f)
I
+ 0(1) can be made smaller than d by proper choice of
lit
Zl(6), ... , Zm(6) ( and hence is t in m ), using the Kolmogorov inequality for martin-
I
I
I
I
,-
I
I
,-
I
66m } forms
E~Z*(6),
~
m
C
1
(E). Again, since {Z:(6) -
a martingale sequence, where aD. is the a-field generated by
m
~
-2
(E) that
1
P {max
IZ*(6) - E Z*(6) 1 >d } < d- 2V (Z*(6))
ep 1 <m <n
m
ep m
ep n
2
2
2
= d- {n 6 I(f) [1 + o(l)J } = d- c (E)I(f){ 1+ o(l)} 2. E/2,
l
gales ( see [11, p.386J ), one gets by taking n
(2.13)
by proper choice of
6
C
(E). This leads to ( for all ep E I )
1
-2
-2
Pep{ c l (E) 6 .:. N(6) .:. C 2 (E)6
} ~ 1 - E , as 6 +
(2.14)
C
Define now Y,(6)=Y,=X, -612, i> 1, U (6) =
J.
J.
J.
m
O. Q.E.D.
l:,m lh (y,(6)). Then, for small 6 ,
J.=
J.
Am =
(2.15)
l:i:l log {f(Y i -- 6/2)f(Y i + 6/2)}
2
-1 m
= 6Um(6) + m(6 /8).[m l:i=l{h'(Y i + a 1 6/2)
A
o <aI'
a
2
<1. Again, when S is the true location, let
~m
be the MLE of it ; by the
assumed strong unimodality of f, the MLE exists, is unique and is a strongly consistent estimator of S. Now,
(2.16)
0=
m- 1 l:,m heX, _
J.=l
+
(6/2 -
J.
SA)
m
= m-ll:,m heX ,-612) + (612 - ;
J.=l
J.
~m){ m-1E,m1[h'(X,-6/2+
n(6/2J.=
J.
m
)(m-1l:'~lh'(X'
J.J.
em) - h'(X,- 6/2)J},
J.
- 612))
0 < n <1.
I
..'I
I
I
I
I
I
I
Ie
I
I
I
5
e
</J E I, 6 small, on using (2.2), (2.3), the strong consistency of e
m
-1 m
and the Kintchine law of large numbers, one gets that m ~. 1h'(X.- 6/2) + I(f) a.s.
For
</J6 ,
=
1.=
as m +
00
and on defining W (6)
,
m
(2.17)
= (~m - 6/2){m-1~.m1h'(X.1.=
1.
m
m
Lemma 2.2. For every
E >0, there exists a
6
0
= 6 0 (E),
O.
such that
(2.18)
</J
E
1.
}
+
Proof.The 1hs of (2.18) is equal to
(2.19)
P</J {
AN(6)-6N(6)WN(6) (6)
~ E,
N(6)
2 , c(2) 6- 2 )
~ ( c(1)6E
E
P</J{
AN(6)-6N(6)WN(6) (6)
> E
N(L1)
E
(
(1)6- 2 c (2) 6-2) }
cE
' E
by lemma 2.1, the first term is bounded by E , while the second term is bounded by
(2.20)
1 - P~{IA - L1mW (6)1 < E for all m E(~1)6-2, c(2)6- 2 ) }
~
m
m
E
E
2
2
< 1 - P~{IA - 6mW (6)1 < m06 for all m E(c(1)6- , c(2)6- 2 )}
-
provided m06
that for all
(2.21)
m
~
2
m
< E. Again,
6
~
-
E
E
(2.17) implies that for any 0 >0,
6 (E) can be so chosen
0
6 (E),
0
P~{IA - m6W (6)1 < om 2 , for all m E(c(1)6- 2 , c(2)6- 2 )} > 1 - E •
m
~
m
E
E
-
From (2.19), (2.20) and (2.21) the lemma follows directly. Q.E.D.
Consider now another stopping variable N*(6) defined to be the first integer
for which the following inequalities are vitiated
(2.22)
(log B) / (m 6)
< W (6)
m
< (log A)/ (m6 );
I
excess over the boundaries can be neglected. If
I
+
We now have the following lemma.
I
,e
6/2)} ,
2
A - m6W (6) = oem 6 ) a.s., as 6
a.s. ;
if
I
I
1.
N*(L1)WN"~(6fL1)<6-110gB we accept HO' while H1 is accepted
1
i f N* (6)W * CL1) (6»6- 10gA.
N
When 6 is small, W (6) is also infinitesimally small for each m ~1, so that the
m
L~F{</J6) and L~F)C</J6) denote the DC
functions of the two stopping rules N(6) and N*Cl1) respectively, by lemmas 2.1 and 2.2,
(2.23)
Since
C2.24)
1im
6
+
0
I~F)C</J6)
F
- Li )c</J6)I
~F(O) ~ 1- a and ~F)C6)~ 8, as 6
lim 6
+
0 P {Type I error } =a,
=0
+
for all </J E
I.
0, we have immediately for N*(6),
lim6
+
0 p{ Type II error }
=8.
Finally, we shall see in section 6 that this procedure retains asymptotically C as 6
+
0) the Wa1d-optima1ity of the SPRT.
I
..I
I
I
I
I
I
I
•I
I
I
6
3. "'JV'\.,
The ~
sample mean
procedure. The procedures considered above assume the knowledge
~- ~
of the df F. In the absence of this prior
in~ormation
on F, we may consider the
following procedure based on the sample mean and variance. It remains valid for a
broad class of df with finite mgf in the neighbourhood of O.
m-l~.mlX.
is an unbiased estimator ( in fact, the BLUE) of e. In the
1=
1
definition of W (~), {m-l~.mlh'(X. _~/2)}-1 can be interpreted as an estimate of
1
m
1=
1
k
{I(f)}- , which is the variance of the asymptotic distribution of m2(e - e). Hence,
Now, X
m
A
k
-
noting that the variance of m2( X - e) is
m
the sample variance s;
(m-l)-l
0
2
which can be unbiasedly estimated by
~i:l(Xi - Xm)2, and keeping in mind the stopping
variable in (2.22), we may define the following procedure based on the sample means:
Continue sampling as long as
(3.1)
(s
2
m
-
~)
log B)/(m
~/2)
< (X m
(s2 log A)/(m ~);
<
m
2
if for the first time at the mth stage, X - ~/2 ~(smlog B)/(m ~) accept H ' and
O
_
2
m
accept HI when X - ~/2 > (s log A)/(m ~); the corresponding stopping variable is
m
m
denoted by NM(~).Had 0 2 been known, then the SPRT of Wald[19J reduces to (3.1) when
F is normal, provided we replace s2 by
m
2
0
•
In that case, Wald's technique for the
SPRT remains equally applicable for the mean procedure as here also we deal with
L~F)(¢~) and ~ (¢~). The OC and
iidrv's; we denote the OC and ASN functions by
NM(~)
L~F{%~)
ASN for the actual stopping variable
pectively. Then, since, for 0 2 < 00, s2 +
~M(¢~)'
are denoted by
and
res2
0
a.s. as m + 00, along the same line as
m
in lemmas 2.1 and 2.2, it follows that whenever 0 2 exists,
L(F)(¢~)I
o
(3.2)
o , \:f
¢
E:
I,
which ensures that N (6) has asymptotically ( as ~ + 0) the strength (0'.,13). I t will
M
be seen in section 6 that i f F has a finite mgf, then
(3.3)
~M(¢~ ) / ~ 0 (¢~) = 1, V¢
lim ~ + 0
I,
E:
I
and we have the following theorem on the termination probability for NM(~)'
Theorem 3.1. If 0 2 < 00, then for every (fixed) e (= ¢~), ~ not necessarily small,
I
(3.4)
t
I
.e
I
limn+
Proof. P
e
If
e
=
{NM(~»
~/2, n
k2
(
oo
e {
P
NM(~) >
2
e
n
O.
=
k
k
-
2
k2
2
{(snlogB)/(n2~ ) < n ( X - ~ /2) < (s 10gA)/(n
n
n
n }~ P
X -
n}
~/2)
~ N(0,02 ) (as n +
converge to 0 (a.s.) as n +
1
00
00
),
~)
}.
while the two limits both
hence the probability can be made arbitrarily small
1
1
k
Xn - ~/2) = n~( e - ~/2) + n~( Xn - e) = n 2(e - ~/2)
;
-
for large n. If e # ~/2 , n~(
+ 0 (1), and hence the proof follows trivially by noting that the two limits in
p
k
(3.1) converge a.s. to o while for e > ( or <) 612, n 2(e - 612) + 00 t>r _00 )as n+
The two procedures based on X and
m
em will be compared in section 6.
00 •
I
..I
I
I
I
I
I
I
II
I
I
I
I
I
I
I
••I
7
4. The proposed rank order procedure. Because of the vulnerability of the sample
'V'V"
,,~
""JI./V'I "'V"VV'\V
~
mean and variance to gross errors, outliers, and their inefficiency for df's with
heavy tails, the procedure in section 3 is not so robust as compared with the alternative rank procedure to be posed below.
For each n > 1 and real b (
(4.1)
Rn1. (b) = !.:2
where c(u) =
l,~
_00
< b <
00),
define
+
or 0 according as u >, = or <0. Consider now n scores J (i/(n+l»
n
{= EJ(U ) }, i=l, ••• ,n, where U <.•. < U are the ordered random variables in a
ni
nl
nn
sample of size n from the rectangular (0,1) distribution, and J(u) : 0 <u <1 is the
score function characterized by a known symmetric df G with a finite Fisher information
l(g),(defined as in (2.2) with f replaced by g,) in the following manner:
(4.2)
l
-g'(G- «1+u)/2»/g(G- l «1+u)/2», 0 <u <1.
J(u)
For example, when G is normal or logistic, J(u) is the inverse of the chi distribution
with 1 degree of freedom or u, and the corresponding J (i/(n+l», i=l, ••. ,n are
n
known as the normal or the Wilcoxon scores. Assume that
(4.3)
is t in u : 0 <u <1 and is not a constant
J(u)
this holds when g is strongly unimodal, as is the case with normal or logistic or
many other G. Let then
1
(4.4)
foJ(u)du
j.l
1
f o J 2 (u)du = leg) <
and
00
For each real b, consider the usual one sample rank order statistic
(4.5)
n
seX. - b) J (R . (b)/(n+l»
1= l
1
n n1
I.
T (b) =
n
By (4.3), T (b) is
n
+ in
b :
_00
< b <
00
•
(4.6)
translation-invar~nt
e * = sup{ b: Tn(b»
nl
Now, asymptotically ( as n
(4.7)
C(F) = 2
JO
and it is assumed ( as in
(4.8)
lim x
+
00
2c(u) - 1.
When e is the true parameter, T (e) is
distributed symmetrically about O. Hence,
median-unbiased and
; s(u)
n
as
in [lO~we consider the following robust,
estimator S* of e
n
0 }, e * = inf {b: Tn(b)< 0 };
n2
1
+
00
),
n~(
en*-e)
'\" N(0,T 2 ), where
(d/dx)J(2F(x) - l)dF(x)
(> 0),
[12J) that the following holds
(d/dx) J(2F(x)-1) is bounded.
* + e n2
* )/2.
en* = ( Snl
T
2
= V 2 /C 2 (F),
I
..I
I
I
I
I
I
8
Also, we assume that both f(x) and f'(x) are bounded almost everywhere. Finally,
we assume that for some finite positive K,
J'(u) ~ K(l - u)
(4.9)
-1
Ie
I
I
I
I
I
'I
I
,.
I
fa exp {t J(u)}du
<
for all t <t (> 0).
00,
-0
Note that (4.9) is comparable to Wald's requirement of finite mgf for
Z.(~),
~
defined
after (2.8), and it holds when G is normal, logistic, double exponential or many
other df.
Since, when 8 = 0, the distribution of T (0) is known and is symmetric about 0,
n
we can always select an
(4.10)
a
n
P { IT (8)1 <T
8
n
8
* =
U,n
- n,a
(~a,
specified ) and a T , such that
n,a
Po {IT (O)j' < T
}
n
-
n,a
}
=
1 - a
n
~
1 - a .
Let then
(4.11)
inf { b: T (b) <-T
n
A
I
1
~
,0 <u <1,
(4.12)
C
n
=
2T
n,a
}
n,a
/{ n( 8 *
U ,n
8
- 8
~'(
L,n
= sup{
b: T (b) > T
}
n
n,a
* )}.
L ,n
A
I t is shown in
[I5J that C
n
is a strongly consistent estimator of C(F). Thus, a
strongly consistent estimator of
T
2 is
A
T
A
2
n
=
V2 /C n2
•
Then, analogous to the procedures
described in the preceeding two sections, a stopping rule may be formulated as follows:
Continue sampling as long as
(4.13)
6/2
A2
< ( v 2log A)/(m6 C ) ;
m
accept H ' while if 8 * ~
O
m
if 8* <
m-
~/2
+ (
v2logA)/(~
accept HI' Now, with a view to simplifying further the above procedure, consider the
asymptotic ( as
(4.14)
~ ~
0 ) linear relationship proved in Sen and Ghosh [15J :
- m:k2 ( 6/2 - 8 * )C(F) = 0 a.s., as m
m
~
00
,
which along with (4.6) and some simplifications lead to the following proposed rule:
Continue sampling as long as
A
1
A
-1
( v 2log B)( ~C)- < T ( ~/2) < (v 2log A)(~ C) ;
(4.15)
m
m
m
if T ( 612) < (V 2logB)( 6~ )-1, accept H ' while if T ( 6/2) > (v2logA)(~ C)-I,
O
m
m
m
m
accept HI; the corresponding stopping variable is denoted by N (6).In this as well as
J
in the next section, where there is no confusion, we shall write
NJ(~) =
N(6).
We are particularly interested in the above procedure when 6 is small, 8 =
¢~ ,
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
10
the OC function of N (6) by
J
L1 F) (ep6).
Then, we have the following theorem.
Theorem 5.1. Under (4.8) and (4.9), the proposed sequential rank tests have asymptotically ( as D.
(5.1)
the same OC function as of the Wald SPRT i. e. ,
+ 0 )
I L1 F)(epD.)
lim D. +0
-
~F)(epD.)1
0 for all ¢ E I.
=
Remark. Note that from the results of Wald [19J , we have
(5.2)
and hence, from (5.1) and (5.2), we have
(5.3)
limD.
+
(F)
0 L
(0)
J
=1 - a
and
. .
l1mD.
+
(F)
0 L
J
(D.) = B ,
a property which may be termed as the aSymptotic ( as D.
+ 0)
consistency of the propo-
sed sequential rank order tests (SROT).
Proof of the theorem. By definition, ITn(b)I <n~ , for all n> 1 and b. Also, it follows
from the results of Sen and Ghosh [15Jthat as n
(5.4)
+
for every (fixed) positive s,
00,
~n - 11 > O(n-~(log n)3} -< O(n- s ), where we take s> 2.
P { IC(F)/
Hence, on defin1ng d = V2{~ C(F)}-l( min [-10gB, 10gA J), one gets from (4.15) and
(5.4) that
N(D.)
(5.5)
>
D.-ld (+
00
),
in probability, as D.
Again, from theorem 4.3 of [15J, we have for n
(5.6)
sUP
where
I
(5.7)
*
n
b
In-~ h
n*
n
(8) - T ( 8
n
~
n*(D.)
O.
s
with probability> 1 - O(n- ),
00,
+ n-~b)} - bC(F)1
=
O(
n-~(log n)2),
~l} • Further, on defining
{b : Ibl «log n)k, k
we have for every
(5.8)
n
+
+
K D.- 2 (-10gD.),
K <
00
,
as D.
0,
+
ep E I,
*
A
2
Pep{V (10gB)/(D.C n *(D.)) <Tn *(D.)(D./2)<
Pep{ N(D.) > n (D.)} <
*
A
2
V (10gA)/(D.Cn *(D.))}'
-k:
where on using (5.4)-(5.7) and the asymptotic normality of {n (D.)} 2Tn*(D.)(epD.) ( under
1
Pep , this has the same distribution as of
{n*(D.)}-~Tn*(D.)(O) under H ), it readily
O
follows that the rhs of (5.8) converges to 0 as D.
+
O. Thus, as D.
+
0, dID.
2
N(D.) <
n *(D.), in probability, and since in this interval (5.6) holds,
(5.9)
{N(D.)}
k:
-k:
{N(D.)PD.(¢ - ~)C(F) = o (1), as D.
2
Since, when 8 holds,
p
+
O.
{T (8), n> 1 } forms a martingale sequence ( cf. [15J ), it
n
-
I
..t
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
11
follows from theorem 4.4 of Strassen [17J
(after verifying the needed regularity
conditions as in [16J ) that on writing
~t =
(5.10)
and denoting by {
~(t),
t -~ '"'"
T =
t
(5.11)
2
(n+l - t)Tn(S) + (t - n)Tn+l(S) , for n
t
~O
vt -!.:2
t <n+l, n> 0, TO(S) = 0,
} a standard Brownian motion on (0, 00), we have
~(t)
+
0(1) a.s. , as t
+ 00 •
!t;
3
Finally, noting that for all n> d/6 , the rhs of (5.4) is O( 6 (-10g6 ) ) = 0(1) as
6 + 0, we have
(5.12)
= 0 , for all ¢
I,
€
V/C(F), and
where
6' = 6/T,
T=
(5.13)
P ,( S, ¢) = P {A standard Brownian motion
6
line (10gB)/6' + t(~-¢)6'
(10gA)/6'
+
t(~-
~(t)
: 0 <t< 00, first crosses the
before crossing the line
¢)6' } •
Since, { Z* (6), n >1 }, defined after (2.8), also forms a martingale sequence,
n
-
proceeding in an analogous way, we have
(5.14)
= 0 , for all ¢
Note that if we let
6
+
€
I.
0, for the SPRT, we obtain that
(5.15)
lim 6
+
° P6 (
~,
¢) = p( ¢) exists for all ¢
€
I,
S. Hence, the proof is completed from (5.12)-(5.15).
where p(O) = 1- a and pel) =
Next, we investigate the asymptotic ASN function of the proposed SROT. We
consider the case of S = ¢6, ¢
I,
€
6 small, and first, we let ¢ #
~.
Then, we
have the following theorem.
Theorem 5.2. For every ¢ ( #
(5.16)
~)
€
I, under (4.8) and (4.9),
~i:0{62E¢(NJ(6»} = T2 {P(¢)10gB + [l-P(¢)JlogA}{( ¢_~)-l}=
Proof. For some arbitrarily small positive
€
,
0/( ¢,T ).
define
(5.17)
where C will be chosen later on. Then, we have
(5.18)
2
6 E¢{N/6)} =6
2
{
En~nl (6) + E (6)< n <n (6) + En> n (6) np¢[ NJ (6) = nJ } ,
nl
2
2
where the first term on the rhs of (5.18) is bounded by
€
•
Also, by using (5.4),
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
•-
I
12
we obtain for small
~,
A
p¢{ N(~) >n2(~)}
2
p{ IC(F)/cn2(~) - 11 > E } +
A
A
A
P¢{(V2logB)/(~Cn2(~)) < Tn2(~)(~/2) < (V2logA)/(~Cn2(~)),IC(F)/cn2(~)-11 < E}
(5.19)
P¢{(V2l0gB)(1+E)/(~C(F)) <Tn2(~)(~/2)< (v 2logA) (l+E)/( ~C(F))} + O({ n2(~)}-s).
1
Again, for any ¢ E I, 1¢~-M21 =I¢ - ~I~, and {n2(~)}~ I¢ -~I~ = CV{(10gAB- ) (-log ~)}\
2
k:
~C(F))
(5.20)
k:
k:
= {O( -logA)} 2 = O(log n2(~))2. Hence, on using (5.6) and noting that({n2(~)}2.
/¢ -~I
-1
= O«-log
~)
-k:
2) = 0(1), as
2
p¢{ N(~) >n2(~)}
~ +
0, we have
k:
p¢{[n2(~)J2(~ -¢)~C(F)(l+ 0(1)) < [n2(~)J
-~
Tn2(~)(¢~)
[n2(~)J~~C(F)(1+ o(l))(~ -¢) } + O([n2(~)J-s)
= PO{CV(~
-¢){(logAB
-1
k:
)(~10gn2(~) + O(1))}2(1
< CV(~ -¢){(logAB
-1
)(~10gn2(~)
-k:
<[n2(~)J 2Tn2(~)(0)
+ 0(1))
k:
+ O(1))}2(1+ o(l))} +
Now, it follows from Sen and Ghosh [16, section 3J that for c
n
2
-k:
-k:'
2
O([n2(~)J
-s
).
k:
= c(logn) 2, c> 0,
_c 2 /2v 2
-s
POi n 2T (0) >c } = Po{n 2T (0) <-c } <exp(-c /2v ) = n
< n
nn
n
n
n
2
by choosing c ~2sV2 . Thus, it follows that by proper choice of C in (5.17), the
(5.21)
rhs of (5.20) can be made smaller than o({n2(~)}-s), for ~ sufficiently small. Now,
~n>n2(~) nP¢{N(~) = n}= (n2(~) + l)P¢{ N(~» n2(~)} + ~n:(~)+l p¢{ N(~»
(5.22)
00
= ~n2(~)+1 P¢{ N(~) >n}
Consider now n:
kn2(~)
2
n
{
+ O(
«k+l)n2(~)'
n2(~)
}
-s+l
),
~ +
as
n }
O.
k being a positive integer. The same proof
as in above leads to
(5.23)
p¢{ N(~»
s
n } < p¢{ N(~) >kn2(~) } = O({kn2(~)}-s) = O(n- ), for k=1,2, ... ,
which leads to
(5.24)
~n>n (~) p¢{ N(~) >n}
2
2
O( {n2(~)}
-s+l
), s> 1, as ~
+
O.
Thus, using (5.17), (5.18), (5.22), (5.24) and the fact that
n (~)
n2(~)
(5.25) ~nl(~)+lnp¢{ N(~) = n} = ~nl(~)+lP¢{ N(~) >n } + {nl (~)+l}P¢{N( ~»n1(~) }
- {n2(~) + I} p¢{ N(~»
it suffices to show that for every E > 0,
I'
2n2(~)
(5.26)
1m ~+ 0 I ~ ~nl(~)+l p¢{ N(~» n} - ~(¢,T)
Now, from (5.6), one gets that as
~
+
0,
I
n2(~) } ,
< E \f¢( ~~)E
I .
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
,-
I
13
-k
k4
sup
3
(5.27) nO (A)
<
<
(A) n 2/ T (MZ) -T (¢6) -n(ep -~)6C(F)1 = 0(6 (-10gM )
u
_ n _n
u
n
n
Z
s-l
-·s+l
with a probability ~ 1 - o({nO(M}
) = 1 - 0(6
)
1 - 0(1), where
(5.Z8)
nO(M =
v 2{min(-10gB, 10gA )}(l+ E)/{C(F)6]J}
(+
00
0(1),
as 6 + 0 ).
Note that for 6 sufficiently small, nO(6)~ n (6). Since, IT (6/Z)I <m]J , for all m> 1,
m
l
by (5.19), (5.Z8), and the definition of N(6), we have for all n (6)<n <n (6) , as 6+ 0,
Z
l
A
A
2
2
Pep{ N(6» n } ~ P¢{(v logB)/(6Cm) < Tm(6/Z)< (V logA)/(6Cm), nb(6)~m <n}
s
~ Pep{(v 2 (1+E)10gB)/(6C(F» <T (6/Z) «V 2 (1+E)10gA)/(6C(F», n (6)<m <n (6)} +0(6 )
m
Z
O
(5.Z9)
2
2
1 <m <n} + 0(6 s ),
= P {v (1+ E)logB <T (6/Z) < V (1+ E)logA
ep 6 C(F)
m
6 C(F)
which, by the use of (5.Z7), reduces to
(5~30)
where
(5.31)
'\;
!.:
Z (6) = m2 { T (¢6) + m(ep m
m
~
)6 C(F)} , m> 1 .
-
Essentially retracing steps backwards, one gets that for n (6) <n <n (6) , as 6 + 0,
Z
l
v2(1-~)10gB
v 2 (1-E)10SA
s l
(5.3Z) Pep{N(M>n } ~ Pep{ 6 m~ C(F)
< Z'm(6) < 6m~C(F)
,1 <m <n}
+ O( 6 - ).
'\;
'\;
Thus, if we define two stopping variables N (6) ( and N (6) ) by the least positive
Z
l
'\;
~
integers for which Z (6) first crosses either of the two curves (V 2 (1+E)10gB)/(m 6C(F»
m
k2
or (V 2 (1+E)10gA)/(m 6C(F»
k
( and (v 2 (1-E)10gB)/(m 2 6C(F»
k
or (V 2 (1-E)10gA)/(m 2 6C(F»
),
we have from (5.Z9) - (5.3Z) that.. as 6 + 0,
(5.33)
62(E~~~~~+lPep{ ~l (6)
>n}
+ 0(1)
n (6)
2
~ 62(E~f~~~+1 Pep{
N(6»
n }
{ '\;
6 (E nZ(6)+1 Pep NZ(6) > n})+ 0(1).
1
n (M
'\;
l
Note that 6 2 (E n=1 Pep{ N ( M > n} < E , and for n ~ nZ(6), as 6 + 0,
i
'\;
V2 (1-(-ljiE)10SB
'\;
V2 (1-(-1)iE)logA
>
(5.34)
Pep{ Ni (6»
n} < Pep{
6 n~C(F)
i
= POi n2(~ -ep)6C(F) +(v(l-(-l) E)10gB)/(n 26C(F»
k
2
.
6 n~ C(F)
< Zn(6) <
k
1
< n
•
-~
}
Tn(O) <
1
n~(~ -ep )6C(F) + ( V2(1-(-1)lE)10gA)/(n~6C(F»}
= O( n- s ) , by (5.Z1), for i=l,Z.
Hence, from (5.Z6), (5.33) and (5.34), it suffices to show that for every n > 0, there
exists a positive E, such that for all ¢ (
~ ~)
E I,
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
14
lim6 ~ 0
(5.35)
I 62
00
~n=l
p~
{'V
I
- 'l'(~. T)
Ni (6) = n}
'V
< n • for i=1.2.
'V
We shall prove (5.35) only for N (6).as the proof for N (6) is identica.L
2
l
'V
IT (~6)1 < n~ • and
that
n
(5.36)
lim inf ![N>nJ'
k
.
I~
Izn (6)1 -<n~(~ +
-
~n(6) I dP~
-
).
~I
Hence. using (5.34). we have
k
;: 0 , lim inf
Note
! [N>nJ
11
2 ITn (~6)
I
dP <jJ
;: o.
,~
Let now
~
n
(S) be the a-field generated by sgn(X.-S) and R .(S). i=l •••.• n; n> 1.
~
Then. from theorem 4.6 of [15J • we note that for S =
-
n~
<jJ6. { T (~6). ~*(S); n >1 }
n
n
-
forms a martingale sequence with E¢{Tn(~6)} = O. for all n. Thus, using the elegant
result of Chow. Robbins and Teicher [6J • we obtain that
T~ (6)(¢6)} =
EO {~ (6)(0) }
1
1
which along with (5.31) implies that
(5.37)
E<jJ{
(5.38)
'V
E ¢{ Nl (6)}
= O.
'V
k'V
1
= E~[{ N l (6)}2 z~l (6) (6) J[ (~ - ~M C(F)J- .
Now. neglecting excess over the boundaries ( permissible for 6 ~
~an only assume the two values (V 2 (1+E)logB)(6C(F))-1 and
(V
2
'V
Mv
0). {Nl(6)}2Z~1(6)(6)
(1+E)logA)(6C(F))-1
* . and P*(<jJ6). By the method of proof of
with respective probabilities. say. Pl(¢6)
2
theorem 5.1. it follows that
lim6 ~ 0 PI*(¢6). = p(¢) = lim 6 ~O{l- P2*(¢6)}.
(5.39)
where
(5.40)
Since
p(~)
is defined in (5.15). Hence. from (5.37) - (5.39). we obtain that
2
'V
} =
lim6~0 {6 E~[N1(6)J
T
2
[(l+E)P(¢)logB + (l+E)[l-P(¢)JlogAJ
(<jJ
-~
;: (1+E)'l'(¢,T).
)C(F)
E is arbitrarily small. (5.35) follows from (5.40). Q.E.D.
The above proof fails when<jJ = ~. nor the Wald technique ( see [19, P.176J ) seems
readily applicable. However. one may note that if
(5.41)
(dP(~)/d¢ ) 11
~-o
=(
dP(¢)/d¢ ) 11+ = P' (~) exists
~
then considering a sequence of <jJ values. say
0
~
±
E • E (> 0)
r
r
~
0 as r
~
00.
using
theorem 5.2 and the L' Hospital rule, one gets that
(5.42)
T
2
P'
(~) logAB
-1
•
6. ARE results. We want to compare the performance of the proposed SROT with that of
~
the MLE and the mean procedures proposed in sections 2 and 3. For two sequential
procedures Q and Rfor testing H against HI ( as in (2.1) ). denote by N (6) and
O
Q
-I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
•e
I
15
NR(~)
the corresponding stopping variables. Then, if both the procedures are
~-+-
asymptotically ( as
procedure R when
e
0 ) of strength (a,S), the ARE of the procedure Q wrt the
= ¢~
, is defined by
(6.1)
2
Note that under the assumption of finite mgf of XIS, all moments of X exist, and then
E Is
2
n
-
021
2k
inequality, p¢{
on the true
=
k
O(n- ). Taking k = 2 + 0, one finds that on applying the Markov
Isn2
- 02
I
> £ } ~ £ -4- 20.O(n-2-0 ). For the mean procedure based
0 2 , again neglecting excess over the boundaries for small ~ , Wald's
equations ( [19J, (3.57) on p.53 and (A.99) on p.176) hold. Hence, proceeding as in
section 5, we obtain that
(6.2)
Also, when ¢ = ~, in the same way as in (5.41) one gets that lim ~ -+-OE{~2 NM(~)} =
0 2 P'(~)log(AB-l). Thus, p'(~)
(6.3)
=
l
-logAlogB /(logAB- ). Then, we get from (5.42) that
lim~ -+- 0 (~2 E~ {NJ(~) })=
2
- T 10gA LogB.
Hence, from (5.16), (6.2) and (6.3), the ARE of the proposed SROT wrt the sample
mean procedure is given by
(6.4)
e( J, M )
=
02C2(F)/V2
The above is the Pitman efficiency of a general rank order test wrt the Student
t-test. In the particular case of J(u) = u , i.e., of the Wilcoxon signed rank
statistic, (6.4) equals to
2
12 02( /0000 f (x)dx )2, and this is (i) ~
in the class of df with finite second moment, (ii) equal to
3/TI
=
0.864 uniformly
0.955 when F is
normal and (iii) greater than 1 for many non-nornal F, including the class of heavy
tail df's. Again, when J(u) =
<I>-1«1+u)/2),
<I> being the standard normal df, Le.,
for the normal scores statistic, (6.4) is bounded below by 1, where the lower bound
is attained only when F is also normal. This, clearly indicates the asymptotic
supremacy of the normal scores procedure over the standard normal theory procedure .
Now, since E¢[ log {f(X - ~)/f(Xl)}J
1
=
f(X )} J = ~2I(f) + o(~2), neglecting excess
l
~2(¢ -~)I(f) + 0(~2) and V¢[ log{f(Xl-~ )/
over the boundaries and denoting by W
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
16
the Wald SPRT procedure, i t follows from Wald [19J that
(6.5)
.
2
_ {{P(.</>)lOgB +(l-P(</»)logA}/{(</>
11m 6 ~o [6 E</>{ NW(6)}J _( 10gA 10gB)/I(f),
-~)I(f)}
From (5.16), (6.3) and (6.5), we obtain that
(6.6)
where
e( J, W )
p = { JI
= {I(f)
T 2}-1
= C2 (F)/{I(f)
Vi}
=
p2 ,
~(u)J(u)du}/{(Jl ~2(u)dU)(JOIJ2(u)dU)}~, and ~(u) = -f'(F- l «1+u)/2»/
0 0 .
f(F- 1 «1+u)/2), 0 <u <1. For the equivalent representation oLthe central term in
(6.6) in terms of
when F
=G (
p2, we may refer to H;jek and S;i.d~k [7,p.236 J.ln this situation
up to a scale variation ), i.e., the true and the assumed df's differ
only in scale parameters,
and Wilcoxon
si&ned.ran~
p = 1. and hence {6.6) equals to 1. Thus, the normal scores
statistics lead to SROT which are
a~totica1ly
( as
~ ~ 0)
optimal when the underlying df is normal and logistic. In fact, when F is normal, the
SPRY , the mean procedure and the normal scores SROT aLe all asymptotically optimal.
We conclude this section by the following theorem which reveals the asymptotic
optimality of the MLE procedure ; in fact, it shares asymptotically the same properties
as of the Wald SPRT.
Theorem 6.1. Under the conditions in section 2, the MLE procedure satisfies the
with N (6) replaced by N*(M.
W
Proof. In the definition of the stopping rule (2.22), upon replacing m6W (6) by
ASN equation in (6.5)
m
6Um(6) , defined just before (2.15), we get a parallel stopping rule whose stopping
variable is denoted by N (6). We start with </> ~ ~, and let
U
(6.7)
n1 = n (6,E) = [ £6- 2J, and n = n (6,£) = [ K£~~2 J,
l
2
2
where E ( >0) is arbitrary and K
£
is so chosen that (6.11) (to follow) holds.Note that
(6.8)
where U (6) involves sum of iidrv's. We define r' ~ 6- 2 as 6 ~ 0, and let X =
m
k
kr'
6. (Ei=(k_l)r'+lh(Xi - 6/2»), k=1,2, ••• , then E</>( Xk ) ~ (</> -~)I(f) +0(1) and V</>( Xk )
~
I(f) as
6
~
O. Since
X involves sum of iidrv's with finite variance, by the
k
1
classical central limit theorem, { X - E</>(Xk)}{ V</>( Xk)}-~ ~
k
(6.9)
108A - 10gB )} ~ n > 0, for all
o
N(O, 1) as 6 ~ O. Hence,
<
6 < 6 .
-
0
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.-
I
17
Hence, by using the same technique aa in Stein[l7] , it followa that for all n:
kr' < n «k+1)r', p¢{ Nu(~»
~2 ~n>n
(6.10)
p¢{ Nu(~)
2
where
kr'} ~
(1 - n)k, for k=1,2, .••. ; hence
> n } < ~2r' ~k:K +1(1- n)k ~ (1- n)K~n~l ~
€
n-1 (1- n) K€
~ €
and
K€ > (1 +€) { max( -10gB, 10gA)} /{/¢ - ~Ir(f)
Since ~2~n<n1 n p¢{ NU(~) = n} and ~2~n<nl n p¢{
it suffices to show that
N*(6) = n} are both ~ € , and by·
for every
€
> 0,
~ +0 (~2~n:+l P¢{ N*(~) > n }) < € ,
(6.12)
lim
(6.13)
lim ~ +0 (~2~~~+ln[p¢{Nu(~) = n}- p¢{ N*(~) = n}J )
2
00
~ ~n2+l
P¢ {
A
n , len - 81 < C~}- p¢{ N*(~) >n,
where C is some positive constant. Then,
A
P ¢{ N;', (~) >
(6 .15 )
~
p
= O.
.
> n } can be written
as
N*(~)
~2Ln2:l[ p¢{ N*(~»
(6.14)
}.
2
-k
~Um(~)' ~ E¢{ NU(~)} + ~(¢,r 2(f», as ~ +0, to prove the theorem
the Wald technique on
Now,
€ ,
€
is so chosen that
K
(6.11)
n} ~ p¢{ NU(~»
Ien
n,
eI >
-
A
I
C~ } ~
o{ ~i~lh(Xi- c~ ) > O}
P ¢{ 8n
A
18n -8 I
> C~}],
- 8 I > C~ }
+ P O{ ~i~lh(Xi + C~ ) < 0 } .
~
+ 0, E { h(X - C~ )} ~ -C~r(f) + o(~), VO(h(Xl - C~» ~ ref) + 0(1),
O
1
and by Wald's assumption on SPRT, the mgf M(t) of h(Xl-C~) exists, we have
Since for
(6.16)
inf {exp [nEOh(X - C~ )t + nlogM(t) J }
0 } ~ t>O
l
n
PO{ ~i=lh(Xi-C
~»
~
[ exp(
-ntC~r(f)(l+
0(1»
+ nlog( 1 +
+ oCt 2 ) ) )J It=Cn-~
~t 2 ref)
= exp {- n~ ~C2r(f)[ 1 + 0(1) + o([n~~J -l)J } •
Since for n > n , n~~ ~
2
-3/2
-3
(6.17)
6n
~
~ the rhs of (6.16) ( for
K€,
~
+
0) can be made smaller than
C-6 r 3 (f) , where 0 <C < C.
l
l
A similar bound holds for P {
O
~i~lh(Xi + C ~) <0 }. Hence, from (6.14), (6.15) and
(6.17), it follows that the second term on the rhs of (6.14) can be made smaller than
24(K~€ C16r 3 (f) )-1 < €'(> 0), where €' ( depending on ~ ) is also arbitrarily small.
Also, by (2.3) and (2.16),
A
(6.18)
p~{ N*(~»
~
~
n, Ie
n
-
91
A
<C~ } <
p¢{ 10gB -n~~ < ~Un(~)
-
P~{logB < n~W (~) <logA, Ie
< 10gA -
n
n
~
n~~},
¢ € I,
- el < C~ }
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.-
I
18
where 0 < ~~ «c + ~)~2n~
and n~
0 as ~
+
+
O. Since E¢( ~Un(~»
= n~2(¢ -~)I(f)
+ 0(~2), and
10gB ( or 10gA ) ± n~~
= (~ -¢ )I(f)n~2{1 + o(n~) +
+ o({n~2}-1)} = ( ~ -¢ )I(f)n~2{1 + 0(1) + O(K- l )} , as n~2 > K
I::
'I::
we can proceed again as in (6.16) (as
U (~) - E~{ U (~)} has finite mgf ), and
(6.19)
- E¢ (~ Un(~»
n
n
't'
~ +
obtain a similar bound as in (6.17). This leads to the asymptotic ( as
0 )
negligibility of (6.14).
To prove (6.13), we note that
n
n
~2(~ 2
(6.20)
n p~{ N*(~) = n}) =~2(~ 2
't'
n1+l
n 1+1
~2(nl+l)p¢{ N*(~) >n }
-1
1
2
second term on the rhs of (6.20) is bounded by
The
n
~2 (n +l)p¢{ N*(~) >
-
l
n
, n
~i=lh(Xi - ~/2)
-!.:2
2
1::
n
2
} •
2+ O(~2 ), while upon noting that
logA both converge to 0 as
~
+
0, and
A
- ~/2) ~ N(
as ~
n2
(6.21) ~2(n2+l)p¢{ N*(~) > n 2 } ~ (KI::+ O(~2»
n%22 ( a
1
= ( KI:: + 0(~2) ) p¢{ n;~logB<
+
as
0
n
2
+
(i.e.,
00
~ +
n
(6.23)
0, we readily conclude that
P¢{logB <n 2 ~Wn2(~) < 10gA }
A
1
n
_!.:
2
an - ~/2)(n; ~i=lh(Xi~/2»<n22logA }
, 2
0).
n
It remains only to consider the term
(6.22)
1
ni(
+
~2~n ~lP¢{ N*(~) >n} . One can write
n
1
A
~2~nl~lP¢{N*(~»nn} = ~2~n1~lP¢{ N*(~»n'n1~s~~ ~nl an,-al> c~ }
+ ~2~ n +2lP~{
N*(~) > n, n < ns~p< n Ie,
n - al < c ~ } ;
1, 't'
A
-,
p¢{n ~u~,< nlan' - al> c~ } = Po{ ~i~lh(Xi -c~ ) > 0, for some n':n1~n'~ n}
1-
n'
-
+ P { ~i=lh(Xi+
O
c~)
< 0 , for some n' : nl~ n'~ n }.
Write Y = h(X - c~), i ~l,and ~= EOY . Then,
i
I
i
n'
n'
(6.24)
P { ~i=l Y > 0, for some n1< n' <n} = PO{ ~i=l(Yi- ~»
O
i
Since for small
~
non-increasing in
,
ti,
~ =-C~I(f)
+
o(~),
VO(Y ) = I(f) +
l
O(~),
-n'~,for some n1<n'<n}.
and cn =
(-n~)
-1
is (> 0)
by the well known H£jek-R~nyi inequality, we have from (6.24)
that the sum of the two terms in (6.23) is bounded above by
(6.25)
~-2 {
2 '"
VO(Y 1 )
}
-1
(-1 _ -1»
(n1 + n l
n
which can be made smaller than
E: ,
< 4 ~-2 -l{V (Y )} < 4 (1 + 0 (1) )
<"
n1
0 1 c 2I(f) ,
-
by proper choice of C. It follows from (6.22) and
I
..I
I
I
I
I
I
I
--I
I
I
I
I
I
I
.e
I
19
(6.25) that
A
p¢{ N*(~) > n'n <s~~< n1en'- el <C~ } + 0(1).
(6.26)
1A
Again, if n <sun~ < I e , - el <C~ , (1- E)logB < ~U ,(~) «1- E)logA => 10gB <n'~W ,(~)
1n
n
n
n
< 10gA => (1 - E)logB < ~Un'(~) < (1+ E)logA, for all n 1 ~n' ~ n 2 • Hence, on defining
N~i)(~) ,i=1,2, as two stopping variables analogous to NU(~)' with 10gB, 10gA replaced
by (1+ (_l)i E)logB and (1+ (-l)i E )logA respectively, for i=1,2, one gets
nz
{ (1)
~ ~n1+lP¢ NU (~»
sup
n, n1<n'~ n
Z
(6.27)
~z~
<
nz
n +l
1
~z~ nn+Z l
<
1
Now, for each
'V
A
- e
1
}
<C~
A
p { N*(~»
¢
P,J..{
Ien'
n'n
NU(2)(~»
n,
1
sup
Ie
el < C~ }
<n' <n
n'-
~u~ < I~ ,
n 1 _n _n n
-
el <
c~ }
N~i)(~), retracing backwards the steps (6.20) - (6.26) one obtains that
•
the first and the third terms in (6.27) converge ( as ~
+
1
0 ) to (1+ (_1)1E )~(¢,I-~(f))
i=1,2 respectively, where we note that for each of them the Wald technique holds for
E¢N~i)(~) . Since E is arbitrarily small, the proof of the theorem is completed.
7. Concluding remarks. An alternative test procedure for the testing problem in (2.1)
~
has been proposed by Albert [1]
following a suggestion of H. Robbins based on the
dual problem of bounded length confidence interval for the parameter under test. This
procedure has the strength (
a(~),
S(~)
a(~)<
) with
S(~).
An alternative test
procedure of asymptotic strength ( a,S) can be formulated as follows :
Under H : e = 0, n -~crn (0) ) is
O
HI: e =
~ , n -~crn (0) - ~ C(F) ) is
tV
N(O, V Z
),
while for small values of ~ , under
tV
N( 0, V Z
).
Since
V
is known, had C(F) been
known, by considering the one sided test and equating its first and second kinds of
errors to a
and S respectively, we obtain that the required sample size n is equal to
(7.1)
where
T
E
is defined in (4.16).
But, in practice, the df F as well as the functional
C(F) is unknown. As in Sen and Ghosh tIS] , we have under the conditions of section 4,
1
(7.2)
A
(V T )/{ n~( e
Y
U,n
A
e ) } a.s. C(F)
L,n
+
as
n +
00,
where 1 -2y is the confidence coefficient of the corresponding bounded length
confidence interval problem. Hence, one can propose a sequential procedure where the
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
•e
I
20
stopping variable N is the first positive integer n ( >n ) for which
-0
A
A
eU,n - eL,n
(7.3)
<
-k
accept H if N 2T (0)
l
N
6T
v
~
I(T
y
+
a
T
13
) ;
,and accept H otherwise.
O
T
a
Both the procedures considered in this section are based on the dual problem of
confidence interval for
e . However, they suffer from the drawback that their asymptotic
ASN are the same as the corresponding fixed sample size test had C(F) been known ,
but not of the sequential tests proposed here or the SPRT. Thus, these procedures
are usually less efficient than the ones considered in the previous sections. For
example, when F is normal, the ARE of the procedures considered in this section with
respect to the Wald
SPRT or our normal scores SROT will be only about 50% for the
usual levels a =13 = 0.05. For brevity, the details of this aspect are omitted.
REFERENCES
[ IJ ALBERT, A.(1966). Fixed size confidence ellipsoids for linear regression
parameters. Ann. Math. Statist.
~,
1602-1630.
[ 2J BERK, R.H., AND SAVAGE, I.R.(1968). The information in rank order and the stopping
time of some associated SPRT's. Ann. Math. Statist.
~,
1661-1674.
[ 3J BRADLEY, R.A., MARTIN, D.C., AND WILCOXON, F.(1965). Sequential rank tests, I.
Monte Carlo studies of the two-sample procedure. Technometrics,
2,
463-483.
[ 4J BRADLEY, R.A., MERCHANT, S.D., AND WILCOXON, F.(1966). Sequential rank tests,
II. A modified two-sample procedure. Technometrics
~,615-623.
[ 5J BRADLEY, R.A., AND WEED, H.D.(1969). Sequential one-sample grouped rank tests.
Abstract, 37th session of the Internat. Statist. Inst., London.
[ 6J CHOW, Y.S., ROBBINS, H., AND TEICHER, H.(1965). Moments of randomly stopped
sums. Ann. Math. Statist.
""
v
~,
789-799.
,.
[ 7J HAJEK, J., AND SIDAK, 2.(1967). Theory of Rank Tests. Academic Press, New York.
[ 8J HALL, W.J.(l969). A sequential Wilcoxon test. (Abstract). Ann. Math. Statist.
40, 1879.
[ 9J HALL, W.J., WIJSMAN, R.A., AND GHOSH, J.K.(1965). The relationship between
sufficiency and invariance with application in sequential analysis. Ann. Math.
Statist. 36,
575- 615.
[lOJ HODGES, J.L.JR., AND LEHMANN, E.L.(1963).Estimates of location based on rank tests •
, Ann.
Math. Statist. 34, 598 - 611.
[llJ LOEVE, M.(1963). Probability Theory. ( 3rd edition). Van Nostrand, Princeton.
I
..I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
21
[12J PURl, M.L.(1964). Asymptotic efficiency of a class of c-sample tests. Ann. Math.
Statist. 35, 102-121.
[13J SAVAGE, I.R., AND SETHURAMAN, J.(1966). Stopping time of a rank-order sequential
probability ratio test based on Lehmann alternatives. Ann.Math. Statist.
~,
1154-1160.
[14J SEN, P.K.(1970).On some convergence properties of one-sample rank order
statistics. Ann.Math. Statist. 41, No.6, in press.
[15J SEN. P.K., AND GHOSH, M. (1969). On bounded length sequential confidence intervals
based on one-sample rank order statistics. Inst. Statist., Univ. North Carolina
Mimeo Ser. Rep. No. 648 .
[16J SEN, P.K"
AND GHOSH, M.(1970). A law of iterated logarithm for one-sample rank
order statistics and an application. Inst. Statist., Univ. North Carolina
Mimeo Report No.
687.
[17J STEIN, C.(1946). A note on cumulative sums. Ann. Math. Statist. 17, 498-499.
[18J STRASSEN, V.(1967). Almost sure behaviour of sample sums of independent random
variables and martingales. Proc. 5th. Berkeley Symp. Math. Statist. Prob.
( ed: L. LeCam and J. Neyman).
~,
315-343.
[19J WALD, A. (1947). Sequential Analysis. John Wiley, New York.
[20J WEED, H.D.(1968).Sequential one-sample grouped rank tests for symmetry. Ph.D
Dissertation, Florida State Univ., Tallahassee.
[21J WILCOXON, F., RHODES, L.J., AND BRADLEY, R.A.(1963). Two sequential two-sample
grouped rank tests with applications to screening experiments. Biometrics.
19, 58-84.