Internat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 549-554
549
A NOTE ON EXPONENTIALLY BOUNDED STOPPING TIMES
OF CERTAIN ONE-SAMPLE SEQUENTIAL RANK ORDER TESTS
Z. GOVINDARAJULU
Department of Statistics
University of Kentucky
Lexington, Kentucky 40506
(Received May 20, 1985)
ABSTRACT.
A simple proof of the exponential boundedness of the stopping time of the
one-sample sequential probability ratio tests (SPRT’s) is obtained.
KEY WORDS AND PHASES.
Nonparametric; finite sure termination; test for symmetry.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 60G40, 62G10, 62L10.
i.
INTRODUCTION AND SUMMARY.
Stein
[i] gave a very mild condition under which the stopping time of a SPRT
based on a sequence of i.i.d, random variables is exponentially bounded.
It is well
known that exponential boundedness of the stopping time implies finite sure termination and the finiteness of the moment-generating function of the stopping time.
Weed, Bradley and Govindarajulu [2] propose one-sample sequential probability ratio
tests
(SPRTs) based on Lehmann alternatives and also study their finite sure terminaExtending the work of Sethuraman [3], Govindarajulu [4] has obtained a very
tion.
mild Stein-type of condition (namely, that a certain random variable U(Z) is not
identially zero) for the exponential boundedness of the stopping time of the one-
sample SPRT’s.
Here, using a general result of Wijsman [5] a much simpler proof of
the same assertion is obtained.
2.
NOTATION
Let
ZI,Z2,...
be a sequence of i.i.d, random variables having a continuous dis-
tribution function (d.f.) F(x).
a time.
Assume that we observe the Z’s sequentially one at
We wish to test
F(z) + F(-z)
HO:
(2.1)
for all z.
i
Let
P(IZI < zlZ < O)
P(Z<zlZ_>O) {F(z)
H(z)
and G and H are zero for
z< 0.
H0:
G(z)
F(-z)}/F(O),
{F(O)
G(z)
F(O)}/{I-F(O)},
a
for all
Alternatively, we write
H(z)
for all z and F(0)
and take the alternative hypothesis to be
H
and
G(z) # H(z)
for some z.
1/2
z_>O,
Z. GOVINDARAJULU
550
Consider a subalternative hypothesis given by
HA:
GA(z)
H(z)
for all z, A> O,
%0’ %0
A is specified F(O)
A# i,
specified.
Let t denote the number of stages to which the experiment has proceeded.
is
ZI,Z2,...,Z t
Z’s among
absolute values [values] of those
Let
denote the
Also note that m is binomially distributed with parameters t and %
Let V
i
IY l
IZil
Yn ].
i
i .-’,t.
Let G
m
(l
et
Further
[Hn]
where
i or 0 according as V
i
corresponds to a negative or positive Z respectively where
ordered
V1,-’’,V t.
F(O), O< %< i.
denote the empirical d.f. based on X
"’’’t
(That
XI,...,Xm [YI,--.,Yn]
ZI,---,Z t that are negative [positive].
stages.)
will be observed in t
_<
Vlt _<
Vtt
I
’’-,X
m
it
denote the
Also, let
Pt(A=61A)/Pt (A=61A=I) 2tPt(A=IA)"
Ut(A’6)
(2.2)
From Weed, Bradley and Govindarajulu [2] or Govindarajulu [4, (2.6)] we have
Lt(A,6
2
t t!
t
m
%0(I-%0 )n.=
IHI{ %tGm(Vi)
A(l-%t)Hn(Vi)}
+
-1
(2.3)
where
k
k
%t
m/t,
I if
where l(c,d)
Gk(Z)
l(Xi;z)/k
1
c < d and zero elsewhere.
[I(Z;0) +
W0(Z;z)
(z)
and
j--1
(2.4)
l(Yj;z)/k,
Let
AI(0;z)]I(IZl;z)
I(X;z) + AI(Y;z)
and
(Note that
EWo(Z;z)
+
%tGm(Z)
Wt(z)
A(l-%t)Hn(Z).
%G(z) + A(I-%)H(z).)
W(z)
With the above notation one can write
t
L
t(A,)
2tt!t-t(l-o)nAn/ i--iWt(Vi)-
(2.5)
Hence
t (A’) nLt (A’)
+
n
t
n
n
n (i-%0)
2
+
t!
+
t
n
n
n
t
+
m
n %0
t
A
I n Wt(Vi).
i--i
Then the rank order SPRT (ROSPRT) for testing H 0 against HA is given by:
if
< b[ t >_ a] and take one more observation of
Accept H0 [reject H
constants. The stopping time is:
fixed
<a
are some
<a where b< 0
b
O]
<t
T
r if b <
<a for t
l,’’’,r-i and (r)<b or
We will be interested in the termination properties of T.
T*
r if
I_t
< d
max(-b,a) for t
Then one can easily see that T >r -->T* >r.
to show the exponential boundedness of T
3.
,
>a
(r=l,2,’’’).
(2.6)
Also define
l,’’-,r-I and
tr
Hence P(T >r) < P(T*> r).
>-d"
(2.7)
So it suffices
which will be done in the sequel.
CERTAIN RESULTS PERTAINING TO EXPONENTIALLY BOUNDED STOPPING TIMES.
[5] has obtained some general theorems which assert that the stopping
Wijsman
EXPONENTIALLY BOUNDED STOPING TIMES OF RANK ORDER TESTS
551
time of a sequential procedure is exponentially bounded unless a certain function of
In the following we
the underlying random variable is zero with probability one.
will state only the results that are relevant to our needs here.
(Wijsman [5, (Corollary 2.2)]).
LEMA 3.].
Let
Z,ZI,Z2,’’-
be i.i.d, random
Let L*
variables taking values in i and having common distribution F.
t
L*t(Zt),Z
t
be a sequence of real valued statistics and let T* be defined by (2.7).
(ZI,---,Z t)
Assume, further, that there is a measurable function f: i+R, a measurable subset S of
with P(ZeS)I0 and
f(z) # 0 whenever zeS, and a set
SO S
such that for every
positive integer r the function
K
is measurable.
n
sup{iL t+r (Z
If K
0
t
’Y
r
r
et Ii f(Yj)l:
Yl,’’"
,YrESo
then T* is exponentially bounded
n exp
(See Wijsman [5, p.298]).
PROOF.
In the following we will state a useful
lemma which is more germane to nonparametric needs.
LEMMA 3.2.
(Wijsman[5]).
Let
Z,ZI,Z2,---
variables taking values in a measurable space
Further, let
be a sequence of i.i.d, random
([,A) and having the common d.f. F(z).
be a real interval (perhaps infinite) and let f be a bounded, real
valued function on
[, such that f(x,-)
is A-measurable for every xel and
is non-decreasing for every ze[ or non-increasing for every
I1
Jn(X)
J(x)
f(x,Zi)/n,
zel.
f(.,z)
Also let
ff(x,z)F(dz)
and
R
n
sup{lJ n (x)
J(x)
l:
xel}
Then R
O.
n exp
PROOF. (See Wijsman [5, p. 308]).
4.
CERTAIN USEFUL LEMMAS.
In this section we give some results as lemmas which will be used in the
sequel.
LEMMA 4.1.
(i)
For
(Sethuraman [3, p. 1326]).
< x <
n(l+x) 2 x/(l+
and for 0 <
Ix
Let -i < 8 < O<a.
we have
a(a-8)/[(l+cO(l+]
(4.1)
J
(4.2)
<i
n(l+x)
DEFINITION 4.1.
x.
The stopping time T of a sequential procedure is said to be
exponentially bounded if for any
P(T > t)
there is c < and 0 < 0< 1 such that
_< co t,
t
1,2,-.-
(4.3)
Notice that T* is exponentially bounded implies that T is
exponentially bounded.
DEFINITION 4.2. A sequential procedure is said to have property Q if
it terminates finitely and the stopping time possesses a finite
moment-generating function
(m- g f
under P; that is P(T<)
1 and E(exp(NT)) < for some N >0 (here E
denotes expectation under P).
LEMMA 4.2. A sequential procedure has property Q if T is exponentially bounded.
PROOF. Easy to construct.
Z. GOVINDARAJULU
552
LEMMA 4.3.
For every
g
>0, and sufficiently large t, and 0 < %< i, there exists
0 < O(g,%) < i such that
P[%t n
n
Xt
1-%
p[n{ (l_%t)
PROOF.
5.
! pt(,%)
X > c]
t/(l_)l-}>
] <
pt (,).
See Lemma 4.3 in Govindarajulu [4].
MAIN RESULTS.
In this section we prove the main result pertaining to the exponential bounded-
ness of T*.
THEOREM 5.1.
Define
n2 +
U(Z)
n {A(1-%O)} n{A(1-%O)/%O}I(Z;O
n W(IZI) f{Wo(Z;v)/W(v)}(dv
(5.1)
where
kG(v) + (l-%)H(v).
F(v)
(5.2)
Then T is exponentially bounded under F unless
PF(U(Z)
PROOF.
I.
O)
Because of Lemma 5.1 it suffices to show that
r
t+r t
j=l
0 as t-o.
U(Zj)-exp
Towards this we can write
t+r t
n2
r
+ BI + B2 + B3
(5.3)
where
B
B
nt!
n(t+r)!
1
(t+r)n(t+r) +
t
n
t
t+r
fn{A(1-%O)} n{A(1-%o)l%0
r
2
[
i=t+l
I(Z i;O)
and
t+r
B
t
n Wt+r (Vj)- [ n{Wt+r(Vi)/Wt (Vi)}
i=l
j=t+l
3
B
4
+ B 5 (say).
After using Stirlings approximation to factorials, we have
B
Denoting
Zt+ i
by
B
B
since W
t(-)- exp
4
as
t/
(5.4)
oo.
r
r
[ I(Yi;O)
n{A(1-O)/%
O} j=l
r
fn{A(1-%0)}
r
=-lj ln Wt+ (lYjl)
r
exp
"
j=l
n W(lyj[)
Next consider B 5.
W(’) uniformly in (-).
W (v)
t
we can write
-r
we have
Yi’
2
1
t
-i
i=l
Wo(Z i;v)
Since, for fixed v
(5.5)
(5.6)
553
EXPONENTIALLY BOUNDED STOPPING TIMES OF RANK ORDER TESTS
t+r
i__[lWo (z
-fn{(t+r)/t} + Zn
fn{Wt+r(V)/Wt(v )}
;v)
t
iIWO
(Z ;v)
i
(5.7)
i=[1Wo (Yi ;v)
n
-fn{ (t+r)/t} +
i
tW (v)
t
Now,
t
y
gn{ (t+r)/t} +-r
(5.8)
as t +.
i=l
Consider
Zn{l +
i Wo(Yj;Vi)
(Vi)
tW
I
i=l
t
Define
suplW t(v)
{(0:
A(t,g)
<
W(v)
c},
v
Note that A(t,e) is exponentially bounded.
where m denotes an outcome.
On A(t,r_)
r
n{l +
i=1
,Vi)
ZI Wo(Yj
t
_<
;Vi)
i Wo(YJ
(Q-?)
t
I
i=l
1
n{l +
;Vi)
iI Wo(Yj
.-)
(5.9)
t
(W(Vi)-E)
0<8.< i.
Wo(v)
r
I
[
<
tW(Vi)
i=l 1
il
l+ei E (w(vi)-e)
Now
L.H.S.
-
t
Wo(Y j ;Vi) W(V.)-
i=l
t
I 1
i
W(Vi)
r (l+A)
i=l t
e r(l+A)
+
2
t
[
i=l t
[_
(5.10)
r(l+A)
r(l+A)
+
(W(Vi)-e)
g
1
L w
(w (v)-E)
(vi)
i--1
i
(W(Vi)-E)
2
F(dv)
f0 W(v) [W(v)-e)
r(l+A)F(d(v))
2
(W(v)-)
Since e is arbitrary, R.H.S. tends to zero as t+oo.
By taking
Wt(v
W(v) +
,
one can bound the difference from the other side.
Also note that
r
i--i
[ W0(Yj;Vi)
w(vi)
j=l
r
/o
j=l
W
0
(yj ;v)m(dv)
W(v)
Q.E.D.
Z. GOVINDARAJULU
554
6.
GENERALIZED RANK ORDER SPRTs.
L*t(A,)
L
(A,). Considering *(t)
m/t and obtain
by A t
and using Lemma 4.3, we can establish that for large t
Suppose we estimate
%0
n
r
+r
I U*(Zj) exp O,
where U*(Z) is obtained by replacing
j=l
by % in U(Z) given by (5.1).
%0
Then, one can
readily establish:
THEOREM 6.1.
on
7.
0) # I, then the generalized rank order SPRT based
If P(U*(Z)
has property Q.
CONCLUDING REMARKS.
(i)
Inversion of model H
A
via the transformation Z
(z)
Lehmann’s class of alternatives given by i
associated with the random variable Z, A I/A and
(ii)
{i
%0
-IZ
(z) A}
,
produces the other
where
F(O) =A/(I+A) if
%0
are d.f.s
I/(I+A).
Although for simplicity, continuity of the distribution of Z is assumed,
it is not necessary for the validity of the results.
If F has a denumerable number
of jumps, then F can be made continuous by the continuization process described in
Govindarajulu [6, Remark 3.14.7].
Alternatively we can take F(z)
z[p(X!z)+p(X<z)
as suggested by Wijsman [5, Equation (4.2)] and all the results will be valid.
I thank Professor Robert Wijsman for suggesting that the use of
ACKNOWLEDGEMENT.
the general result in his 1977 paper will provide a simpler proof of the main
assertion of this paper.
*This author’s research is supported by a grant from the Office of Naval
Research, Grant No. NOOOI4-84-K-DI84.
Reproduction in whole or in part is permitted
for the purpose of the U.S. Government.
REFERENCES
A note on cumulative sums, Ann. Math. Statist. 17 (1946), 498-499.
i.
STEIN, C.
2.
WEED, H.D., Jr., BRADLEY, R.A. and GOVINDARAJULU, Z. Stopping times of some
one-sample sequential rank tests, Ann. Statist. 2 (1974), 1314-1322.
3.
SETHURAMAN, J. Stopping time of a rank order sequential probability ratio
test based on Lehmann alternatives, II, Ann. Math. Statist. 41 (1970),
1322-1333.
4.
GOVINDARAJULU, Z. Stopping times of one-sample rank order sequential probabil(1984), 305-320.
ity ratio tests, J. Statist. Planning and Inference
5.
WIJSMAN, R.A.
6.
GOVINDARAJULU, Z. The Sequential Statistical Analysis of Hypothesis Testing,
Point and Interval Estimation, and Decision Theory, American Sciences
Press, Box 21161, Columbus, Ohio.
A general theorem with applications on exponentially bounded
stopping times without moment conditions, Ann. Statist. 5 (1977), 292-315.