OPTIMALITY WITHIN THE CLASS OF SEQUENTIAL PROBABILITY
RATIO TESTS
by
Gordon Simons
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 904
J anUJlJUj 1914
ABSTRACT
This paper shows there is a very strong optimality property within
the class of sequential probability ratio tests (SPRT).
This property,
which was discovered by B.K. Ghosh in a weaker form, is completely analogous to the classical optimality property found by Wa1d and Wo1fowitz.
The only real differences are the following.
It is weaker in the sense
that nothing is said about tests which are not SPRT's.
But it is strong-
er in the sense that absolutely no assumptions are made about the data
(such as being 1. i. d.) and in the sense that conclusions about the
numerical ordering of expected sample sizes are replaced by conclusions
about the stochastic ordering of stopping variables.
The property
may be roughly paraphrased to s tate that one can not reduoe the erpor
probabiZities of an SPRT without additionaZsampZing.
Although intui-
tive1y reasonable, this does not seem to be mathematically obvious.
AMS 1970 Subject Classifications: Primary 62L10
Key Words and Phrases: Sequential probability ratio tests, Optimality.
OPTINALITY WITHIN THE CLASS OF SEQUENTIAL PROBABILITY
RATIO TESTS
f
by
Gordon Simons
1.
Introduction.
B.K. Ghosh (1970) has described a type of optimality -
"uniformly most efficient" - which is possessed by every sequential probability ratio test (SPRT) irrespective of assumptions on the data.
It
will be recalled that for data which is i.i.d. under both (simple) hyotheses and for a given SPRT, one can not reduce either error probability
without resorting to a test of larger expected sample size (under both
hypotheses).
This expresses the well-known optimality property dis-
covered by Wald and Wolfowitz (1948).
Ghosh's result essentially says
that this same property holds without the i.i.d. assumption within the
class of SPRT's.
Specifically, there is no other SPRT with smaller error
probabilities and a smaller expected sample size.
This paper strengthens Ghosh's result and circumvents a questionable step in his proof.
a given SPRT.
Let
I.e., a (/3)
a
and
a
be the error probabilities for.
is the probability that the test eventually
terminates with the rejection of the null (alternative) hypothesis when
it is true.
(N
=~
Let N be the sample size when the test is terminated.
if no termination occurs.)
Finally, let
the corresponding quantities for a competing SPRT.
a', /3'
and N'
be
We prove the follow-
ing two results:
(i)
If a'
S
a, 13'
hypotheses.
S
/3, then
N:S N'
almost sureZy under both
F'uT'ther, the terminaZ decisions are the same
- 2 -
when
(ii)
=N
N'
almost supeZy under both hypotheses. 1 /
< co
If., in addition~
[N < N' J
a' < a
or
then the event
13' < 13,
occurs with positive probabiZity under both
hypotheses.
That such strong results should hold is not completely surprising.
They follow fairly easily for the i.i.d. case from the classical optimality property previously mentioned.
However, they can be derived -
even for the general case - from elementary considerations.
We shall begin by establishing a basic result which applies to
any randomly stopped sequence of likelihood ratios.
With this, we
produce some identities which relate pairs of SPRT's.
are used to prove
2.
(i)
and
(ii).
Randomly stopped likelihood ratios.
ability measures on a measurable space
of
F.
(1)
f
A dP
= Q(E,
E
(where
Let
(Q,
P
-1
A
=0
random mapping
A ~ co),
when
X,
E if it is
f
A-I dQ
E a sub - a - field
A will be called
E - measurable and
= P{E,
E
A = co).
Q be fixed prob-
and
and
F)
An extended non-negative random variable
a ZikeZihood ratio for
These, in turn,
Moreover, if
A ~ 0), E
E
E
E is generated by a
A will be called a ZikeZihood ratio for X as
1/
The terminal decisions could disagree when N = N' <
continuation regions for the two SPRT's are disjoint.
co
if the
- 3 -
~ must necessarily exist, and it is unique up to a P and
we11.!/
Q equivalence.
El , E ,
be a nondecreasing sequence of sub - a - fields
2
and let N be a stopping variable relative to this sequence.
Let
F,
of
1. e., the event
[N
hood ratio for
En
E for which E[N
= n]
E
and
EN
= n]
partitioning the event
f
(2)
AN dP
E [N <~]
= Q(E,
E , n
n
N
<
< ~])
Further, let
~
be the likeli-
n
F - measurable events
a-field of
~ 1.
n
[N
1.
be the
E , n
E
~
It easily follows from
(1)
(by
that
~, ~ ~ ~),
f
E [N <00 ]
~;l dQ = P(E,
N <
~,~?O),EEEN.
Thus:
(3a)
P(E)
=0
~
Q(E)
=0
for E
E
[N < 00,
(3b)
Q(E)
=0
=> P(E)
=0
for E
E
[N
3.
Identities.
larize
N
~ ~ 00] ~.
< ~, ~N
; 0]
~.
We continue using the notation of Section 2 but particu-
in two different ways - one with a prime.
but not essential, to think of
A
n
It is convenient,
as a likelihood ratio for a sample
of size n, taken from an infinite sequence of potential observations.
Let S(A,B),
0 S A
<
BS
~,
be an SPRT.
Formally, it can be
This convenient defintion is equivalent to the more conventional
definition involving the ratio of densities. A = co occurs when the
denominator is zero and the numerator is positive.
- 4 -
~air
described in terms of a
N = first
n
~
~
and
\i
where
An i(A,B), •
1 such that
D • 0 if N <
(N,D)
~
A, • 1 if N <
The error probabilities are a· peN <
Further, let
abilities
~
~,
if no such n exists;
00
and
~ ~
B•.
D • 1) and a· Q(N <
~,
D • 0).
(N', D') = S(A', B') tie a se'cond SPRT with error prob-
a'
and
Proposition 1.
a'.
If A ~ A'
(4a)
A(a'- a) + (13'- a)...
(4b)
(a'-a) + B-l(a'-a)
and B' s B,
then
I
(\i,-A)dP + I
(A-\i)dP + AP(N'<N ...co ),
[N'<N,D'=O]
[N'<N<~,D-O]
= I (~;_B-l)dQ
+
I
(B-l_~l)dQ + B- 1 Q(N'<N= co).
[N'<N<~,D=l]
[N'<N,D'=l]
(We interpret B-1 as zero when B = 00.)
Proof
[N'<N, D' = 0]
~
EN'.
By (2), the first integral of (4a)
equals
Q(N'<N, D' = 0, \it -I co) - AP(N'<N, D' = 0) • Q(N'<N,D'=O) - AP(N'<N,D'=O).
Likewise, the second integral equals AP(N'<N<co,D=O) - Q(N'<N<oo, D=O).
Thus it remains to check that
(a' - a
=)
P(N'<co,D'=l) - P(N<co,D=l) = P(N'<N<oo,D=O) - P(N'<N,D'.O) +
and
(a' -a
=)
Q(N'<co,D'=O) - Q(N<oo, D=O) • Q(N'<N.D'=O)- Q(N'<N<co,D=O).
P(N'<N=~),
- 5 Both of these follow easily from geometrical considerations.
lar~
(4b)
one needs to observe that
N' s N
and that
N' = N < co
In particuII(>
D'=D.
o
is proven similarly.
Notice that each term of the right hand sides of the identities
(4a)
and
(5)
A s A'
(4b)
~
is non-negative.
Thus
~ O~
A(a'-a) + (B' -B)
B's B =>
(a'- a) + B-
This means that it is impossible for the inequalities
to hold with one of them strict.
case in practice), then
Proposition 2.
(6a) a-B' •
l
If
(6b) a'-a =
(~~
[N'<N]
a
~e's
~
a
a + B.
B~
then
(A' - ~,) dP + A' {P{N<N'.co) + P{N<N'<co,D'=l)}
-
D'.l, D=O) + Q{N'<N<co,D=O),
f
B-l)dQ +
(B- l _
~l)dQ+B-l{Q(N'<N.CO)+Q(N'<N<oo~D=O)}.
[N'<N<oo, D-l]
+P(N'=N <co, D'=l, D=O) + P(N<N'<co~ D'=l).
Proof
This proof is similar to the proof for Proposition 1.
The geometrical implications needed for
that
N' <00 ~ D' =0
O.
A s 1 s B (which is usually the
A's A and B's
f
s
(a'-B)
[N <N' <00 ~ . D' =0]
Q(N'·N<co~
f
~
a '+ B'
(A'N-A') dP +
[N <N ' ]
+
If
a'
l
=> NSN' ~ D=O.
(6a)
are
N<N'
=f>
DcO, and
o
- 6 -
These identities yield the implication
A's A, B'S B
=l>
a'
~
a,B' S B.
But this is obvious from geometrical considerations.
4.
The theorem.
We are now in a position to prove the theorem described in
Section 1:
Theorem
(i)
=> P (N S N') .. Q(N S N') - 1, P (N ' == N <00, D'
a'S a, a' S 13
= Q(N'
(ii)
Proof. (i)
and (ii)
A SA', B'S B, a'S a. S's 13
(8)
A'SA, BS B'
.(9)
A's A, B SB', a'Sa, B's a
(10)
0, Q(N<N'»
O.
are immediate from the following implications:
(7)
=l>
D)
- N < 00, D' =I D)- O.
7J)ith at Zeast one st'Piat =>P(N<N'»
a' S a, a's 13
=I
=> a' = a.
13'= 13, P(N'=N) .. Q(N'=N)= 1.
peNS N') = Q(NS N') .. 1.
A'S A, B'SB,a'sa
with one strict
=l>
P(N<N'»
O,Q(N<N'»O.
=> a'=a. P(NSoN') .. Q(NSN') = 1, P(N'=N<oo, D' .; D)
= Q(N'=N<oo, D'=I D) .. O.
- 7 -
(11)
A'SA, B'SB, a'Sa, e'<e
(12)
ASA', BSB', a'se
P(N'>N»
=c>
0, Q(N'>N»
=> e'-a, P(NSN') .. Q(NSN'):I 1,
P(N'=N <'OCl,D' =! D)
(13)
A SA',
B
SB', a' < a, a's
(8)
Implication
from Proposition 1,
o.
13
=I>
peN'> Ii»
0, Q(N'> N) >
is obvious, implications
and
(10)-(13)
serve that the right hand sides of
= Q(N'=N < OCl,D' =! D) = O.
(7)
o.
(9)
and
follow
follow from Proposition 2.
(4a), (4b), (6a) and (6b)
Obconsist
of non-negative terms and that the first integral of each has a
strictly positive integrand on its range of integration.
Proof of (7).
The "givenll of
(7), (4a)
and
(4b)
readily imply
a' .. a" e' De, peN' < N, D' = 0) .. Q(N'< N, D' = 1) .. O.
(3a)
and
(3b)
(with N replaced by N')
.. P(N '< N, D' .. 1)
In turn,
imply Q(N'< N, D' .. 0)
= O. Thus P (N SN') - Q(N SN') - 1 and, hence,
0
P (N' - N) - Q (N' .. N) - 1.
Proof of (9).
(9')
A SA'
(9)
is
,B'SB,~Sa',
equivalent
to
aSfS' with one strict IO:i>P(N'<N) > 0, Q(N'<N) > 0,
o
which obviously follows from Proposition 1.
Proof of (10) and (11). The "given" of
and
(6b)
(10)
(and hence of (11»
imply a'-a, Q(N'<N) - P(N'-N<OCl, D'=! D) - P(N<N'<OCl,D'-l) - O.
The remaining conclusions in
(10) follow from
(3a)
and
(3b).
- 8 -
Furthermore,
f
13-13'-
(6a)
simplifies to
(~- A') dP +
[N<N']
from which
(11)
Finally, (12)
(10)
and
Remark
(11).
f
(A' - ~,) dP + A' P(N<N' = 00),
[N<N'<oo, D'-O]
o
easily follows.
and
(13)
are proven in the same way we prove
Notice that the value of
(a,S)
tion of N under both hypotheses.
P(N'-N) = Q(N'=N)
is possible.
0
This completes the proof of the theorem.
= 1.
determines the distribu-
For if
a'-a
and
S'=S,
This is not to say that every value
Neither does
a'= a
and
a'- a
imply
then
(a,S)
A'=A,B'=B
in
general.
REFERENCES
.
[1]
.
. [2]
Ghosh, B.K. (1970). Sequential. Tests of StatistiaaZ
Addison-Wesley •
Hypotheses~
Wald, A. and Wolfowitz, J. (1948). "Optimum character of the
sequential probability ratio test," Ann. Math. Statist.
19, 326-39.