·e
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
JanU4lUj 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 Wald and Wolfowitz.
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 1.1. 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 tos tate that one can not r-eduae the er-r-or-
pr-obabiZities of an SPRT without additionaZ scrmpZing.
Although intui-
tively reasonable, this does not seem to be mathematically obvious.
AMS 1970 Subject Classifications: Primary 62L10
Key Words and Phrases: Sequential probabil.ity ratio tests, Optimality_
OPTIMALITY WITHIN THE CLASS OF SEQUENTIAL PROBABILITY
RATIO TESTS
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 Wa1d and Wo1fowitz (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.
I.e.~
Let
a (e)
a
andB
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
=w
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', B'
and N'
be
We prove the follow-
ing two results:
(i)
If a'
~
a, 13'
hypotheses.
~
a,
then
N ~ N'
almost sUr'eZy under' both
FuT'ther', the teminaZ deaisions aPe the same
- 2 -
when
If~
(ii)
N'
=N
almost supeZy undep both hypotheses. 1 /
< co
in addition J
[N < N']
O<JauPS
ex' < ex
OP
(3' <
a.
then the event
with positive ppobabiZity undep 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 SPRI's.
are used to prove
2.
(i)
and
(ii).
Randomly stopped likelihood ratios.
ability measures on a measurable space
of
f.
An
(1)
f
A dP
= Q(E.
E if it is
A f co).
E
(where
Let
(n.
P
f
A-
l
dQ
Q be fixed prob-
and
and
f)
extended non-negative random variable
a ZikeZihood patio for
These. in turn.
E a sub - a - field
A will be called
E - measurable and
= peE.
A f 0). E
€
E
E
-1
A
=0
random mapping
when
X.
A = co).
Moreove r, if
E is generated by a
A will be called a ZikeZihood patio for
X as
1/
The terminal decisions could disagree when N = N' <
continuation regions for the two SPRT's are disjoint.
00
if the
- 3 -
well. 1 /
A must necessarily exist, and it is unique up to a P
and
Q equivalence.
Let
F,
of
be anondecreasing sequence of sub - a - fields
El , E ,
2
and let N be a stopping variable relative to this sequence.
1. e., the event
[N
hood ratio for
= nJ
~
and
~
En
E for which E[N
= nJ
partitioning the event
(2)
f
E[N<~]
~ dP
= Q(E,
E , n
n
~
1.
be the
c E , n .~ 1.
n
[N <
~J)
Further, let
A
be the likeli-
n
F - measurable events
a-field of
It easily follows from
(1)
(by
that
N < ~, ~ ~ ~),
f
E[N<~]
-1
AN dQ
= P(E,
N < ~,~?O),E~EN'
Thus:
=0
(3a)
P(E)
(3b)
Q(E). 0 => P(E)
3.
Identities.
larize
N
=l>
Q(E) = 0 for E c [N <
=0
for E c [N <
Let
~,
AN
~
0] EN.
We continue using the notation of Section 2 but particu-
in two different ways - one with a prime.
but not essential, to think of
of size
~, ~ ~ ~] ~.
A
n
It is convenient,
as a likelihood ratio for a sample
n, taken from an infinite sequence of potential observations.
S(A,B),
0
~
A< B
~ ~,
be an SPRT.
Formally, it can be
1:./
This convenient defintion is equivalent to the more conventional
definition involVing the ratio of densities. A= ~ occurs when the
denominator is zero and the numerator is positive.
- 4described in terms of a pair (N,D)
N" first n
~
1 such that
D .. 0 if N <
~
and
where
A, .. 1 if N <
~ ~
The error probabilities are a = peN <
Further, let
abilities
and
a'
and
~ ~
B•.
D .. 1) and S .. Q(N <
~,
If A S A'
and B'
I
S
B,
~,
D .. 0).
I
(a'-a) + B-l(S'-S)"
(~;_B-l)dQ +
as zero when B =
[N'<N, D' = 0]
EN'.
€
(A-~)dP + AP(N'<N =~ ),
[N'<N<~,D=O]
[N'<N,D'=l]
(We interpret B-1
then
I
(~,-A)dP +
[N'<N,D'-O]
Proof
~
a'.
A(a'- a) + (S'- S) -
(4b)
if no such n exists;
(N', D') - S (A', B') tie a second SPRT with error prob-
Proposition 1.
(4a)
~
An t(A,B), ..
I
(B-l_~;l)dQ +
B- 1
Q(N'<N- ~).
[N'<N<IlO,D=l]
00.)
By (2), the first integral of (4a)
equals
Q(N'<N, D' .. 0,
~, ~ ~)
- AP(N'<N, D' .. 0) - Q(N'<N,D'=O) - AP(N'<N,D'=O).
Likewise, the second integral equals
Thus it remains to check that
P(N'<~,D'=l)
and
(a' -a
- P(N<oo,D=l) ..
(a' - a
AP(N'<N<~,D"O)
-
Q(N'<N<~,
D=O).
-)
P(N'<N<~,D=O)
- P(N'<N,D'-O) +
=)
Q(N'<co,D'=O) - Q(N<co, D=O) .. Q(N'<N.D'-O)-
Q(N'<N<~,D=O).
P(N'<N=~),
- 5 -
Both of these follow easily from geometrical considerations.
lar, one needs to observe that N'
(4b)
~
N and that N' - N <
In particu-
00
10(>
D'-D.
o
is proven similarly.
Notice that each term of the right hand sides of the identities
(4a)
(5)
and
(4b)
is non-negative.
Thus
A(a'-a) + (B' -B) ~ 0, (a'- a) + B-1
A s A', B's B II(>
This means that it is impossible for the inequalities
to hold with one of them strict.
case in practice), then
Proposition 2.
(6a)
a-a'" f
a'+
a'
f
O.
a' s a ,B's B
If A s' 1 s B (which is usually the
~
a+ B.
If A's A and B'S B,
(A'N- A') dP +
(a'-a) ~
then
(A' - ~,) dP + A' {P(N<N'=oo) + P(N<N'<oo,D'=l)}
[N <N ' <00, . D' =0]
[N <N' ]
+ Q(N'=N<oo, D'=l, D=O) + Q(N' <N<oo,D-O),
(6b) a'-a ""
f
(~~
[N ' <N]
-
f
B-l)dQ +
(B- I _ ~1)dQ+B-I{Q(N' <N=oo)+Q(N'<N<oo,D=O)}
[N ' <N <co, D= I]
+P(N'=N <00, D'=l, D-O) + P(N<N'<oo, D'=l).
Proof
This proof is similar to the proof for Proposition 1.
The geometrical implications needed for
that N'<oo, D'-O
II(>
NsN', D=O.
(6a)
are N<N'
II(>
D=O, and
o
- 6 -
These identities yield the implication
A's A, B'sB
:Q
a'
~
a,S' sa.
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)
a'S a, /3's a
P(N s N') • Q(N s Nt) .. 1, P(N' .. N <oo,D' =I D)
=0
= Q(N'
(ii)
a'
S.
Proof. (i)
.. N <
00,
D' =I D)= O.
a, /3's /3 with at Zeast one striat =>P(N<N'»
and (ii)
0, Q(N<N'»
are immediate from the following implications:
atS
(7)
A SA', B' s B, a' Sa,
(8)
A'SA, BS B'
.(9)
A's A, B SB', a'Sa, a's a with one strict => P(N<N'»
(10)
II(>
o.
/3
=>
a' = a, a' - a, P(N ' -N ) .. Q(N ' =N )= 1.
P(NS N') = Q(NS N') = 1.
A's A, B'SB,a'Sa
=>
O,Q(N<N'»O.
a'=a, P(NSN') - Q(NSN') - 1, P(N'=N<co, D' =I D)
.. Q(N'=N<co, D'=I D) - O.
- 7 -
(11)
A'sA, B'SB, a'Sa, 13'<13 =t> P(N'>N»
(12)
ASA', B SB', f!'s13
==>
0, Q(N'>N»
13'-13, peN SN')
== Q(NSN') = 1,
:f D) - Q(N'-N< e»,D' .,; D) = O.
P(N'=N <:e»,D'
(13)
A SA', B SB', a' < a, 13's
Implication
(8)
from Proposition 1,
13 =t> peN'> N) > 0, Q(N'> N) > O.
is obvious, implications
and
O.
(10)-(13)
serve that the right hand sides of
(7)
and
(9)
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' 13' =13, peN' < N, D' - 0) = Q(N'< N, D' - 1) - O.
(3a)
and
(3b)
(with
== P (N '< N, D' - 1) = O.
N
replaced by
Thus
N')
imply
In turn,
Q(N'< N, D' == 0)
P (N SN i) = Q(N S N') = land, hence,
0
P (N' - N) - Q(N' - N) == 1.
Proof of (9).
(9')
(9)
is equivalent to
A SA' ,B'sB,asa', aSs' with one strict =l>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)
imply
(10)
(and hence of (11»
a'=a, Q(N'<N) - P(N'-N<oo, D'''; D) = P(N<N'<co,D'-l) - O.
The remaining conclusions in
(10) follow from
(3a)
and
(3b) •.
- 8 -
Furthermore,
f
a-a'-
(6a)
simplifies to
(~: A') dP +
[N<N'l
from which
(11)
Finally, (12)
(10)
and
Remark
(11).
eas~ly
and
f
(A' - ~,) dP + A' P(N<N'
[N<N'<c», D'=O]
o
follows.
(13)
are proven in the same way we prove
Notice that the value of
= Q(N'=N)
is possible.
0
This completes the proof of the theorem.
(a,S)
tion of N under both hypotheses.
P(N'=N)
= ~),
= 1.
determines the distribu-
For if
a'=a
and
a'=a,
This is not to say that every value
Neither does
a'= a
and
a'=
a
imply
then
(a,a)
A'=A,B'=B
in
general.
REFERENCES
Hypotheses~
[1]
Ghosh, B.K. (1970). SequentiaZ Tests of StatistiaaZ
Addison-Wesley.
[2]
Wald, A. and Wolfowitz, J. (1948). "Optimum character of the
sequential probability ratio test," Ann. Math. Statist.
19, 326-39.