1 1bis research was supported by the Air Force Office of Scientific Research
under Contract #AFOSR-75-2796.
AN IMPROVED STATEMENT OF OPTIr~LITY
FOR SEQUENTIAL PROBABILITY RATIO TESTS
by
Gordon D. Simons 1
Department of Statistics
university of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series #1037
November, 1975
ABSTRACT
An improved version of the
opti~ality
property of sequential probability
ratio tests is stated.
MIS 1970 sub,fect cZassifications.
Key words and phrases:
Primary 62L10
Optimality, sequential tests, sequential probability
ratio tests, KUl1bad-Leibler information number.
An Improved Statement of Optimality
for Sequential Probability Ratio Tests
by Gordon Simons
l
Univeristy of North Carolina at ChapeZ HiU
Wald and
~lo1fowitz
(1948) first proved that sequential probability ratio
tests (SPRT's) possess a strong optimality property.
Improvements on their
statement of optimality have been given by Burkholder and Wijsman (1960 and
1963) and by J. K. Ghosh (1961).
anot~ler
The intent of this paper is to discuss
improvement.
1Ve shall assume the usual iid model (See Lehmann (1959), page 97.)
with respect to two probability measures
observation has the density
an SPRT
aI'
S(AO' AI)'
o<
PO under
E'O·~
0'
competing test with error probabilities
EN'
o
and
or
In particular, every
under
A < 1 < Al < "", with error probabilities
O
and with expected sample sizes
sample sizes
Po and Pl'
and
a'
0
Further, let
ElN.
and
I
aI'
Let
a
O
T'
",I
"'0
<_
'"
"'0
and
",I
:<:;
be a
The optimality property stated by Wald and
E·'11'·
"'J' •
"'I
and
and with expected
Wolfowitz says that if
(1)
be
T
N
"'1'
and if
lThis research was supported by the Air Force Offioe of Scientific
Research under Contract # AFOSR-?5-2?96.
-2··
E
(2)
oj~'
<
00
and
then
(3)
We shall show that (1) can be replaced by the weaker assumption
(4)
(3 1
o
~
and
(3'
1
and the quantities
defined analogously.
The conditions in (4) imply that
",I
"'0
B'o and B'I
+
are
< '" + '"
"'1 - "'0
'-'"1
",I
but permit one of the inequalities in (1) to be violated.
As an aside, it appears to us that these ratios (the B's) may be more
useful measures of error than are error probabilities.
For instance, they
appear in the fundamental result
and
(5)
(See Lelmann (1959), page 99.)
l'oreover, they appear in various places in
{'Ie shall refer to t;leJ11 as nOT'maZized error
Po(rejecting EO)
probabilities . (Observe that S =
o PI (rejecting H ) if the test always
O
terminates. Thus, the error probability in the numerator is being norr.lalized
the literature on sequential tests.
by a probability of non-crror in the denominator.)
Besides the result we
have mentioned, we shall discuss another 1vell known result which can be
strengthened by replacing assumptions about error probabilities with assumptions
about normalized error probabilities.
could be found.
We suspect that r.1any other examples
-3-
Proof that (4) and (2) imply (3)
For the sake of brevity, we shall draw
rather freely from the notation and results appearing on pages 105 and 106
of Lehmann (1959).
The conditions in (1) are used by 1'!ald and Wolfowitz only to conclude
that
(6)
+
where 7T., 0
+
(l-7T )w 1 a I '
is the prior probability that
< 7T < 1,
density, and where w
o
>
0 and wI
type II errors, respectively.
>
PO
is the correct
0 are losses due to type I and
Thus, it suffices to show that (4) implies (6).
According to Figure 3 (page 106 of Lehmann),
7T '
where 1]. I
and rr n
R·
WI
\'I
are values in the open interval (0,1) which satisfy
I_1T"
1T
7T
l-7T
l-rr
Thus,
(1-7T)W
1
which, in view of (5), implies
(7)
_
+w
O 1
1-7T I
rr '
-4-
Finally, (6) follows from (4) and (7).
This is nost easily seen by first
showing that the conditions in (4) are algebraicly equivalent to
and (6) is algebraicly equivalent to
o
We shall now discuss another result which can be strengthened by the intro-duction of normalized error probabilities. Let
X represents an observation
Ku11back-Liebler information number), where
appearing in the random sample.
nQmbers whose sum
is
(8)
N
0.
0 + ai
< 1.
Suppose
0.*
o
o
and
0.*
1
are fixed positive
Then, for any test whose stopping variable
and whose error probabilities are
E
I(po,1J )=E lor,(po(X)/[ll(X)) (a
O
1
and
a1 <
- 0.*l '
1-a.*
o
log - 0. *
1
N
This follows from the well known result that
a
0.
EO N
~
0
log - o
I-a
1
+
1-0. 0
(1-0. 0) log - 0.
1
I (PO' PI)
-5-
and from the fact that:the:function
g(x,y)
=x
x
log --l-y
I-x
(I-x) log--Y
+
is convex in'~;q,,~h of its arguments.
when the conditions
aO
~
a
o<
x, Y
<
1,
It is not difficult to show (8) holds
O and
a
and
13 1
l
~
ai
are replaced by the weaker
conditions
(9)
13 0
vlhere
i3*
0
13*
0
~
= a*/(l-a*)
o
1
g(a O' a l )
~
and
13*
1
=
~
13*
1
a*/(l-a*)
1
0
One must show that (9) implies
g(a(j, a*)
1
References
[1]
Burkholder, 0.1.. and Wijsman, R.A. (1960).
admissibility of sequential tests.
[2]
[3]
Ann. l1ath. Statist. 31, 232 (abstract).
Burkholder, D.L. and Wijsman, R.A. (1%3).
admissibility of sequential tests.
Ghosh, J.K. (1961).
Optimum properties and
1l0ptimum
properties and
Ann. Math, Statist. 34, 1-17.
On the optimality of probability ratio tests in
sequential and mUltiple sampling.
Calcutta Statist. Assoc. Bull. 10,
73-92.
Testing Statistical Hypotheses.
[4]
Lehmann, E.L. (1959).
[5]
Wald, A. and Wolfowitz, J. (1948).
probability ratio test.
Wiley, New York.
Optimum character of the sequential
Ann. Math. Statist. 19, 326-339.