T
HEORY
P
ROBAB.
A
PPL
.
c
°
2006 Society for Industrial and Applied Mathematics
Vol. 50, No. 3, pp. 463–466
Translated from Russian Journal
ON THE DVORETZKY–WALD–WOLFOWITZ THEOREM ON
NONRANDOMIZED STATISTICAL DECISIONS
∗
E. A. FEINBERG
†
AND
A. B. PIUNOVSKIY
‡
(
Translated by the authors
)
Abstract.
The result by Dvoretzky, Wald, and Wolfowitz on the suﬃciency of nonrandomized
decision rules for statistical decision problems with nonatomic state distributions holds for arbitrary
Borel decision sets and for arbitrary measurable loss functions.
Key words.
decision rule, nonatomic measure, nonrandomized decision rule, equivalent decision
rules
DOI.
10.1137/S0040585X97981937
Consider the following statistical decision problem studied by Dvoretzky, Wald, and
Wolfowitz [5], [6]. Let (
X,
X
) be a Borel space and let Ω =
{
μ
1
,μ
2
,...,μ
N
}
be a Fnite
number of probability measures on (
X,
X
). The true probability distribution
μ
on (
X,
X
)
is not known, but it is known that
μ
∈
Ω. There is also given a Borel space (
A,
A
) whose
elements
a
represent the possible decisions. Let
A
(
x
)
∈A
be the sets of decisions allowable
at states
x
∈
X
.
We assume that (a) the graph
Gr
(
A
)=
{
(
x,a
):
x
∈
X, a
∈
A
(
x
)
}
is a measurable subset of
X
×
A
, and (b) there exists at least one measurable mapping
ϕ
:
X
→
A
with
ϕ
(
x
)
∈
A
(
x
) for all
x
∈
X
. In particular, (b) implies that
A
(
x
)
6
=
∅
,
x
∈
X
.
A decision rule
π
is a regular transition probability from
X
to
A
such that
π
(
A
(
x
)

x
)=1
for all
x
∈
X
. A decision rule is called nonrandomized if for each
x
∈
X
the measure
π
(
·
x
)
is concentrated at one point. A nonrandomized decision rule
π
is deFned by a measurable
mapping
ϕ
:
X
→
A
such that
ϕ
(
x
)
∈
A
(
x
) and
π
(
ϕ
(
x
)

x
,
x
∈
X
. We call such a
mapping a decision function and denote it by
ϕ
.
The loss is deFned by a vectorfunction
ρ
(
μ,x,a
)=(
ρ
1
(
)
,...,ρ
M
(
))
,
where
μ
is the true value from Ω and
a
∈
A
(
x
). ±or each
μ
n
∈
Ω, the function
ρ
m
(
μ
n
,x,a
),
m
=1
,...,M
, is measurable in (
) and takes values in [
−∞
,
∞
].
±or any decision rule
π
and any
μ
n
∈
Ω, consider the risk vector
R
(
μ
n
,π
R
1
(
μ
n
)
,...,
R
M
(
μ
n
))
,
where
R
m
(
μ
n
Z
X
Z
A
ρ
m
(
μ
n
)
π
(
da

x
)
μ
n
(
dx
)
,m
,...,M.
Here and in what follows, all the integrals of a measurable function
f
are deFned as
Z
f
(
y
)
ν
(
dy
F
+
+
F
−
,
where +
∞
+(
−∞
−∞
,
F
±
=
R
f
±
(
y
)
ν
(
dy
),
f
+
= max
{
f,
0
}
, and
f
−
= min
{
0
}
.
The decision rules
π
1
and
π
2
are called equivalent if
R
(
μ
n
1
R
(
μ
n
2
)
for all
n
,...,N.