ON LEAST FAVORABLE SET
FUI~CTIONS
by
Olaf Krafft
University of North Carolina
and
Teclll1ische Hochschule Earlsruhe
Institute of Statistics ~limeo Series No. 528
This research was supported by the
Air Force Office of Scientific
Research Contract No. AF-AFOSR-76o-65
Department of statistics
University of North Carolina
Chapel Hill, IT. C.
l'
Introduction and
Summar~
The concept of a least favorable distribution was introduced by
[l4l.
~'Jald
Lehmann and Stein [12J showed its use for testing a composite
hypothesis against a sinple alternative.
level a
In order to study tests of
which minimize the maximum power Lehmann [18J introduced the
notion of a pair of least favorable distributions at level a.
In [8J and
[9J it is shown that least favorable distributions can be considered as
solutions of certain infinite linear programs.
Whereas the existence of an optimal test can be proved under mild
conditions which guarantee the weak compactness of the class of all test
functions, the problem of the existence of a least favorable distribution
at level
a requires quite a few additional considerations. The reason
for that consists essentially in the fact that the action space of the
statistician contains only two points whilst the parameter space in
general contains infinitely many points.
discuss some aspects of this problem.
The main aim of this paper is to
We will restrict ourselves to the
case of testing a composite hypothesis H at level
alternative K.
a against a simple
The results can be generalized to the case that K is also
composite and they are similar for the syrmnetrical version of the test
problem described in [8J, but those extensions would give no better insight
into the problem.
Since least favorable distributions can be useful also
for non-parametric problems, see for instance Hoeffding [6J, vie vlill rnal:e
as few restrictions on the parameter space as possible.
The basic lemraata for our discussion will be the representation
theorems for linear functionals on certain -Linear topological spaces and a
convergence theorem for measures which can be regarded as a generalizat:i.::m
1) This research was supported in ~art by the Air Force Office
of Scientific Research Contract No. AF-AFOSR-76o-65
of Belly's theorem.
All of these lemmata guarantee only the existence of
least favorable distributions so that our theorems ,-Till give no nev ueth::Jds
to construct least favorable distributions.
In section 2 we will introduce the definition of a least favorable set
function which is more general than the traditional one in the sense that
vle
do not require the cr-additivity of a least favorable measure.
This
generalization does not perrnit the usual interpretation of a least favorable
distribution as an optimal
strategy of Nature.
However, this approach
seems to be justified by the fact that the existence of a least favorable
set function can be shown under rather mild conditions.
(The advantage of
such a generalization for stochastic processes is remarked and used also by
Dubins and Savage [3J).
The principal tool in this section will be the
theorem on a separating hyperplane for convex
sets.
In a corollary
He
will give conditions under which the least favorable set functions arc also
cr-additive so that the cornmon conclusions can be derived.
In section 3 "lie introduce the concept of a least favorable support in
order to weaken the condition of the compactness of the parameter space used
by Hald [14J.
A least favorable support is essentially the set in the
parameter space on which a least favorable distribution is concentrated.
Under the assumption that there exists a compact least favorable support,
a least favorable measure can be shown to be the weak lirni t of a sequence
of measures each of I'lhich is concentrated on finitely many points of the
parameter space.
The topological and measure theoretic notions used in this paper
be found in the bool~ by Dunford and Schvlartz [4].
2
n~y
2.
Least favorable set functions.
Let W =
space (I,'n),
Lwe : e
€
8J be a' c lass of probability measures on a laeasurable
we' a probability measure on (X/\) not belonging to H.
assume that W + [We 'lis dominated by a
~-finite
measure
densities with respect to ~ are denoteQ by De (x) and
~.
He
The corresnonding
Pe'(x);
throughout the
paper Pe (x) is assur.~d to be continuous in e(for ~-almost all x) in the
topology which is indicated in the conditLms of the respective theorer:lS.
By Q we denote the c}.ass of all test functi:ms i:;:
= [;p(x):;J(X)
~
measurable,
1(" -
° '5 (.;,(x) -s:
1 for all x € ~'~}.
The problem of testing the hypothesis H: e€ 8 at level cx ,cx €(O,l),
against the alternative K:e
cp*
€
Ii
= e'consists
ex, H'
~ex H
, = Ly(x):
y (x) € ':Ii,E
such that E e,cP* (X) ::: \ I';? (X)
Let
in the determination of a function
}I
be a
e cp
(X) < ex
..... '
for all <p €<;P
_
for all
e €8}
,
rv
"', H.
~-field over 8 ,"A a probability measure on (8 ,Jl) and assume
Pe (x) to be 11 x ~'-measurable.
If CP~ denotes an opt:imal test for testing
the simple hypothesis H"A that the true density is P"A
~!e?e(X)d
"A (e) against
K at level ex , then "A* is called a least favorable diS tribution at level ex iff
A measure "A* which is defined in this way
pretat~on.
requirement that "A
Definition 1.
is
'If
a direct statistical inter-
For mathenatical purposes, hOI-rever , it seems to be adequate to
use a more general definition.
re (x)
ha~
is
This will be done by departing from the
~-additive:
Let 8 be a topological space and Jf be a field on8 such that
x '" -measurable.
addi ti ve set functi::ms :m 11.
By ba (:I{) vie denote the class of all bounded
Then an l-X- €
least favorable set function (1. f.
So
ba (H), 11(-? 0, is called a
f.) for testing
3
)I
at levela against
or
iff for the optimal test :p* and 1* the
follo~Iing
c::mditions are satisfied:
(i)
(ii)
To show that when
'!
~-field
is a
~
and 1* is
-additive definition 1
is
equi valent with that given in connection uith formula (1) we will use the
follovring
Lemma
~.
Necessary and sufficient for that ' .i
*
-a,H is optimal and
is a least favorable distribution is the existence of a number c > 0
€ q,
~*
such that
A* - e.• e.,
(2)
p ,( x)
e
> c f p e(x)
8
d
A*
IJ.-a. e,
p
e' (x) < c
fp
e
(x) d A* •
8
The
proof of this lemma which is an extension of corollary
5
by Lehmann
[10], p. 92, is based on the fact that a least favorable distribution can
be shown to be equal to
,...
!\ =
A * / 7\* (8), I-There 7\* is that f'inite measure 7\
for w'hich tIle function
(4 )
achieves its minimum.
The statement of lemma 1 then fo llows from the
equality
where the infimum is to be taken VTith respect to all finite measures A
which are concentrated on finitely many points of S.A proof of leNJm 1
and of formula (5) I·lhich vT:Lll be used also in the proof of the orem ) rrny
be found in [9].
4
Theorem 1.
Let
O:-additive.
'H
be a cr-field and the 1. f. s. f 1* be non-degenerate and
ThenA-K- =1* / 1*(8) is a least favorable distribution as
defined in connection vath
(D.
If, conversely, A* is a least favorable
distribution and c is the constant of lemma 1
then 1* = CA* Hi 11
satisfy defini tioD 1.
Proof.
The proof foUm'Ts immediately from lenuna 1 observing that because
of the cr-additivity of P condition (ii) in definition 1 becomes
J
(6)
,
(Pe'(x) -Jpe(x)dl*)cr*(x)dfl=ma~
e
I
9~
so thatcp*(x) satisfies (3) 'withe
= l*(e).
Conditions (i) and (ii) of our definition Jf a l.f.s.f. are closely
related to formulae used in the proof of the generalized fundamental
lemma by Dantzig
and Wald [2], the numbers k. given there correspond
l
to the values of I *.
We will prove the existence of a 1. f . s. f. along
the lines of that proof and of the proof of theorem 1 by Hurwicz and
Uza'va [7].
Theorem 2. Let
e
be a closed subspace of a normal topological space T,
be the field generated by the
measurable.
~.
closed sets of 8
p e(x) be
and
1!
11 X"'-
Then there exists a regular l.f.s.f.
Since a closed subspace of a normal space is itself normal 8 can
be assumed to be a normal
topological space.
Let
c
(8) be the space of
all bounded continuous real functions on 8 "Tith norm /If /I = sup jf(e)
eEe
I.
Let q;-* be an optLnal test for testing H at levela against K and the subsets
P, Q. and R of the Cartesian product
P
Q.
={
(a,h(e)): a ~ Ee~' h(e)
~
It-
x (; (e) of
a-E aC!( for some
ffl-
CD, € 9?
[ (a,h(e)): a > Ee,cp*, h( e) >0 for aLLG E 81-,
R = [ (a,h(e)): a ::- EetcF*, h(e)
> 0 for aUe
5
€
eL
and e (8) be defineci o..s
1
. I
) and (8. ,11 ) are two points inP"I'rlth
2 2
If (al,tt
Q
the point (p a
p .:?
1
+
(1.
-
1
p)
1
in R·- x C (e) •
'5
1
E8 ':1 ' ~ (8)
+ ( 1- p) a
€
1"2
cD.
2
,p h
'5 a- Ze Ci1 '
+ (1 - p) h 2 ), 0 :::: P -:: 1, "rill be in P since
l
Hence P - and similarly Q. and R - is a convex set
Moreover P has a non-empty interior:
example the point (a
= -l,h(e);;: -1)
a point of Q. and assume (a", h")
Since h" (8) :::- 0 for all
e
€
e
€
It contains for
as an interior point.
P.
Let (a",h") be
Then there will be a
y"€ ¢
such that
i t follows that cp" if of level ex and hence
its power does not exceed the power of ,p* : E 8 "?" :::: E e\;*.
The last in-
equality implies a" :::: Ee,cp* so that (a",h") VTould not be an element of Q.
This proves that P and Q. are disjoint sets.
conditions of theorem V. 2.8 in
Therefore P and Q. satisfy the
[4] which states that P and Q. can be
separated by a non-zero continuous linear functional.
Theorem Iv.6.2 in
[4] the dual space of c(e) can be represented by the
space rba (e),i. e. by
functions on 11 •
According to
the space of all regu:i.ar bounded additive set
Hence there exist numbers b and
r
and an l-l<-
€
rba (8)
such that
bar + fh'dl* ::::
e
r ::::
ba"+f h"dV,;-
e
holds for all (a', hI) € P and (a", h") € Q..
The continuity of (b, H-) implies
that
(8)
r <
ba" +
f
e
h" d l *
for all (a",h")
6
€
Q. ,
~ denoting the closed hu~_l of Q..
So (8) iliU hold for all (al,h")E E.
Since the point (a'':- = Ee(,,*,h*(e) = a - Ee,"-l<-) lies in the intersection
of P and R the number
r will
satisfy
dl* .
e':'.~-Y.- +8f (a - E,.w*)
t:I
The left side of (7) show·s that 1 *" is non-negative since otherwise for a
1- bE
-
sufficiently small h' "Te ,'lOuld have ! h' d1:>:- >r.. Similarly b can be
8
sho,'ffi to be non-ner;ative. The same inequality (7) can be used to sh:n,
that 1* is non-der;enerate if E e'CP* < 1..
(If Ee'~·* = 1
of definition 1 are trivially satisfied
"rith 1.*= 0.)
the conditions
For in the case
that 1 * would be degenerated b would be :t;)ositive and by choosing a'
i t H~uld follow that 1
'5
Ee,r;d(-.
=1
Assume now b would be equal to zero.
It
,'!Quld follow that 1* is non-degenerate and by choosing h'(e)sa, h"(e)=O
f~r:nula( 7) would imply
a 1* (8) < 0 ,
which contradicts
a
€
(0,1).
Observing that for all cp
€ <I>
the points (a = Ee,cp, h( e)= a-E
elements of P and dividing by b
10le
e cp) are
get from (7) and (9)
N
+ !(a-E(jP*)dl* for all 'fiE ¢,
8
Hence 1* satisfies condition (ii) of
N
",here '1* is
definition 1.
It also satisfies condition (i) since for (a",h") =(E ,'07i-,O)
e
we get from (8) and (9)
b Ee,'P*
=
f(a-Ee~')*)dl* ::: b Ee,q-l<8
so that
f8(a-E~?*)di* :::
o.
(i) follows from the fact that 'P* is of level a.
7
In the case that there exi sts a 1. f. s. f. 'l'Tl1icl1 is cr-additi ve, lemma 1.
shoyrs that the structure of an optimal test '.-* can be described by means
of a 1. f • s. f.
A test of the f rm (3) will be called alike lihood
test with respect to 1*
,~ere
1* = CA*.
l'
at i:)
In general the knowledge of a
1.f.s.f. 1* does not permit to decide \vhether an optimal test is a
likelihood ratio test w.r.t. 1* or not. The reason for that is that
Fubini's theorem can not be generalized to the case when one of the involved set functions is not cr-additive, see for example Yosida and Heuitt
[16J.
However, under certain additional conciitions on
c(e)
possible to represent the dual space of
set functions.
corollary.
e
andpe(x) it is
by the space of thecr-additive
Sone of these conditions are given in the following
There the subsets
o(e)
respectively as the set of all h(e)
e(e)
and
€
c(e)
subsets of e and as the set of all h( e)
€
of
C(e)
are defined
yn1ich vanish outside compact
c(e) which tend to zero at
infinity.
Corollary 1. Let H be the cr -field generated by the closed sets of
Pe(x) be
11
x C\3-measurable.
If
e
e and
is a closed subspace of either
(a) a compact Hausdol·ff space and Pe(x)
€
c(e)
f:::>r fl-almost all x
€ ':,
(b) a locally compact space and Pe(x) €,\)(e) for fl-almost all x € :£, or
(c) a locally compact space and P e(x)
then there exists a regular
Proof.
0--
€
e(e) for Il-almost all x
additive 1. f. s. f.
The statement follows from the proof of theorem 2
in case (a) from theorem Iv.6.3 in [4J,
in case (b) from theorem D in [5J, sec. 56,
in case (c) from exercise 9 in
8
[lJ, p. 67.
and
€
",
or
Part (c) of corollary 1 is related to assumption B used by Leh~r.['lm
Remark:
[nJ.
e
Concerning our condition that
informed by Prof. Lellmann that
Under any of the
sh::mld be a closed set the author -..'as
such a condition is necessary in his naper too.
conditions of coroLLary 1. the optimal test l'liLL be
a lil<elihood ratio test H.r.t. the corresyonding Lf.s.f.
If none of the
conditions holds an optimal test is at least approximable by a likelihood
ratio test w. r. t. a least favorable measure uhich is concentrated on
finitely many points of
Theorem
3. Let
Then for every€
e
as it is shown in the following
:p* be an optimal test for testing H at level
>
0 there exists a measure A' on (e,Jf) which is concentrated
e
on finitely many points of
cp'(x) =
(n)
and a test cp I,
{l'
Pel (x)
> f Pe(x) dA
0,
Pe' (x)
Pe(x) dA'
fEe;> IdA'
(12 )
a against K.
e
=
e
< f
e
I
a A' (e) ,
such that
E e d'
( 13 )
Proof.
I
-
E e,CP *
'S
€
•
Formula (5) shows that for everyE
concentrated on finitely many points of
e,
>0
lie
can find a A" Hhich is
say on (e ,
l
e2 ,·.·, en )
such that
(14 )
Since
f
(e ,
l
e2 , ••• , e
( ....1\11)
n
for testing HI:e E (e ,
l
)
-
E e ,CP * <
_
E.
is a compact set in
e2 , ••• , % J
at level
e
(X
optimal test and Ala l.f.s.f. for HI against K.
so that (13) is satisfied.
The
vle can apply corollary 1
against K.
From
(5)
Let ep
and
I
(14)
be an
we get
equalities (n) and (12) follow from lemma L
9
3.
Least favorable supports.
In this section we '.Till discuss one important case for which a least
favorable distributi::m exists but where in general the conditions of the
corollary in section 2 are not satisfied.
It is the case in which a
uniformly most pOvrerful test turns out to be a test for two simple
hypotheses.
That . .' lill be true for example if the class IV is such that
the corresponding densities have monotone lil:e1ihood ratio and one-sided
hypotheses are consioered.
Let
1,07
As a simple exanple take the following:
be the class of normal distributions vTi th ImovTn mean a
o
and un}<:nOi-m
~
01·
2 2 _ 2
222
variance er, ~ the Lebesgue measure, H: er:::- 00' K: er - er , er <
o
l
Here a least favorable distribution exists (it is concentrated on the
point
er~
) but none of the conditions of the corollary in section 2 is
satisfied.
To study problems of that kind the follOWing definition may
be useful:
Definition 2.
Let Jf be a er-field over 8.
favorable support iff for every
* ~
e cPG(X)
(15)
is called a least
€ 11
""
G there is a e
G such that
€
* (X) ,
E e C?G
E
*
e~
An G
where ~G(x) denotes an optimal test for testing HG:e
€
G at level
a
against K.
Definition 2 is suggested by the following consideration:
It can be
shown that there exists ahmys an optimal test c; * such that
(16)
sup
e€8
-othe~~ise
EEfP
*
(X)
=a
the test
""*
cp (x) = ((1- a):p * (x)
+
a - sup
e€8
*
E~p * )/(1 - sup EeCf- )
e€8
vrould satisfy (16) and Hould have at least as nru.ch pmrer as cp:.
of (2) no least favorable distribution . .·r.'ula. exist if
10
Because
E f.P
( 17)
~~.
e
(X) < 0: for all e
€
e.
By definition 2 we have characterized a set G for l'lhich the equality
EdP* (X) =
0:
holds as it is seen by
If G is a least favorable support then the test
Lemma 2.
optimal for H: e
Proof.
Since
€
*
~G
e
at level 0:
is at
-J(-
Ee~G (X)
(18)
0:
is also
against K.
level 0: for
<
*
~G
HG:
for all e
e
€
G we have
€
G.
Condition (15) implies that (18) holds for
all e
€
e.
Since <D
0:,
H is a
f
...
* is optimal in <Do:,I\r' c::: G-¥.- will be optimal also in <Do:~
subset of <Do: H and:;)G
, G
ExanWles of least favorable supports are for instance the sets on \lhich
the least favorable distributions derived. in [10],3.8,3.9 and. 8.2 or in [6J
are concentrated.
The following
simple example shows that a least
favorable support nny not be determined uniquely.
Let VI be the class of distributions which have densities p (x) =e x(l_e)l-::
l'lith respect to the counting measure Il , ~. =( O,l), H:e
K : e' = 1/2.
Here every G(1)) = (1), 1- 11h 1)
€
e ,
€
e
e
= (0,1/2)~)(1/2,1),
will be a least favorable
*
support since 'YG( 1)) Trill be identically equal to 0: for every 11
€
e .
An interesting example
for that (17) holds and hence no least favorable
distribution exists is
given by Lehmann [11], p. 413; another typical
case is discussed in [15], example 2.23.
In the proof of the existence theorem of this section we
will use the
following
Lermna 3.
Let (f (x))
n
be a sequence of Il-integrable functions which
converges pointwise Il-a.e to the Il-integrable function f (x).
o
follows that
11
Then it
lim
n ~IO
Is
f
=
n
~.
= Is
(x) dlJ.,
f
0
where the sets S
S
(x) dlJ.
0
n
n
and S
0
0
are defined by
(x:f (x) >O}and S
n
=( x: f (x) >O}.
0
0
= { x:
Let T be defined by T
f :)(x)
_
and let I sex) be the
o € lim sup Sn I i . e. if f n (x0 ) > 0
it follows that lim f (x ) = f (x ) > o.
characteristic function of a set S.
for infinitely many
= O}
0,
If
X
0
0
0
0
-
Hence we have
(] 9)
lim sup
By similar arguments
(20)
1'le
Sri
S::>UT.
can show that
lim inf S
and that for x
~
n
~
S
0
o ~ SoUT
(21)
lLn inf IS.
n
(xo )
= lin
sup IS
0
(x ) = O.
o
From formulae (19)-(21) it follows that
lim sup fo(x)I
S
n
f
o (x) IS,
(x)::::
(x) :::: f::> (x) lim sup IS
(x):::: f 0 (x) IS llT (x), and
n
o'
(x) IS (x).
0
n
n
(x) is bounded by the IJ.-integrable function I f (x)
(x) < lLll inf f
fo(x) lim inf IS
o
Since f o (x) IS.
o
can apply the lemma ::>f Fatou and get
I
I
f
0
I fo
(x) d IJ. ::::
n
(x) lim sup Is (x) dIJ. <
lim sup
(x) IS
n
f
0
(x) lim inf Is
lim inf
I
(x) d IJ.
4.
I
< I
lim sup f o (x) IS
f o (x) IS
(x) d IJ.::::
n
f o (x) IS UT (x) d IJ. = I fo(x)
::>
lim inf f 0 (x) IS (x) d IJ. ::::
n
(:c) dlJ.
n
(x) dlJ. ,
n
Hhich ShovlS that lim
Theorem
I
°
I 1m
n~tO
I
f o (x) IS
(x) d IJ.
n
=I
f
(x) I
::>
(x) d IJ. .
S
Let 8 be a complete, separable metric spaCeOand)f
field generated by the closed sets of 8.
Pe(x) is assumed to be
measurable and to be bounded by a ~-integrable function
12
be the ~_
If
x
c,'._
g (x) such that
<
Pe(x) ~ g (x) for all e
€
If under these conditions there exists a
8.
compact least favorable support G
additive l.f.S.f.A
-¥--
on Jf.
€ 1f
then there exists a regular
Moreover
(j-
A*(8) "rill be less than or equal
to Ija .
Proof.
to the remarks made in c:nnection ·with formulae (4)
According
and (5) A* will be a l.f.s.f. if
Ee,;p-l<- =f (A*).
Because of lemma 2 it '''ill be
sufficient to ShOil that a A* exists
Hhich satisfies
0= Ee,cpa = f(A*).
From (5) it follows
that there exists a sequence (An )
of finite
measures An such that
o =n~i:
(aA n (G) +!
Since we can choose (~)
r.
(Pe,(X) -! Pe(x) dA n )+
d~ ).
in such a way that A n is concentrated on
finitely many points of G, i.e. A n for fixed n is concentrated on a
compact surset of G, (A n) can be assumed to be a sequence of regular
measures.
For such measures a weak convergence is defined in the
following way: (A rt) converges weakly to A 0 iff! f(e) dA
8
! f( e) dA
for all bounded continuous functions f.
0
n
converges to
By Prohorov [13],
8
theorem L 12, i t is sho'm that (A
n
) will have a vleak limit A
iff there
0
exi sts a constant M such that
(22)
A
n
(G) ~ M
for all n ,
and for every € > 0 there exists a compact set K
(23 )
Since f (A
A
n
n
€
such that for all n
(G - K ) < € •
€
) ~ A n (G) holds for all n vle can choose {A n )
13
so that
(24)
and hence (22)
that
wi.'_l
(23) is also
be
satisfied.
From the compactness of G it folL:nls
satisfied so that {'I\ n} has a weak limit "- o.
Formula (24) shows that '1\ o(G) is bounded by
lja.
It remains to show that
lim J~ (Pe'(X) -Jp e(X) dAn )+ d \-l
(25)
= J (Pe,(x)-J Pe(x)d'l\o (
d
Il·
n~eo
The weak convergence of {'I\ n} to'l\ 0 implies that
lim J Pe(x) d '1\ n =J Pe(X)d '1\ 0
n ~ eo
Let fn(x) and fo(x) be defined by
Il -a.e.
fn(x) = Pe,(x)-J Pe(X)d\ ' fo(x) = Pe,(x)-J Pe(x) d '1\0
and Sand S be defined as in lemma 3.
n
Since by assumption
0
I
(x) I :::Pe' (x) + g(x)j a ,
n
vre Can for f (x) apply the theorem of the dominated convergence.
f
n
JS
n
Hence
f (x)d Il will tend to Jf' {x):lll uniformly in Sn as n tends to infinity
n
S :)
so that for every e >
n
0
I S·J f n (x)d
(26)
n
the inequality
Il -
Jf
S. 0
n
< e/2
(x)d Il
holds for sufficiently large n.
Since lewna 3 implies that for sufficiently large n
I Jf
S 0
n
(X)ull -
Jf
S'
(x)d Il
0
I<
e/2 ,
-
o
(25) can be shown to be true with the help of the triangle inequality.
By defining '1\*( C)
= '1\ 0 (CG), C
€
1', we get a measure which has the
desired properties.
Remark 1.
It is likely that the compactness condition on the least
favorable support G can be weakened by a similar condition as it is
gi ven in (23).
14
Remark 2.
Clearly the methods of section 2 can be applied also in
connection with a least favorable support.
On the other hand if 8
itself is compact the method used in the proof of theorem
3 is also
applicable to prove the existence of a least favorable measure.
15
References
~
Act. Sci. et Ind. 1175.
[2]
On the fundamental lemma of
---
Ann. Math. Statist. 22
Dubins, L. E. and Savage, L. J. (1965).
must.
[4]
Hermann et Cie., Paris.
Dantzig, G. B. and Wald, A. (1951).
Neyman and Pearson.
[3]
/
Elements de Mathematique, VI, Integration.
[1] Bourbaki, N. (1952).
87-93.
How to gamble if you
McGra1'i-Hill, New York.
--
Linear Operators, 1.
Dunford, N • and Schwartz, J. T. (1966).
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[5]
Halmos, P. R. (1964).
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[6]
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[7]
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[8]
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[10]
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~biete.
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[11]
Lehmann, E. L. (1952).
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On the existence of least favorable
Ann. Math.
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408-416.
~
[12]
Lehmann, E. L. and Stein, C. (1948).
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Ann.
~futh.
Host -powerful tests of composite
Statist. ..............
19 495-516 •
16
[13J
Prohorov, Yu. V. (1956).
Convergence of
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i Primenen.
1
~
random processes and
(Russian) Teor. Verojatnost.
177-238 (English translation in Theor. Probab. and
~.vl~ 157-214).
[14 J Wald, A. (1950).
statistical Decision Functions. Wiley, New Yor1:.
[15J
Witting H. (1966).
[16J
yosida, K. and Hei'litt, E. (1952)
Mathematische Statistik.Teubner, Stuttgart.
Trans. Amer. i.1ath. Soc. 72
,~
46-66.
17
Finitely additive measures.