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SUFFIcmrcy AND MODEL-PRESERVING TRANSFORMATIONS
by
D. Basu
University of North Carolina
and
Indian Statistical Institute
Institute of statistics Mimeo Series No. 1~20
January 1965
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This resp.t.Tch 11B.S supported by the National Science
Foundation Grant No. GP 16-60.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Cha.pe1 Hill, N. C.
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SUFFICIENCY AND MODEL-PRESERVDfG TRANSFORMATIONS
D. Ba.su
University of North Carolina ml0. Indian Statistical Institute
Introduction and Su.'lllllarY
1.
(b!'~), I-Tllere
Given a statistical model
measures on tile cr-field
~
~
is a family of probability
of sub-sets of the sa..'1IPle space
!'
a mappinc
T
of
X onto itself ir.i..ll be called model-l)reserving if
i)
T is a one-to-one bimeas1.1rable map of the measurable space Q~, ~)
onto itself.
ii)
Let
1: =
PT-1(A)
E
peA) for all A€
A and
P€
!:.
(T) be tIle family of all model-preserving transformations.
Since the
identity map is always model-preservinc, the family T is never vacuous.
easy to check tl1at T constitutes a
STOUP
It is
i'litl1 the group operation tat.en as the
composition of maps.
For example, consider a sample
~,~, •••
1. '-
xn
of
n
independent observations on
a normal variable "litll mean zero and unlmOim variance ~.
family
!
includes all linear orthogonal transformations and also includes other
non-linear tra..'1sformations.
For instance, if we define
Yi = '{'(Xi)
'ofhere
In this case, the
IXi I ,
(i = 1,2, ... n) ,
* function
9 is an arbitrary skew-symmetric
on the real line tal;:ing only the
t"ro values -1 and +1, then it is not difficult to see that
independent normal variables with means zero and variances
Yl'Y2, ••• ~rn are
02.
Again, any
orthogonal transformation of the y i •s ,·r.i.ll leave the model invariant•
*(r(x)
= ""({'( -x) for all x
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Two measu..rable set.s A and B are said t.o be P - equivalent if
P(A A B) == 0
Where
for all P€ !: ,
A is the symmet.ric difference operat.or.
T\ro real valued measurable functions
l'
and
g
(b~)
defined on
are
said to be !:-equivalent if the set
( xl f(xH g(x) }
is E,-equivalent to the null-set.
The set A€
Let
~
!
-1
'-rill be called T-imrariant if A and T 'A are !:-equivalent.
be the class of T-invariant sets.
all set-operations, it follows that
intersection of the family(
Thus, A*
~}
~
-1
Since the set-function T
is a sub-field* of A.
Let
preserves
A* be the
of sub-fields corresponding to different T€
! .
is tIle sub-field of all rreasurable sets that are T-invariant for
The sub-field A*
each rodel-preserving transformation T.
maximal invariant statistic generated
b~;
corresponds to the
the group! of model-preserving trans-
formations.
Since the statistician is in tlle same situation (vis a vis the model)
Whether he observes
x
or
Tx, the principle of invariance demo.nds t"i1a.t the
decision tha.t he takes should be invariant. "lith respect to eac11 model preserving transforma.tion T.
This leads to the invariance reduction of the data. by
the substitution of t.he rodel (b ~ Eo) llith the simpler model
(X, A*, Eo)
On
•
the otl1er hand, in the presence of a sufficient sUb-field
of sufficiency leads us to the sufficiency reduction
(b ~ E,)
*abbreviation for
..
sub-rr-field.
2
~
the principle
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If
~
Ilappens to be the minimal. sufficient statistic then tIle above Hill be
the maximal sufficiency reduction.
In many situations the maximal sufficiency reduction of the data is the
same as the invariance reduction.
For instance, consider the exaxJPle of
n
independent observations on a normal variable with mean zero and unl:nOim
variance.
Tllis paper atten:q>ts to prove tl1B.t the maximal sufficiency reduction
e~:tensive
cannot be less
ing two
than the invariance reduction.
We prove tIle fo1101'l-
theoren~
Tl'leorem 1: If S be a boundedly complete (and hence minimal) sufficient sUbfield then
S cA
Theorem 2:
I f the family
*
Eo of probability measures be dominated and M be the
sufficient sub-field then
l~nimal
*
MeA
2.
Proof of Theorem 1
Let S be a boundedly complete sufficient sub-field.
We need the follovl-
ing well-lmo1'n1 lemma.
Lennna. 1:
If
z
be a bounded A-measurable function such that
E(z Ip)=: 0
then for
an~r
bounded
for 0.11 P€ P
~-measurable
E(zf Ip): 0
fu.nction
for all
f
PE: P
We omit tIle proof of the lennna..
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NOi."
let
S
and
T be arbitrary but fixed members of
and let
3
S
and
T respectively,
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S
Since
T
= T-1S
o
•
lea.ves the model (~ !, E,) invariant, we have
for all
i.e.
Where
P€ P
,
I p) == 0, for all PE P ,
S
o
stands for the indicator of the set A.
E(ls-I
I
A
Noting that tIle two functions
IS-IS
and
satisfy the conditions of
IS
o
Lemma. 1, we at once lJave
° for
E[(Is-IS )IS Ip] ==
o
P€ P
all
or
p(s) s p(SS 0)
for all PE P
But
peS) == peso)
for all P€ P
...
p(5.6.
In other words,
Since
S
and
so )
==
°
for o.ll PE P
S is T-invariant.
T i·rere arbitrary mer-ibers of
~
and
!
respectivel:,', l·re finally
have
S cA
3.
*
•
Proof of Theorem 2
Let us asstUl1e that P is dominated by a
cr-finite measure.
For tile proof
of Theorem 2 ue need tl1e follo'\oring tiro lemmas.
Lemma 2:
?ilere exists a countable selection P 'P ' •••• of measures from P
l 2
such that
t:1C ,:lec,S'llre
l'1here
o < c. < 1 and
~
1: c.
~
= 1,
do:::.inates P •
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Lemrn
:3:
tnat
fp
If
is
fp
=:
I.
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!1
be the smallest sub-field of
M-measurable for every Pe: ;E., then
!1
!
such
is the minma.l sufficient
sub-field.
We omit the proofsof the above t'·10 well-known lennnas.
Let
member of
T be an arbitra.ry but fixed member of
P
invariant it follows at once that
T.
Since
T leaves
T
leaves each
Q invariant
also, i.e. the two measures
are identical.
NOir, for any A€ A and
P(A)
PE:!:,
= Jf
A
=
I_
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and if
=
Jf
(x)dQ
P
A P
(a" f p
(X)dQT- l
=
(~-Q
dP
d" )
""
= QT-l )
4_l AfpT (X)dQ,
the fina.l step being a standard result in integration theorJ.
AGain,
Since
P(T-~) = f_~fp(X)dQ.
P(A)
= P(T-lA)
Remembering that
T
, we now have the identity
is one-to-one and bimeasurable we can re-write the last
identity as
J fp(x)dQ
A
Thus, for each
==
J fpT(X)dQ,
for all Ae: A •
A
PE:!:, the tllO functions
and llence tl1ey must be ;E.-eQuivalent.
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fp(X) and fpT(x) nmst be Q-equivalent
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Let
Mp
sub-field of
be the sub-field Generated
~
~,
Ylcnber of
sucll that
fp
is
f p'
i. e.,
Mp-measurable.
then tllcrc exists a Borel se"t
B
Dp
is the smallest
If U be a typical
on tIle real line such tjlat
•
Since
~'c
and
fp
f'pT
are
P- equivalent it folloi'TS tilat ti'lO sets
-P-eauivalent.
~
T"lat is,
1-1
Sillce
and
llaVC
H
is
E.-equivalent to
T-~ In other irords,
T were arbitrary members of
for all
Re):1er:lberinG t11at
iJe
Mp
and
!
I·1
is
T-invariant.
respectivel~r,
ue
established that
M
PE: P
is the smBJ.lest sub-field that contains all the
nOil have
-MeA*
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b~l
6
!1p
f
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References
1.
Bahadur, R. R. (1954), Sufficiency and Statistical decision functions,
Ann. Math. Statist. 25, 425-462.
2.
Hall, W. J., Wijsman, R. Q., and GhoSh, J. K. (1964), The relationship
bet~~en
sufficiency and invariance, Institute of Statistics Mimeo
Series No. 403.
3.
Lehma.nn, E. L. (1959), Testing Statistical ltyPotheses, 'V1i1ey, New York.
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