A Proof for Lehmann-Sche↵e Theorem Kun Meng
[email protected] Department of Biostatistics Brown University, Providence, RI, 02903, USA
Lehmann-Sche↵e Theorem: Suppose that P is a family of populations, T is a sufficient and complete statistic for P, and ✓ is an estimable parameter functional on P. If S is an unbiased estimator of ✓, then EP (S|T ), which does not depend on the choice of P 2 P, is the unique UMVUE of ✓.
1
for
A
Lehmann
proof
( the
is
-
also
Swheffe
theorem
'
the
estimable
$
7-
,
Ea $
functional
parameter he
for
( I )
=V
I
such
statistic
a
=E
#
( IIT )
P
-
a.
s
.
)
important
⇒
Since
$*=E#l$*lT )
⇒
.
E
( IIT )
is
the
unique
for parameter functional
is
UMVUE V
.
*
that
HLPEP
.
T.IE#($IT)
proof From
the
sufficiency
which
does
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is
unbiased
an
Var
If
$*
he
then
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⇒
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#
for Holt
,
estimator
#
÷
of
E#l$k ) HIEP
for
H #
EP
completeness
of
T
{ F↳$*H
Van
with
,
*
the
implies
$
#
It
of
$IT )EVarp$*eVar#E#l$k )
Ea
From
=
Van
for
#
Van
V
unbiased
an
Vary 's'*e
Var
of
,
that
such
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e
choice
the
on
estimator
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#
7-
depend
of
-
E# E# (
-
VII )
.
,
E. HIT ) } HIEP
for
=o
( $*lT ) =E§($IT ) $*K )
=
Var
#
$*
for
fas
H I
.
.
,
