Idempotence is a well-known property of functionals of location. It means that the value of the functional at a singular distribution must be identically to the mass point of this distribution. First, we explain the role of idempotence in the known axiomizations of location functionals. Then we derive the distribution of idempotent and sufficient statistics. In the special cases of parametric families of location we get the so-called power-n-distributions. Power-n-distributions again are distributions with a parameter of location and can be derived from every location family for which the density is constrained. Additionally we show that the completeness of the populations family insures the completeness of the family of power-n-distributions. And at last, we give a further, now very easy proof that the normal distribution is the only one for which a idempotent, sufficient and unbiased estimator attains the Cramer-Rao-lower bound. --

Klein, Ingo

English

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Klein, Ingo
Working Paper
On Idempotent Estimators of Location
Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für
Statistik und Ökonometrie, No. 23/1998
Provided in cooperation with:
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Suggested citation: Klein, Ingo (1998) : On Idempotent Estimators of Location,
Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Statistik
und Ökonometrie, No. 23/1998, http://hdl.handle.net/10419/29596Friedrich-Alexander-Universit at Erlangen-N urnberg
Wirtschafts-und Sozialwissenschaftliche Fakult at
Diskussionspapier
23 / 1998
On Idempotent Estimators of Location
Ingo Klein
Lehrstuhl f ur Statistik und  Okonometrie
Lehrstuhl f ur Statistik und empirische Wirtschaftsforschung
Lange Gasse 20  D-90403 N urnbergIDEMPOTENT ESTIMATORS OF LOCATION
Ingo Klein
Lehrstuhl f ur Statistik und  Okonometrie
Universit at Erlangen{N urnberg
Lange Gasse 20
D{90403 N urnberg
Germany
E-mail: ingo.klein@wiso.uni-erlangen.de
Abstract
Idempotence is a well{known property of functionals of location. It means that
the value of the functional at a singular distribution must be identically to the
mass point of this distribution. First, we explain the role of idempotence in the
known axiomizations of location functionals. Then we derive the distribution of
idempotent and sucient statistics. In the special cases of parametric families
of location we get the so{called power{n{distributions. Power{n{distributions
again are distributions with a parameter of location and can be derived from
every location family for which the density is constrained. Additionally we show
that the completeness of the populations family insures the completeness of
the family of power{n{distributions. And at last, we give a further, now very
easy proof that the normal distribution is the only one for which a idempotent,
sucient and unbiased estimator attains the Cram er{Rao{lower bound.
21 Introduction
Idempotence is a well{known property of functionals of location. It means that the value
of the functional at a singular distribution must be identically to the mass point of this
distribution.
First, we explain the role of idempotence in the known axiomizations of location functio-
nals. Then we derive the distribution of idempotent and sucient statistics. In the special
cases of parametric families of location we get the so{called power{n{distributions. Power{
n{distributions again are distributions with a parameter of location and can be derived
from every location family for which the density is constrained. Additionally we show
that the completeness of the populations family insures the completeness of the family of
power{n{distributions. And at last, we give a further, now very easy proof that the nor-
mal distribution is the only one for which a idempotent, sucient and unbiased estimator
attains the Cram er{Rao{lower bound.
2 Statistical measures of location
2.1 Statistical functionals
Let D be the set all univariate statistical distribution functions and F 2 D. Furthermore,
let X1;:::;Xn be a sample from the population with distribution function F and Tn =
Tn(X1;:::;Xn) be a statistic. If Tn can be written as Tn) = T(Fn), where T does not
depend on n and Fn is the empirical distribution function of X1;:::;Xn, then T will be
called a statistical functional (see Fernholz (1983), p. 5). As domain of t will be considered
a set of distribution functions F that contains the empirical distribution functions Fn for
all n  1 and the population distribution function F. Instead of T(F) we write T(X) if
X is distributed according to F.
32.2 Idempotence
Let F contain the singular distribution x with mass point x 2 R what means x(y) =
I[x;1)(y), y 2 R with indicator function
IA(y) =
(
1 for y 2 A
0 for y = 2 A
for A  R. T will be called idempotent if T(x) = x.
Instead of idempotence Fleming & Wallace (1986) speak about "reexivity" and Acz el
(1990) about "agreement". Reexivity or agreement are one of the characterizing proper-
ties of so{called merging functions that aggregate dierent ratio-scaled measurements.
Idempotence can also be used for the characterization of measures of location. If all the
measurements of a variable have the same value x a location measure should have the
value x, too (see Klein (1984), p. 136).
2.3 Idempotence and the axiomatization of location
There are several systems of conditions (axioms) to characterize the specic properties of a
statistical functional of location (see f.e. Bickel & Lehmann (1975), Oja (1981), Dabrowska
(1985)). All these approaches are very similar.
The axiomatization of Bickel & Lehmann starts with two special relations on D: The rst
binary relation is the well{known stochastical ordering . If F;G 2 D F will be called
stochastically not larger than G (shortly F  G) if F(x)  G(x) for all x 2 R.
The second relation R  D  D also is binary. Let X be the random variable with
distributon functions F. We say (F;G) 2 R if G is the distribution function of the random
variable  X.
Due to Bickel and Lehmann T : F  ! R is a measure of location, if
(1) F;G 2 F and F  G =) T(F)  T(G)
(2) F;G 2 F and (F;G) 2 R =) T(F) =  T(G)
(3) F;F  g
 1 2 F for g(x) = a + bx, a;b 2 R =) T(F  g
 1) = g(T(F))
4The third property will be called "equivariance". Bickel and Lehmann (1975), p. 1047
show the independence of these axioms.
For symmetric distributions the symmetry point determines the value of every measure
of location (see also Bickel & Lehmann (1975), p. 1047, Theorem 1.1). Because singular
distributions are symmetric with the mass point as symmetry point location functionals
have to be idempotent (see Bickel & Lehmann (1975), p. 1047, Theorem 1.2).
There a two alternative ways to show the idempotence of functionals. First, we can prove
this property in a direct way. Second, we only have to show that a functional is a measure
of location in the sense of Bickel & Lehmann.
Obviously, a lot of popular distributional measures are idempotent. The most import-
ant is the arithmetic mean
R 1
0 F  1(p)dp. This also is a measure of location on the set
of distribution functions for which the integral exist. F  1(:) is the well{known quantile
function
F
 1(u) = inffx 2 RjF(x)  ug for u 2 [0;1]:
The quantiles F  1(p), 0 < p < 1 are idempotent but not neccessarily a measure of location
in the sense of Bickel & Lehmann. Further idempotent functionals are considered in Klein
(1999).
In this paper we will show the consequences of idempotence on such important statistical
concepts as suciency, completeness and the Rao{Cram er-inequality.
3 Idempotence and suciency
An estimator Tn will be called sucient for a parameter , if the conditional distribution
of X1;:::;Xn
fX1;:::;XnjTn
does not depend on  (see for example Casella & Berger (1990), p. 247).
Due to the factorization theorem in the case of i.i.d. random variables X1;:::;Xn, Tn is
sucient for a parameter of location  if and only if
n Y
i=1
fX1(xi;) = fTn(t(x1;:::;xn);)hn(x1;:::;xn)
5for all x1;:::;xn belonging to support D(X1)n if X1;:::;Xn and  2   R. hn(:) may
not depend on .
If Tn is idempotent the factorization theorem gives
fTn(x;) =
fX(x   )n
hn(x;:::;x)
for x 2 D(X1) and  2 . Instead of hn(x;x;:::;x) we write hn(x).
The following examples consider some parametric family of distributions for which the
sucient statistic is well{known.
Example 3.1 Let X1;:::;Xn be independent and identically bernoulli{distributed with
parameter p then it is easy to see by the factorization theorem that
Pn
i=1 Xi is a sucient
statistic. The 1{1-function Xn of
Pn
i=1 Xi also is sucent such that
fXn(x;p) =
pnx(1   p)n nx
hn(x)
for x 2 f0;1g and p 2 (0;1). For fXn to be a probability function it is
hn(x) =

n
nx

:
hn(:) depends on x.
Example 3.2 Let X1;:::;Xn be independent and identically distributed with the density
f(x;) =
1

e
 x
for x > 0 and  > 0.  is a scale parameter that can be estimated suciently by the
sample mean Xn with density
fXn(x;) =
1=ne nx=
hn(x)
:
It is well{known that
Pn
i=1 Xi has the density of a gamma{distribution
fPn
i=1 Xi(x;n;) =
 n
 (n)
x
n 1e
x=
6for x > 0. Hence, it is
fX(x;n;) =
 n
n (n)
(nx)
n 1e
 nx=
such that
hn(x) =
n (n)
(nx)n 1
is a function of x.
Example 3.3 Let X1;:::;Xn be independent and identically distributed with
fX(x;) =
1

I(0;(x)
for  > 0. The smallest order statistic X(1) is sucient and idempotent. The corresponding
density is
fX(x;)
n=h(x) =
1
nI(0;)(x)
n=hn(x):
It is well{known that the density is
fX(1)(x;) = nfX(x;)FX(x;)
n 1 = n
1

x

n 1
I(0;)(x)
such that hn(x) = 1=(nxn 1) depends on x.
Example 3.4 Let X1;:::;Xn be independent, identically distributed with density
fX(x   ) = e
 (x )I(;1)(x)
for  2 R. In this case , it is
fX(x   )
n=hn(x) = e
 n(x )I(;1)(x)=hn(x):
If we set hn(x) = n, we get an exponential distribution with parameters  und 1=n. The
smallest order statistic is an idempotent function of X1;:::;Xn that has this distribution
and is sucient for . Notice that hn(:) is independent of x.
74 Power{n{distribution
The last example is special because we consider a a parametric family of location. In this
case the sucient and idempotent statistic also has a distribution with location parameter.
This is due to the fact that hn(:) is constant.
This result can be generalized to every parametric family of location. Let X be distributed
with density fX(:   ) having support D(X). Then we dene
fn(x;) := fX(x   )
n=
Z
D(X)
f
n
X(y   )dy
for x 2 D(X) and  2 R. This only can be a well{dened density if
hn :=
Z
D(X)
f
n
X(y   )dy = E(f
n 1
X (X   )) < 1:
A sucient but not very restrictive condition is that fX(x;) < 1 for all x 2 R. Because
Z = X    is pivotal, (D(Z) and fZ do not depend on ), we get
E(f
n 1
X (X   )) = E(f
n 1
Z (Z))
what does not depend on . Therefore, fn is a density with location parameter . We
write fn(: ) and call fn "power{n{density of fX. If there is a sucient and idempotent
estimator of a location parameter this estimator is distributed according to the power{n{
distribution.
Example 4.1 Ferguson (1962) showed that under some conditions of regularity the on-
ly parametric family of location belonging to the exponential family distribution { and
therefore admitting a sucient statistic { is given by
fX(x   ) = jj
rr
 (r)
exp
 
 re
(x ) + r(x   )

:
Special cases are the normal, the exponential{ and as a limit case the uniform distribution.
This parametric exponential family of location can be derived from an  {distribution with
parameters r and  via a logarithmic transformation. Therefore, we call the members of
this exponential family of location loggamma{distributions. The parameters r,  und  are
related by
 = 1= log(r=):
8For loggamma{distributions, it is
fX(x   )
n = jj
n rnr
 (r)n exp
 
 nre
(x ) + nr(x   )

:
Hence, we have
h =
Z 1
 1
fX(x   )
ndx = jj
n 1n
nr (nr)
 (r)n
and the power{n{density is
f

n(x   ) = jj
(nr)nr
 (nr)
exp
 
 nre
(x ) + nr(x   )

:
This is a loggamma{distribution with parameters ,  and nr.
Power{n{distributions can be derived from every location familiy, not only from families
for which a sucient and idempotent estimator exists. In the following example we con-
sider the t{distribution for which the order statistics are minimal{sucient (see Cox &
Hinkley (1974), p. 34).
Example 4.2 Let X be distributed according to the t{distribution with parameter v
fX(x   ) =
 ((v + 1)=2)
 (v=2)
1
p
v

1
1 + x2=v
v=2+1
for x 2 R,  2 R and v > 0. Again, we have
hn =

 ((v + 1)=2)
 (v=2)
n 
1
p
v
n  (v=2)
 ((v + 1)=2)
p
v
with v = n(v + 1)   1 and
f

n(x   ) =
 ((v + 1)=2)
 (v=2)
1
p
v

1
1 + x2=v
(v+1)=2
:
If Y 
n again is the random variable belonging to this distribution then Y 
n obviously is
not t{distributed. Instead, the transformation Z
n =
p
v=vY 
n gives an t{distribution with
n(v + 1)   1 degrees of freedom. Hence, the distribution of Y 
n only is closely related to a
t{distribution.
95 Completeness of the power{n{distribution
A parametric family of location ffX(x   )j 2 Rg will be called complete if for any
function u(X) with E(u(X)) = 0 for all  2 R there must be P(g(X) = 0) = 1 for all
 2 R (see f.e. Casella & Berger (1990), p. 260). An estimator is complete if its parametric
family is complete.
Now it is easy to see that the completeness of the underlying parametric family of location
implies the completeness of the location family of power{n{distributions: Let X be a
complete random variable and Let Yn be the random variable corresponding to the density
fn. Then, we have
E(u(Yn)) = E

u(X1) 
fX(X)n 1
Kn

:
Now set v(x) = u(x)fX(x)n 1=hn for x 2 R. If E(v(X)) = 0 for all  2 R then P(v(X) =
0) = 1 for all  2 R because of the completeness of X. It is fX(x)n 1 > 0 for x 2 D(X)
Hence, we have P(u(X) = 0) = 1 for all  2 R. This shows the completeness of Yn for all
n 2 N.
If Tn is a sucient location estimator and especially idempotent then Tn is complete and
sucient (and therefore minimal sucient) if the populations distribution is complete.
Example 5.1 Due to Lehmann & Sche e ((1950), p. 314) the family of Cauchy{
distributions with location parameter  is complete. Therefore, the family of distributions
with density
fn(x   ) =
 (n)
 ((2n   1)=2)
1
p


1
1 + x2
n
:
is complete.
Example 5.2 Due to Patel et al. ((1976), p. 166) the exponential distributions with lo-
cation parameter are complete. As we have shown the rst order statistic has the corre-
sponding power{n{distribution that therefore has to be complete. Hence, the rst order
statistic is sucient and complete. Additionally, it is unbiased. Then, due the theorem of
Lehmann and Sche e the rst oder statistic is an "uniformly mininum variance unbiased
estimator" (UMVUE) for the location parameter.
106 Cram er{Rao{lower{bound and idempotence
It is well known that under some regularity conditions the variance of an unbiased su-
cient statistic Tn attains the Cram er{Rao{lower{bound if there are functions K(;n) and
t(x1;:::;xn) with
n X
i=1
@ lnfX(xi   )
@
= K(;n)(t(x1;:::;xn)   )
(see Mood, Graybill & Boes (1974), pp. 315). Hence, we have the unbiased and sucient
statistic Tn = t(X1;:::;Xn). If additionally Tn is idempotent, then it is
 n
@ lnfX(x   )
@x
= K(;n)(t(x;:::;x)   ):
This implies for the populations distribution fX
lnfX(x   ) =  
K(;n)
n
1
2
(x   )
2;
such that
fX(x   ) =
1
2
e
 1=2(x )2
is the density of the normal distribution. This is a very simple proof that under conditions
of regularity the normal distribution is the only one having an idempotent, unbiased and
sucient estimator for the parameter of location that attains the Cram er{Rao{lower{
bound.
7 Summary
Idempotence is a well{known property of functionals of location. It means that the value
of the functional at a singular distribution must be identically to the mass point of this
distribution.
First, we explained the role of idempotence in the known axiomizations of location func-
tionals. Then we derived the distribution of idempotent and sucient statistics. In the
special cases of parametric families of location we got the so{called power{n{distributions.
Power{n{distributions again are distributions with a parameter of location and can be
derived from every location family for which the density is constrained. Additionally we
11show that the completeness of the populations family insures the completeness of the
family of power{n{distributions. And at last, we gave a further, now very easy proof that
the normal distribution is the only one for which a idempotent, sucient and unbiased
estimator attains the Cram er{Rao{lower bound.
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On Idempotent Estimators of Location

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