LIMITING SECOND ORDER DISTRIBUTIONS FOR
FIRST ORDER FUNCTIONALS, WITH APPLICATION
to L- AND M-STATISTICS
by
De nnis D. Boos
Department of Statistics
North Carolina State University
Institute of Statistics Mimeo Series
December, 1977
# 1152
SUMMARY
LIMITING SECOND ORDER DISTRIBUTIONS FOR
FIRST ORDER FUNCTIONALS, WITH APPLICATION
TO L- AND M-STATISTICS
Let T(o) be a real-valued functional defined on the space of distribution
functions (d.f. 's) and T(Fn ) the statistic obtained by evaluating T(o) at F ,
n
the usual empirical d.f. for a sample Xl' •.. , Xn .
Under restrictions on the
X., existence of a nonzero (first) differential for T leads to the approxima:L
tion
In
T(F).
n
(T(F ) - T(F) - n -1 I:;V(X. )) -L-> 0 and to asymptotic normality of
n
:L
Existence of a second differential for T leads to the approximation
n(T(F) - T(F) - n-lI:;V(X.) - n-~Q(X.X.))~ 0, which then yields the limitn
:L
:LJ
ing distribution of n(T(F ) - T(F) - n-~(X. )).
n
:L
The variance of this latter
asymptotic distribution is used to compare estimators with identical first
order approximating sums n-lI:;V(x.).
:L
Application is provided by L- and M-
statistics with spe(;1.al attention given to a-trimmed means and "Hubers. II
1.
Introduction.
The study of statistical functions (real-valued func-
tionals defined on the space of distribution functions (d.f.
IS»~
was pioneered
by von Mises (1947) and continued by Kallianpur and Rao (1955), Fillippova
(1962), Kallianpur (1963), Gregory (1976), and Boos and Serfling (1977).
Most
of these efforts have concentrated on finding the limiting distribution of
n
k/2
(T(Fn ) - T(F», where T
.
~s
the functional of interest, Fn is the usual
empirical d.f. generated by a sample Xl' •.. , Xn from a distribution F, and k
refers to the !lorder!l of the ftmctional.
If T(o) has a nonvanishing generalized first derivative T(F;A), then k=l
and under
suitab~e
regularity conditions on the sequence of r.v. 's (X.} from
~
F, it follows that
Jii
(1.1)
where
n
(T(F ) - T(F)
n
_!n
L
. 1
~=
~ 0 ,
V(X.»
~
n-.= ,
Vex) = T(F; 0 -F) and 0 refers to the d.f. degenerate at x.
x
x
In other
contexts T(F;o -F) is called the influence curve of T (see Hampel (1974».
x
We will use ~ and ~> to refer respectively to convergence in probability
and convergence in distribution.
Clearly, (1.1) provides an approximation to
the error T(Fn ) - T(F) which yields asymptotic normality of T(Fn ) under the
restrictions of appropriate central limit theory.
Many common statistical
functions such as the mean, T(F) = SxdF, the pth qua.ntile, T(F) = F-l(P), and
the variance, T(F) = jtx-JXdFJ2dF, satisfy (1.1).
functionals is Andrews, et al. (1972).
A rich source of location
In addition to asymptotic normality,
Boos and Serfling (1977) provide a law of the iterated logarithm for first
order functionals.
If the first nonvanishing derivative is the second, designated
·2
T (F;A ,A ), then under suitable regularity conditions
l 2
n
(1.2)
l
\
n(T(F)-T(F)--2 L.
n
n
n
-
L
i=l j=l
-2-
Q(x.,x.»~O,n-.=,
~
J
where 2Q(x,y) = i-(F' 0 -F 0 -F).
'x ' y
d
~
«»
Under further restrictions we then have
2
«»
ti=ltj=l ~iZi ' where the ~i are solutions to an eigen-
value problem
associated with the "kernel" Q(x,y), and the Z. are i.i.d.
J.
standard normal variables. Goodness of fit statistics often satisfy (1.2)
(see Filippova (1962) and Gregory (1977) for examples).
Extension to the case
that the first nonvanishing derivative is the third or higher is straightforward in principle.
The intent of this present investigation is to study "second order"
properties of first order functionals.
That is, for functionals satisfying
(1.1) we seek to find the limiting distribution of
n(T(F ) - T(F) - ! \. n V(X.))
n
n LJ.= l
J.
Our approach will make use of the first and second Frechet-type differentials
2
(derivatives) T(F;6) and T (F;~1'~2)' In particular, the existence of such
differentials will imply, under suitable restrictions, that
(1.4 )
n(T(F ) - T(F)
n
n
1
n
n
[V(X.) - ---2 [
[Q(X.,X.)) -E-> 0 , n~ .
n 1=1
J.
n i=l j=l
J. J
_!
The limiting distribution of (1.3) is then available from existing results
regarding quadratic forms n -2 f'. n 1 \'. n lQ (X. ,X. ) .
'-1.= L. J =
J. J
In Section 2 we present the relevant definitions and theorems to implement
the approach.
In Section 3 we briefly examine the connections between our
results and those of C. R. Rao's on first and second order efficiency
(e.g., Rao (1963)). In Section 4 we will establish (1.4) for L- and M-estimators.
In particular, we compare the "Huber" M-estimators and the corresponding
a-trimmed means with the same asymptotic variance.
2.
Differentials.
set % of d.f.
IS.
Denote
Let T be a real-valued functional defined on a convex
by~(%)
the linear space generated by differences
-3-
H-G of members of
%,
~(%) = (A: A=c(H-G), H,G~, c real}.
is equipped with a norm \ \.\
DEFINITION.
Assume that ~(d)
I.
T has a differential at F€d with respect to the norm
1\·\ I
and the set ~F if there exists a quantity T(F;A) defined on A~(d) and linear
in A which satisfies
lim
\\ G-F!!-.o
(2.l)
T( G) - TU'l...::-r(F;G-F) = 0 . 0
II G-FII
G~F
T(F;A) is called the "differential."
Relaxed versions of (2.l) were introduced
in Boos and Serfling (l977) to make verification easier.
In particular, if
T(Fn ) - T(F) - T(F;Fn-F)
(2.2)
\\ Fn -Fr!
then we say that T has a weak stochastic differential at F with respect to
\\.\\ and the sequence (Xi} which generates F .
n
The vRlue of (2.2) can be
seen from the following lemma whose proof is trivial.
Let
~(T,F)
= EF(T(F;OX- F )}
and ~2(T,F) = Var(T(F;OX-F)}.
LEMMA.
l.
Let (Xi} be
with distribution F.
~ respect to
I1·\ I
~
sequence of r. V.
Suppose that T has
and (Xi}'
~
rS
(not necessarily independent)
weak stochastic differential at F
Suppose further that
op (l) , n-tCD .
Then
(2.4 )
~ (T(Fn ) - T(F) - T(F;Fn -F)) ~ 0 , n~ .
Moreover, if the Xi's are independent and
-4-
~(T,F)
2
= 0 and 0 < a (T,F) <
=, then
C
d
2
~n (T(Fn ) - T(F)) ~ N(O,cr (T,F)), n~.
EXAMPLE.
2
cr .
JD.
Iet T(F) = 1/SXdF(X).
Suppose F has mean }.I.
Then T(F;6) = -SXd6/}.I.2 , T(F;Fn-F) =
(T(F) - T(F))
n
=,/ii (x-l_~-l)
-4
f
0
° and variance
n-l~~1(}.I.-Xi)/~2,
N(O,cr2/}.I.4).
Analogous to the definition of the (first) differential, we have
DEFINITION.
I1·\ I,
T has a second differential at F€%with respect to the norm
the set ~F' and the functional T(F;6), if there exists a quantity
2
T (F;!lP!l2) defined on
(2.6)
lim
II G-FI\....o
(!l1'~)~(J")~(%)
and bilinear in
T(G) - T(F) - T(F;G-F) -
(!ll'~)
1 2
2T
(F;G-F,G-F)
(I IG-FI I )2
=0
.
which satisfies
o
G€~F
T2(F;!l1'~) is called the "second differential." Note that we do not assume
anything about T(F;6) (T(F;!l) =
(2.1) or (2.2) to hold.
° is
allowed), though we implicitly expect
The analogous definition of a weak stochastic second
differential follows for G=W .
Ll
A simple leIIDIJa will show how to find the candidate for ;(F; !ll'~) in most
problems.
Iet DGT(F) be the usual right-hand derivative of T(F+t(G-F)) with
respect to t evaluated at t=O.
In Boos ((1977a), Iemma 3.1) it was shown that
if T has a (first) differential at F with respect to
F =
t
F+t(G-F)€~F
T(F;G-F).
I1'\ I
and ~F' and
for all sufficiently small t, then DGT(F) exists and equals
The following lemma provides the analogous result for second
Let D~T(F) be the second right-hand derivative of T(Ft )
differentials.
evaluated at t=O.
LEMMA 2.
respect to
small t.
Suppose that T has
II ·11
~
first and second differential at F with
and ~F'
Iet G€J" be given such that Ft S.6F for all sufficiently
2
Suppose that D~T(F) exists.
Then D~T(F) = T (F;G-F,G-F).
-5-
PROOF.
for fixed G.
Note that Ft-F = t(G-F), so that 11Ft-F! 1 = tl IG-FI I~ as t~
For small t T(F;Ft-F) = tT(F;G-F) = tDGT(F) by Lemma
3.l of Boos (l977a).
The definition of the second differential,
can then be rewritten as
By
Taylor expansion of T(Ft) about t=O, we have
(2.8 )
Combining (2.7) and (2.8), we get D~T(F) = ~(F;G-F,G-F).
EXAMPLE.
0
Let T(F) = l/SXdF as in the previous example.
Then
and
Setting t=O, we find DGT(F) = -SXd(G-F)/~2 and D~T(F) = 2[SXd(G-F)]2/~3
For a sequence [X.} of i.i.d. r.v.
o< SlxldF < =,
1
IS
with distribution F such that
it is not difficult to show that
T(F;~) = SXd~/~2 and
T2(F;Al'~2) = 2SXdAlSXd~/~3 are weak stochastic first and second differentials
with respect to 11·11= and (Xi}'
-6-
The usefulness of T(F;~) in obtaining asymptotic normality of T(F ) for
n
first order functionals stems from the representation of T(F;F -F) as an
n
average n-l[i~lV(Xi)' Likewise, the usefulness of ~(F; ~l ,t. 2 ) stems from the
2
representation of T (F;Fn -F,Fn -F) as a simple quadratic form
n
-2r~=1~j=12Q(Xi,Xj)'
n \" n
Both representations follow from linearity and the
simple form of the empirical d.f. Fn = n
-I} n
~=lOX.:
1.
1 n
1 n
= T(F;- [ (OX -F)) =
[T(F;o -F)
n i=l
i
n i=l
Xi
1
= n2
1
~(F;F n -F,Fn -F) = T (F;-
n
L (0
n i=l
n ~
L
r(F;o
1
=-
n i=l
1
n
1
n
n
LV(X,) ,
. 1
1.
1.=
1
rn
-F), (0 -F))
Xi
n j=l Xi
Xi
-F
, n-1 .Ln1 (0 X. -F))
J=
J
n ~
.r - (F;oX.1. -F,oX.J -F)
= 2" .I.
n 1.= l J=l
n
= -2 L L 2Q(X. ,X.) ,
1. J
n i=l j=l
2
where for simplicity we have set T(F;o x -F) = Vex) and T (F;o x -F~6 y -F) = 2Q(x,y).
The asymptotic properties of such quadratic forms have been investigated
by numerous authors including Filippova (1962) and Gregory (1977).
gives a general result for contiguous sequences.
for the simple i.i.d. situation.
The latter
We reformulate his result
Let Q(x,y) be a symmetric, nonzero kernel on
R such that
2
-7-
2
SQ (x,y)dF(X)dF(y) < ~
and
(2.10)
SQ(o,y)dF(y) = 0
a.e. [F]
Let t~k,k ~ o} denote the finite or infinite collection
for a fixed d.f. F.
of eigenvalues of Q which satisfy SQ(x,y)fk(y)dF(y) = ~kfk(x)
Sfk(X)fj(X)dF(X) = 0
if
k~j,
and
Sf~(X)dF(X)
= 1.
Let f
O
The next lemma follows from Theorem 2.3 of Gregory (1977).
LEMMA 3·
F.
~
Suppose that 'Ek=l~k <
1
n
(2.11)
where
~
Let (Xi} be
Zl,Z2' '"
n
~.
correspond to
~o=O.
sequence of independent r. v. 's with distribution
Then
n
d
L. L.Q(X.,X.)
i=l j=l
~ J
~
a.e. [F] ,
-;>
~
[~kZ~,
k=l
n.-.c=,
i.i.d. standard normal variables.
In light of (2.9) and (2.10) and the assumed independence of the X. IS,
the mean and variance of n
-1
nVarFQ(X,x) + 2(1-n
-1
-1
2
n
n
~=ltj=lQ(Xi,Xj) are given by EFQ(X,X) and
)EFQ (xl
~
,x2 ).
Let L (R ,F) be the function space consisting of all real-valued functions
2 l
g
2
such that Sg dF <~.
The eigenvalues defined above pertain to the Fredholm
operator Ag(x) = SQ(x,y)g(y)dF(y) which maps L (R ,F) into itself.
2 1
If Q(x,y)
is symmetric and (2.9) holds, then these operators are called Hilbert-Schmid
operators.
The theory of such operators is well-developed (see [22], Chs. 2
and 3, for a survey of results) and yields the representation
<Xl
Q(x,y) = tk=l~kfk(x)fk(Y)'
SQ2(x,y)dF(x)dF(y)
=L:=l~~'
This latter representation leads to
and if further,
-8-
L:=ll~kl
<
~
, then
Clll
SQ(x,X)dF(X) = Lk=l"'k'
Of special interest are "degenerate" kernels having
the form Q(x,y) = Lk~lgk(x)~(y) for functions gk,~€L2(Rl,F) ,k=l,t.
The
operators defined by such kernels have only a finite number of eigenfunctions
and eigenvalues.
In Section 4 we will show that the kernels associated with
certain M-estimators have this simple form.
Combining Lemma 3 with our differential theory leads to the following
theorem.
~(Xi}
THEOREM 1.
with distribution F.
with respect
be
~
sequence of r. v. 's (not necessarily independent)
Suppose that T has
1£ 11·1 \, (Xi}' and
~
T(F;~).
weak stochastic second differential
Suppose further that (2.3) holds.
Then
(2.12)
tT2 (F;Fn -F,Fn -F))
n(T(F ) - T(F) - T(F;F -F) n
n
if -the
X. are independent
and Q(x,y) =
Moreover, - ~ -
-E-> 0 , n~·
t~(F;6 x -F,o y -F)
Clll
is symmetric and satisfies (2.9) an': (2.10) andIk=l"'k < Clll, then
d
2
n(T(F ) - T(F) - T(F;F -F)) ---> [AkZ k ' n~ ,
k=l
n
n
where Zl,Z2' ...
~
Clll
i.i.d. standard normal variables.
PROOF. (2.12) follows trivially from (2.3) and the definition of weak
stochastic second differential.
REMARKS.
(i) If T(F;Fn -F)
Then (2.13) follows from (2.12) and Lemma 3· 0
= 0,
then (2.13) just expresses the usual
convergence of second order functionals.
preted as an "influence curve" for T.
In this case Q(x,y) might be inter-
(U)
If (2.4) and (2.5) hold, then
(2.13) expresses the rate at which T(Fn ) - T(F) is approximated by T(F;Fn -F).
We see two potential applications of this latter case.
For estimators which
are first order efficient as defined by Rao (1963), we suggest
-9-
2EFQ2
= 2SQ2(x,y)dF(X)dF(y)
smaller the better).
as a measure of second order efficiency (the
We compare this idea with Rao' s definition of second
order efficiency in the next section.
For two estimators (or test statistics)
whose approximating sums T(F;F -F) are equal, though neither is first order
n
2
efficient - as in the case of many robust estimators, we suggest 2E Q as one
F
cri~ion to use in choosing between the two.
In Section 4 we will compare Land M-estimators on this basis.
Continuing our previous example, we find that ~ (F; ~l' ~)
EXAMPLE.
2SXd~lSXd~/~3
yields Q(x,y)
= (x~)(y_~)/~3,
and SQ(x,y)f(y)dF(y)
is satisfied by only functions of the form f(x)
get f(x)
= (x~)/cr
Note that 2E Q2
F
3.
and A =
cr2/~3.
= c(x-~).
=
= Af(x)
Normalizing, we
Hence
= 2cr4-6
~
= Var (cr232
f.l. - Z ).
Second Order Efficiency.
In a series of papers in the early sixties
(Rao (1960), (1961), (1962), (1963», c. R. Rao developed a theory of asymptotic
efficiency which is intimately connected with the likelihood function and
Fisher information.
More recent papers on the subject are Ghosh and
Subramanyan (1974) and Efron (1975).
(a)
Guiding themes in Rao's development were:
flEfficient fl estimators should in some sense summarize data
without loss of information.
(b)
Superefficiency should not oceur.
(c)
There should be a way to choose between efficient estimators.
Rather than attempt to assess Rao's contribution, we will give the essential
definitions and show some relationships between his theory and that of Section 2.
-10-
Let L(X_;S) be the likehood function of a sample X
-
=
(Xl' ..., X ), each X.
having d.f. Fe and density f(x;e), where e is real-valued.
L(X;S)
-
Tn
n
= n._lf(X.;e).
ll
= Tn(X i ,
~
and
l
For independent Xi'
accord with (a) and (b) above, Rao (1961) defines
... , X ) to be "first order efficient" for estimation of e if
n
In
where
In
n
(~
n
jL
oe
log L(X;S) - ~ - ~(T -e))
Jri
n
p ~ 0, n~ ,
are constants possibly depending on e·
~
If the X. are independent and Fisher information is finite,
l
J(e)
= EF
e
(%e log f(x;e)]2 <
~ , and ~ =
° , ~ = J(e)
, then (3.1) leads
to the often-used asymptotic variance criterion of efficiency
r.:
d
",n (Tn -e) -~ N(O,
1
Jill)
,
n-+= .
Of course, (3.1) is stronger than (3.2) and eliminates the usual superefficient
pathologies.
The question posed by (c) seems to have originated in a multinomial
model where a number of methods produced estimators satisfying (3.1).
These
methods included maximum likelihood and a number of minimum distance methods.
In a motivating discussion, Rao ((1961), p. 538) first suggested the asymptotic
variance of
n(~ ~ log L(X'e)
n oe
-'
as a measure of the rate of convergence in (3.2).
instead the asymptotic variance of
C3 .4)
n(~ o~e
log
L(~; e) -
-~
-11-
However, he decided to use
The minimum asymptotic variance o~ (3.4), say E ,
2
minimized with respect to X.
is called the second order
.
e~~iciency o~
T.
n
The motivation
seems to be that under certain reguJ.arity conditions E
2
ni &D.d niT are the
respectively.
i~ormation
n
re~ers
this measure
= limn...(ni-niT ),
where
contained in the sample and in the statistic
Thus E is not so much a measure
2
the estimator T , but
~or
o~
the rate
o~
convergence
instead to some intrinsic property
.
o~
T
n
o~
as a
substitute ~or the whole sample (see Rao (1963), p. 200).
The ~ollowing theorem shows how the asymptotic variance o~ (3.3) relates to
E in the case a
2
y
n
= (~I I
Let U (!;8)
n
-e) + C(T _S)2 , C e (--,at) .
THEOREM 2.
F S'
= 0, a = J(8).
= Cn~(8)] -10/08
log L(!;8) and
n
Let [Xi} ~ ~ sequence o~ independent r.v. 's with distribution
Suppose ~ J( 8) <
\lit
~ E
F
{ala 8
log ~(X; 8n
=0
and
8
(3.6)
,:n....
-Then
'n( y n -Un(X;
- 8»
fl/u.
....I'-..;.
0 , n-- ,
and
Note that
(3.5) and (3.6) are conclusions of the form (2.4) and (2.12).
Here Un(K;e) plays the role of T(Fe;Fn-F ).
e
and (2.10) and ~k:1Ak <
If Q(x,y) is symmetric and satisfies
=, then (3.6) and Lemma 3 tell us that the asymp-
totic variance of n(Tn-e-Un(K;e»
is Var
(~:lAkZ~) = 2Lk:1A~ = 2E Q2.
wise, if the same assumptions apply to the kernel
-12-
F
e
Like-
Q*(X,y) = Q(X,y) + C(.fe log f(X;e»)(te log f(y;e»,
then (3.8) and Lemma 3 tell us that the asymptotic variance of n(Y -U (x;.e»
n n*2
is 2E FeQ
.
*2
and E2 = nuncE F eQ
Thus E2 is given by
(SQ(X,y)fe-10g
f(X;6)~.lOg f(Y;6)dFe6QdFe(y»)2]
[J1(6)]2
PROOF OF TBEOREM 2.
-
n
_.1.
The basic assumptions and (3.5) imply that
and Jnun(!;e) = n 2!i=1[ J(e)]
to N(O,(J(e)]-l).
-1
In(Tn -e)
eac~ converge in distribution
a/oe log f(xi;e)
Then
In
(Y -U (X;s»
n n-
= In (T -S-U (x;e»
'"
n
n+
in C(Tn _S)2
•
The first term on the right-hand side of (3.9) ~ 0 by (3.5), and the second
term ~ 0 by the asymptotic normality of T.
n
For (3.8) it is enough to show
(3.10 )
Factoring (3.10), we have
In(T -S + U (X;S»· lil(T -e-u (X;9».
'"
n
n'"
n
n-
The first factor is 0p (1) by the asymptotic normality of ~(Tn -e) and
I~U (X;S).
'"
n-
The second factor is 0 (1) by (3.5). 0
p
.-13-
EXAMPLE.
2
= [ SxdG].
Let T(G)
the standard normal.
Then T(F )
e
Suppose that Fe(x)
= e,
J(e)
= 1/4e
(x-Je)/8je, Un(~)e) = (8je n)-l~:l(XI-Je), Q(x,y)
Q* (x,y)
= Q(x,y)(l +
Thus 2E F Q2
CJ(e)].
e
=2
= ~(x,je),
where ~ is
for efO, %e log f(x;e)
=
and E2
=
(x-ie)(y-Je), and
= minc
EF Q~
e
= 0.
L- and M-Statistics.
4.
4.1
Introduction.
Let F-l(t)
= inf(x:F(x) ~
t}.
Define the L-functional
1
(4.1)
TL(F)
= S F-l(t)J(t)dt
°
where J(t) is a given "score" function.
In Boos (1977b) conditions are given
for T to have a differential
L
(4.2 )
and for
-
where
d
2
In(T (F ) - TL(F)) ---~ N(O,cr (J,F)), n~ ,
(4.3)
L
2
cr (J,F)
n
= EFCS(I(X ~
t) - F(t))J(F(t))dt]2.
The defining equation (4.1)
allows for both location functionals and scale functionals.
J(t)
= I(a
J(t)
= t-l/2
~
t
~
For example,
l-a)/(1-2a) yields a form of the a-trimmed mean, and
yields a form of Gini's mean difference.
The location M-functional T is defined to be the solution of
M
(4.4 )
~F(C) = SV(X-C)dF(X) =
-14-
°,
~
where
is real-valued and often skew-symmetric about O.
If (4.4) has more
than one solution, then some additional rule must be given to define T
M
uniquely.
In Boos and Serfling (1977) conditions
are given for
~
to have
a strong stochastic quasi-differential (another variant of the first differential) given by
SHX-TM(F) )d6(x)
-",' (TM(F))
and for
(4.6 )
where
applies to simple location problems (scale known).
~(X)
can be replaced by ~(x/&), where & is some estimate of scale.
When F and
V are suitably restricted, Jaeckel (1971) showed an interesting
relationship between L- and M-functionals:
cr
2
If scale is unknown, then
(J,F).= cr 2 (~,F).
approximating sums
setting J
= ~' (F-l(t))
The differential approach shows even more:
TL(F;Fn-F) and TM(F;Fn-F) are equal.
and
J~ (TM(Fn ) - TM(F) - TM(F;Fn -F)) ~ 0 , n~,
-15-
the first order
More specifically,
Lemma 1 and (4.2) and (4.5) lead to
(4.8)
yields
where
1
T (F ;F -F)
L
n
= n-
n
~ (-jtI(X.st)-F(t)]J(F(t))dt)
. 1
~
~=
and
(4.10)
Let F be symmetric about O,continuous, and strictly increasing.
skew-symmetric
-A~(O) =
about 0, i.e. *(-x)
S*' (x)dF(X) = 1.
Set J(t)
= -*(x),
Let * be
and differentiable a.e. [F] with
= *' (F-l(t)).
Then TM(F)
= TL(F) = 0,
n
(4.10) becomes n- l .tl*(X.), J(F(t))= *' (t), and (4.9) becomes
~-
~
n-~i~l(-jtI(Xi~t)-F(t)]*' (t)dt).
Integration by parts yields term by term
equality between (4.9) and (4.10),
= V(X.).
~
4.2.
Second order theorems forL- andM-statistics.
compare the estimators T (F ) and
L
distributions.
n
~(F ),
~
n
In order to further
we need to look at their second order
Theorem 3 establishes the existence of a second differential for
T under mild restrictions on F and J. A corollary then gives the limiting
L
distribution of n(TL(Fn ) - TL(F) - TL(F;Fn-F)). Theorem 4 provides directly the
limiting distribution of n(TM(Fn ) - TM(F) - TM(F;Fn-F)) without proving the
existence of a second differential.
Let.r J
of G.
=
(F:! SF-l( t )J(t )dt I< =} and.&- F
For a bounded positive function
q
-16-
=
(G: SG
C
SF} , where SG
= support
on (0,1) we define the norm
(4.11)
where [x ,x J is the smallest interval (possibly infinite) containing SF'
l 2
Suppose that J is Lipschitz of order 1 ~ [O,lJ and has
l
bounded derivative J' at all but a finite set B of F- measure zero. If
THEOREM 3.
~
(4.12)
then TL has ~ second differential at Fe:TJ with respect to
given
11'11 q(F)
and JtF
~
(4.13 )
PROOF.
Since (4,13) is clearly bilinear in (6 ,6 ), we need to show
1 2
(Inside the integrals we suppress the fact that F and G depend on x.)
In Boos
(1977b) it is shown that T(G) - T(F) = S(K(F)-K(G))dx, where K(y) = J;J(U)dU.
= J(y).
Note that K' (y)
Thus
S-
IT(G) - T(F) -
=
I
S
[K(G) -
ts - (G-F)2J'(F) dx l
(G-F)J(F)dx -
K(F) - (G-F)J(F) -
Rl:B
(4.15 )
s
IIG-Fll~(F)RJ_Bq2(F(x))W(x)dx
1
-17-
,
~(G-F)2J'
(F)]dxl
where
w(X) = K(G(X))-K(F(X))-rG(X)-Fe X)]J(F(X));-KG(X)-F(X))2J' (F(X)) , G(x)fF(X) ,
[G(x) - F(X)]
= 0 , G(x) = F(x) .
Note that for each xe~-B we have lim! !G-F!!q(F)~oW(X)
expansions (see Cartan (1971), Theorem 5.6.3).
= 0 by simple Taylor
To show that (4.14) follows from
(4.15), it is only necessary to show that lim and
S can be
interchanged in (4.15).
This interchange is allowed via dominated convergence by (4.12) and a bound on
W(X):
for G(x)
lw(X)!
f
F(X)
= !K(G(x) )-K(F(x) )-[G(x)-F(x)]J(F{x) )-~[G(X)-F(x)]2J' (F(x) )\/[G(X)-F(X)]2
G(x)
= \ \ [J(u)-J(F(X))] du- ~[G(X)-F(x)J2J' (F(X))\/[G(X)-F(x)]2
Ftx)
~
G(x)
2
I \ Alu-F(x)l du l/[G(x)-F(x)J +
Ftx)
REMARKS.
(i) If
J
~IIJ' 11=
is trimmed in neighborhoods of 0 and 1, then (4.12) is
unnecessary and the conclusion of Theorem 3 holds for II 'llq(F) replaced by
(ii) A popular choice of q is
(4.16 )
q(x)
1..-6
= [x(1-x)J2
-18-
1
,0 < 5 < 2 .
In Boos (1977b) it is shown that
(4.17)
for q in a certain class (including (4.16»
sequence from F (F arbitrary).
(iii)
and for F generated by an i.i.d
n
We reemphasize the lack of restrictions
on F, such as continuity, in the statement of Theorem 3.
Remark (ii) and Theorem 1 provide an application of Theorem 3 in the following corollary, whose proof should be transparent.
COROLLARY.
(4.16).
Suppose that the conditions of Theorem 3 hold with q given
Let (Xi} be ~ sequence of i.i.d. r.v. 's with distribution F.
Moreover, if QL(x,y) = -tStI(~t)-F(t)][I(~t)-F(t)]J' (F(t»dt
~
Then
has eigenvalues
(4.19)
where Zl,Z2' .... are i.i.d. standard normal variables.
EXAMPLES.
Lipschitz.
(i)
The trimmed mean, J(u) = I(~USl-a)/(1-2a), is not
However, a smoothly trimmed version can be defined as follows. Fore >0 let
J(u) = 0 for uSa-e, let J linearly slope up to [e + (1-2a)]
-1
at u = a, let
J(u) = [e + (1-2a)]-1 for ~USl-a, and let J linearly slope back to zero at
u'= I-a + e and remain zero for U>l-a + e.
Then J is Lipschitz (of order
on [O,lJ and has a derivative except at a-e, ~, I-a, and l-~+e.
1)
Remark (i)
applies here, so that (4.12) is unnecessary and the other hypotheses of
Theorem 3 are satisfied for any F with unique quantiles a.t F-l(a-e), F-l(~),
-19-
F-
1
(l-a),
and F
-1
(l-~e).
Note that as €~, this functional approaches the
trimmed mean, and we anticipate the validity of Theorem 3 under relaxed conditions on J and strengthened conditions on F.
(ii)
Gini's mean difference,
J(u)
Turning to M-statistics, we establish directly the analogues of (4.18)
and (4.19).
The origin of the M-estimator T is left unspecified, subject
n
only to (4.24) and (4.25), in order to allow flexibility of definition.
See
Boos and Serfling (1977) and Collins (1976) for two different methods of
Choosing Tn'
Let
11'11 V
be the usual variation norm
n
Ilgllv
= lim
sup
a-t-=
LIg(x.
. 0
l=
l+
1) -
g(x l·
)1,
b-t+-=
the sup being taken over all partitions
a
= Xo <
Xl < ... < x
< xn+l
n
=b
of [a,bJ.
THEOREM 4.
X;(T o )
= O.
Let F and ~ be such that AF(To )
= 0,
lim Il~(x+b) - ~(x)l Iv
=0
b-+o
(4.21)
f:
0, and
Assume that ~' exists everywhere and is continuous.
satisfy
(4.20 )
A~(To)
lim
b-+o
I I~' (x+b)
- ~' (x)ll v
(4.22)
(4.23)
-20-
=0
Let ~ ~ ~'
1£1 (x) be
~
sequence of i.i.d. r.v. 's with distribution F.
~t
the sequence
Tn = Tn (XI' ... Xn ) satisfy
-
P(~F (Tn)=O) ~
n
(4.24)
1,
n~ ,
and
T ~T ,n~.
n
0
(4.25 )
Then
2
neT -T -T (F;F -F) - !T (F;F -F,F -F)) ~ 0, n~ ,
noM
n
n
n
M
(4.26)
where
(4.27)
(4.28)
where
Zl'Z2""
REMARKS.
and ~~' (0)
=0
~
(i)
i.i.d. standard normal variables.
If *(x) : -*(-x) and F is symmetric about 0, then To=O
(provided ~~tO) exists).
(ii) (4.22) and (4.23) are used to
show that certain sums involving * and *' satisfy the central limit theorem.
(iii) (4.20) and (4.21) are essential in our method of proof and arise from use
of the following lemma found in Boos (1977a).
-21-
LEMMA 4.
( -QI:l, 01:1 )
•
-Let -the
-
function h be continuous and -of bounded
variation on
- - - -......;..;~.;;;;....;"'"
Le t G and F be d. f.
PROOF OF THEOREM 2.
IS.
Then
By the mean value theorem
*
A (T ) - AF(T) = (T -T )A'F(T ) ,
F non 0
n
(4.3 0 )
n
where T* lies between T and T (and hence T* ~>
T , n-+=).
n
n o n
0
* f:
we can choose Nllarge enough so that P(A~(Tn)
choose N large enough so that P(A (Tn)
2
F
= 0)
n
Since A I (T )
F 0
0) > l-€Vn>Nl .
> l-€VrJ>N .
2
* f
arguments are all conditional on the event {A~(Tn)
O}
The following
n {A F
(Tn)
From
(4.30)
= O}
which
we have
We can express TM(F;Fn-F) in a congruent manner by adding and subtracting a
, remainder term,
Then
(4.3l)
T
n
-22-
0,
Likewise we can
n
has probability> l- €Vn>max(N ,N ).
l 2
f:
1
At this point we show that (4.31) is
(n-2 ).
p
0
The first term on the right-hand
side of (4.31) is bounded by IIFn -Fill
1~(x-Tn
) - *(x-T
)1 I /IA
' (T*)I which is
CD
o
F
vn
1
open -2) by (4.20) and (4.25) and the fact that I IFn-FI
I
CD
1
is 0p(n- 2 ).
1
remainder term R In is 0p (n -2) by the central limit theorem applied to
~n (T 0 ) = n-l~*(X.-T0 ) and the convergence A'F(T*M) ~ A'F(T0 ).
~
The next step is to express ~M2(F;Fn -F,Fn -F) in a suitable form:
=
(T
T)
n-
0
-A' (T*)
F
n
~'( T ) + R
2
non
where
and
Thus
- To
- TM(F;F
I Tn
n -F)
$1 A' 1(T* ) lipn (Tn )
F
-
-
2
tTM
(F;Fn -F,Fn -F)1
(T ) - (T -T ) p' (T ) I
non 0 n 0
,rD
n
-23-
+ R
3n
,
The
It is sufficient to show that each of these 4 terms is 0 (n- l ).
By the mean
p
value theorem
where T** lies between T and T. Since In(T -T ) is asymptotically normal
n
non 0
(Via (4.31) and the central limit theorem), we need to show
\P~(T;*)
op(n-~).
- Pn ' (T o )\ =
\p'nn
(T**)
By
Lemma 4
- ¢'no
(T )1 ~ II Fn -F\I II *' (x-T**)
n
=
f
(x-T0)\"
Iv ·
Condition (4.21) coupled with T**~
T0 yields the result. By similar arguments
n
l
it is easy to show that -2n
R_ and R
are 0p (n- ). However, it is not clear that
3n
R
is 0p(n-1). This will follow if In(A~(T:) - A~(TO» ~ O. Write
ln
1
The first factor ..E-> A~' (To) = 0 and the second factor is 0 p (n -2). 0
EXAMPLE.
W(x) = arctan x.
* is strictly increasing.
The solution Tn of A (c) = 0 is unique since
F
n
Conditions (4.20) and (4.21) are easy to verify,
and (4.25) follows in the i.i.d. situation from the results of Huber (1964).
Any symmetric F will satisfy the other requirements of Theorem 4.
The "degenerate" form of the kernel Q,M lends itself easily to calculation
of the associated eigenvalues.
symmetric about 0, and
~(x,y)
For the special case that *(x) = -*(-x), F is
S*' (x)dF(x)
= *(x)(l-*' (y».
= 1, the kernel reduces to
Since our theory requires Q,M to be symmetric, we use
QM =
C*(x)(l-*' (y» + *(y)(l-*' (x»J/2 in place of ~, noting that
n
n
(
)
n n", (
)
~i=l~j=l~ Xi'X j = ~i=l~j=lQ,M Xi'X j .
-24-
LEMMA 5.
Let
~!L1..
and that S~' (x)dF(X) = 1.
F be symmetric about O.
Suppose that
~(x)
= -~(-x)
Then the eigenvalues and eigenvectors of ~ ~ given
(4.32)
and
( ) -f IX
1i!.2.
1-1\1'_
_+
J2C l
where
c l = S*2(X)dF(X)
PROOF.
and
c
2
(x)
JZc 2
=
tW- _ (1-1)1' (x)) ,
(x) =
f
2
S(~' (x))2dF (x)
-
J2 c l
-
J2c 2
- 1 .
All eigenvectors must be of the form
a~(x)
+ b(l-~' (x)).
Normalizing yields the constants. 0
Under the conditions of Theorem 4 and Lemma 5, (4.28) becomes
(4.34)
Note that Var
The L-statistic kernel Q is much harder to handle. For F uniform (0,1),
L
De wet and Venter (1973) show that the eigenvalues associated with Q satisfy a
L
Sturm-Liouville-type equation, which simplifies calculations.
In general we
shall be satisfied with calculating the anticipated variance of the limiting
2
second order distribution, i.e., 2E FQL' where
2
EF~ =
i:-
CD
j
2
CD
::~ [min(F(s),F(t)) - F(s)F(i)] J' (F(s))J' (F(t))dsdt.
This last expression can be justified under the assumptions of Theorem 3 with q
given by (4.16) via a version of Fubini's theorem.
-25-
In the special situation that TL(F) = To and TL(F;Fn-F)
a:
TM(F;Fn-F),
results (4.18) and (4.26) allow calculation of the limiting distribution of
n(TL(Fn ) - Tn)'
Jaeckel (1971) has given the weaker result TL(Fn ) - Tn
THEOREM 5.
increasing.
Let the d.f. F be symmetric about 0, continuous, and strictly
Let Vex)
=
~
Suppose that (Xi} is
~
= Open -1).
Iv' (x)dF(X)
-v(-x),
= 1,
and set J(t)
= V' (F-l(t)).
sequence of independent r.v. 's with distribution F such
(4.18) and (4.26) hold.
Then
(4.35 )
Moreover, if
~
eigenvalues
t.. k
of Q,D
= Q,L
-
,..,
~
satisfy
CD
;'=It.. k
<
=, then
(4.36 )
where Zl,Z2""
PROOF.
TL(F;Fn-F)
~
i.i.d. standard normal variables.
The conditions on F and V iI1<:!ure TL(F)
= TM(F;Fn-F).
= To
and
Subtraction of (4.26) from (4.18) gives (4.35).
Application of Lemma 3 then gives (4.33)· 0
4.3.
vex)
Hubers~: ~-trimmed
= max(-k,min(k,x»,
means.
The "Huber" family of M-estimators,
and the ~-trimmed means, J(t)
are asymptotically equivalent to first order if F(-k)
= I(~
=~
~ t ~ 1-~)/(1-2~),
for any d.f. F
symmetric about 0 with f(-k)
= f(k) >
order asymptotic variances.
Although Theorems 3 and 4 do not apply directly
O.
We would like to compare their second
(since V' doesn't exist at -k and k and J is discontinuous at ~ and l-~), we
expect the results of these theorems to remain true for F sufficiently regular
in neighborhoods of -k and k.
as we want by
t* and !
In fact we can approximate V and J as closely
which do satisfy Theorems 3 and 4.
trimmed means mentioned previously provide
mean.
-26-
su~h
For example, the linearly
an approximation for the
~-tr~ed
If J is of bounded variation and if F has density f such that fW-l(t)J
is positive on the support of J, Q may be expressed as
L
Q (x,y) =_~
~I (
J-
XSF
-1
0
L
(t))-tJ[I(~F
-1
(t))-tJ. dJ(t)
f[F-l(t)J
For the a-trimmed mean (4.37) becomes
(I (
xs: 1
F (a))-a)(I(y s: F-1 (a))-a)
(4.38 )
[
f[F-l(a )J
_ (I(x s: F-1 (l-a))-(l-a))(I(y s: F-1 (l-ct))-(l-a))
f[F
-1
(I-a)]
Note that the kernel Q is symmetric and degenerate, allowing potentially
a
easy calculation of eigenvalues.
If f[F
-1
(a)J
= feF -1 (l-a)J
, the variance
of the limiting second order distribution is given by
2
2EFQ~ = 2SQ~(X,Y)dF(X)dF(Y) = -(1---2a-)-(-f~-F--':""1(-a-)J-)~2
Similarly, letting ~k(x)
we have Qk(x,y)
= max(-k"min(k,x))/(1-2a)
= [Wk(x)(l-~~(Y))
so that S~~(X)dF(X)
= 1,
+ ~k(Y)(1-~~(x))J/2 and
(4.40)
2
where we have written Ak(F) for the first order asymptotic variance E~k.
In Tables 1-4 we have computed (4.39) and (4.40) for a variety of
situations.
In almost all these situations the HUber outperforms the corres-
ponding a-trimmed mean.
However, in most real life problems, scale is un-
known, thus necessitating a scale invariant version of the M-functional T .
M
One such version is the solution TM,S of A.F(C/S(F»
-27-
= 0,
where S(F) is some
J
scale functional.
Boos (1977a) has given conditions for TM,S to have a strong
stochastic quasi-differential TM,S(F;~) when S(F;~) exists.
In particular,
if F is symmetric and S(F) = 1, then TM, S(F;~) is given by (4.5) and ~S(Fn )
-M,
has essentialJ¥ the same first order properties as TM(Fn ).
Although we have
not proven the existence of a second differential for
straightforward
calculation of
~
,S'
D~TM,S(F) allows us to anticipate the form of TM,S(F;~1'~2).
Under the additional assumption that S(F;5 -F) is an even function about 0 and
x
s~' (x)dF(X) = 1, we expect
(4.41)
where B = 1 + S~" (x)xdF(x).
This leads to the unsymmetric kernel
(4.42)
and second order asymptotic variance
(4.43)
where AT(F) and AS(F) are the first order asymptotic variances of TM,S(F n )
and S(F).
n
F
k
using
(F- l (3/4) - F- l (1-4»/d , a normalized interquantile range, where
F
is chosen for each F so that S(F) = 1. Now the a-trimmed mean appears
S(F)
d
In Tables 1-4 we have included (4.43) for ~~
=
very competitive.
In the middle values,
~lO ~
a
~
.25, the a-trimmed mean
tends to outperform the Huber scale version for the first three distributions.
This same pattern persisted in a number of other distributions not displayed.
-28-
However, for F = 0.75 ~ + .25~(x/10) (not displayed) the Huber scale version
was better than the a-trimmed mean for all values of k and a except k=l,
a:.234.
For the Cauchy (also not displayedbwith density f(x) = ~-1(1+x2)-1,
the a-trimmed mean wins at only 4 values of k and a.
Thus it appears that
for long tails one might prefer using the Huber scale version.
For the
double exponential the a-trimmed mean appears to trounce the Huber scale
version for k ;:.: 7, a:so .248.
However, when we tried using the mean absolute
deviation as a scale estimate, the results were reversed.
scale version easily won for all k and a computed.
That is, the Huber
This suggests that with
judicious choice of scale, the Huber scale version might perform better in a
variety of situations.
It would be interesting to try other scale estimates
or use a simultaneous estimation scheme such as Huber's Proposal 2 (see
Huber (1964), p. 96).
-29-
TABLE 1.
F =
~,
standard normal
.k
a-=F( -k)
~(F)
AS(F)
a--trimmed
mean
Huber
Huber -Scale
version
0.1
0.2
0·3
0.4
0.5
0.6
0·7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1·9
2.0
2.1
2.2
2·3
2.4
2·5
0.460
0.421
0.382
0.345
0·309
0.274
0.242
0.212
0.184
0.159
0.136
0.115
0.097
0.081
0.067
0.055
0.045
0.036
0.029
0.023
0.018
0.014
0.011
0.008
0.006
1.492
l..423
1.362
1.309
1.263
1.222
1.187
1.156
1.130
1.107
1.088
1.072
1.058
1.047
1.037
1.029
1.023
1.018
1.014
1.010
1.008
1.006
1.004
1.003
1.002
1.360
1·360
l..360
1.360
1.360
1.360
1.360
1.360
1.360
1.360
1.360
1.360
1.360
1.360
1.360
1.360
1·360
1.360
1·360
1·360
1·360
1.360
1.360
1.360
1.360
16.871
7·303
4.256
2.816
2.006
1.500
1.164
0.928
0.757
0.630
0.532
0.456
0.395
0.347
0·307
0.274
0.247
0.223
0.203
0.186
0.171
0.158
0.146
0.136
0.127
17·242
7·555
4.415
2·902
2.035
1.484
1.113
0.850
0.658
0.515
0.405
0·320
0.254
0.202
0.160
0.127
0.100
0.079
0.062
0.048
0.037
0.029
0.022
0.017
0.013
16.306
6.864
3·932
2·597
1.877
1.450
1.179
0·997
0.868
0·772
0.695
0.632
0.576
0·525
0.477
0.431
0.386
0.344
0·303
0.264
0.228
0.1940.164
0.136
0.112
-
-30-
TABLE 2.
F=
k
a=F( -k)
0.1
0.2
0·3
0.4
0.5
0.6
0.7
0.8
0·9
1.0
1.1
1.2
1·3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2·3
2.4
2.5
0.464
0.428
0·393
0·359
0.326
0.294
0.265
0.237
0.212
0.189
0.168
0.149
0.132
0.117
0.104
0.093
0.083
0.075
0.068
0.063
0.058
0.054
0.051
0.048
0.046
.9~+
.H (x/I0), contaminated normal
Ak(F)
AS(F)
1.811
1.737
1.672
1.618
1.571
1.532
1.501
1.476
1.457
1.443
1.435
1.432
1.433
1.439
1.448
1.462
1.479
1.499
1.523
1.550
1.579
1.611
1.646
1.683
1.723
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
1.464
-3l-
a-trimmed
mean
22.736
10.018
5·959
4.038
2·957
2.284
1.840
1.534
1.319
1.167
1.060
0.989
0.949
0.936
0·950
0·994
1.074
1.201
1.389
1.663
2.061
2.640
3.487
4.735
6.583
Huber
Huber -Scale
version
23·174
10·301
6.120
4.099
2.936
2.195
1.692
1.335
1.073
0.876
0.724
0.607
0.514
0.440
0.381
0.334
0.296
0.265
0.241
0.222
0.206
0.194
0.185
0.178
0.173
22.003
9·441
5.524
3·729
2.756
2.176
1.806
1.560
1.388
1.264
1.172
1.102
1.048
1.006
0·975
0·953
0.940
0·935
0·939
0.951
0·972
1.001
1.037
1.081
1.130
~BLE
3.
F = t , t distribution with 3 degrees of freedom
3
k.
a=F(-k)
Ak(F)
AS(F)
a-trilIlmed
mean
0.1
0.463
0.427
0·392
0·358
0.326
0.295
0.267
0.241
1.768
1.700
1.645
1.602
1.570
1.546
1.530
1.522
1.519
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
21·955
0.2
0·3
0.4
0·5
0.6
0·7
0.8
0·9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2·3
2.4
2·5
0.217
0.196
0.176
0.158
0.142
0.128
0.115
0.104
0.094
0.085
0.077
0.070
0.063
0.058
0.052
0.048
0.044
1.521
1·527
1.537
1.550
1.565
1.582
1.601
1.621
1.642
1.664
1.686
1.709
1.731
1.754
1.776
1.799
-32-
9·771
5·917
4.110
3·103
2.484
2.078
1.801
1.607
1.468
1.369
1.299
1.250
1.218
1.199
1.191
1.192
1.201
1.216
1.237
1.263
1.294
1·329
1·368
1.410
Huber
22·334
9.965
5.963
4.038
2.934
2.232
1.756
1.417
1.167
0.976
0.828
0.711
0.616
0.539
0.474
0.420
0·375
0·336
0·302
0.273
0.248
0.225
0.206
0.188
0.173
Huber-Scale
version
21.202
9·198
5.499
3·823
2·923
2·388
2.046
1.816
1.653
1.534
1.443
1.372
1·315
1.267
1.227
1.192
1.161
1.133
1.108
1.084
1.062
1.041
1.021
1.002
0.983
TABLE 4.
F
= D-EX,
k
a=F(-k)
0.1
0.2
0.3
0.4
0·5
0.6
0·7
0.8
0·9
1.0
1..1
1.2
1·3
1.4
1·5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2·3
2.4
2·5
0.452
0.409
0·370
0·335
0·303
0.274
0.248
0.225
0.203
0.184
0.166
0.151
0.136
0.123
0.112
0.101
0.091
0.083
0.075
0.068
0.061
0.055
0.050
0.045
0.041
double exponential with density rex)
= !e-!x!
Ak(F)
AS(F)
a-trimmed
mean
Huber
Huber-Scale
version
1.033
1.067
1.100
1.133
1.165
1.198
1.230
1.261
1.292
1·323
1·352
1·382
1.410 .
1.438
1.465
1.492
1.517
1.542
1.566
1·589
1.611
1.633
1.653
1.673
1.692
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
2.081
10.508
5·517
3·858
3·033
2.542
2.216
1.986
1.816
1.685
1·582
1.499
1.431
1·375
1·327
1.287
1.253
1.224
1.198
1.176
1.156
1.140
1.125
1.111
1.100
1.089
9·825
4.817
3·143
2·303
1.796
1.457
1.213
1.029
0.885
0·770
0.675
0.596
0.528
0.471
0.421
0·377
0·339
0·305
0.275
0.249
0.225
0.203
0.184
0.167
0.151
10·351
5·388
3·758
2·960
2.495
2.195
1.988
1.839
1.728
1.641
1.573
1.517
1.470
1.429
1·393
1·359
1·328
1.298
1.268
1.239
0.209
1.180
1.149
1.119
1.088
-33--
REFERENCES
[1]
Andrews, D. F. et al.
Univ. Press.-
[2]
Boos, D. D. (1977a). The functional approach in statistical theory
and robust inference. Unpublished dissertation, Florida State
University.
[3J
Boos, D. D. (1977b). A differential for L-statistics. FSU Statistics
Report 416 (ONR Technical Report No. 116), Florida State University.
[4]
Boos, D. D. and Serfling, R. J. (1977). On differentials, asymptotic
normality and almost sure behavior of statistical functions, with
application to M-statistics for location. FSU Statistics Report
415 (ONR Technical Report No. 115), Florida State University.
[5J
De wet, T. and venter, J. H. (1973). Asymptotic distributions for
quadratic forms with applications to tests of fit. Ann. Statist. 1
380-387.
[6J
Cartan , H.
[7J
Collins, J. R. (1976). Robust estimation of a location parameter
in the presence of asymmetry. Ann. Statist. ~ 68-85.
[8J
Efron, B. (1975). Defining the curvature of a statistical problem
(with applications to second order efficiency). Ann. Statist.
1189-1217.
[9]
Filippova, A. A. (1962). Mises theorem on the aSJ~ptotic behavior of
functionals of empirical distribution functions and statistical
applications. Theor. Probability~. 1 24-'57·
[10]
Ghosh, J. K. and Subramanyam, K. (1974). Second order efficiency of
maximum likelihood estimators. Sanko/a Ser. ~ 32. 325-358.
[11]
Gregory, G. G. (1976). Large sample theory of differentiable
statistical fUnctions. Unpublished manuscript.
[12]
Gregory, G. G. (1977). Large sample theory for U-statistics and tests
of fit. Ann. Statist. 2 110-123·
[13]
Hampel, F. R. (1974). The influence curve and its role in robust
estimation. J. Amer. Statist. Assoc. 69 383-393·
[14]
Huber, P. J. (1964). Robust estimation of a location parameter.
Ann. Math. Statist. 35 73-101.
[15J
Jaeckel, L. A. (1971). Robust estimates of location: symmetry and
asymmetric contamination. Ann. Math. Statist. 42 1020-1034.
(1971).
(1972) .
Robust Estimates of Location.
Differential Calculus.
Princeton
Hermann, Faris.
l
REFERENCES (Continued)
[17]
Rao, C. R. (1960). Apparent anomalies and irregularities of the method
of maximum likelihood. Proceedings of the 32nd Session of the Int.
Stat. Inst., Tokyo. Reprinted with discussion in Sankhya 24
73-102-.-
[18]
Rao, C. R. (1961). Asymptotic efficiency and limiting information.
Proc. Fourth Berkeley ~. Math. Statist. Prob. ! 531-546.
[19]
Rao, C. R. (1962). Efficient estimates and optimum inference procedure
in large samples, with discussion. J. Royal Statist. Soc. Ser.
~ 24 46-63.
[20]
Rao, C. R. (1963). Criteria of estimation in large samples.
Ser. ~ 25 189-206.
[21]
von Mises, R. (1947). On the asymptotic distributions of differentiable
statistical functions. Ann. Math. Statis. 18 309-348.
[22]
Zabreyko, P. P. et al. (1975). Integral Equations - a Reference Text.
Noordhoff International Publishing, Leyden.
-35-
Sankhya