A CONSISTENT ESTIMATE OF A
NONPARAMETRIC SCALE PARAMETER
by
Gerald L. Sievers
Department of Mathematics
Western Nfichigan University
Department of Statistics
University' of North Carolina
SUMMARY
A consistent estimate is proposed for the scale parameter
the model Y.1
=1
l.l. +1
e., 1 ~ i
e
~
If 2
in
n, where the l.l.1 are \IDknown location
parameters and the e i are independent, identically distributed random
errors with density f\IDction f. This parameter arises in the
variance fonnula for rank estimates of location.
The proposed estimate
is based on differences of residuals Yi-D i , where Di is an estimate
of l.li' When the l.li follow the structure of the general linear model,
the estimate is shown to be consistent under the usual asstunptions on
the design matrix.
The estimate does not require the s)'JJD1letry of
the density f.
Key Words:
Nonparametric, scale parameter, rank estimates, linear model
This research was supported by the Office of Naval Research Contract No.
NOOOI4-78-C-0637 (NR 042-407). Distribution of this document is \IDlimited.
Reproduction in whole or in part is permitted for any purpose of the
United States Government.
1.
1
~
INTRODUCTION
Let n observations Y1 'Y Z, ... ,Yn follow the model Y.1= 1
~.+ e.
1
i ~ n, where the 1..1.1 are unknown location parameters and the e.1 are
independent, identically distributed random variables with density f.
Of particular interest here is the case where the
~i
follow the
structure of a linear model.
Consider the problem of estimating the scale parameter
y
= !fZ(x)dx. This parameter arises in various statistical procedures based
on ranks.
For instance, in one-sample, two-sample and linear model problems
it appears in the variances of rank estimates when
Wilcoxon scores are used.
,
In linear model problems, Hettmansperger and MCKean (1976), (1977) proposed
test statistics that are based on the drop in a dispersion function between
full and reduced models and this difference must be scaled by y.
For the one-sample and two-sample problems, Lehmann (1963) proposed a
consistent estimate of
y
based on the length of a distribution-free confidence
interval for a shift parameter.
For the linear model problem, under the
assumption that f is synmetric about zero, Hettmansperger and McKean (1976),
(1977) proposed using Lehmann's estimate of y computed from the residuals
and proved it was consistent.
In this paper an estimate of
y
is proposed that is applicable for linear
models and more general situations without asslUlling symmetry for the density f.
To describe this estimate consider residuals
estimate of
~i'
1~ i
~
n.
Let
Hn (t)
Z.1= 1
Y. -1
0·,
where
D·1
is an
be a weighted empirical cdf of the
absolute values of differences of residuals,
-2-
Hn (t)
•
=
~ .a.. </>(IZ.-Z.I, t)
l<J 1J
J
1
where </>(u,v)
= 0,
I.
The proposed estimate of y is
a ..
i<j 1J
=
1.
~,
1 as u > v, U = v, u < v and a ..
1J
~
0 with
Y = Hn (i/ m )/ (2t/ m )
(1.1)
th
where t is the a-- quantile of Hn(t) , 0 < a < 1.
A
A
Another similar estimate is
(1.2)
s WM are the M = (~) ordered absolute differences
IZ.-Z.
I for i < j and k is chosen so that m Wk = 0P (1).
J 1
The unweighted (or equal weight) case provides the basic estimate
where WI
•
s;
while weights can be used for special pUrPOses.
For example, the use
of 0-1 weights would restrict the number of differences used in
reduce computational effort.
Hn and
In analysis of variance models and stratified
sampling models where the data arise naturally in groups, the use of weights
a.·
1J
= 0 if Z.,
Z. are from different groups would leave the estimate
1 J
dependent only on the within-group variation of the data.
This intuitively
appealing estimate has apparently not been considered in the literature.
For the one-sample problem with
~.
1
= ~ for all i and under the assumption
.
of a symmetric distribution, Antille (1974) considered an estimate of
y
based
on the number of averages (Z.+Z.)/Z in the interval (0, l/m) for i < j.
J
1
This estimate is basically a window estimate of the density of (Yl+YZ)/Z
at
~.
The window is not symmetric about zero and the window width does not
adjust to the spread of the data.
-3-
Schweder (1975) proposed a Lehmann-type estimate of y (in formula
(3.1)) which is basically a window estimate and recommended a uniform window
•
synnnetric about zero.
Assuming
lJ. =
1
0 for all i so that no nuisance para-
meters need be estimated, his estimate is
2
I.1,J.<HIY.J -Y.I,
h /2)/n h
1 -n
n
=
((n-l)/n)Hn (hn/2)/hn
+
(l/nhn)
Hn (hn/2)/hn
where hn is the width of a unifonn window and Hn is the empirical cdf of
IY.-Y. I for i < j. The estimate (1.1) is seen to be similar to this type of
J
1
estimate with a window width of 0p (n-~). In proving consistency and asymptotic
normality, Schweder assumed the window width satisfies ;n hn +
ruling
out the O(n -~ ) cases, and recommended hn = O(n -1/3 ) .
The estimate y differs from Schweder's estimate by allowing for the
00
•
,
estimation of nuisance parameters in using residuals rather than iid
variables.
Also it uses a window width'estimated from the data.
In an
actual application of Schweder's estimate the choice of an appropriate s for
hn = sn~1/3
is difficult to make and his MOnte Carlo study indicates that the
bias of the estimate is sensitive to the choice made.
The estimate y* of (1.2) can be viewed as a nearest neighbor type
estimate based on the absolute values of the differences of residuals.
Nearest neighbor density function estimates are discussed in MOore
and Yackel (1976), (1977).
Cheng and Serfling (1981) considered estimates of efficiency-related
functionals, including
y
as a special case.
-4-
2,
MAIN RESULTS
The main results are presented in this section,
•
The proofs can be
found in the Appendix,
For each integer n
(2,1)
Y
~
where Y
"'Il
= (Y l n ' ""
1,2, '"
=
= u"11
+
, assume
e"fi
Ynn )' is an n xl random vector, ""Jl
u
is an n x 1 parameter vector and e
~n
= (~ln ' ""
~nn )'
is an n x 1 vector of independent,
identically distributed random variables, each with density function f,
At times the n will be suppressed to simplify the notation,
vector
n , define
For an n x 1
an n x 1 vector
•
Let an indicator function be defined by
u
= v, u
<
given a..
v.
1Jn
Suppose for each integer n
= a1J
.. for 1 ~
i < j ~ n with
~(u,v)
= O,~,l
= 1,2, '" a set of weights is
L'1<).
a ..
1)
= 1, Consider the weighted
empirical cdf of the absolute values of the differences in
II (t,)J,Y) =
n
~
L'1<J. a1J.. ~(IZ.(n)-z·(n)
I,t)
J
1
This is not quite a proper cdf since
~
as u > v,
~(n)
given by
,
= ~ in case of ties.
REMARK 2,1
•
(i)
Hn (t,..u,Y)
~
(ii)
fin (t , !J.,.X)
(iii)
=
ll • Z(lI))
Hn (t,n~~
~I:;
'}d 1 fin (t , n-g ,X) IQ
Hn(t,Q(X),Y) I~ 1 Hn(t,~a),D
if ~Q') is an estimate of B based on
IQ
y such
that
QO: -l-}) = BQ')
- B '
-5-
The remark shows that with a translation-invariant location estimate,
it is sufficient to deal with the case
•
For each n
~
= Q•
= 1,2,
.•. let Mn be a measurable subset of n-dimensional
Euclidean space and consider the following assumptions for the sequence
{Mn}oo •
n=l
In jn.-n in I +
max
0
as n
+
00
•
Isi<j~n
ASSUMPTION (Az)
~
For any 0 > 0, there exists integers NI and L and
partitions Mn(l) , ..• , Mn(L) of ~ such that for each l=l, .•. , L, if
•
u
then (lIn)
dij(l)
= sup{~jn-~in: ~n
d~.(l)
1J
=
I·l<J.(d~.(l)
1J
E
Mn(l)}
inf{~.In -~.1n :.Bn EMn (I)}
- d~.(I))2
1J
<
0 for all n ~ Nl .
Two other assumptions that will be needed are as follows.
ASSUMPTION (~)
There exists constants No and B such that
O
n2
\ . .2
a .. S2
B
{.l<J 1J
O
for all n
~
NO.
If C(t) denotes the cdf of the difference of two independent
random variables, each with density f, then G has a density g = G' with
•
get) continuous at t = 0 and 0 < g(O)
Note
g(O) =
If 2 = y
<
00
•
To study the asymptotic properties of the estimate in (1.1) consider
the function
-6-
IEM\1A 2.1
•
Let {Mn } be a sequence of subsets satisfying assumption
and let assumptions (A3) and (A4) hold. Then for anY,Dn E ~,
(~)
n = 1,2, ... and any real number t > 0
This lennna shows that
rn Hn (t/vn,n
~
,Y )/2t converges in probability
"'11
to Y but the result is not strong enough since an appropriate t must be
chosen to fit the data and the nuisance location parameter
estimated when
Jdn must be
Jdn is not zero. The following theorem shows that the
convergence in the lennna is tmifonn and this will allow the construction
of a suitable estimate of y.
•
rnEOREM 2.1 Let {M } be a sequence of subsets satisfying assumptions
n
(AI) and (AZ) and let assumptions (A3) and (A4) hold.
number. Then
a
as n
Suppose that ~ = ~(Xn) is an estimate of
assumptions.
For all n = 1,2, ...
•
-+-
00
Mn
Let to be a positive
•
satisfying the following
-7-
ASSUMPTICN (A )
6
For all €:
>
0 there exists a sequence of subsets {M }
n
satisfying (-\) and (A ) and an integer N such that
Z
PO(Bn
lliEOREM Z. Z
Mu)
€
(~),
Let asslU1lptions
a posi tive number.
for all n ~ N.
> I-€:
(A4 ), (AS) and (A ) hold.
6
Then
p
sUPO~t~toIRn(t'Mn'Yn) I I
l6n
+ 0 as n +
Let the n x I residual vector be Z
41
for H(t)
•
LE~
o<
= G(t)
Z. Z
Hn will
be
=Y
41
u
"'11
00
•
and the empirical cdf of
Z be Hn (t) = I.<.a
.. ~(Iz. - Z.I,t).
1 J 1J
J
1
denoted by to.' Similarly the quantile
the absolute differences in
of order a for
Let to be
"11
- G(-t), t > 0, will be denoted by t
(~)
Let asslUTIptions (A ), (AS) and
3
a
The quantile
of order a
•
hold.
Then for
a < 1,
P
A
tl
al!n
For a given a
+t
a
asn+
oo
(0,1) there would exist to so that
E
arbitrarily high probability for large n.
to.
~ to with
Then Remark 2.1 (iii), Theorem
Z.2 and Lenma 2.2 yield the main result that follows.
OOROLLARY 2.1 Let assumptions (A3), (A4) , (As) and
o < a < 1 and
y = ftn (ta /Iii
•
P
Then
+
The estimate
Y as n +
y arises
from
(~)
hold.
Let
)/(2£ /Iii ) .
a
00
•
Rn(t,~,xn) by using t = to. = W[lli]+l ,
-8-
W denote the M = (~) ordered absolute differences
M
/2.-2.
I. The estimate y* of (1.2) arises by using t = In Wk , where k=kn
J 1
converges to zero at an appropriate rate. Specifically, if 0 < a < a 2
l
and k satisfies a < Tn k 1M < a 2 for all n, then rn Wk is bOlmded away
l
n
n
n
from zero and from above in probability and Theorem 2.2 gives the following
where WI
•
$
•••
::;;
result.
COROLLARY 2.2 Let assumptions (A3), (A4 ), (AS) and (A ) hold.
6
Assume
there are posi tive cons tants aI' a Z such that 0 < a < Jf kn/M < a for
l
2
all n. Let
p
Then
•
•
Y*lk!n
+
Y as n
+
00
-9-
3.
APPLICATION TO LINEAR MODELS
In this section it is shown that the assumptions of the previous
•
section are satisfied for a linear model under mild conditions and as
a result,
y or
y* can be used as a consistent estimate of y.
using rank procedures, a consistent estimate of
y
Thus when
is available without
assuming symmetry for the error distribution.
Consider the model (2.1) with
(3 .1)
e
l:!n = B01 + !
~
/ rn ,
where 1
X = (x1J
.. ) is an n x p design matrix
.- is an n x 1 vector of ones, ,..,
,
and k = (~l' ... , Ap) is a p x 1 parameter vector. Let ~O be a positive
number and V = {~:I~kls ~O for k = 1, ... , p}.
For each integer n = 1,2, ... ,
let
ASSUMPTION (B l )
For each k = 1, ••• , p,
where xk is the average of the kth column of ! .
asn ....
oo ,
where '1: is a p x p positive definite matrix and ~c
centered design matrix.
= (1 -
(l/n)J)~ is the
-10-
REMARK 3.1 Let assumptions (B ) and (B 2) hold. Then the sequence of
l
subsets Mn in (3.2) satisfies assumptions (AI) and (A2) .
•
Proof:
For ""J1
n
E
Mn write n·J n - n·In = LkP= l(x'kx'k)6
k/;n.
J
I
max,<.I <J'< n In.J n -noIn I ~ 26 0 LPk=lmax,.....<
{Ix'I k - xkl/1Ii }, if ~
IS n
(B l ) implies (Ar).
J,::>
.J.,:>
Next let 6'
for all i
Then Mn(i)
>
O.
=
{SO.!
+
V.
Then
Then there is a partition VI' ... , VL of V such that
1, ... , L, ~, k'
=
E
Then
.c. ~/ IIi
€
:
VI implies 16k-6kl
~
€
Vi} for i
==
< 6' for each k = 1, ... , p.
1, ... , L, is a partition of
M
.
n
Now
2
(1/n ) L
i<j
==
=
(p6,2/ n 2) Lk=l nL~=l (x ik - xk) 2
p6 ,2 trace (~~~/n) •
Since 6' ]s arbitrary and (B 2) holds, it follows that (A2) holds. 0
'"
Suppose that ~'" = ~(y)
is an estimate of~. The intercept parameter
8 has no effect on matters here since it can be absorbed in the error
0
variables of model (2.1).
-11-
ASSUMPTION (B 3) Let ~ satisfy ~(X-~ ~/ Iii) = ~(D
bounded in probability when ~ = Q in model (3.1).
- ~ and let ~ be
This assumption is satisfied for the usual rank estimates of
~
proposed by Jaeckel (1972), Hettmansperger and McKean (1977) and others.
REMARK 3.2
Let asslUl1Ptions (B 1) and (B 2) hold and let ~ satisfy
assunption (B 3).
of
Then assunption (AS) holds for!Jn in the co1tm1l1 space
~
and asslUl1Ption (A6) holds with M given by (3.2).
n
Proof: Asstnnption (AS) follows directly from (B ). For £ > 0, the
3
bOlmdedness in probability of ~ implies that a sufficiently large !!.o can
be chosen in defining V for (3.2) so that PO(~
€
V) > 1 - £ for large n.
Then (A6) follows since ~ € V implies Mn= eO! + ~ ~/rn € Mn .
0
These remarks show that (B ), (B ) and (B 3) can replace (AS) and (~)
1
2
in Corollaries 2.1 or 2.2 in the case of linear model. Thus under typical
asstnnptions,
y or
y* provides a consistent estimate of y.
-12-
4.
LEHMANN'S TWO-SAMPLE ESTIMATE
Lehmann (1963) proposed a consistent estimate of
•
y
for the two-sample
problem based on the length of a confidence interval for a shift parameter.
It is interesting to note that the estimates
y and
y* are computationally
similar to Lehmann's estimate if the set of residuals is duplicated to form
a second sample.
LehmaIm' s work showing consistency is based on the
assumption of two independent samples and would not apply in the present
context.
To describe Lehmann's estimate let two samples be denoted by
Yl , .. 0' Ym and Xl' 0.0' Xn0 Define U(~) = Li Lj ~(Xi'Yj-~).
~U and ~L by U(l:I u)'; da - (1/2) and U(~L) ;, mn - da +(1/2), where
(mn/2) - za/2(mn(m+n)/12)1/2 and za is the upper a th quantile
Define
da =
of the standard
is 2z a /zll3(m+nHflU-l\).
Consider the case where the two samples are identical, that is
normal distribution.
m = n and Yi
becomes
=
Xi.
Then Lehmann's estimate of
Then
U(-~) =
n2 -
y
U(~), ~L = -~U
and Lehmann's estimate
(4.1)
2.1
Now suppose that the residuals
two "samples".
of section 2 are duplicated to form
n
Use equal weights a ij = 1/(2).
~
n -1 \ \
Hn(t) = 1 - (2)
~
Then for t > 0,
~
L'L' ~(t,Z.-Z.)
J
1
J
1
= 1 - (~)-lU(t)
Using this in the equation defining
~U
yields the equation
-13-
1hus for large n,
~U
satisfies
(4.2)
and with (4.1) it follows that Lehmann's estimate is approximately
Hn (~u) 12~U' This is the same form as the estimate y in Corollary 2.1.
Also note that in the two -sample model, Lehmann showed rn ~U converges in
probability to a constant and this matches the behavior of t a in y. The
probability levels a used in each estimate are not related in any useful
way.
In Corollary 2.2, suppose kn is defined by kn1M = 2z a 12 I vOn.
(4.2) shows that ~U corresponds to Wk and the estimate y* is then
n
directly in the form of Lehmann's estimate (4.1).
Then
-14-
5.
APPENDIX
Proof of LEMMA. 2.1
Write
)
Rn Ct,n"'Jl ,Y
~n
where
WI)
..
=
=
vn . I . a.1) CWo1)
1<)
0
0
-
2ty/1ii)
CPCIY.-Y.-d
.. l, t/vn) and d..
= n· -no . Using
)
1
I)
l ) ) n In
EOCW..
)
I)
=
GCd..
+ t/Iii) - G(d.. - t/m)
I)
1)
= g(~ .. )(2t/m)
I)
I~·
where
.-d..
I
1)
1)
<
t/Iln and y
= g(O), it follows that
EO(Rn(t,nn,Xn))
= 2t.L.
aij(g(~ij)-g(O)) •
1<)
Dn
Since
E
Mn , Assumption (~) implies that
max I~ .. I -+ 0 as n -+
1.......
I)
.i$1<)~n
I
00.
max
l~i<j~n
I d.. 1
I)
-+ 0 and then
The continuity of g at zero gives
Ig(~ .. )
- g(O)
large.
Hence for any £ > O,IEO(Rn (t,n~n ,Y"'fi))1 < 2t£ for large n and it
I)
follows that
< £ for all };::;i<j~n for any £ > 0 if n is sufficiently
EO(~(t'nn'Yn))-+ 0
Using a,s sLUllPtion
(A3)
as n -+
00.
and IW·.I
~ I, a standard argument shows that
I)
varO(Rn(t,nn,Xn)) -+ 0 as n -+
ClO.
With the mean and variance tending to
zero the lemna follows. []
Proof of lHEOREM 2.1
This proof is somewhat long and it will be divided into pieces with
some 1emnas.
Begin by choosing any £ > O.
Choose 0 > 0 sufficiently small
so that 4g(O)B 0/6 < £/6, where BO is specified in
C~).
For this 0 we have
-15-
N1 (take N1 > NO of (A3)) , the partitions Mn(l) , l=l, ... , L and the
extreme differences d~.(l)
and d~.(l)
of (A_).
Choose 8' > 0 so that
1J
1J
Ul
Partition the interval T
6g(0)6' < £/6.
= [O,t O] into subintervals
T1 , ... , TM of width less than 6'. For each m = 1, ..• , Mlet
t rnU = sup{t:t € Tm} and t rnL = inf{t:t € Tm} .
Then for each rn = 1, ... , M
(5.1)
... ,
For each l=l,
5 .. (l,rn)
1J
L and m = 1, ... , M, define
1, if Yj -Yi
€
(dL (l) - t~/rn
or Yj -Y i
€
eJ;.j (l)
=
+
, d~j (l) - t;/rn )
t;/vfi , d~j (l)+ t~/rn )
0, otherwise
for Bi<jsn and Q"~,m =
LEMMA 5.1
For each l
Proof:
=
rn . L.
I<J
a IJ
.. 5..
(l,m) •
IJ
1, ... , L and m = 1, .•. , M
Fix l and m and wri te
=
rn i<jLa..IJ (WIJ.. -W!IJ.)
- 2y (t - t ' )
where W.. = </>(IY.-Y. - d··I, t/m), W!. = </>(IY.-Y. - d!.I, t'/rn) ,
IJ
d..
I)
=
n·
In
J
- n·
In
1
I)
and d!.
I)
=
1))
n!
)n
-
n~
III
1
IJ
. Inspection of possible cases shows
-16-
that
Iw..
-w~·1
1)
1)
t,t'
E
T •
m
~
for all Bi<j~n, all D, D' € Mn(i) and all
s..
(i,m)
1)
Then the lennna follows.
[]
Using asslDllption (A ) , there is <5" > 0 such that Idl < 0" implies
4
g(d) <
2 g(O). Then from AsslDllption (AI) there exists N2 > Nl such that
Injn-ninl < 0"/2 for all
(5.2)
(-<5"/2)
Dn
M , all l~i<j~n and all n ~ N2 •
n
E:
This implies
~ d~.(i)
~ d~.(i)
~ (0"/2)
I)
I)
for all i = 1, ••• , L, all
~i<j~n
and all n
~
N •
2
LEMMA 5.2 For all i = 1, ... , L and m = 1, ... , M,
(a)
there exists an integer N such that
3
(b)
EO(Gl,m)
~
(e/6) + 4 g(O)o'
varO(Qi,m)
+
0
Proof:
= vn
L
i<j
=;n L
i<j
= vn
as n
+
00
for all n ~ N3 ,
•
a .. Eo(S .. (i,m))
I)
I)
a .. {[G(d~.(i)
1)
1)
L a I).. {g(~I).. (i,m)) [dUI).. (i) -
t~vn) - G(d~.(i)-tU/rn)]
I)
L·
U
U
L
I)
I)
U
L
- t )/vnn
m
m
i<j
L
d .. (i)+ (t - t )/vn ]
I)
m
m
+ g(v .. (i,m))[d .. (i)- d .. (i)+ (t
I)
where d~.(i)-tU/vn ~ ~ .. (i,m) ~ d~.(i)- tL/vn and d~.(i) + tL/vn ~
m
I)
I)
I)
m
v .. (i,m) ~ d~. (i) + tmU/vn for all l~i<j~n.
I)
1)
m
I)
m
-17-
Now there is an integer N3 > N such that
Z
to/rn < O"/Z
for all n ~ N •
3
This along with (5.Z) implies
-0" <
E;: ••
1J
(l,m) < 0"
-0" < v .. (l,m) < 0"
1J
for all 15 i <jsn and all n
~
N •
3
Then by the choice of 0" it follows that
rn 4 g(O) i~j\ a ij [dUij (i) rn i<jI a 1J.. [d~.
(i)
1J
s 4 g(O)
from (5.1) for all n
assumptions (Az),
~
(~)
N •
3
L
U
L
d ij (i) + (tm - t m)/
-
d~. (i)]
1J
rn ]
+ 4 g(O) 0'
Applying the Cauchy-Schwarz inequality and
yields
s4 g(O) B I6 + 4 g(O) 0'
O
s (E/6) + 4 g(O) 0'
from the choice of 0 •
The second part of this lemma follows from assumption (A ) and the
3
fact that Is.. (l,m) lsI with a standard variance argtDllCnt. 0
1J
Choose an arbitrary £' > O.
choose a sequence nl E M (i) , n
~
LEMMA 5.3
n
Proof:
= 1,Z,
= 1,
... , Land m = 1, ... , M
.... and a number t
m E Tm •
There exists an integer N4 such that
Po(IRn(tm'~'Xn)1 >
for all l
For each l
= 1,
£/3)
~
£'/ZLM
... , L, all m = 1, ••. , M and all n
~
N4 .
Apply Lemna Z.l for each of the finite number of pairs (l,m).
0
-18-
.
With these preliminaries completed, the proof of the theorem will
For each l = 1, ... , Land m = 1, ... , Mwrite
now be completed.
E )
I~Ctm'~'Xn)1
~
POCQl,m
=
Po CQ.e. , m - EO CQl ,m)
+
S'
PoCQ.e.,m - EO CQl ,m)
+
e
+
2yo' +
EO CQ.e. , m)
+
~
E)
I~Ctm,n;;'Yn) I
~
E)
IRn CtmNJ1
,nl,Y
. . .n) I ~ 2E/3)
POCQo~,m - EOCQo~,m ) ~ E/3)
+
l
POCIRn Ctrn ,n
,Y)I ~ E/3)
Nll "'ll
E' /1M
where Lemma 5.1 was used in the third line, Lemma 5.2Ca) and the choice
of 6' were used in the fifth line, Chebyshev's inequality and Lemma 5.3
were used in the seventh line, and the last line follows from
1eImna
5.2Cb),
since the first tenn in the seventh line above can be made less than E'/21M
for alll,m for n sufficiently large, say for n
~
N5
>
N4 .
Finally,
-19-
..
PaC
sup
t$;t
(J:;
IRnCt,nn,Yn) I
~
£ )
o
n"e:Mn
,M ,L £'/LM
Lm=lLl=l
for all n
~
= E'
NS and this completes the proof.
[]
Proof of THEOREM 2.2
With assumption CAS) and Remark 2.1 it is sufficient to prove the
result when
Bn
=
Q.
PaC sup
Let
£
> O.
Then
IR Ct,u,Y ) I >
()::; ts ton
£ )
"'fl "il
Then Theorem 2.1 and assumption CA6) imply that this can be made arbitrarily
small for n sufficiently large and the result follows.
0
Proof of LEM-1A 2.2
The proof is basically straightforward and brief details will be given
here.
Assumption CAS) implies that it is sufficient to deal with the case
-20-
.
A
A
A
A
Noting that Z.-Z. = (Y.-Y.) - (lJ· -lJ· ),.assturrption (A6) (only
J I
J I
In In
part is needed) implies that max I~. -~. I will be arbitrarily
1 .... • •
In In
.!51<J~n
small with arbitrarily high probability for large n and the empirical cdf
of
z.J -Z.
I
can then be bmmded by empirical cdf' s of Y. -Y..
J
I
Assturrption (A_)
-3
is used to show the weighted empirical cdf of Y.-Y. converges in probability.
J
I
-21-
..
REFERENCES
ANTILLE, A. (1974). A linearized version of the Hodges-Lehmann Estimator.
Ann. Statist. 2, 1308-1313.
CHENG, K.F. and Serf1ing, R.J. (1981). On estimation of a class of
efficiency-related parameters. Scand. Act. J., 83-92.
HEITMANSPERGER, T.P. and M:Kean, J.W. (1977). A robust alternative based
on ranks to least squares in analyzing linear models. Technometl'ics
19, 275-284.
JAECKEL, L.A. (1972). Estimating regression coefficients by minimizing
the dispersion of the residuals. Ann. Math. Statist. 43, 1449-1458.
LEHMANN, E.L. (1963). Nonparametric confidence intervals for a shift
parameter. Ann. Math. Statist. 34, 1507-1512.
MCKEAN, J.W. and Hettmansperger, T.P. (1976). Tests of hypothesis based
on ranks in the general linear model. Comm. Statist. A5, 693-709.
M)()RE, D.S. and Yackel, J.W. (1976). Large sample properties of nearest
neighbor density function estimators. StatisticaZ Decision TheopY and
Related Topics. Academic Press, New York.
M)()RE, D.S. and Yackel, J.W. (1977). Consistency properties of nearest
neighbor density function estimates. Ann. Statist. 5, 143-154.
SCHWEDER, T. (1975). Window estimation of the asymptotic variance of rank
estimators of location. Saand. J. Statist. 2, 113-126 .
•
UNCLASSIFIED
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IECURITY CLASS,,.CAT'ON OF THIS "A"'L (",,,,,. ".,. h.,.., .. J)
READ INSTRUCnONS
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1Jl~I'OR'~
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TITLoIt (_d Subl/,'.)
TYPE OF REPOR T II PERIOD COVERED
5.
A Consistent Estimate of a Nonparametric Scale
Parameter
FCIPI EN r's CATALOG NUMBEA
Technical Report 1981
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..- : ) - - - f
Gerald L. Sievers
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11. CONTROLoL.ING OFFICE. NAME AND ADDAESS
12. REPORT DAlE
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- .... _ . ,
.-.----~~----~~._
..
II. SUPPLEMENTARY NOTES
~
,.
----- ._.
-_._--_._-----~
Nonparametric, scale parameter, rank estimates, linear models
....._- . _ ------_._---------1
~-----~----------------.------20. A.ST~ACT (Con,'nulJ on
.1.10
t)' 81111
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,.V.t."
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DO
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1473
EDITION OF I NOV
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UNCLASSIFIED
SECU"ITY C\'ASII"ICATION 0' THIS I"AGlt (W'7l. .
20.
a••
btt.... t>d)
ABSTRACT
A consistent estimate· is proposed for the scale parameter ff2
.
in the model Vi
= ~i
+ ei , 1
~
i
~
n, where the
~i
are unknown location
parameters and the e.1 are independent, identically distributed random
errors with density function f.
This parameter arises in the
variance formula for rank estimates of location.
is based on differences of residuals
of
~i.
When the
~i
vi -0 i , where
The proposed estimate
~i is an estimate
follow the structure of the general linear model,
the estimate is shown to be consistent under the usual assumptions on
the design matrix.
The estimate does not require the symmetry of
the dens ity f.
e
.....
~
:;::. _=.
j :
J. ~
c V~
UNCLASSIFIED