•
•
ON SEQUENTIALLY ADAPTIVE ASYMPTOTICALLY
EFFICIENT RANK STATISTICS
by
Marie Hu~kova
Department of Statistics
Charles University, Czechoslovakia
and
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1475
December 1984
ON SEQUENTIALLY ADAPTIVE ASYMPTOTICALLY EFFICIENT RANK STATISTICS
•
v
,
Marie Huskova
and
Department of Statistics
Charles University
Prague 8, Czechoslovakia
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina
Chapel Hill, NC 27514, USA
Key Words and Phrases: Adaptive rank statistics and estimates;
asymptoticaUy optima'l score function; contiguity; Fishel'
information; Legendre po'lynomia'ls~ 'linear runk statistics;
orthonorma'l system; R-estimators; sequentia'l estimation;
stopping ru'le.
ABSTRACT
Adaptive rank statistics arising in the context of (asymptotically) efficient testing and estimation procedures for a regression
(as well as two-sample location) parameter are considered.
An
orthonormal system based on the Legendre polynomials is incorporated
in the adaptive determination of the score function, and the proposed sequential procedure is based on a suitable stopping rule.
Various properties of the stopping rule and the sequentially
adaptive procedure are studied.
Asymptotic linearity results
(in a regression parameter) of linear rank statistics are studied
with reference to the Legendre polynomial system, and some improved
bounds are derived in this context.
1.
Consider
n
INTRODUCTION
independent random variables (r.v.)
with continuous distribution functions (d.f.)
Xl"",X
n
Fl, ••. ,F ,
n
respectively, all defined on the real line
E
=
(-~,~).
It is
assumed that
(1.1)
e
where
F
8 are unknown (location and regression) parameters,
and
is of unspecified form, and the
1
are known (regression)
H : 8 = 0, against
O
typically, a rank test statistic is of the
constants, not all equal.
8 > (or « or
c.
:f) 0,
For testing
form:
n
Sn (~) = Ei =1cn ian (Rn1.)
where
scores
=
of
Xi
a (l), ... ,a (n)
n
n
a
~
=rank
-
Rn i
(i)
n
{~(u),
(1. 2)
Xl'· .• 'Xn (i = l, ••• ,n),
are (usually) defined by
the
among
= E~(Un i)
(1. 3)
•
is the score generating function, U < ••• < U
nl
nn
are the ordered r.v. 's of a sample of size n from the uniform
O<u<l}
(0,1) d.f., and
c
ni
= (ci-Cn )!en
1 < i < n
c
n
n
- 2
E.1= l(c.-c)
•
1
n
1 n
= -E
1c.
n i= 1
(1. 4)
If
F
possesses an absolutely continuous probability density
function (p.d.f.)
!(fl!f)2dF
f
with a finite Fisher information
« ~), where f'
is the first derivative of
ref)
f,
=
then
for a local (contiguous) alternative of the form
H :
n
(1.1)
holds with
8 = C-l~
the asymptotic power of the one-sided size
based on
Sn(~)
~
n
q
fixed
(1. 5)
test (0 < a < 1)
is given by
(1. 6)
•
is the standard normal d.f.,
where
<I>(T )
ex
=1
- ex,
and
(1. 7)
(1. 8)
-1
F
is the quantile function corresponding to the d.f.
= J~
<a,b>
a(t)b(t)dt,
~
generality) that
=
and it is assumed (without any loss of
<~,l>
= O.
Thus, for every
asymptotic power in (1.6) is maximized when
~
=
so that
~F'
Sn(~F)
and
~F
F,
~ > 0,
P(~'~F)
the
= 1, i.e.,
is an aSymptotically optimal score function
is an asymptotically optimal rank test statistic.
The
same conclusion holds for the left-hand side (B < 0) and two-sided
(B
~
0) alternatives.
For the estimation of the regression parameter
B,
we may
consider an aligned rank statistic
•
n
Sn (t,~) = L.1= lC n ian (Rn i(t» ,
(1. 9)
tEE
Rni(t) = rank of Xi-tc i among
i = l, ••. ,n. Then, for nondecreasing
defined as in (1.2)-(1.4), with
Xl-tc , ..• ,Xn-tc , for
l
n
S (t,~) is nonincreasing (in
n
of B is defined by
....
B
n
Bn(~)
of
B,
(~) = ~(Sup{t:
S
n
~,
t), and the so called R-estimator
(t,~)
> O} + inf{t: S
n
(t,~)
< O})
• (1.10)
is a robust, translation-invariant and consistent estimator
and, under quite general regularity conditions, as
n
increases,
(1.11)
Thus, by reference to the Cramer-Rao information limit, we may con....
clude that
when
Bn(~)
P(~'~F)
= 1.
is an asymptotically efficient estimator of
Therefore,
the estimation problem too.
~F
B
is asymptotically optimal for
procedure rests on a suitably posed stopping rule, and various
properties of the stopping rule and the procedure are then studied
in Section 3.
Some additional results of general interest are
presented in the concluding section.
2.
SEQUENTIALLY ADAPTIVE ESTIMATOR OF
<P
F
First, let us introduce the Legendre polynomials
k~O}
{Pk(u), U€ [0,1],
in
L (0,1).
2
Pk(u)
wbich form a complete orthogonal system
We define
= (2k+l)~(_1)k(k!)-1«dk/duk){u(1_u)}k) ,
u€[O,l] ,
k>O.
(2.1)
Note that
= 1,
Po
Pl(u) = 13(2u-l),
the standardized version of
P (u) =
2
15(1-6u(1-u» (= P2 (1-u», P (u) = v7(20u -30u +12u-l) (= -P (1-u»,
3
3
and so on. For k~l, Pk(u) is of bounded variation inside [0,1],
the classical Wilcoxon socre function (and is
3
r-;
though not necessarily monotone.
k~O,
u
€
t
in
u),
2
In general, we have for every
[0,1],
(2.2)
Also, note that
(2.3)
Further, by integration by parts,
F is itself a polynomial of degree q (~l), then it
can be equivalently written in terms of Pl, .•. ,P , so that by
q
(2.3) ,
Further, if
<P
(2.5)
We shall term (2.5) a finite-case.
logistic d.f.,
Moreover, if
<P
f
F
= PI'
[For example, when
so that (2.5) holds with
is a symmetric d.f.,
<P
F
q
F
is a
= 1.]
is a skew-symmetric
function (i.e., <Py(u) + <PF(l-u)
0, V O<u<l) , so that by (2.2),
;0;
(2.6)
We shall term (2.6) a symmetric-case.
In general, we may not have
a finite or symmetric representation.
With respect to
{PO,P , .•• }, for
l
Fourier series
2
<PFE L (0,1),
P
=
as has been assumed, we have the
uE[O,l)
k>l
[In the symmetric case, we have
(2.7)
(2.8)
Y =0
o
<PF(u) ~ Ek>OY2k+lP2k+l(u),
Side by side, we define for every
O<u<l.]
m~l,
Then, note that
(2.10)
(2.1l)
Now the
P
are specified functions, while the Y are
k
k
unknown. We proceed to estimate the Y
from the sample, and
k
then with a suitable choice of m in (2.9), we estimate <P
by
F
Ek<iYkPk' where Yk and ~ are the estimates of Yk and m,
respectively.
In this context, the asymptotic linearity results
S (t,<p)
n
that the X.
[on
in
that
B
First, we note
are independent but not necessarily i.d.
~
estimate
t] have been exploited fully.
in (1.1), we may note that
Sn(t,P )
l
is \
in
t (EE).
Pl(u)
is
t
To
in
u, so
Let then
(2.12)
be the R-estimator (based on Wilcoxon scores) of B in (1.1) and
2
let C be defined as in (1.4). For some (fixed) t (>0), define
n
F? and hence, cf>F are rarely known, while, in
view of the asymptotically optimality results stated above, estimaIn practice?
tion of
is of genuine interest.
cf>F
Note that
= 1(f),
<cf>F'cf>P>
so that
(1.12)
Thus, one may naturally be interested in incorporating an
orthonormal system
tion of
cf>F.
{cf>0,cf>1,cf>2' •.• }
in a Fourier series representa-
If one assumes that
cf>F
is expressible as the
difference of two nondecreasing, absolutely continuous and square
integrable functions (inside
for every
£>0,
cf>F(U)
o
where
~F,E
(0,1», then [viz., Hajek (1968)],
there exists a decomposition:
= ~OF (u) + ~(l)(u) - ~(2)(u)
,E
F,E
O<u<l
F,E
,
(1.13)
is a polynomial, and
(1.14)
Also, for a strongly unimodel p.d.f.,
cf>F
is
I.
logf(x)
is convex, so that
Motivated by these, we are tempted to incorporate an
orthogonal polynomial system. and the Legendre Polynomials provide
the desired solution.
A series representation of
the Legendre polynomials
series
{Yk
= <cf>F'Pk >;
{Pk(u),
k~O}
UE
[0,1]; k~O}
cf>F
in terms of
involves the
of the Fourier coefficients.
The
estimation of these
Y
is greatly facilitated by making use of
k
the Jure~kova (1969) -linearity results on S (t,cf» (in t), as
n
further extended by Ghosh and Sen (1972), Hu~kova and Jure~kova
(1981), Hu~kova (1982, 1983) and others.
In the context of asymp-
totically efficient adaptive R-estimates of location, some
alternative procedures are due to Beran (1974), Eplett (1982) and
and others.
Along with the preliminary notions and basic regularity
conditions, estimation of
cf>F
through the use of Legendre's poly-
nomials is considered in Section 2.
The proposed (sequential)
y,n,k ...
(2t) -lis
n
(6n -tC-l,P
) - s (8 +tC- l ,P )}
k
k
n
n n
n
k>l . (2.13)
By the basic results of Jure~kova (1969), the
Yn,k are translationinvariant, consistent and robust estimators of the Y • We may
k
remark here that for the testing problem (H : 6=0 vs. 6#0), we
O
do not need to estimate
6,
and hence. may consider an alternative
(simpler) estimator:
Yn,k
(2t)-1{S (-tC-l,P )
n
n
k
=
-s n (tC-l,P
n
k
)}
k>l
(2.14)
...
Note that under
H as well as {Hnl in (1.5), the Yn,k are
O
consistent estimators of the Y , though they are not translationk
invariant. But, for (a fixed) 6#0, they may not perform that
well.
Hence, for the estimation problem, they are not recommended.
We plan to incorporate these estimators
(Yn, k
or
Yn. k)
in (2.9)
to estimate ¢F'
Note that in the symmetric case.
•
for every
k~l.
need not be \
EY
Yn , 2k should estimate 0,
Moreover, even in the aSYmmetric case, the Yk
in
(~l).
k
though (2.11) ensures that the series
2
converges, so that for every
k
(= m(E,F», such that
E>O,
there exists an
m
(2.15 )
Motivated by this, we choose a sequence
{r}
"0
n
of positive
I 00 but n- r n
as n-¥JO. and another
n
of positive numbers, such that En" 0 as n-¥JO,
integers, such that
sequence
1
{En}
r
and let
(2.16)
K
n
where
choose
K
r
(~2)
n
estimator of
is a prefixed positive integer.
to be larger than 2.
¢F
is
Typically, we
Then, our proposed (adaptive)
(2.17)
Also, the adaptive estimator of
l(f)
is
(2.18)
Note that by (2.7) and (2.17),
1
I I~n
-~FI O
12 = J n
{~(U)-~F(U)}2dU
(2.19)
where, by (2.15), as
K +r
goes to 00 (in probability or almost
n n
surely (a.s.», the second term on the right hand side converges to
o (in probability or a.s.).
Hence, the stochastic behaviour of the
stopping number
{Kn } and the mode of convergence of the
(to 0) play the vital role in our study. Theorems 3
.....
Yn,k - Yk
and 4 (in Section 3) pertain to this study, while the main results
are presented in Theorems 1 and 2 (in Section 3).
In the remainder
of this section, we present the basic regularity conditions.
Note that
l(f) <
00
ensures that
~
00
J-oolf'(x)ldx ~ 1 (f) <
00.
We strengthen this to:
f(x)
and
flex)
are bounded almost everywhere (a.e.) (2.20)
Next, we note that by (2.1), there exists a positive and finite
constant
D (independent of
k), such that for every
k~l,
(2.21)
for
every
j =
0,1,2,3,4.
Therefore, by (2.7), (2.9) and (2.21), for
m~l,
(2.22)
so that whenever the series
side of (2.22) converges to 0
!.:
Ek>lk2lYkl
as
converges, the right hand
m+oo; however, EYk2
<
00
does not
•
~
(0,1),
itself not a bounded function on
may not hold.
..
is also not needed for our study.
ever the
Yk
is
this "max-norm" convergence
I I~n-~FI 100
+
0
in probability
!.;
Lk>lk2lYkl < 00
n+oo), and hence, the condition that
2
~F
For our subsequent analysis, we do not need this
stronger mode of convergence (i.e.,
or a.s., as
and, in fact, when
Lk>lk Iykl < 00,
necessarily imply that
However, we may note that when-
are dominated by a monotone sequence
{y
*2
}, such
k
*2
*2
2
that LY
< 00, then kY
(and hence, kY ) converge to 0 as k
k
k
k
increases. Having this in mind, we use a variant form of (2.11)
as a part of our assumptions:
kY
2
k
converges to
as
0
k+oo
(2.23)
[It is possible to obtain better results if we assume that
l 2
k Y + 0 as k+oo, for some l>l. We shall make comments on it
k
in Section 4.] Regarding the regression constants {c }, following
i
Hajek (1968), we assume that as n+oo,
(2.24)
•
(2.25 )
Finally, in (2.16), regarding the choice of
and
{e: }
n
{r},
we
o) ,
(2.26)
n
will consider the various possibilities:
(i)
e:
n
=
e: (>0)
(ii)
and/or
r
"0
and
e: n
= r
n
for every
(';::2),
rn /
00 with
n (.:::n
(2.27)
n+oo
We shall elaborate on this later on (in Section 4).
3.
THE MAIN THEOREMS
For every positive integer
1(£,a)
so that
1(£,a)
I 1(f)
as
we conceive of a sequence
a,
2
n
(3.1)
= Lk<aYk
aIm.
{a }
let
Defining
of
r,e:
n
n
as in (2.16) ,
positive integers, such that
the follQwing conditions hQld:
o
lim
-1(1ogn )3( r + a )15/ 2
~ n
n n
o
n~
lim
n-kX>
(r IE a )
n
n n
(3.2)
(3.3)
lim 13
4 2
(a r (logn) In E ) = 0
= 0 and ~
n n
n
(3.4)
Note that (3.3) implies (3.2), and (3.4) ensures that
(3.5)
We shall make some comments on the choice of
Theorem 1.
Assume that the d.f.
satisfying (2.20).
F
{En ,rn,a}
n
later on.
in (1.1) has a density
f
Then, under (2.27) and (3.2),
"-
l(f, K+rn ) -< 1n <- l(f, a n +r)
n
a.s., as
n-kX>
(3.6)
and hence,
lim
n-kX>
If, however,
£.
n
In
= 1(£)
(= E>O)
and
a. s., as
r
n
(=r)
n-kX>
(3. 7)
are held fixed, then
(3.7) may not hold.
Theorem 2.
(1.5) hold.
For the model (1.1), assume that (2.20) and
{H }
n
in
Then, under (2.23) , (3.3) and (3.4),
ISn(¢n) - Sn(<P F a +r )
, n n
I
-+
0 ,
in probabili ty, as
,
n-kX>
(3.8)
and, for
<PF,a +r'
the asymptotic efficiency results of
n n
"Section 1 hold. Thus, ¢n is an asymptotically efficient score
function, in probability.
For the stopping variable
{K}
n
in (2.16), we have the
following.
Theorem
r
~.
If, in (2.16),
En = E (>0),
for all
(=r) is any (fixed) positive number for every
n
satisfies
n,
n,
and either
or
{r}
n
•
lim { r 7 (logn) 2/n
}
n
=0
n~
then, for every
(3.9)
there exists a positive integer
£>0,
k£
such
that
K < k
n -
a.s., as
£
( 3.10)
n~
2
Lk>mYk = 0,
satisfy the following: for some n>O,
In the finite case, i.e., under (2.5), when
V m_>q+l,
if
{£},
n
{r}
n
4
lim{
-l( max {6
~ £
r 0 n-l+n , r n14n -2 (logn)})
n-.-o·
o
(3.11)
then
< q+1
n-
K
a . s., as
n-+09
(3.12)
In the general case, i f (2.23), (3.2) and (3.4) hold, then
Kn -< an
a.s., as
(3.13)
n~
li, in addition, there exists a sequence {rnO} of positive
•
integers, such that
V r <r
o
(3.14)
n
then, under the hypothesis of Theorem 1,
a.s., as
K
n
(3.15)
n~
The proofs of these theorems rest heavily on the asymptotic
behaviour of
{Yn , k;
k~l},
and towards this, the following
theorem is of prime interest.
Theorem 4.
Let
{X.; PI}
-
satisfying (2.20).
exist a
d>O
~
be Li.d.r.v. with a -p.d.L
Then, for every
and a positive integer
s>O,
n '
s
0>0
and
a>O,
f
there
such that for every
n>n ,
-s
-s
>dun,k } ~ n
,
(3,16)
k < k = 0 (n
- n
for every
2/13
-2
where
(logn»,
2
u
= max {k 5/2 n -~o , k13/2 n-1(1 ogn )3/ } ,
n,k
k>l
(3.17)
The proof of this theorem is presented in the Appendix.
We may note that
Sn
in (2.12) is based on
Sn(t,Pl).
Thus,
proceedings as in (10.3.60) through (10.3.65) of Sen (1981), we
obtain that for every
s>O,
there exist positive constants
and a positive integer
p{c
n
18n -S/
n ,
s
such that for every
> c l(logn)} < c 2n s
-
n>s
)
-s
s
(3.18)
s
From (2.13), (3.16), (3.17) and (3.18), we obtain that for every
k<k = 0(n 2/13 (10gn)-2) and n>n ,
-n
-s
I
p{ IYn, k-Yk > (2a)
A
-1
n- s
dun, k} < c*
s2
(3.19)
-
c: 2 = 1 + c s2 • The same inequality applies to the Y k in
n,
(2.14) when H
(or {H }) holds. In the sequel, we take s>2.
n
O
Now, returning to the proofs of the Theorems (l, 2 and 3») we
where
m~K,
note that by virtue of (2.16), for every
k+r
2
n: j = kn+1 yn,J. > E: n ,
{K >m} =
n
"~k2.m}
(3.20)
Also, note that for every positive integers a,b
a+b
2
2
Ir j =a (yn, j-Y.)
J
A
I
a+b
2
a+b 2
(Y
.-y.) +r. y.}
J=a n,J J
J=a J
< Hr.
-
A
With these, we consider first (3.13).
r
n
= ora )
n
=
0(n 1/ ?),
by letting
s'
.
(3.21)
Since by (3.4) and (3.12),
s - 1/7 (>13/7),
from (2.23), (3.19) and (3.20) that for
we obtain
a "" an + 1 and b = r n'
the right hand side (rhs) of (3.21) is bounded from above, with
probability greater than
1 - c: n-s'
2
by
a +r
a +r
2
n n 2
n n
}
2 {rj=an +1 Y. + r j =a +1 0 (un,J.)
J
n
(3.22)
as by (3.4), (r /£ a ) ~ 0 as
-2
3 14
n n n
n (logn) a
~ 0
as n~ and
!
n
0 (>0)
choosing
n~.
By (3.2) and (3.4),
7
n (logn)a ~ 0 as n~. Thus,
n
.
-1+20 6
adequately small, we conclude that n
an ~ 0
-1
Hence, the rhs of (3.22) is 0(£) as n~. Since, b.y
n
an+rn 2
(2.23), as n~, r j =au+1Yj = o(rn/an ) = o(£n)' we have
as
n~.
P{K >a
n
<
-
n
n~nO}
for some
r n~nO P{Kn >an }
a n +rn
< r
p {r .
- . n>n
- O J=an +1
,.., 2
y. > £
n,J
n
[by (3.20)]
}
•
0.23)
as
Sl
-1 > 3/7.
Hence, (3.13) holds.
The proof of (3.11) is quite analogous.
yk=O, V k > q + 1.
-
Thus, in (3.22) for
I
a
n
Note that under (2.5),
= q + 1,
we have the
> 1 - c: n-s )
2
bound (with probability
q+r +2
=
~j=q~2
{0(j13n-2(10gn)3)
+ 0(j sn- 1+ 20 )}
= 0(n- 1+ 2Q r 6 ) + O(n- 2 (10gn)3 r 14)
n
n
0.24)
Thus, (3.12) follows from (3.24) and (3.23).
To prove (3.9), we define
(3.25)
Then, Ie t ting
e:n =e:
and
r
as in (3.9) we have by (3.19) and
n
(3.20),
for some
P{K >k,
n
e:
n~nO}
k £ +rn
2
<
L
>
P{L
k
+1
Y
- n_n O k = £
n, k>
A
k +r
d
2
A
< L: >
-
P{Ek£k +n (y k-Yk)
n_n
= E: 1 n,
O
> E:/8}
(by (3.21)
and (3.25»
(by (3.19»
•
(3.26)
where we have made use of the fact that by (3.17) and (3.19), with
-s
A
2
-2 2 2
n , (Y k-Yk) < 4a d u k and
probabili ty > 1 - c*2
s
n,
n,
k +r
k +r
k +r
~ £
n u2
-1+26
E:
n 5
-2
3
E: n
13
uk=k +1 n k = O(n
)Lk=k +1 k + O(n (logn) )L:k=k +1 k
=
£
'
£
-1+26 6
-2
3 14
O(n
r ) + O(n (logn) r )
o
n
(>0)
n
adequately small).
E:
+
a as
n~
.
(by (3.9) and lett1ng
Tois completes the proof of (3.10).
For the final part of Theorem 3, i.e., (3.15), note that
(3.27)
so that by (3.14) and (3.27)
-!.:
\Y \ = o(j 2),
where by (2.23),
and by (3.13), we may take
j
o
r <
a n , V n -> nO. Hence, using (3.17), (3.19), we claim that with
-s' (for some s~>7/3), the
probability greater than 1- c*2n
n -
~Q
s
•
0 2
second term on the rhs of (3.28) is bounded by o(n
r (r +r ) ) +
-1
2
0 6
_1
0
2
nn n
o(n (logn) r n (rn+rn )- ) = o(n ~-E n a n (rn+a n ) ) +
0(n- 1 (10gn)2£ a (r +a }6) = 0(£), by (3.2) and (3.4).
n n n n
n
Further note that
o
P{K <r,
n- n
L
n>n
-0
n~nO}
for some
P{L
r+r
n "2
<E
k= r+1Yn k '
n'
r: K<r,<r o}
--n
for some
(by (3.23»
0.29)
•
Hence, (3.15) holds.
This completes the proof of Theorem 3.
Next, we consider the proof of Theorem 1.
Note that by
(2.16) and (2.18),
fn
>
-
Lk<K+r y2n k
n
'
with probability
1
0.30)
By (2.23), (3.19) and (3.27), we have under the assumed regularity
conditions, with probability > 1- c:2n
=
Thus,
"
I
I (f, K+r ) - 0(1)
n
> l(f, K+rn )a.s., as
n-
n~.
-s~
,
s'>7/3,
(by (3.2»
Also, by 0.13)
0.31)
1n
< ~
- k<a +r
- n
.?n,k
n
= 1 (f, a +r ) +
n
n
0
(1)
a. s., as
n~
(3.32)
where in the ultimate stage, we have made use of (2.23) and (3.19)
along with the bounds in (3.17) and (3.4).
proof of (3.6).
This completes the
Under (2.27), (2.11) and (3.6) imply (3.7).
For
the final part of the theorem, consider the following hypothetical
model:
O<u<l
where
(3.33)
is a finite positive integer and choose Y
and
2
2
1
Y
such that Yq~d>e: (>0) and l2Yl> (2q+l)q (q+l) Y ' Then
q
q
we have <PF(u) I in u, l(f) = Yl +Y q and 1(f, a+b) = Y =
1
1(£) -Y22.1(f) -d, \'a,b: a+b~q-l. As such, by (2.16) with
E
n
=e:
q (> K+r)
(given) and
r =r(given), using (3.19), we conclude that
n
Kn = K a.s., as
while
1(£, K+r) = l(f) -d<l(f)
-E.
(3.34)
n~
Hence (3.7) does not hold.
This completes the proof of Theorem 1, and points the desirahility
of letting
E
n
and
rn,
dependent on
n,
so that (3.7) holds.
For the proof of Theorem 2, we note that by (1.2), (1.3) and
(2.19),
K +r
n n"
= ~. 1 Y .S (P )
J=
n,] n j
where the anj(i)
is defined hy (1.3) with
(3.35)
<p(.)
replaced by
•
Pj (. ) ,
1<i<n
and
j~1.
m~l,
Simi,larly, for every
(3.36)
Keeping (3.l3) in mind, we first prove the following:
Lemma 5.
Under
{H } in (1.5) and the regularity conditions of
n
Section 2, when (3.4) holds, defining r*n = an +r n ,
(3.37)
A
~n,k(·)
where
Proof.
'"
= Lj2kYn,jPj(.),
for
k>l.
Proceeding as in (10.3.60) through (10.3.64) of Sen (1981,
Ch. 10) and using (2.21), it follows that under (1.5), for every
k (>1),
C
(if
s
(2)
,cs
for every
s (>0),
and an integer
there exist positive constants
nO'
such that for every
n~nO'
(3.38)
.
Therefore, choosing
s>l,
we conclude that under {H }
n
max ISn (P ) - ~Yk I = 0 (log n)
k<r*
k
-
n
p
in (1.5),
•
(3.39)
Next, we note that
(3.40)
•
By (3.3), (3.4), (3.17), (3.19) and (3.39), with probability
greater than 1 - (c(2)+c*2)n- s ' (where s'>l), the rhs of
s
(3.40) is
s
0(1) ,
when
0 (>0)
Thus, (3.40) is 0p (1)
Lenuna 6. Under {H}
n
in (3.17) is chosen
as
n~,
(3.41)
1/30
<
and hence, (3.37) holds.
in (1.5) and the assumed regularity conditions,
.
(3.42)
Proof.
every
First, consider the case of
~O.
H :
O
Note that for
q >it,
(3.43)
where, by the orthonormality condition in (2.3) - (2.4) and the
standard moment formulae for the
S (P.),
n J
it follows that under
(3.44)
the fourth central moment may also be easily computed and verified
that
(3.45)
where
c (>3)
is a positive constant.
For every
t
€
10,1],
we
write
k(t)
= max{k:
(3.46)
L
Then, note that by (2.11) and (2.27),
2
Eo>
*Y.J
J r
2
j>K+r Yj
o as
and
n
o
as
n~.
So, for every
<5
>0,
there exists an
n
such that for every
<5>E .>K+
J
Thus, writing
wn =
{W (t)
n
r
n
2
YoJ2>E
J.>r *Y.J -> 0
n
(3.47)
by
...
Theorem 12.3 of Billingsley (1968), we conclude from (3.45) and
(3.46) that
[O,lJ,
W
converges weakly to a Brownian motion
n
W on
and the implied "tightness" part of this weak convergence
ensures that for every
and an integer
nO'
£>0
and
n>O,
such that under
there exist a
(0<0<1)
<5
HO'
(3.48)
This ensures (3.42) under
H ' We invoke the contiguity of the
O
probability measures under {H } to that under H
to extend
n
O
this under {H } as well. Q.E.D.
n
Returning now to the proof of (3.8), we note that by (2.16)
and (3.13),
K+ r
< K +r < a + r
n- n
n- n
n
a. s., as
n-+oo
As such, (3.8) follows from (3.37), (3.42) and 0.49).
0.49)
Also,
note that (3.16), (3.17), (3.l8) and (3.8) ensure that as
n-+oo~
(3.50)
•
Further, by (1.7), (1.8), (2.7) and (2.11), we conclude that
(3.51)
and hence, the asymptotic efficiency results in (1.6) and (1.11)
are shared by
{¢F
,r
*}
as well.
n
A
(3.50), we conclude that
¢n
(adaptive) score function.
4.
•
Combining this with (3.8) and
SOME
is also an asymptotically efficient
Q.E.D.
GENERAL REMARKS
We have noted [after (2.8)] that for a symmetric
k~l,
F,
l'2k(')'
do not provide any basis for the projection in (2.9), while
l'2k+l(')'
k>O
are themselves skew-symmetric functions and form a
basis for the skew-symmetric
we need to choose
r>2.
¢F'
This also explains why in (2.l6),
To eliminate the pathological cases. as in
(3.33), one also needs to choQse
r=r
;f in n.
n
However, in our
main theorems (viz., Theorems 1,2 and 3), we have a variety of
conditions on
these.
(r,£),
n
and we need to have a closer look on
n
Ideally, one would like to choose
increasing function of
less cumbrous.
to choose
On
{r}
n
so that
n,
"
~
{r}
as a slowly
n
becomes computationally
n
the other hand, by (2.11), one would als.o like
adequately large, so that the residual term is
negligible (up to the desired extent).
Keeping these opposite
pictures in mind, we may suggest the following.
En' and a n
if we let ·r
n
further that
..
need to satisfy (3.2) (or (3.3»
Note that
r
and (3.4).
n
,
Thus,
= 0 (logo) , then a £ / (logn) + 00 as n+oo and
13 -1
S 2
n n
/n + 0 as n+oo. In particular, if we
an £n (logn)
.
let
r
n
= O(logn) ,
then note that
a
n
,.,. n2/l5(logn)-~
an
£ n
= ·(logn)
3/2
E ) + 0 as
, so that r n I(ann
n+oo, and (3.3) - (3.4) hold. It is also possible to choose
a
-al
r n = O(n l ) for some a : 0<a <I/15. In that case, anEnn
+00
l
l
as n+oo and (3.3) and (3.4) have to be true for {a}, {£}. If
~
~
n
n
we let a ~ n a n d £
n 3, then we have a - a - a > 0
n
2
n
3
•
1
a 2 > a l +a , and al>o, a >0. Further, by (3.4),
3
3
l3a2 - a l +a < 2, and by (3.3), 15a2/2 < 1. Such a choice may
3
easily be made, e. g., 0 < a = a < ~2' a < 2/15. In the above
l
2
3
discussion, if instead of (3.3), we use (3.2), then 2/15 may be
Le.,
replaced by 1/7 as well.
Also, in (2.23), if we make a more
restrictive assumption that
kiy~
+ 0
as
k+ oo
,
for some
then, in (3.4), we may use the condition that
as
n+oo,
i>l
(4.2)
(rn/£ ai) + 0
and this in turn will reduce the order of
n n
a. In the
n
then we need
extreme case, if we let rri;: r (~2) for every n,
only that £ a +00 but (a 13 E: -1 )/(n 2 (logn) -4 ) + 0, as
n n a n .~
-a
for this, we may let a ~ n (lognfZ,
£
'" n
where
n
n
and conclude that (3.10) and (3.12) hold.
n+oo,
and
0 < a_< 2/15,
However, as· has heen
•
remarked in Theorem 1, we may not desire to set
n,
r =r
for every
n
and hence, such a solution will not be of much interest.
We may also remark that for the given model (1.1), under
H
n
in (1.5), or for the estimation problem, the
Xi
are not
A
Hence, for the study of the properties of the
1. 1. d.
~n
(Z.13), involving the estimator
Yn,j
in
in (Z.lZ), noting that the
A
residuals
Xi -
Bn c.
are not necessarily independent or identically
~
distributed.random variables, we are not in a position to adapt
directly the results of Beran (1974) and Ep1ett (198Z), among
The Jure~kova-1inearity results and contiguity of
others.
probability measures provide the necessary tools in this respect.
In this context, it is not necessary to use the R-estimator in
(Z.lZ).
holds.
We may as well use any other estimator for which (3.18)
Further, one may also consider a general regression model
Bc.
where in (1.1),
is a
q
(~l)-vector
and the
A
Again an estimator
B'c~ , P1,
~'= (B1""'~ )
~
q
i
are specified q-vectors too.
is replaced hy
~
B
~n
of
=i
~
B satisfying (3.11) (in the Euclidean
v
~ (
norm) may be used, and the R-estimators of Jureckova
1971) are
particularly appealing in this context.
In the finite case, Le., under (Z.5), using (3.1Z) and the
asymptotic (joint) normality of the
the asymptotic behaviour of
Y
. ,j < q + 1,
n,Jn ~(ep (u) - ep (u», O<u<l;
1
one may study
'"
F
n
however,
this becomes quite complicated in the infinite case.
APPENDIX:
Note that for each
k
PROOF OF TIIEOREM 4.
(~1),
S (c
n
-1
t,P )
k
n
difference of two nonincreasing functions in
we let
t .
nJ
=n
-1
j,
j
= O,+l,
••• ,+[alogn],
--
is expressible as a
t (EE).
Hence, if
then the supremum
(over
t , t E (-a log n, a log n» in (3.16) can be dominated by the
1 z
maximum (over the t nj ) on the grid-points (plus a non-stochastic
remainder term O(n- 1 ». Hence, to prove (3.16), it suffices to
choose an
s* > s + 1
and verify that the probability inequality
in (3.16) (without the supremum over
t ,t ) holds with s heing
1 Z
replaced by s*, uniformly in (t , t ) E (-a log n, a log n). For
1 Z
the sake of the simpHci ty of this proof, we take t = - t = t
z
1
and
a=l,
i.e.,
t E (-logn, logn).
we have, for every
k~l,
real
Then, by the Taylor exvansion,
t,
where
*
TnO,k(t)
n
Ei =l cni { Pk [
=
ELR.
[E[R.
d1 (.. tic)
n Ix.1 ]]"
n1 (tic)
n IX.1 ]]}"
n+ 1
-P k
n+l
-2tYk '
(A.2)
and, for
j = 1,2,3,
every real
T*. k(t)
nJ,
=
T . k(-t) .. T . k(t),
nJ,
nJ,
where for
u,
T . k(u)
nJ,
for
j = 1,2,
and
(3) (a E[R i(u/e )Ix.] +(l-a)R . (u/e
T ,k(j) = En c P
n3
i=l ni k
n
n
n
1
n+1
n
n1
x (n+1)-3{R .(u/e) -ELR i(u/e )IX.J}3
n1
with
a E (0,1).
n
n
n
n
1
CA.4)
Note that
3
+ IRn1.(u/en )-Rn i(O)-E[Rn i(u/en )-Rn1.(O)lx.]
1 1 }
Since
have
)J
n"
Rn1.(0) - ELRn1.(0) Ixi ] = n[Fn (X.)-F(X.)]
.. [l-F(X.)J,
1
1
1
(A.5)
we
max I
-1{ R. (O)-ErR . (0) X. J} I < (n+l) - 1 + sup{ IF (x) .. F(x)
l<i<
__n (n+1)
n1
n1
1
n
I:
xEE}.
CA.6)
Further, by the Dvoretzky, Kiefer and Wolfowitz inequality, for
every
c>O
and
n~l,
2
sup I
I -1
~
-2c logn
p{ x Fn(x)-F(x) ~c(n logn)} ~ 2e
(A.7)
Therefore. by (2.21). (2.24) - (2.25) and (A.5) - (A. 7), we have
2
with probability greater than 1 - 2n- 2c ,
~ Dk13/2n~[n-3/2c3(10gn)3/2 + 0(n- 3)]
= 0(n- l k 13 / 2 (10gn)3/2)
where
c (>0)
in (3.16),
(A.8)
can be so chosen that for the chosen value of
2
2c > s*.
s
Next. we note that
2 Xi}
E{(Rn iCcn-1 u) - Rn1·
. (0) - E(Rn iCCn-l u) - Rn i(O) I X.])
1
1
= I:nj =lVar{I(X.J -<X.1 -u(cn1.-cnJ.)
-I(X.<X.)IX.}
J- 1
1
»
E{(I(Xj -<Xi-u(cn1.-cnJ
-< I:~J= 1 ('.J.')
Jr 1
x
-1(X.<X
J- i »2I 1.}
•
(A.9)
leX. < X. - u(c .-c .» - I(X.<X.)
J - 1
n1 nj
J- 1
are independent bounded r.v., by using the exponential
Since for a given
(l~j~n)
Xi'
the
inequality [viz. Theorem 2 of Hoeffding (1963)], we obtain that
P{IR i(C-lu) -R i(O) -E[R . (C-lu) -R .(O)lxiJI > n\10gn)2/3Iul~lx.}
n
n
n
n1 n
n1
1
~
2 exp{-~(log n)
1
(1
= {2n-~ . ogn
•
where
~(logn)
1/3
adequately large.
[1 + o(l)]}
}{l + o(l)}
can be made larger than
(A. 10)
s*+l,
by choosing
Since the unconditional probability can be
obtained by integrating over
lui ~ logn,
)1/3
4/3
X.,
we have on noting that
-s*
with probability :: l-n
,
1
n
I
I
max
-1
~1
"i" < R i( C u)-R. (0) -E [R . (C u)-R. (0) Xi]
n1
n1 n
n1
n
1< n n
~
2/3
k
=O(n 4Clogn)
(logn) 2) •
and hence,
E.n 1 I c . I (n+l)
" 1=
n:r
-31 Rn1.(C-1
u) -R .(0) -E[R iCC-1 u)
n
n1
n
n
3
-R i(O) I x.l 1
n
1
o(n!j) O(n--3n 3/ 4 (logn) 2 (logn) 3 / 2)
2
= 0(n- 71\10gn)2(logn) 3/ )
0(n- 1 (logn)3/2)0(n- 3 / 4 (logn)2)
0(n- 1 (logn)3/2)
(A.12)
Combining (A. 8), (A.12) and (2.21), we conclude that for every
(~s+I),
for every
there exist positive numbers
d*
and
n*,
s*
such that
n~n*,
(A. 13)
T~O.k(t).
Consider next
n
Note that
.»,
1 - F(X.) + E. IF(X.-u(c .-c
1
J=
1
n1 nJ
5 2
O(n-!j) and
p~l)
2 Dk / .
II
ILX)
where
E[R .(u/C )lx.J
n1
n
1
max{lc i- c
l<i:fj<n}::;
n
nJ
-
•
.1:
Hence for every
t
E
(-logn, 10gn).
we may rewri te
n
T*O k(t) = E. l c . {U .-ED i}
n ,
1= n1 n1
n
+ O(n -!jk 5/2 10gn)
(A.14)
where the
n
=
Uni
(I-F(X i ) +E j =1 F(Xi +t(c ni -cnj
Pk
n +1
n
») _
(I-F(Xi )+I: j =I F (X i -t(Cni-Cnj
Pk
n +1
}»)"
(A. IS)
...
1
(12i2n)
are independent (bounded by
2
Var(U .) < ED .
n1 n1
uniformly iil
l<i<n
positive integer
m.
and
t
E
we have
2(2k+l)~)
= O(n-1 (logn) 2k 5 )
(-logn. logn).
r.v. and
(A.16)
For any (arbitrary)
•
n
E{L
1 c (U i..-EU .)}
i = ni n
n1
Zm
•
(A.17)
)
Consequently, for every
d>O
and
6>0,
~ (dn-~+(\S!Z)-ZmO(n...m(logn) Zm kSm )
= O(n- Zom (logn)Zm}
;: O(n -s* ) ,
when
(A.1S)
Zorn> s*
From (A.14) and (A.IS), we obtain on noting that
•
for every
0>0,
that for every
such that for every
n~n*,
t
E
s*
(>
s+l),
n-~logn;: o(n~},
there exists an
n*,
(-logn, logn),
(A.l9)
Finally, we consider the case of
We deal only with
T~Z,k(t),
T~l,k(t)
TnZ,k(ti)
T:2 ,k(t).
as the other case follows more simply.
E[Rni(u!Cn)lxi ]
in (A.3) as
Using the expression for
may rewrite
and
as in after (A.l3), we
n
-Z{
(n+1)
(Z)(l-F(X)+L _IF(X.-u(c .-c
L
c .p
r=
1
n1 nr
1~i"j".t~n n1 k
n+1
x [I(X.<X.-u(c .-c ) - F(X.-u(c .-c .»]
J- 1
n1 nj
1
n1 nJ
x [I(Xo<X.-u(c .-c .) - F(Xi-u(c i- c_ o)]
~- 1
n1 nJ
n
l~
n
+
E
c.P ( Z)(1-F(X)+L r ;:IF(Xi-u(cn1.-cnr »J"
1<i f j <n n1 k
n+1
-
-
x [I(X.<X.-u(c .-c
J- 1
n1 nr
» - F(Xi -u(cn1.-cnr »]z}
»)
(A.20)
say
It
is easy to show that
EA(u)
n
= 0,
'Vu ~ E, and using (2.20),
'.
(2.21) and (2.24) - (2.25), we can show that for every
u, u'
€
(-logn, logn)
(A.2l)
Thus, we may write
= [A(-t) -A(t) ] + [B (-t) -EB (-t) -B (t) +EB (t) ] + O(n-lk 13 / 2 logn) •
n
n
n
n
n
n
(A.22)
Parallel to (A.l7), here also, we may compute, for an arbitrary
m (~l), E[A(-t)_A(t)]2m and E[B(-t)_B(t)_EB(-t)+EB(t)]2m. Using
n
n
n
n
n
n
the conventional technique (as in the case of higher order moments
of Hoeffding's (1948) U-statistics
but with some straightforward
modifications) along with (2.20) - (2.21) and (2.24) - (2.25), it
follows readily that
(A.23)
E[B (-t) -B (t) -EB (-t) +EB (t)] 2m =O(n -2~9m(logn) 2m + n-3~13m(10gn) 2m) .
n
n
n
n
(A.24)
Hence, proceeding as in (A.18), we claim that for all
k: k = 0(n 2 / l3 (10gn)-2), for every d>O, 0>0,
= o(n-s* )
where the
(A.. 25)
u
are defined in (3.17) and m is chpsen adequately
n,k
large, so that the last step follows by using (3.17). Therefore,
.
parallel to (A.19), we have here
-s*
<~
P{IT*2
n , k(t)1 >un, k}
'
..
for every
n>n*.
With a similar inequality for
(A.26)
T~l,k(t),
we
obtain from (A.l), (A.13), (A.19) and (A.26) that for every
s* (> s+l) ,
there exists an
n*,
such that
s
p{jS (-tC-I, P ) - S (tC-I, P ) - 2tY i > u k} < n- *
k
k
n
n
n
n
n,
k
for every
n~n*,
and uniformly in
t
€
(-logn, logn).
,(A.27)
The proof
for Theorem 4 is complete.
ACKNOWLEDGEMENT
The work of the second author was partially supported
by the National Heart, Lung and Blood Institute, N.I.H.,
Contract No. NIH-NHLBI-7l-2243-L •
•
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•
·c,