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ON BOUNDED LENGTH CONFIDENCE INTERVAL
FOR THE REGRESSION COEFFICIENT BASED ON
A CLASS OF RANK STATiSTICS
by
Malay Ghosh and Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 680
April 1970
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ON BOUNDED LENGTH CONFIDENCE INTERVAL FOR THE
REGRESSION COEFFICIENT BASED ON A CLASS OF RANK STATISTICS*
BY MALAY GHOSH AND PRANAB KUMAR SEN
University of North Carolina, Chapel Hill
SUMMARY.
coefficient (in a simple linear regression model) based on a class of robust
rank statistics are considered here, and their various asymptotic properties
are studied.
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In this context, several strong convergence results on some simple
and weighted empirical processes as well as on a class of rank order processes
are established.
Comparison with an alternative procedure based on the least
squares estimators is also made.
1.
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Bounded length (sequential) confidence intervals for the regression
INTRODUCTION
For the simple linear regression model Xi = SO+Sci+e i , i=1,2, ••• , where the
c
i
are known regression constants and the e
i
are independent and identically dis-
tributed random variables (iidrv) with an absolutely continuous (unknown) cumulative distribution function (cdf) F(x), defined on the real line (_00,00), we desire
to provide a robust confidence interval (of confidence coefficient I-a, O<a<l)
for the regression coefficient S, such that the length of this confidence interval
is bounded above by 2d, for some specified d>O.
The proposed procedure rests on
the use of a class of regression rank statistics [due to Hajek (1962, 1968)] for
the derivation of robust confidence intervals for S [cf. Sen (1969, section 4)],
as extended here to the sequential case along the lines of Chow and Robbins (1965).
Several results, needed for this sequential extension, are derived here.
First an elegant result of Jureckova (1969) on the weak convergence of a class of
*Work supported by the National Institutes of Health, Grant GM-12868.
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rank order processes to some appropriate linear processes is strengthened here to
almost sure (a.s.) convergence under more stringent regularity conditions on
the c. and the score-functions underlying the rank statistics.
1.
with a martingale (or semi-martingale) property of the regression rank statistics,
deduced in section 4, guarantees the "asymptotic (as d+O) consistency" and
"efficiency" [see Chow and Robbins (1965)] of the proposed sequential procedure,
and also enables one to study its asymptotic relative efficiency (ARE) with respect
to the procedure by G1eser (1965) and Albert (1966), which are based on the least
squares estimators.
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In this context, several well-known rank statistics are con-
sidered and the allied ARE results are briefly presented.
In particular, for the
so called normal scores statistic, it is shown that the ARE is bounded below by
1, uniformly in a broad class of {F}.
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This result along
2.
NOTATIONS, ASSUMPTIONS AND THE MAIN THEOREM
In accordance with the model of section 1, consider a sequence {x 'x ' •••• }
1 2
of independent random variables for which
(2.1)
where So is a nuisance parameter.
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= is:
A
A
A
We intend to determine a confidence interval
A
SL,~S<Su,n} (where SL,n and SU,n are sample statistics) such that
P{SEI }
n
= 1-0.,
the preassigned confidence coefficient,
"
A
O<Su
-SL
<2d, for some predetermined d(>O).
,n
,n-
(2.2)
(2.3)
Since F is not known, no fixed-sample size procedure sounds valid for all F.
It is therefore desired to determine sequentially a stopping variable N (a
"
"
positive integer) and the corresponding (SL,N'
SU,N)'
such that (2.2) and (2.3)
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hold.
Our proposed procedure is based on the following class of regression
rank statistics.
In a sample ~n
=
(X1 ,· •• ,Xn ) of size n(~l), let Rni
= Lj~lu(Xi-Xj)
u(t) is 1 or 0 according as t > or < 0] be the rank of Xi' l<i<n.
c*.
n~
=
(c.-c
~
n )/Cn , l~i~n, where cn
=n
-1 n
L c.
1 ~
and C2
n
Let
n
= \'L(Ci-C1
[where
n
)2•
(2.4)
Then, as in Hajek (1962, 1968), a regression rank statistic is defined as
Tn
= T(X-n ) = L.n1c*.J n (Rn i/(n+1»,
~=
where the "scores" J (i/(n+1»,
n
{J(u):
l~i<n,
O<u<l} in the following manner.
(2.5)
n~
are generated by a "score-function"
We let In(u)
= In(n~l)'
for (i-1)/n<u<i/n,
and define
l~i<n,
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(2.6)
where Un l<"'<Unn are the n ordered random variables in a sample of size n from
the rectangular (0,1) distribution.
~
-1
(u):
~(x)
O<u<l, where
The score-function J(u) is defined as
is an absolutely continuous cdf satisfying the two
conditions
(i)
M(t)
f
=
00
etxd~(x)<oo for all O<t<t '
(ii)
where \jJ(x)
by (2.8) ,
lim
-1
u+O{\jJ[~
(2.7)
o
_00
-1
(u)]/u, \jJ[~
= d~(x)/dx.
(l-u)]/(l-u)} ~ K>O,
Thus, we have M(t)
1 tJ( )
= feu
du<oo
°
(2.8)
for all O<t<t , and
!J(u)1 ~ K [-log u(l-u)], IJ'(u)! ~ K [u(l-u)]-l, O<u<l; Ko<oo.
o
o
-
0
(2.9)
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These assumptions appear to be more restrictive than those in Hajek (1962, 1968)
and Jureckova (1969) [where the weak convergence results are only studied], but
are somewhat needed for our purpose, as we require, in the sequel, to establish
certain a.s. convergence results where the exponential bounds [to follow from
(2.9)] are quite useful.
It may be noted that (2.7) and (2.8) hold for the
entire class of normal, logistic, double-exponential, exponential, rectangular
and many other cdf's.
In fact, if we use J(u) = u [i.e.,
~
rectangular cdf] or
J(u) as the inverse of the standard normal cdf, we obtain respectively the
Wilcoxon and the normal scores; the corresponding T are termed the Wilcoxon
n
and the normal scores regression rank statistics.
For both of these, (2.7) and
(2.8) hold.
Regarding {c } [where c
-n
-n
= (cl, ••• ,c )], we make the following assumptions,
n
where the first one is due to Hajek (1968):
(i)
max
l~i<n
Ic * I
ni
= O(n -~ ),
(2.10)
(2.11)
(ii)
(iii)
Q(n) is a strictly increasing function of n, such that for every
a>O,
lim Q(a n)/Q(n) = sea) whenever lim
n~
n
(2.12)
n~
where sea) is strictly monotone (increasing) with s(l)=l.
The condition (2.10)
is again more stringent than the classical Noether-condition [cf. Hajek (1962,
1968)], but is satisfied in the majority of practical situations.
less restrictive than the parallel condition:
(2.11) is
2
l 2
lim n- Cn
=o
C >0' made by GIeser
n~
(1965) and Albert (1966) in connection with least squares theory.
For example,
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if c. = a+(i-1)h, h>O, i=1,2, ••• , (2.10) and (2.11) hold but not the G1eser~
condition.
2 2
3
Also, in this case, Q(n) =nh (n -1)/12, so that s(a) = a •
Finally, regarding the cdf F in (2.1), we assume that F£ ~('l'), where
~ (If)
is the case of all absolutely continuous F for which the density function f(x)
and its first derivative f'(x) are bounded for almost all x(aoaox) and further
~ f(x) J' [F(x)] are bounded.
From (2.1), it follows that under H :
o
iidrv,
T(~n)
(2.13)
S=O, implying that Xl"'"
X are
n
has a completely specified distribution generated by the nl equally
likely realizations of (R ,···, R ) over the permutations of (1, ••• , n).
nn
n1
Hence,
there exists two known constants T(l) and T(2) (depending on c ), and an a
n
-n
n
n
(known) such that
P{T~l) ~ T(~n) ~ T~2)IHo}
[For small n, a
n
A2
n
may not be equal to a].
=
1
(n-1)
\' n
Li=l
[J
= 1-a (+1-a as n+oo)
n
If we let
(--L)-J ] 2. J
n n+1
(2.14)
n
'
n
=
1 \'
n
n Li=l
J ( i )
(2.15)
n n+1 '
then from the classical Wa1d-Wo1fowitz-Noether-Hoeffding-Hajek permutational
central limit theorem, (cf. [8, p. 160]), we have
lim {T(2) /A } = lim {_T(l) /A }
n+oo
n
n
n+oo
n
n = La / 2 ,
where
~(L )
£
= 1-£ and
~(x)
(2.16)
is the standard normal cdf.
It follows from Sen (1969, Section 6) that T(X -aoc ) is
-n
-n
+ in
a:
-oo<a<oo.
Hence, if we let
Q
= Sup{a:
fJL,n
A
T(X-aoc »T(2)} SU,n = inf{a:
-n -n
n
'
T(X-aoc )<T(l)}
-n -n
n
'
(2.17)
it follows as in Sen (1969, Section 4) that
P{SL
<S<SU IS} = 1-a (+l-a as n+oo).
,rr-,n
n
(2.18)
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We are now in a position to define our sequential procedure.
d>O, let N(d), the stopping variable, be the smallest positive integer
an initial sample size (>3)] for which SU,N(d)-SL,N(d)~ 2d.
A
the stopping variable N(d).
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[~no'
Then our proposed
A
SL,N(d)~S<SU,N(d)} and is based on
confidence interval for S is IN(d) = {S:
We justify the proposed procedure on the ground of
its robustness (for outliers or gross errors etc) and its asymptotic properties
considered in the following theorem, sketched in the same fashion as in Chow and
Robbins (1965).
Theorem 2.1.
Under the assumptions made above, N(d) is a non-increasing function
of d(>O), it is finite a.s., EN(d) <00 for all d>O,
lim EN(d) = 00.
d-+O
~~~ N(d) = 00 a.s., and
Further,
(2.19)
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For every
lim P{
} = I-a for all FE ~
d-+O
SEIN(d)
~(~),
(2.20)
~~ [EN(d)]/Q-l(V(d)) = 1,
(2.21)
where
00
v(d) = [{AL /2} !{dB(F)}]2, B(F) =
a
A2 =
Jl
o
J2(u)du_~2 and ~ =
Jl
J (d/dx)J[F(x)]dF(x),
(2.22)
-00
J(u)du.
(2.23)
0
The proof of the theorem is postponed to section 5; certain other results
needed in this context and having importance of their own are derived in the next
two sections.
3.
ASYMPTOTIC BEHAVIOUR OF SOME EMPIRICAL PROCESSES
It has been shown by Jureckova (1969) that under H:
o
and finite b, denoting W(X ,b) = T(X )-T(X -bc*)-bB(F) ,
-n
-n
-n-n
S=O, for all real
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(3.1)
our primary concern is to strengthen this statement to almost sure convergence,
specifying the order of convergence as well as extending the domain of b to
Ibl ~ C(log n)k, k>l, O<C<oo.
=
H (x;b)
n
We define now
1 n
+1 Lu(x+bc*i-X,),
n
1
n
~
for -oo<x<oo and _oo<b<oo.
n
= 1Lc*.u(x+bC*.-X
),
i
n~
n~
S*(x'b)
n '
(3.2)
Then, by (2.5) and (3.2), we have
00
T(X -bc*)
-n -n
= J
-00
J
n
[Hn (x;b)]dS*(x;b).
n
(3.3)
It will be convenient for us to study first the asymptotic behaviour of the two
processes in (3.2).
Let {Yl'Y ",,} be a sequence of iidrv having the rectangular (0,1) distriZ
bution.
For every t:
O<t<l, define
(3.4)
It follows that for every i: l<i<n and b: -oo<b<oo,
(i)
(ii)
(iii)
t+dni(t;b) is t in t:
O<t+d n~. (t ,'b)<l'
-'
O<t<l,
(3.5)
d .(t;b) and bc*. have the same sign,
(3.6)
dn (t;b)
(3.7)
n~
n~
Consider then the two stochastic processes
n
= 1LC*iu(t+d
.(t;b)-Y )
i
n
n~
n
..
- Lc*.d .(t;b), -oo<b<oo, O<t<l;
1
n~
n;
(3.8)
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1 n
L*(t'b) = (n+1)-~ fU(t+dni(t;b)-Yi), -oo<b<oo, O<t<l.
n
'
(3.9)
Simple computations yield that
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(3.10)
(3.11)
L*(t'b) = (n+1)~ H (F- 1 (t);b), O<t<l, -OO<b<OO.
n '
n
For every h(>O), there exists two positive constants (K ,K ) and
1 2
Theorem 3.1.
an n* (which all may depend on h). such that for n>n*,
(3.13 )
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(3.12)
(3.14)
where 0>0, k>l and 1* = {b: Ibl ~C(log n)k, O<C<oo}.
-
Proof.
Let a
-- n
k
n
= C(log n) , r
n
= [n
°1
] and n
r,n
= ra /r , r=O,±l, ••• ,±r , where
n n
n'
0<0<01<~
and [s] denotes the integral part of s(>O). Then , nr+1,n-n r,n =
-0 1
k
~o
O(n
(log n) ), for all r(=-r n , .•• ,rn-1). Also, let s n = [n 1] and ~ s,n =
sis n , s=O,l, .•• ,s.
n
c*. [u(t+d . (t;n
n~
n~
Now, for be:[n r,n ,n+
r 1 ,n ], we have
r,n
)-Yi)-u(t-Y )] < c*. [u(t+d . (t;b)-Y. )-u(t-Y i )]
i
- n~
n~
~
.(t;nr+1
-< c*.[u(t+d
n~
n~
, n )-Y.)-u(t-Y.)],
~
~
(3.15)
(3.16)
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(3.17)
for all r = -r , .•. ,r -1, O<t<l.
n
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r,n
,n +1 ]
r ,n
IG~(t;b)-G~(t;o)1 ~ r:~~l IG~(t;nj,n)-G~(t;O)1
+ O(n
-°1
k
(log n) ), -rn<r<r
-1.
-- n
(3.18)
We denote by 8 (n) = {i: c* > 0, l<i<n}, 8 (n) = {i: c*. < 0, l<i<n},
ni nJ.
-1
2
U(s) = ~
c* [u(~
+d (~
·b)-Y )-u(~
+d (~
·b)-Y )] and
b 'ni £L8 (n) ni
s "n ni s n'
i
s+l , n ni s+l , n'
i
2
v(s) =
b,n
b£l •
n
2. 8 ()
n
c*.[~
nJ. s+1 ,n+d n i(~ s +1 ,n ;b)-~ s,n-d n i(~ s,n ;b)], for s=O,l, ... ,s n-1,
J.£ 1
Then, we get after some simple algebraic manipulations that
~ n c*
Li=l nJ.·
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Hence, for b£[n
+d (t"
·b)-Y) - (~
+d. (~
,.b»] _ V(s) _U(s)
ss,n nJ.· ss,n'
J..
s ,n nJ. s ,n
b "n b n
[u(t"
n
-< Li =1c*nJ.. [u (t+d nJ.. (t; b) - Y.)
J. - (t+d nJ.. (t; b ) ) ]
+ V(s) + U(s).
b,n
b,n
Thus,
1~.n1c*.[u(t+d
LJ.= nJ.
n i(t;b)-Y.)
J. - (t+d nJ..(t;b»]
I
max I~ n c *
-< j =s,s+1 L·J.= 1 nJ..[u(~.J ,n+d n i(~j ,n ;b)-Y.)-(~j
J.
,n+d nJ..(~j ,n ;b»
+ V(s) + U(s) •
b,n
b,n
Hence, by (3.18) and (3.19), we have,
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(3.19)
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< max
- O<s<s
--n
+ max
O<s<s
--n
max
-r <r<r
-<\ (log
+ O(n
n- - n-
1
V(s)
+ max
nr ,n ,n O<s<s
- - n
max
U(s)
-r n<r<r
1
nr,n ,n
- n-
k
n) ).
(3.20)
Hence, it suffices to prove that the first three terms on the right hand side
of (3.20) are bounded above by O(n-h (log n) k ) with probability
~
l-O(n-h ) for
every fixed h>O.
Consider any fixed r (say
~
0) and any fixed s.
Then ,
G*(~
n
s,n'·n r,n )-
G*(~
.-EZn i)' where, Zn~. = c*i[u(~
n s,n ;0) = t.nl(Z
L~=
n~
n
s,n+d n~.(~ S,n ;n r,n )-Y i ) u(~
-Y.)], i=1,2, ••• ,n are independent and Z . can assume the values 0 and
s,n ~
n~
ICn*i l respectively with probability l-d n i(~ s,n ;n r,n ) and d n i(~ s,n ;n r,n ) respectively.
Hence, for every g>O, and h>O,
p{t.nl(Z .-EZ .»g} < exp{-h[g+t.nlEZ .]}
L~=
n~
n~
L~=
n~
< exp{-h[g+~ (~
,n )]} n.n
;n )}
· {l+(exp[hlc*il]-l)d .(~
n s,n r,n
~= l
n
n~
s,n r,n
=
~(g,h),
Upon taking now g = gn = Ki n
0.21)
(say).
-0
k
(log n) , h = hn = n
O<Ki<oo, we obtain after some simplifications that
01
,where
0<0<01<~' k~l,
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log X (g ,h )
n n n
(3.22)
uniformly in s=O,l, .•• ,sn and Ir\<r •
n
Thus, for n adequately large, we have,
uniformly in rand s,
p{G*(~
n s,n ;n r,n ) -
< exp{-Kin
01-0
Gn*(~s,n;O) > Kl'n
-0
k
(log n) }
k
(log n) }[l+o(l)],
(3.23)
and a similar bound is easily obtained for the left hand tail.
p{ IG*(~
;n
) n s,n r,n
< 2 exp{-Ki n
Thus, for large
G*(~
;0) I> Kl'n-O(lOg n)k}
n s,n
01-0
k
(log n) [l+o(l)]}.
(3.24)
Also, it follows obviously that
-0
V(s)
= O(n
nr n
n'
1) uniformly in
Irl~rn' O<s<sn.
(3.25)
0
By similar techniques as in (3.21), we get after putting h=n 1 that for fixed
rand s,
( )
p{u s
nr
n'
>g} < exp[-Kl'n
n n
01-0
k
(log n) (l+O(n
0-0 1
(log n)-
k
»].
(3.26)
From (3.24)-(3.26), we obtain (by use of the Bonferroni inequality) that for
large n, the left hand side of (3.20) is bounded above by Kin
probability greater than or equal to
-0
(log n)
k
with
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.-
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1-2(s +1)2r
n
Since, sn
=
~+61
[n
n
exp{-Kin
61-6
k
(log n) (1+0(1»}.
(3.27)
6
], r n
1
= nand
6 >6, Ki can always be so selected that for
1
n>n*,(3.27) is bounded above by 4 exp{-h(log n)k} ~4n-h (since, k~l). This
completes the proof of (3.13).
It can be shown by the same technique as in above that
< max
- O<s<s
--n
max
[L*(~
+ O<s<s
n ss+l n; 0) - L*(~
n Ss n; 0)]
- - n-1
'
,
(3.28)
where
w~~~
•
=
(n+1)-~ I i E
(n)[u(~s
n+dni(~s n;n r n)-Yi)-u(~s , n+dni(~s , n;n r +1 , n)-Y i )]'
S'"
2
Using (3.9) we obtain that for every g >0, h >0,
n
n
log P{L*(~
n s,n ;n r,n )-L*(~
n s,n ;O»g}
n
k
_< -(n+1)2hngn
+ {to
L
(n) + 'tL
'
log E exp[hn {u(~
(n)
ss,n+d ni (~ss,n''n r,n )
~ES1
iES 2
-Y.)
~
1
= -(n+1)~h
u(~
s,n-Y.)}]
~
h
10g[1+(e n_1)d .(~
;n
)]
g + I
n~
s;n r,n
n n
iES(n)
1
-h
+ LiES(n)
2
10g[1-(1-e
n)(_d .(~
n~
s,n
;n
r,n
»].
(3.29)
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13
Ix I<1,
On using the inequality that for
hn = n
-0 1
,g n = KIn
-0
k
(log n) ,
log (l± Ix I) .s. ± Ix I, and on taking
(0<0<01<~)'
we get for adequately large n that,
-0
k}
log P{L*(~
n s,n ;0) > KIn (log n)
n s,n ;n r,n )-L*(~
~-0-01
~ -KIn
=
~-0-01
-KIn
-20 1
k
-0 1
n
{Ii=l
d
.(~
n~
s,n
;n
r,n
)}
-0
-1
2k
+ n 1 [n{O(n ) (log n) }]
k
1
01.s.~,
~-0-01
-KIn
(log n)
+n
)·O(n~(log n) ), by (2.10), (3.4) and (3.7).
+ O(n
Since 0<0 1 , if we let
k
(log n)
(3.30)
the right hand side of (3.30) is
k
-k+o
(log n) [l+O(n
2
k
-(0 1-0)
(log n) )+O(n
)] ,
(3.31)
and hence, for n sufficiently large,
P{L*(~
n
-0
k}
s,n ;n r,n )-L*(~
n s,n ;0) > KIn (log n)
(3.32)
and the same bound holds for the left hand tail.
inequality, for large n,
Hence, by the Bonferroni
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14
.e
max<
{ max
p O<s<s
--n
r
I r 1-n
IL* (~
n
s,n
;n
r,n
)-L* (~
;0) I > Kn-0 (log n) k}
n s,n
~-o-o
1 (log n) k }
< 4r (s +l)exp{-K'n
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I
I
--I
-
n
n
4n-h ,for k>l and n>n*,
~
(3.33)
where h is any (fixed) positive number.
In
Since, L*(~ +1 ;O)-L*(~
;0) = (n+1)-~I [u(~ +1 -Y.)-u(~
-Y )] involves
n s ,n
n s,n
1
s ,n ~
s,n i
iid (0,1) valued random variables, proceeding as in (3.21)-(3.24) and using the
Bonferroni inequality, we obtain
< 2s
< 2n
~-o-o
exp[-K n
n
2
-h
0-0
l(log n)k{1+0(n
1)}]
for any fixed h>O by proper choice of K (since k>l).
2
(3.34)
Similarly it can be shown that
~-0-01
< 2 exp[-K n
3
uniformly in Irl~rn and O<s~sn.
k
(log n) [l+O(n
0-0 1
)]],
(3.35)
As before, on using the Bonferroni inequality,
we get
p{ max
O<s<s
--n
~-O-O
< 4r (s +1)exp{-K n
3
n n
~
1
k
(log n) (1+0(1»}
-h
4n ,for any fixed h(>O), as
k~l.
(3.36)
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(3.14) then follows from (3.28), (3.33), (3.34) and (3.36).
Q.E.D •
Theorem 3.1 will be utilized to prove the following basic result which
strengthens (3;1) to a.s. convergence.
Theorem 3.2.
Under the assumptions of Section 2. for every s(>O). there exists
(1)
positive constants ( c
, c(2»
s
s
and a sample size n , such that S=O and all
s
n>n
-s
p{ sUPIW(X b)/ > c(l)n-O(log n)k+l} < c(2)n- s •
bEI*
-n'
s
s
n
Hence, W(X ,b)+O a.s., as n+oo.
-n
-Proof.
Note that if we let I n (n1l) = EJ(U nJ..), l<i<n [cf. (2.6)], we have
R •
R •
c * [J ( nJ.)
I"n
Ll ni n n+l
_ J( nJ.)] /
n+l
(3.37)
by (2.10) and theorem 2 of Chernoff and Savage (1958).
it suffices to work with I n (n1l) = J(n1l)'
l<i~n.
Hence, for our purpose,
If we define
R(~)
l<i<n, -oo<b<oo, we have
nJ. = Ljnlu(X.-bC*i-[Xj-bC*j]),
=
J.
n
n
T(X )-T(X -bc*) =
-n
-n -n
f
f
00
_00
J[H (x,O)}dS*(x;O)
n
n
00
-00
J[Hn (x;b)]dS*(x;b)
n
(3.38)
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16
where
00
f
=
J[H (x;O)]d[S*(x;O)-S*(x;b)],
n
n
n
(3.39)
{J[H (x;O)]-J[H (x;b)]}dS*(x;b).
n
n
n
(3.40)
_00
00
f
=
_00
In (2.1), without any loss of generality, we may let SO=O and assume that
O<F(O)<l.
We select x(l) and x(2) such that
n
n
F(x
(1)
n
) = l-F(x
=
f
n
+
f
_00
x
+
(1)
2
k
(log n) ,
(3.41)
(2)
x
o
n
-(k:+cS)
) = 4K n
l
Then, for n sufficiently large, say for n>n
,
-0
where K is defined in (3.14).
l
we have x(1)<0<x(2) and
n
n'
x (1)
(2)
f
n
00
+
o
f
x
n
{J[H (x;O)]-J[H (x;b)]}dS*(x,b)
(2)
n
n
n
n
(3.42)
Since, by (3.2), IS*n(x;b)I<{(n+l) maxI.
<J.<n IC*il}H
n
n (x;b), we obtain by some
standard computations that
x
max< Ic*.1 f
i
l ~_n
nJ. 0
(2)
n
IH (x;O)-H (x;b)IIJ'[H (x;b,8 )]ldH (x,b),
n
n
n
n
n
where H (x;b,8 ) = 8 H (x;O) + (1-8 )H (x;b), 0<8 <1.
n
n
nn
n n
n
(3.14), we obtain for large n
On using (3.12) and
(3.43 )
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sup
sup
-oo<x<oo bEI*
1
Hn(x;b)-Hn(x;O)
I
= O(n
-~-o
k
(log n) ),
(3.44 )
n
s
with probability ~ l-O(n- ), for every (fixed) s(>O).
Hence, on using (3.44),
(2.9) and (2.10), we obtain that
x
(2)
Jon
with probability ~ l-O(n- s ).
that F (x;O) =m/(n+U)F(x).
(3.45)
Let us define F (x;b) = (n+l)-lI.nlF(X+bC*.), so
n
~=
n~
Note that by (2.10)-(2.13)
n
sup
bEI*
1
{H (x;b,e )[l-H (x;b,e )]}- dH (x,b),
n
n
n
n
n
1-Fn(x;b)-Fn(x;O)
I
= O(n-1 (log n) 2k ).
(3.46)
n
Since H (x;O) = (n+l) -II . n lU(x-X.) involves average of iid (bounded valued)
n
~=
~
rv's, using the results of Hoeffding (1963), we obtain that for large n,
s
p{IH (x(2)'0)-F (x(2)'0)1 > K -3/4-0/2(1
)k}<Kn n'
In
og n
2n ,
n n'
where K and K are the same as in (3.14).
l
2
it follows that for all x<x(2)
-
n
Using then (3.12), (3.14), (3.41),
'
l-H (x;O) > (1/3)[1-H (x;b)], V bsI*,
n
with probability
~
l-O(n-s ).
(3.47)
-
n
n
(3.48)
Again, by using the same results of Hoeffding
s
(1963), we have P{IL~(F(O);O)-rnF(O)I>Klv'logn} 2.. K n- , for every s(>O), and
2
hence, by (3.12), (3.14) and the fact that F(O»O, we obtain that for all x>O,
as n-+<x>,
(3.49)
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18
s
with probability> l-O(n- ).
-
From the definition of H (x;b,e ) and (3.48) n
n
(3.49), it follows that for all 0<x<x(2) ,
--n
Hn (x;b,e
·n )[l-Hn (x;b,e n )] -> C[l-Hn (x;b)] ,
(3.50)
(where C>O) , with probability ~ l~O(n-s). From (3.45) and (3.50), for large n,
with probability
~
l-O(n-s ),
(2)
x
sup J n
-1
SUPII(3)(b)1 ~ O(n-0 (log n) k ) {bcI*
{l-H (x;b)} dH (x;b)}
bEI* n2
~ nOn
n
n
00
~ O(n
-0 (log n) k ) {sup
bEI*
n
f· {l-Hn(x;b) }-ldHn(x;b)}
_00
_
-0
k { sup
-l~ n
_ -!- -I}
- O(n (log n) ) bEI*(n+l) Li=l(l
n+l)
n
I} = O(n-0 (log n) k+l ).
= O(n-0 (log n) k ){l~~+ .•• ~
(3.51)
n
It follows similarly that for large n, with probability
-0
k+l
O(n (log n)
). Again
~
s
l-O(n- ),
b~~~II~;)(b)1
n
00
II(42)(b)\
n
~ {(n+l)l~ai~
IC*il}{f
IJ[H (x;b)] \dH (x;b)
__n n
(2)
n
n
x
n
00
+ f(2) IJ[Hn(x;O)] IdHn(x;b)} =
x
I~~)(b)1+I~;)(b)2'
say.
(3.52)
n
On using (2.9), (2.10), (3.41), (3.47) and (3.48), we obtain that for large n,
with probability
~
l-O(n-s ),
=
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19
SUPII(4)(b) I
O( ~){ sup
bEI* n2
1 <
n
bEI*
{I
If
(2)
- og u
n H (x
;b)
n n
n
1
< O(n~){K
1
f
[-log(l-u)]du}
l-O(n
-~-o
Also
k2
n)
00
k+l
1
)
= O(n. -0 (log
I
I
I
I
I
I
I
--I
k+l
).
(3.53)
00
)
s
~ l-O(n- ) (note that it does not depend on bEI*) , while integrating by parts
n
and using the same trick as in the earlier cases, it follows that the second
term is also O(n-O(log n)k+l) for all bEI*, with probability> l-O(n- s ).
n
Ie
n)
f(2)IJ[H n (X;O)] IdHn(x;O)+O(~) f(2)IJ[H n (X;O)] 10d[Hn(x;b) x
x
n
n
By (3.53), the first term is O(n-O(log n)k+l); with probability
I
(b) = O(n
'n2
2
Hn (x;O)].
k
(log n) )
= O(n~ )oO(n-~-o (log
(4)
(1 -u )}d}
u
-
Thus,
as n-+oo,
(3.54)
s
Similarly, it follows that for large n, with probability ~ 1-0 (n- ),
b:~;II~~)(b)1
=
O(n-O(log n)k+l).
Consequently, for large n,
n
sup II (b)
bEI*
n2
I = O(n-0 (log
n) k+l ), with probability
~
l-O(n-s ).
(3.55)
n
In (3039), we now write Inl(b) = I(l) (b) + I(2)(b) where
nl
nl'
00
f
_00
J[F(x)]d[S*(x;O)-S*(x;b)],
n
n
(3.56)
00
f · {J[Hn (x;O)]-J[F(x)]}d[S*(x;O)-S*(x;b)].
n
n
_00
(3.57)
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By (2.10), (3.2) and (3.57), we have
sUP/ (2)
b £ 1* I n 1
n
max IC*i}{
I sup
1* IJ[Hn (Cn -~x;O)]-J[F(Cn-~x)] I} (n+l)·
-< {l<'<
1 n n
X£
'I
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I
I
--I
n
. _J[F(X.)] }{ u( _X.) _u(bc * . _Xi) }I
(b) I -_ b sUPI
1* L~ c * . {J[H (Xi,O)]
£
i
1
n1
n
1
1
n1
n =
n
.{ IH (Cn-~ (log n) k ;0) - H (Cn-~ (log n) k ;0) I},
n
n
where C(>O) is finite.
(3.58)
On using the well-known result that
(3.59)
where Cl , C are finite· (positive) constants, and that in the neighbourhood of
2
o
(where O<F(O)<l), IJ'(F(x»1 is bounded [by (2.9)], and further that
-k
k
-~
k
-k
k
F(Cn 2(log n) ) - F(-Cn (log n) ) = O(n 2(log n) ), we obtain from (3.58) that
= O(n-~ (log
n) k~~ ), with probability
~
l-O(n-s ).
(3.60)
Further, on denoting by c*n = max <.<
Ic*.I, we have
l 1 n n1
[S*(x;b)-S*(x;O)]J'[F(x)]dF(x),
n
n
(3.61)
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Henee, by (2.22) and (3.61),
where X(l)
I
I(l) (b)-bB(F)
nl
[S*(x;b)-S*(x;O)-bf(x)]J'[F(x)]dF(x)
n
n
'I
(3.62)
I
.JII.
Now, P{F(X(l»
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I
I
I
I
I
I
~
I
~
-k
en 2}= [I-en
similarly, P{ l-F(X(n»
~
-~
]
n
~
O(n
-s
en -~}-s
~ O(n ).
), for every (fixed) s(>O), and
Henee, by (2.13) the last term on the
right hand side of (3.62) is bounded (for all bEI*) above by O(n-~ (log n) k+l ),
n
s
with probability ~ l-O(n- ). Also, by (2.9), (3.11) and (3.13), the first term
on the right hand side of (3.62) is bounded (for all bEI*) above by
n
-k
X(n)+Cn 2(log n)
{KIn-0 (log n) k [l+a(n-~ )]}K
J
{F(x) [l-F(x)]} -1dF(x)
-~
x(l)-Cn
~O(n
-0
(log n»
k
{-log F(x(l)-Cn
-~
(log n)
-~
k
(3.63)
k
-k
k
(log n) )-log{l-F(X(n)+Cn 2(log n) )}},
with probability ~ l-O(n- s ), where e>O.
ean be shown that F(x(l)-Cn
k
k
(log n) )
In the same way as in after (3.62), it
~
O(n
-k
) with probability
~
-s
l-O(n ),
where h is some fixed positive number, and a similar statement holds for
-k
k
l-F(X(n)+Cn 2(log n) ».
Henee, with probability
~
l-O(n
-s
), the right hand
side of (3.63) is bounded above by
-0
k
-0
k+l
O(n (log n) )-O(h log n) = O(n (log n)
).
(3.64)
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The proof of the theorem is then completed by (3.55)t (3.62)t (3.63)t (3.64)
and the definition of W(X tb)t given at the begining of this section.
-n
4•
Let
t in n.
~
n
A MARTINGALE PROPERTY OF T(X ).
-n
denote the a-field generated by R =(R It ••• tR ); note that
-n
n
nn
Also t let T*=C T and assume that H:
n
Q.E.D.
n n
0
S=O holds.
~
n
is
Then t the following
theorem holds; the result is of fundamental use in proving the "uniform continuity in probability" (to be explained in section 5) of
to
T(~n)
with respect
-1
C •
n
Theorem 4.1.
{T*t
n
1n } forms
Proof.
If In(u) = EJ(Uni)t (i-l)/n<u<i/n t l<i<n t then under Ho :
S=Ot
a martingale sequence.
We have T* = I.nl(c.-t )J (R ./(n+l»t and hence t
n
1=
1 n n n1
c
+ I.nl(c.1= 1 n+l)E{Jn+l(Rn+1·/(n+2»I~}.
1
n
(4.1)
fl
-l\'n+l.
} = (n+l) Li=lJn+l (1/(n+2»= 0 J(u)du = ~t and
Since t E{In+l(Rn+ l n+l/(n+2» I~n
also for any l<i<n t
E[Jn+l(Rn+li/(n+2»I~n] ~
«n+l)-lRni)Jn+l«Rni+l)/(n+2»
+
(n+l)-l(n+l-R i)J +l(R ./(n+2»t and finallYt [i/(n+l)]J +1«i+l)/(n+2» +
n
n
n1
1
.
n
1
f
i
n-i
n
f
n
[(n-i+l)/(n+l)]J +l (i/(n+2» = i(i)
J(u)u (l-u)
du + (n-i+l)(i_l)
J(u)u i-I
n
n-i+l
n-l
i-I
0n_i
0
(l-u)
du = n(i_l)
J(u)u
(i-u)
du = In(i/(n+l))t for l<i<n t we have from
fl
o
(4.1)
t
E[T~+ll ~n] = (c n+l-tn +l)~+\,.nl(c.-t
L1 =
1 n +l)Jn (Rn1./(n+l»
= (c n+l-Cn +l)~ + L
\,.nl(c.-t
1 n )Jn (Rn i/(n+l»
1=
= I.nl(c.-t )J (R ./(n+l»
1=
1 n n n1
+
(t n-tn +l)n~
= T*t for all n>l.
n
(4.2)
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Hence, the theorem.
Remark.
Note that the above martingale property does not necessarily hold when
we let J (i/(n+l»
n
S~O,
= J(i/(n+l», l<i<n [unless J(u) = u: O<u<l].
--
Also, when
so that given the ranks of Xl, ••• ,X , X + does not have a constant probn
n l
ability (l/(n+l»
of having the rank i(l<i<n+l),
~he
above margingale property
does not hold, in general.
5.
THE PROOF OF THE MAIN THEOREM
The proof is accomplished in several steps.
First, we prove the following
lemma.
Lemma 5.1.
For Vs(>O),3 positive constants
K~l), K~2)
and a sample size ns'
such that for n>n ,
-s
(5.1)
p{ [C (SL -S)+A'r /2/B (F)] < -K (1) (log n) 2} < K(2) n-s •
n,n
a s s
Proof.
(5.2)
We shall only consider the proof of (5.1) as the proof of (5.2) follows
on the same line.
A
By virtue of the translation-invariance of the estimates Su
,n
A
and SL
,n
,we may, without any loss of generality, assume that S=O.
Then, by
(2.7), we have
p{C Su -AT /Z/B(F) > K(l) (log n)2}
n ,n
a
s
= P
S=O
{Tn-rn >T(l)_T
}
n
n'
(5.3)
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where we write T = T(X -b c*), b
-n o-n
0
n
= AT,a /2/B(F)
and
Tn = T(X-n-[b
0
+K(l) (log n)2]c*) •
s
-n
By theorem 3.2, with probability ~ 1-0(n- s ),
A
(1)
-
T -T = -K
n n
s
Also, using the fact that
0
(log n)2B(F) + O(n- (log n)3), 0>0.
~;: T~l)
= -ATa / 2 and writing Sn(x;b o )
(5.4)
= Ii~lc~iF(X-boC~i)'
we obtain by some standard computations that
00
+ O(n-~)
J J[F(x)]dS n (x;b 0 ) = -b 0 B(F)
-00
= T(l)
n
+ O(n-~).
(5.5)
Hence, it suffices to show that with probability ~ 1-0(n- s ),
00
1
f
-00
T
J[F(x)]dS (x;b ) 1 < K(l) (log n)3/2.
non
s
(5.6)
The left hand side of (5.6) can be written as
00
1
00
f' {J[F(x)]-J[Hn
(x;b )]}dS (x;b ) + f J[H (x;b )]d[S (x;b)-S*(x;b)]
on
0
non
n
-00
I
-00
<
I
00
f' {J[F(x)]-J[Hn
(x;bo
)]}dS
n (x;b 0 )1
-00
00
+
I J [S*(x;b
)-S (x;b )]J'[H (x;b )]dH (x;b )1
non
0
non
0
(5.7)
-00
It can be shown by using theorem 2 of Hoeffding (1963) that sUPIS*(x;b )-S (x;b)
x
non
00
s
O«log n)~), with probability>
'J IJ'[Hn
(x;bo
)] n
IdH (x;b 0
) <
- 1-0(n- ). Also, -00
(n+1)
-1, n
Li=lK(i(n+1-i)/(n+1»
2
-1
= O(log n) [by (2.9)].
1
Hence, the second term
on the right hand side of (5.7) is O«log n)3/2) with probability
~ 1-0(n- s ).
Finally, precisely on the same line as in (3.42)-(3.55), it follows that the first
term on the right hand side of (5.7) is also O«log n)3/2), with probability
~
1-0(n-s ).
Hence the proof is complete.
=
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25
A direct consequence of theorem 3.2 and lemma 5.1 is the following.
Lemma 5.2.
For every s(>O), there exists positive constants
(K~l), K~2» and
a sample size ns ' such that for n>ns '
(5.8)
Also, we have the following lemma whose proof follows along the lines of
Sen (1969, Section 3).
Lemma 5.3.
For every real x (-oo<x<oo),
(5.9)
Finally, for the "uniform continuity in probability" (for definition, see
Anscombe (1952»
Lemma 5.4.
of {Su
,n
} with respect to {C- l }, we have the following:
n
For every positive
£
and
n,
there exists a 0(>0), such that as
n~,
(5.10)
and a similar statement holds for {SL
Proof.
,n
}.
By theorem 3.2, lemma 5.1 (where we let s=l+h, h>O), and (2.17), we
h
have, with probability ~ 1-0 (n- ), for all In'-nl<on,
A
= -[C /C ,][l/B(F)][T(X ,-SU ,c ,)-T(X ,-S£ ,)+0(1)]
n n
- n , n -n
-n
n
A
+ [l/B(F)] [T(X-n-SU ,n-n
c )-T(X
-Sc )+0(1)]
-n-n
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.e
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·1
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I
= -[A/B(F»){T0: 12[l-C n Ic n ,)}+(C n Ic n ,)[T(X
-Sc »)/B(F)
-n ,-Sc-n ,)-T(X
-n-n
- T(X -Sc )[l-C IC ,)/B(F)+o(l).
-n -n
n n
By (2.12), I ,su~<~ Il-C Ic ,1<0', where 0'(>0) depends on 0, and can be made
n n
n -nl un
arbitrarily small when 0 is made so.
--I
Also, the asymptotic normality of
T(X-n-Sc-n )/A (with zero mean and unit variance) implies that IT(X-n-Sc-n )1 is
bounded, in probability, as n400 •
Hence, it suffices to show that as n400
P{I ,su~<~ IT(X -Sc )-T(X ,-Sc ,)I>n} < E.
n -nl un
-n -n
-n
-n
(5.12)
Since IT(X
(where T*n is
-n-(3c-n )-T(X-n ,-(3c-n ,)1<1
- (C n ,-C n )/C n ,liTn I + c-7IT*,-T*I,
n
n
n
defined in section 4), it seems sufficient to show that
P(3=O{' n ,su~<~
-nl un IT*,-T*I>nc
n
n
n } < E,
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(5.11)
(5.13)
and in view of the martingale property of T*, see theorem 4.1, from the Kolmogorov
n
inequality (cf. Loeve (1963, p. 386»
P
for martingales, we get,
su~
IT* -T*I>nc }
{
(3=0 In'-nl<on n' n
n
2
The rest of the proof follows from (2.12) and the fact that A
n
= A2+0(1),
for
large n.
We now return to the proof of theorem 2.1.
By (2.17) lemma 5.2 and the
definition of N(d), it follows that for all d>O, N(d) is finite a.s. and is
in d.
We also note that
~
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27
(5.14 )
Hence, in order to show that E[N(d)]<oo, it suffices to show that for large n,
= o(n-l-n),
P{N(d»n}
where n>O.
(5.15)
By definition, the left hand side of (5.15) is equal to
A
= pic n (~-
A
-SL
~U,n,n
»2dC}
n
= O(n-s ),
A
A
(where we can let s>l), as by (5.8), C (SU -SL )
.
n
,n,n
.
1
where as by (2.11), 2dC
fact that
Su ,n-~L ,n>0
> 2dC n~.
n -
0
(5.16)
= 2AT
-0
N
~
/
2/B(F) + O(n
Hence, for every d>O, E[N(d)]<oo.
a.s. for each n, we have dlimO
N(d)
+
lim
monotone convergence theorem d+O E[N(d)]
=
=
00
2
(log n) ),
Using the
and finally by the
00.
Now (2.19) follows directly from (5.8), (2.12) and (2.22); (2.20) also
follows from theorem 1 of Anscombe (1952) after using our lemmas 5.3 and 5.4.
To prove (2.21), we let for each d(>O),
(5.17)
[x] being the largest integer contained in x, and write,
where the summations
~1' ~2
n>n 2 (d), respectively.
and
Since Q(
~3
extend over n<n1 (d), nl(d)<n~n2(d) and
lim
) is +, d+O v(d) = 00 and (2.19) holds, for
every E>O, there exists a d (>0), such that for all O<d<d ,
E
-
E
P{nl(d)<N(d)~n2(d)} ~ P{IN(d)/Q-l(v(d»-ll<E} ~ l-n, where n(>O) is a (preassigned) small number.
Hence, for d<d ,
-E
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28
(5.18)
Also, by the same technique as in (5.16), we have
+ {Q-l(V(d»}-lL P{N(d»n}
3
= {Q-l(V(d»}-1{O(n (d»O[(n (d»-s] + O[(n (d»-s+1]}
2
2
2
= O[(n 2 (d»
-s+l
]
~
0 as
n~,
(5.19)
as in (5.8) we can let s>l and Q-l(V(d»+oo as d~.
Finally,
(5.20)
Thus, the proof of (2.21) follows from (5.18)-(5.20).
Remark:
Using (5.12), the classical Wa1d-Wolfowitz-Noether-Hajek theorem [cf.
Hajek and Sidak (1967, p. 160)] on the asymptotic normality of T (S) for fixed
n
but large n, and (2.19), it readily follows from theorem 1 (p. 601) of Anscombe
(1952) that TN(d)(S) has asymptotically (as
d~O)
zero mean and variance A2 defined in (2.23).
a normal distribution with
Also, on using theorem 3.2, it
follows that for every bEIN(d) , TN(d)(S+b) + bB(F) has asymptotically the same
normal distribution.
These results give simple proofs of the asymptotic
normality of regression rank statistics based on random sample sizes both in the
null (b=O) and the non-null
(b~O)
situations.
The results trivially extend to
any stopping variable N , indexed by a sequence {r} and a sequence of positive
r
integers {n }, such that n +00 and N /n ~ 1 as r+oo.
r
r
r r
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29
6.
ASYMPTOTIC RELATIVE EFFICIENCY
For any two procedures A and B for determining (sequentially) bounded
length confidence intervals for
S
(with the same bound 2d), let PA(d) and PB(d)
be the coverage probabilities and NA(d) and NB(d) the stopping variables
corresponding to the respective procedures.
Then, the ARE of the procedure A
with respect to the procedure B is given by
(6.1)
provided limd+OPA(d) = limd+OPB(d) and either of the limit exists.
GIeser (1965) considered the case when n -1 C2+C>O as n+oo. However, one can
n
easily extend his results when (2.10) and (2.11) hold.
Thus, if G denotes his
procedure, theorem 2.1 also holds with the change that v(d) has to be replaced
by vG(d) = 02L~/d2, 0 2 being the variance of the distribution of F(x) in (2.1).
Writing R for the proposed procedure and using (2.21), we have now,
(6.2)
By definition, VG(d)/v(d) = 02 B2(F)/A 2 is independent of d.
Then, e*=s-l(e) is monotonic in e with e*=l when e=l.
where
V~(d)
as d+O.
-1
-1
= Q (VG(d», v*(d) = Q (v(d».
Both
Write e=s(e*).
Write ¢d=V~(d)/V*(d),
V~(d)
and v*(d) tend to
00
Thus,
(6.3)
Using (2.12) and proving by contradiction, we have,
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(6.4)
The expression 02 B2(F)/A 2 is the Pitman-efficiency of a general rank order test
with respect to Student's t test.
~
In the particular case, when J(u) =
~
-1
(u),
being the distribution function (d.f.) of a standard normal variable, (normal
score) it is well-known (cf. Chernoff and Savage (1958»
that 02 B2(F)/A 2 ~ 1
for all d.f. F with a density f and a finite second moment, equality being attained
when and only when F is normal (0,0 2) d.f.
From monotonicity, it follows now
from (4.4) that in this case e*>l, equality being attained if and only if F is
00
Also, when J(u)=u (Wilcoxon score), e* = s-1(120 2 (
normal.
J
f2(x)dx)2) ~ s-l
_00
(0.864) (cf. Hajek and Sidak (1967, p. 280».
In the case of equispaced regres00
sion line, t. = a+(i-1), i=1,2, ••• , C2 = n(n 2-l)/12, e* = {120 2 { J f 2 (x)dx)2}1/3.
~
n
_00
For normal F, this reduces to (0.955)1/3 : .985, while the infimumn is given by
(0.864)1/3 ~ .953.
In the special case when c i is either 0 of 1, S is the difference in the
location parameters of the two distributions F(x-a') and F(x-a'-S).
classical two-sample problem.
If at the n
zero, C2 = m (n-m )/4 < n/4, for all n>l.
n
n
n
-
th
stage, m of the c i are 1 and rest
n
Looking at the definition of C2 ,
(2.19) and (2.21), we may observe that an optimum choice of m is
n
integral part
of~.
This is the
n
[~n],
the
Thus, among all designs for obtaining a bounded length
confidence interval for S, in this problem, and optimum design (which minimizes
EN(d) for small d) consists in taking every alternate observation for the two
distributions.
Here, n-lC2~ and the ARE reduces to 02 B2(F)/A 2 various bounds
n
for which have been discussed earlier.
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REFERENCES
Albert, A. (1966): Fixed size confidence ellipsoids for linear regression
parameters. Ann. Math. Statist., 37, 1602-1630.
Anscombe, F.J. (1952): Large sample theory of sequential estimation.
Camb. Phil. Soc., 48, 51-58.
Proc.
Chernoff, H. and Savage, l.R. (1958): Asymptotic normality and efficiency
of certain nonparametric test statistics. Ann. Math. Statist., ~,
972-994.
Chow, Y.S. and Robbins, H. (1965): "On the asymptotic theory of fixed width
sequential confidence intervals for the mean." Ann. Math. Statist.,
36, 457-462
GIeser, L.J. (1965): "On the asymptotic theory of fixed-size sequential
confidence bounds for linear regression parameter." Ann. Math. Statist.
36, 463-467.
Hajek, J. (1962): Asymptotically most powerful rank-order tests.
Statist., 33, 1124-1147.
Ann. Math.
(1968): Asymptotic normality of simple linear rank statistics under
alternatives. Ann. Math. Statist., 39, 325-346.
,
y
,
Hajek, J. and Sidak, Z. (1967):
York.
Theory of Rank Tests, Academic Press, New
Hoeffding, W. (1963): Probability inequalities for sums of bounded random
variables. J. Amer. Statist. Assoc., 58, 13-29
Jure~kov~, J. (1969): Asymptotic linearity of a rank statistic in regression
parameter, Ann. Math. Statist., 40, 1889-1900.
Lo~ve, M. (1963):
Probability Theory (3rd Edition), Van Norstrand, Princeton.
Sen, P.K. (1969): On a class of rank-order tests for the parallelism of
several regression lines, Ann. Math. Statist. 40, 1668-1683.