SEQUENTIAL CONFIDENCE INTERVAL FOR TIlE
REGRESSION COEFFICIENT BASED ON KENDALL'S TAU
By
Malay Ghosh and Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 658
January 1970
Sequential Confidence Interval for the Regression Coefficient
Based on Kendall's Tau*
BY MALAY GHOSH AND PRANAB KUMAR SEN
Department of Biostatistics
University of North Carolina at Chapel Hill
SUMMARY
The object of the present investigation is to consider a robust procedure
for the problem of providing a bounded-length confidence interval for the
regression coefficient (in a simple regression model) based on Kendall's (1955)
tau.
The problem of estimating the difference in the location parameters in the
two-sample case may be viewed as a special case of our problem.
It is shown that
the estimate of the regression coefficient based on Kendall's tau [cf. Sen (1968)],
as extended here in the sequential case, possesses certain desirable properties.
Comparison with the procedure based on the least squares estimator [considered
by GIeser (1965) and Albert (1966)] is also made.
1.
INTRODUCTION
Consider a sequence {Xl,X Z, .... } of independent (real valued) random
variables with (absolutely) continuous cumulative distribution functions (cdf)
F (x) ,F 2 (x) , ... , where
l
F.(x)
= F(x-a-8t.),
1
1
i=1,2, ... ;
the t.1 are known regression constants, 8 is the regression coefficient and a
is a nuisance parameter.
It is desired to determine a confidence interval
* Work supported by the National Institutes of Health, Grant GM-12868.
(1.1)
2
I n = {s:
SL ,n -< S -< ~-
} such that (i) P{S£I n } = I-a, the desired confidence
coefficient and (ii) 0 -< ~- ~L ,n -< 2d, (d predetermined). Since the form
~U,n
~u,n
F( ) is not known, we are not in a position to prescribe any fixed-sample size
procedure valid for all F.
Sequential procedures for such a problem, based on the classical least
squares estimators of a and S, are due to GIeser (1965) and Albert (1966).
These procedures, like ours (to follow), are based on the method suggested by
Anscombe (1952) and Chow and Robbins (1965).
However, being based on the least
squares estimates, these procedures are vulnerable to gross errors or outliers,
and may be quite inefficient for distributions with heavy tails (e.g., the Cauchy,
logistic or the double exponential distribution).
For this reason, we consider
here an alternative robust procedure based on the nonparametric estimate of S,
considered in Sen (1968).
The procedure is explained in Section 2.
deals with the main results of the paper.
Section 3
Section 4 is devoted to the study of
the asymptotic relative efficiency (A.R.E.) of the different procedures.
In
Section 5, we consider the two-sample location problem, which is a particular
case of (1.1) when the t.1 can be either 0 or 1.
The appendix is devoted to the
asymptotic linearity of a stochastic process involving Kendall's tau; this result
is used repeatedly in Section 3.
2.
THE PROPOSED SEQUENTIAL PROCEDURE
For every real b(-oo<b<oo), define Z.(b)
= X.-bt.,
i=I,2, ... , and let
1
1
1
Un(b) = (n)-1
L c(t.-t.)c(Z.(b)-Z.(b)),
2
..
J 1
J
1
1~l<J~n
where c(u) is equal to 1, 0 or -1 according as u>, = or <0.
(2.1)
Note that by defini-
tion,Un(S) is distribution-free and its distribution (symmetric about zero) is
3
tabulated (for small values of n) by Kendall (1955) and Smid (1956).
If we let
a
V = (1/18){n(n-l)(2n+5)-L:- nl u .(u .-1)(2u .+5)},
n
J= nJ nJ
nJ
(2.2)
[where among (tl, ... ,t) there are a (>2) distinct values with frequencies u "
n
n~
j=l, ... ,an]' then for large n, (~)Un(8)~ has approximately a nonna1 distribution with mean
a and
variance 1 [cf. Hoeffding (1948) and Kendall (1955)].
such, for each nand !n = (tl, ... ,tn), we can find an
U~
As
t such that
'-n
P{-U*n t -< Un (8) -< U*n t 18} = I-an'
'-n
'-n
where
(2.3)
an~ as n~. [For large n, U~'!n ~ \~./2~/(~), where \]./2 is the upper
100a/2% point of the standard normal distribution.]
As in Sen (1968), we use
(2.3) to derive the following (fixed-sample) confidence interval for 8.
y..
1J
where Sn = {(i,j): t·ft.,
J
1
= (X. -X. ) /
J
1
l~i<j~n}.
(t . - t . ),
J
1
(i, j) E:S ,
n
Let
(2.4)
We denote by n* the mnnber of elements of
Sn (i.e., the number of distinct pairs (t.,t.)
in -n
t ), and denote the n* ordered
1
]
values in (2.4) by
(2.5)
Also, let
M(l) = h[n*-(n)U*
]
n
2
2 nt'
'-n
(2.6)
Then, as in Sen (1968), we have
p{y
(1)
n(Mn )
< 8 < Y
(2)
n(Mn +1)
IS} = I-an .
(2.7)
4
Now, in order to obtain a confidence interval for S of length
~
2d, we define a
n0 (the initial sample size,
stopping variable N=N(d) to be the first integer>
may be 2 or more), for which Y
(2)
-y
(1))
n(Mn +1) n(Mn
(sequential) confidence interval for S is
~
2d.
Then, our proposed
(2.8)
In principle, our procedure is similar to that of Anscombe (1952) and of Chow and
Robbins (1965).
However, they used the sample mean square due to error to set
up an appropriate confidence interval, whereas we use Kendall's tau to derive such
an interval.
The main results are stated in the next section.
PROPERTIES OF TIlE CONFIDENCE INTERVAL IN(d)
3.
In the remainder of the paper, we shall stick to the following notations and
assumptions.
Let
(3.1)
. n1 (t.1 -t-n )( 1.
n = {L1=
p
n+l
- ---,,-)}
~.
I {AnTn };
(3.2)
a
An2 = (1/12){n(n 2 -l) - I.nlu
.(u 2 .-l)}.
J= nJ nJ
(3.3)
Then, concerning {t }, we assume that
-n
max
-
(i) 1·
<l<n It"1-tn IIT n+0 as n-+oo,
(3.4)
(3.5)
n
(1·1·1·) limn inf[,a
L" lU . (n-u. )1 n 2] >0 ,
J= nJ
nJ
(3.6)
5
(iv)
(v)
1·
n~ Pn = P, where Ipl>o,
(3.7)
r n2 = Q(n) is a strictly increasing function of n, such that for
every a>O
1·
lim
n: Q(ann)/Q(n) = sea) whenever n~ an = a,
(3.8)
where sea) is strictly monotonically increasing and continuous
in a, and s(l)=l.
Note that our (ii) and (v) are less restrictive than similar conditions of
GIeser (1965), who requires that lim n- l T2 = c>O. If t.=i, i=1,2, •.. , (i)-(v)
n~
hold, but GIeser's does not.
n
1
Note that by (3.3) and (3.6), we have
(3.9)
Concerning F(x) in (1.1), we assume that f(x) = FI(x) exists (a.e.), and is
continuous in x (a.e.); also assume that
00
Jf 2 (x)dx = B(F)<oo.
(3.10)
-00
Our main theorem of the paper is the following.
Theorem 1.
Under the assumptions made above, N(=N(d)) is a non-increasing function
of d(>O) , N(d) is finite almost surely (a.s.), E[N(d)]<oo for all d>O, lim N(d)=oo
d+O
a.s. , and lim E[N(d)]=oo. Further,
-;....;....<:...-- d+O
lim N(d)/Q-l(v(d)) = 1 a.s.,
d+O
(3.11)
for all F,
(3.12)
lim E[N(d)]/Q-l(v(d)) = 1,
d+O
(3.13)
lim P{S£IN(d)} = I-a
d+O
6
where
(3.14)
Proof.
~
L,n
The proof of the theorem is completed in several steps.
= Yn(M(l)) and
n
~~u,n
First, we let
= Yn(M(2)+1)' which are defined as in (2.6) and (2.7).
n
Then we have the following.
For every 0>0 there exists an no (0), such that for n>n
(0),
- 0
Lennna 3.1.
(3.15)
P{SL
,n
<
S-(log n)/(p T )} < n-(l+o).
n n -
(3.16)
We shall only prove (3.15) as (3.16) follows on the same line.
tion of ~~u
,n
,U
n
P{~.
~u,n
(Bu ,n) > -U~ '-n
t.
By defini-
Hence,
> S+(log n)/(p nTn)} = P{Un (S+(log n)/p nTn) -> -U*n,! }
= P{Un (S+(log n)/pnn
T ) - E[Un (S+(log n)/pnn
T )]
~
-
U~,t
-n
-E[Un(S+(log n)/PnTn)]}·
(3.17)
-~
.?2 n
Using the fact that for large n, U~,t ~ Ta/2v~/(2) = O(n ), and
-n
E[Un(S+(log n)/PnTn)] = - 4An (log n)(2)
write -U~,t -E[Un(S+(log n)/PnTn)] as
-1
[B(F)+o(l)] = O(n
-~
log n), we may
-n
n -1
4An (log n)(2)
B(F) [l+o(l)+O(log n)
-1
].
Now, we make use of a result by Hoeffding (1963; (5.7)) on the deviation of a
(3.18)
7
U-statistic and obtain from (3.17) and (3.18) that for n adequately large
n
A
2
P{£L
> S+(log n)/p nT}
~U,n
n -< exp[-[-2]t n/4]
< exp[-(n-2)tn2 /8],
(3.19)
n -1
-k
where t n = 4An (10g n)(2) B(F) = O(n 2 log n). Thus, (n-2)t~/8 can be made
greater than (l+o)log n, for any 0>0, when n is made greater than no(o). Hence,
for n>n
(0), the right hand side of (3.19) can be made less than
-0
exp[-(l+o)log n] = n-(l+o).
Q.E.D.
Lemma 3.2.
There exists two positive numbers °1 and ° 2 and an n, say no (01,02)'
such that for all n ~ no (01,02)'
-0
p{11.! B(F)PnTn(SU,n-SL,n)/Ta/2-ll
~ O(n llog n)}
-0 -1
<
O(n 2 ).
(3.20)
The proof of the lemma directly follows from lemma 3.1 and the following
theorem whose proof is sketched in the Appendix.
Theorem 3.3.
There exists two positive °1'° 2 and an no(01'02) such that for
n ~ no (01,02)'
P
P{I a ISul
< og n I (n2){Un (S+a/p nTn ) - Un (S)} + 4aB(F)An I~n
~
Lemma 3.4.
O(n
-0 1
log n)} .:. O(n
-1-0 2
).
(3.21)
For every real x (-oo<x<oo)
(3.22)
-00
8
The proof is given in Sen (1968).
For the asymptotic normality of
PNTN(SU ,N- S), we require, as in Anscombe (1952), the 'uniform continuity in
probability" of {it.
"'u,n} with respect to pT.
n n
lerrnna.
Lerrnna 3.5.
For every positive
E:
For this, we have the following
and n there exists a 0(>0), such that as
n+oo
(3.23 )
and a similar statement holds for
Proof.
{6L,n} .
By virtue of lemma 3.1 and theorem 3.3, and the definition of ~T
...u ,n'
= {Ta /2/ 1IZ B(F)}{(PnTn)/(Pn,Tn,)-l} +
[1IZB(F)]-l{[(PnTn)/(Pn,Tn ,)]
(~')/~,Un,(S)-(~)V~~n(S)}+O(l)
+ [(n2)/~]
[Un ,(S)-Un (S)] + 0(1).
n
(3.24)
Since Un(S) is a U-statistic in the independent and identically distributed
random variables Zi(S),
l~i~n,
a reverse martingale sequence.
it follows from Berk (1966) that {Un(S)} forms
Hence, if we write n =no, and let W. = U + . (S) nno-J
o
J
U + ·+l(S), 1~<2n ,W = U
(S), it follows that {WO,Wl , ... ,W2 } possesses
n no -J
- 0 0
n+no
no
the properties of amartingale difference sequence. Hence, by using the well-known
9
Kolmogorov inequality [cf.
Lo~ve
(1963, p. 386)], we have
max
P{l<1<
2n IWl+···+W.J I>tn} -< t n-2 E(Wl +... +W2n )2
~- 0
0
°
(3.25)
But, WI+... +WJo = Un+n - JO Cfn -Un+no ((3), 19<- 2n0 , and the right hand side of (3.25)
o
is equal to t~2{V[Un_n (S)]-V[Un+n (S)]} = (4/9)(2not~2/(n2-n~))(1+O(n-l)).
9J.
Hence, letting t n = nn ~, we have
0
(3.26)
This limnediately leads to (by proper choice of 6)
(3.27)
for n sufficiently large.
Also, by (5.7) of Hoeffding (1963),
p{IUn (S)I>K£ ,,}<£",
where K£ ,,<00.
-
(3.28)
[s(1-6), s(1+6)], and hence, differs from 1 by an arbitrarily small quantity, as
(by (iv) and (v)), n(~)/Vn = 9/2 + O(n-l) (by (iii)), I (~')-(~)1~6(2+6)n2,
3
3
3
2
and IV-~-V-~I
n' n = IVn ,-Vn I/{I~,+~I~,}
n n n n = 0(n- )Ivn ,-Vn I = 0(n- ) [(2/9){n (1+36+0(6 ))n 3 } + 0(6)n 3 ] = 0(6), where the last order follows by using (iii) on the second
ll+oo
term of the right hand side of (2.2).
(3.28) and the above discussion.
Thus, (3.23) follows from (3.24), (3.27),
Hence the lennna.
Returning now to the proof of theorem 1, we note that by lennna 3. 2 and the
definition of N(d) , for all d>O, N(d) is finite a.s., and it is non-increasing
in d.
To show that E[N(d)]<oo for every d>O, we write
10
E[N(d)] = LnP{N(d)=n} = Ln> OP{N(d»n}.
(3.29)
Hence, it suffices to show that for every d>O, Ln>OP{N(d»n} converges; a
sufficient condition for this is to show that as
ll+OO,
P{N(d»n} ~ O(n- l - y), where y>O.
(3.30)
Now, by definition
< P{(~-8 »2d} = P{I.rB(F)p nTn (~-6 »2dl.r B(F)pnTn}
""U ,n L,n
""U, n L,n
-1-0 2
~ O(n
), O2>0,
-
(3.31)
-0
as by (3.20), I.r B(F)p nTn (~_
-8 )-1 = O(n 1 log n) with probability ~ l-O(n
""u ,n L ,n
and by (ii) and (iv) PnTn-t<Jo as ll+OO. Thus, E[N(d)]<oo, for every d>O. Using the
fact that~.
""u ,n
-SL ,n>0
(a.s.) for each n, we have lim N(d) =
d~O
00.
-1-0
Now, from the
Monotone Convergence Theorem, lim E[N(d)]=oo.
d~O
(3.11) is a direct consequence of (3.20), the definition of v(d) in (3.14),
(iv) in (3.7) and (v) after (3.7).
(3.12) follows readily from theorem 1 of
Anscombe (1952) along with our 1errnna 3.2, 1errnna 3.4 and 1errnna 3.5.
To prove
(3.13), we write
E[N(d)]/Q
-1
[v(d)] = {Q
-1-1
[v(d)]} Ll +L 2+L 3 nP{N(d)=n},
where the summation Ll extends over all n<nl(d) , L2 over all
L3 over all n>n 2 (d) , and where
nl(d)<n~n2(d)
Q(n.(d)) = v(d){l+(-l)i€ + (1+(-1)i)/2}, i=1,2.
1
(3.32)
and
(3.33)
2),
11
Since llin v(d) = and llin N(d)/Q-l(v(d)) = 1 a.s., and Q( ) is ,for every
d-+O
d-+O
£>0, there exists a value of d, say d (>0), such that for all O<d<d ,
00
£
-£
P{nl(d) ~ N(d) ~ nZ(d)} ~ P{IN(d)/Q-l(v(d))-ll<£} ~ l-n, where n is arbitrarily
small.
Hence, for d<d ,
-£
{Q
-1
[v(d)]}
-1
LlnP{N(d)=n}
~
(l-£)P{N(d)<nl(d)}
~
n(l-£).
(3.34 )
Also, proceeding along the same line as in (3.31), we have
{Q-1[V(d)]}-l L3nP{N(d)=n} = {Q-1(v(d))}-lnz (d)P{N(d)=n (d)}
z
+
{Q-l(v(d))}-l L3P{N(d»n}
~
{Q
-1
(v(d))}
-1
[L 3P{N(d»n}
+
n Z(d)P{N(d»nZ(d)-l}]
-8
= {Q-1[v(d)]} -l[o(n (d))
z
-8
= O[(nZ(d))
-1-8
Z + O(nZ(d))O([nZ(d)]
Z)]
Z
] -+ 0 as d-+O,
-1
since Q [v(d)]+oo as d-+O and hence, nz(d)+oo as d-+O.
(3.35)
Finally, elementary computa-
tions yield that
(3.36)
Hence, (3.13) follows from (3.34), (3.35) and (3.36).
4.
Q.E.D.
ASYMPTOTIC RELATIVE EFFICIENCY
For two procedures A and B for determining (sequentially) bounded-length
confidence intervals for S (with the same bound Zd), we define the A.R.E. of
A with respect to B as
12
where NA(d) and NB(d) stand for the respective stopping variables.
It may be noted that though GIeser (1965) considered the case where
n-lT~~>O as n~, his results can be easily extended to the case where our (ii)
and (v) of section 3 hold.
As such, if NL(d) stands for the stopping variable
for his procedure (based on the least squares estimators), it follows that our
theorem 1 also holds for NL(d) with the only change that v(d) has to be replaced
by vL(d) =
02T~/2/d2,
where 02 is the variance of the distribution F(x) in (1.1).
Hence, using the result in (3.13) for both the cases and writing NK(d) for the
proposed procedure, we have
eK L = lim {Q
,
d-+O
-1
(vL(d))/Q
-1
(4.2)
(v(d))}.
Note that, by definition vL(d)/v(d) = l20 2p 2B2 (F) = e, is independent of d, and
let us write e = s(e*), so that e* = s-l(e) is monotonic in e, with e*=l when
e=l.
-1
Also, let ¢d = Q (vL(d))/Q
-1
-1
(v(d)), Q (vL(d))
= vt(d) and Q-1 (v(d)) = v*(d).
Note that by (v) of section 3, vt(d) and v*(d) both tend to
00
as d-+O.
Hence, we
have
s (e*) = e = vL (d) / v (d) = 1im {vL(d) / v (d) }
d-+O
= lim {Q(vt(d))/Q(v*(d))}
d-+O
= lim {Q(¢(d)v*(d))/Q(v*(d))}.
(4.3)
d-+O
Using (3.8) and proving by contradiction, it follows that
1im ¢ (d) = e* = s
d-+O
-1
(e) .
(4.4)
13
Hence, from (4.2) and (4.4), it follows that
=s
e K, L
-1
(4.5)
(e)
-00
In Sen (1968), various bounds for e = l2a 2p2(
f
00
f 2 (x)dx)2 are studied, with
-00
special reference to the bounds for p2.
As we shall see in the next section, for
the two-sample problem, p=l, and see) = e, so that (4.5) reduces to l2a
2
f
(
00
f2(x)dx)2,
-00
the usual A.R.E. of the Wilcoxon two-sample test with respect to the Student's
00
t-test.
Since l2a
2
(
f
f2(x)dx)2
~
0.864 for all continuous F, it follows that
-00
(4.6)
i=1,2, ... , so
Consider now the case of equispaced regression line, where t.=i,
1
that Tn2 = n(n 2-l)/12 • In this case, we get at once, that p=l and
00
eK L = (12a 2 (
,
f
1
_
f2(x)dx)2)3
(4.7)
-00
For the case of normal F, this reduces to (0.95)1/3 ~ .985, while the infimum
in (4.6) is given by (0.864)1/3 ~ .953.
This clearly indicates the robustness
and efficiency of the proposed procedure.
For many non-normal cdf (e.g.,
double exponential, logistic, Cauchy, etc.) the proposed procedure is more
efficient than the least squares procedure.
5.
lWO-SAMPLE LOCATION PROBLEM
Consider the special design, where the t.1 can either be 1 or O.
Thus, we
have only two different distributions F(x-a) and F(x-a-S), where S denotes the
difference in the location parameters of the two distributions.
If at the n-th
stage, we have mn of the t equal to 1 and the rest equal to zero, we obtain that
i
14
r n2 = mn (n-mn )/n
< n/4, for all n>l.
-
(5.1)
Looking at the definition of r n2 = Q(n) , (3.11) and (3.13), we observe that an
optimum choice of mn is [~], the integral part of~. Thus, among all designs
for obtaining a bounded-length confidence interval for S, in this problem, an
optimum design (which minimizes the expected value of N(d) for small d) consists
in taking every alternative observations for the two distributions. Here also,
l
p =p=l, and n- r 2 + 1/4, and hence, by (4.5), the A.R.E. reduces to l2a 2 ( ff 2 (x)dx)2,
n
n
00
-00
various bounds for which are well-known.
Looking at (2.4)-(2.7), we observe that
1
n* = m (n-m ) tV ifl2,
and U* t can be computed, for small values of n, from the
n
n
n'-n
extensive tables given in Owen (1963). For large n, note that Vn , defined by
(2.2), reduces to (1/18){2[n3-m~-(n-mn)3] + 3[n2-m~-(n-mn)2]} ~ {2 n 3+O(n 2),
and hence,
U~
t
can be computed by reference to the usual nonnal probability
'-n
tables.
6.
APPENDIX:
PROOF OF TIffiOREM 3.3
We may assume without any loss of generality that S=O (as otherwise, we
may set b=S+b' and work with b').
Also, we shall explicitly consider the case
of a£[O, log n] as the case of a£[-log n, 0] follows on the same line.
b
n
tV
n
Let
°1 (as n+oo), 01>0, and let n
(r/bn )log n, r=l, ... ,bn . Since, Un(b)+
r,n =
in b, for a£[n r- 1 ,n ,nr ,n], we have
(6.1)
and similar inequalities involving their expectations.
few simple steps,
Then, we get after a
15
ae[n
Sup n ] IUn (0) - Un (I
n
n + EUn (I
a PnT)
a PnT)
r-l,n' r,n
I
(6.2)
where W .
n,]
= Un (O)-Un(n.
Ip T )
] ,n n n
,n=0, EUn (0)=0).
nO
+ EU (n. Ip T ),
n ] ,n n n
O~<b
- n
,1<r<b (note that
- - n
Hence,
(6.3)
<
max
O~~bn
[(n)/..f2]
2' n
Iwn,j I
+
Using (3.4), (3.11), nj ,n-n j - 1 ,n
= log n/bn and
A~/Vn
+
3/4 as n+oo [cf. Sen
(1968)], we get,
1
(6.4)
= (n.],n-no] -,n
1 )(A JV:)[B(F)+o(I)]
n n
-6
= O(n \og n), for all j=1,2, ... ,b .
n
Let
Y ..
nl]
</>(Yn
••l,Y]nk
]·)
12.irk2.n,
1~<b
l~<bn'
= X.-(n.
Ip T )t.(l<i<n, l<i<b),
1
],n n n 1 - ~- n
= c(tk-t.)
[c(Yn]
k'- Ynl]
k'- Ynl]
.. )],
. ·+(n·],nIp nTn )(tk-t.))-c(Y
1
1
n]
.
Wrlte,
W . = n -1 (n-l) -1 L\'~
[</>(Y .. ,Y k·)-E</>(Y .. ,Ynk ·) ] ,
n,]
l<i k<n
nl] n]
nl]
]
Note that W . is a U-statistTc minus its expectation (see
n,]
(W 0=0).
n,
e.g., Hoeffding (1948)).
- n
-1 \' n
2n]. = n
L' 12 ., = (n-l)
1= nl]
~
We can write, W .
n,]
-1 \' n
. = E 2n]., 1~ -<b.
Then,
n
n]
=
Lk 1(~·)c(tk-t.)
= r1
1
2(2 .-~ .) + R ., where,
n] n]
n]
[F(X.-a)-F(X.-a-(n·
1
1
],nIp nTn )(tk-t.))];
1
16
1
P[\Wn,J. I [(n2)/~]
n > cln
< P[
-
Izn,J.-~ nJ. I
-0 1
log n]
> ~t ] + P[ IR . \ > ~t ],
n
nJ
(6.5)
n
-~-o
-0
where cl(>O) is not depending on n, t n = cl~(~)-l n 1 log n = O(n
llog n),
as n+oo, since, Vn = O(n 3 ) , (as n+oo), (from (3.9) and the fact that An2 /Vn ~ 3/4
as n+oo).
Also, from (3.4), (3.9) and (3.10), it follows after some algebraic
manipulations that ~ . = 4n. (A /[n(n-l)]) (B(F)+o(l)) = O(n-~log n), uniformly
nJ
J,n n
in j=1,2, ... ,bn . Hence, there exists a positive integer n l such that for n~nl'
O<t'<~
.<l-~nJ.<1, uniformly in j=1,2, ... ,bn (t'-slt).
Also, Z . is the average
n nJ
n n
nJ
of n independent random variables each assuming the values 0 and 2.
Now, using
the inequality (2.1) of Hoeffding (1963), we get, for n>nl ,
P [k
2
< [(~'./(~'.+t')}
nJ
nJ n
IZnJ. -~ nJ. I>t'n]
n(~'.+t')
nJ
n {(l-~' .)/(l-~'.-t')}
nJ
nJ n
.-t')
nJ n]
n(l-~'
(6.6)
+ [{~'./(~' .-t')}
nJ
nJ n
where
~'.
nJ
=
~
..
nJ
.-t')
n(l-~' .+t')
nJ n {(l-~' .)/(l-~'.+t')}
nJ n]
nJ
nJ n
n(~'
It follows after some algebraic simplifications that as n+oo,
nt'2
= - nt' 2 j{ 2 (~ , . ±t ' ) (l-~ , . +t ' ) } + 0 (2 (~I n. ±t I)) .
n
nJ n
nJ n
nJ n
2
-c /4
= O(log n 1 ).
Also,
(6.7)
17
(6.8)
But,
2
E(RnJo) =
n2(n~1)2
~ n2(n~1)2
n2(~-1)2
LL
1~i1k~n
LL
1<i1k~n
V[¢(Y. "Ynk°)]
nlJ
J
E[E{¢(Y .,Y k·)IY °.}-E¢(Y oo'Ynk °)]2
nlJ n J nlJ
nlJ
J
0
L L V[¢(Y.
"Ynk°)]
1~i1k.2.n
nlJ
J
< (4 1)
- n n-
~ nJ.
(6.9)
5 2
= O(n- / log n).
It follows now from (6.5)-(6.9) and the Bonferroni inequality that
b
-c 2 /4
n
P{l~b Iw ·I>t } < b O(n 1 ) + L E(R2.)/(~t2)
.9_ n nJ n - n
j=l
nJ
n
(6.10)
First make a proper choice 01 and subsequently of c1 (>0) to make the right hand
-1-8
side of (6.10) less than or equal to [(const)(n
2)] for any given 02>0. The
result now follows from (6.3), (6.4) and the Bore1-Cante11i lemma.
18
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.
Proc. Camb. Phil. Soc., 48, 600-607.
BERK, R. H. (1966). Limiting behavior of posterior distributions when the
model is incorrect. Ann. Math. Statist., 37, 51-58.
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.
GLESER, L. J. (1965). On the asymptotic theory of fixed-size sequential confidence bounds for linear regression parameters. Ann. Math. Statist.,
36, 463-467.
HOEFFDING, W. (1948). A class of statistics with asymptotically normal
distribution. Ann. Math. Statist., 19, 293-325.
HOEFFDING, W. (1963). Probability inequalities for sums of bounded random
variables. J. Amer. Statist. Assoc., 58, 13-29.
KENDALL, M. G. (1955).
Company: London.
LOEvE, M. (1963).
Rank Correlation Methods.
Second edition, 1955.
Charles Griffin and
Probability Theory (3rd edition) Van Nostrand, Princeton.
OWEN, D. B. (1962). Handbook of Statistical Tables.
Reading, Massachusetts.
Addison-Wesley,
SEN, P. K. (1968). Estimates of the regression coefficient based on Kendall's
tau. J. Amer. Statist. Assoc., 63, 1379-1389.
SMID, L. J. (1956). On the distribution of the test statistics of Kendall and
Wilcoxon's two sample test when ties are present. Stat. Neerlandica, 10,
205-214.