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A LAW OF ITERATED LOGARITHM FOR ONE SAMPLE
RANK-ORDER STATISTICS AND AN APPLICATION
by
Pranab Kumar Sen and Malay Ghosh
Department of Biostatistics
University of North Carolina, Chapel Hill
Institute of Statistics Mimeo Series No. 687
June, 1970
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ABSTRACT
A LAW OF ITERATED LOGARITHM FOR ONE SAMPLE RANK-ORDER STATISTICS
AND AN APPLICATION by Pranab Kumar Sen and Malay Ghosh University
of North Carolina at Chapel Hill
A law of iterated logarithm has been established for one sample
rank-order statistics, and a test procedure for the classical one
sample location problem has been proposed which is of power I and
arbitrarily small type I error.
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A LAW OF ITERATED LOGARITHM FOR ONE SAMPLE
RANK-ORDER STATISTICS AND AN APPLICATION*
BY PRANAB KUMAR SEN AND MALAY GHOSH
University of North Carolina at Chapel Hill
The law of iterated logarithm for
sample sums of iidrv (independent and identically distributed random variables) was first proved by Khintchine [7] for Bernoulli
variables, and, later on, more general results in this direction
were due to Kolmogorov [8], Petrovski [9], Erdos [3], Feller [4]
and others. An excellent exposition of these results is available
in Feller [4], and Strassen [11], while the latter extends these
results to martingales.
More recently, these results have been
generalized to sample quantiles by Bahadur [1] and to U-statistics
by Ghosh and Sen [5].
In the present paper, we find a similar law
for one-sample rank-order statistics, and the details are provided
in section 2.
In section 3, for the classical one-sample location
problem, a test procedure has been proposed along the lines of
Darling and Robbins [2] with zero type II error, and arbitrarily
small type I error.
To achieve this, application is made of results
of section 2, and also of some strong convergence results in connection with one-sample rank-order statistics as given in Sen [10].
*Work supported by National Institutes of Health, Grant GM-12868 •
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Let {Xl' X2 ' •.. } be a sequence of iidrv defined on the measure
spaces (n, A, Pe), having a df(distribution function) Fe(x) = F(x-e),
e ) is an unknown parameter, and F
I
where e (E:
I
about zero, i.e. F(x) + F(-x) = 1, for all real x.
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E:
;j. 0' the class of
all df's continuous with respect to Lebesgue measure, and symmetric
Define for each
positive integer n,
(Z.l)
1
n
X = (Xl'."'X), 1 = (1, ... ,1), R. = -Z + l c(lx·I-lx·I),
~n
n -n
nl
j =1
1
J
i=l,Z, .•• ,n, where c(u) = 1, l/Z or 0 according as u >, =, or < O.
Let J n (i/(n+l)) = EJ(Unl.),
(2.2)
i=l,Z, .•. ,n, and,
1 n
Tn = Tn (X-n) = n- .I 1sgn x.EJ(UnR
), n~l,
1
1=
nl.
where sgn x = Zc(x)-l (x real),
Unl~ ... 'SUnn
are n ordered rv's (ran-
dom variables) from a rectangular (0,1) df, and J(u), 0'Su<l, is a
·
. f·
score f unctlon
satls
ylllg J()
u = u{-l(l+u)
r
--2--'
\1/
r
belng
. a symmetrlc
. df
defined on (-00,00), i.e.,
\f(x)
(2.3)
Define II =
Jl
Oz
degenerate, A
+
\fe-x) = 1, for all real x.
J(u)du, and, AZ =
>
J 2 (u)du. Whenever, \f(x) is non0
2
We assume that 0 < A < 00. Note that if \f(x)
o.
Jl
is uniform over (-1,1) or is the standard normal df, the corresponding Tn is termed the signed-rank or normal-scores statistic.
The
following theorem is proved.
TI-IEOREM Z.1.
If J
E:
LZ+8 for some 0 > 0, then under HO:
F
E:
j.0'
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(2.4)
2
lim supn-+oo III Tn I [A(2 log
. log n) 1/ ] = 1 a.s.
(2.5)
lim inf
III Tn I [A(2 log log n)1/2]
n-+oo
= -1
a.s.
Let Bn denote the a-field generated by §n = (sgn Xl'.'" sgn Xn )
and R
-n = (Rn1" .. ,Rnn), n >
- 1; clearly Bn t in n. Note that for
F E: go' e = 0, the two vectors of signs and ranks of absolute XIS
PROOF.
are stochastically independent (see [6], p. 40).
Then, using the notation EO for
EOCTn )
(2.6)
E~
nTn .
(to be continued later on),
= 0, EO(T-2n ) = nAn2 '
2
-1 n
2
2
I
E[J eu .)] < A <
where A = n
n
1
nl
-
(2.7)
o
-=
Write Tn
00.
+ EO[sgn X +l]EO[Jeu +lR
)
n
n n+ln+l
Also,
IBn ]
R .
R .
\'
[
(
nl)
eu
)
nl EJ eun+lR .+1 ) ]
= .L sgn Xi 1- n+l EJ n+lR. + n+l
1=1
nl
nl
n
_
n
=
I
i=l
sgn
x.
1
EJeunR ) = T
ni
n
where one uses the relation
(2.8)
I - (i/n) "" (i/n)Jn((i+I)/(n+I)) + ((n-i)/n)Jn(i/(n+I)).
n l
-
Thus, (Tn' Bn , n ~ 1) form a martingale sequence.
To prove the theorem, we need now verify only the conditions
of the theorem 4.4 of Strassen ([11], p. 334) which gives us access
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via the extension of the Kolmogorov-Petrovski-Erdos criterion to
martingales ([11], corollary 4.5, p. 337) to the law of iterated
logarithm as given in (2.4) and (2.5).
-
First define, Zl = TI , Zk = Tk-Tk _l (k
~
2).
Then,
R
+
2
sgn X J ( ~)]IB -I}
nn n
n
- J
n-l
n-l.
R .
I
sgn
X.{J ( ~~)
O i-I
1
n n
= E ([
tn-li)}]2IB ) + E [J2(~nn) IB 1 > A2
n
n-l
0 n~+l n-l - n'
(using the stochastic ndependence of sign and rank vectors under
HO and using the fact that EO(sgn XnIBn - l ) = EO(sgn Xn ) = 0).
.
n
2
. 2
'n 2
2
.
n 2 2
Thus, V = I EO(Z. I B·_I) + EO(Zl) ~ l A. + ~. Slnce, I A./nA ~1
n
i=2
1
1
2 1
1 1
as n~, we have, Vn~ as n~. Define now
(2.9)
f(t)
=
t(log log t)2/(log t)4, t ~ 3; f(t)
Then,
(iv) f(t) t, f(t)/t
Again, since
~
in t.
= 1, 0 < t
< 3.
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5
Then, we have,
n-l
l
~
i=1
r.!+1
i n n
[In 'n+l) - I n Cn+l)] + I n Cn+l) = 2Jn Cn+l) •
But,
J (-n ) = EJ (U ) < [EJ2+0 (U ) ] (1/ (2+0))
n'1i+T
nn nn
= [n
Po
1
J 2+0 (u) du](1/2+0)) =
O(n 2+0).
Further, since,
n-l
~
I .L
1=1
R.
R 1.
{J (n1) - J
( n- 1)} sgn X.
n n+l
n-l
n
1
I
n-l
R 1·
R 1·
R 1.+1
R 1·
< I max{ IJ (n- 1) - I - ( n~ 1) I, IJ (n- 1 ) - J ( n- 1)
- . 1
n n+l
n
n+l
n n+l
n 1
1=
n-l
l
R . +1
[J (n-l1
- . 1
n
n+l
1=
<
R 1.
) _ J ( n- 1)]
n n+l
E(2 2 1B
) < J2 (_n ) + A2 < J2 (_n ) + A2 n > 2.
n n-l - n'n+l
-n - n'n+l
'-
I}
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So,
2 n
2
Vn = EZ 1 + L E(Zi!Bi _1)
2
< A
2
+
n 2 i
L J. (T;-:;-)
2 1 1+~'
2
+ (n-1)A
n 2 i
= 2L J.1 (-.--)
1+1
Thus,
f (Vn) = Vn (log. log. Vn ) 2/ (log Vn ) 4
Hence,
Izn 1/[f(Vn )]1/2
n
n
n
2
S 2JnCn~1)[10g{L J~(i/(i+1)) + nA }]2/(L A~ + ~2)1/2[log 10g(L A~ + ~2)].
2 1
2 1
2 1
Now,
n
L J~ (i/ (i+1))
2
I
1
n
S
2
L 0(iT+O)
2
= 0(rf4+oY(2+0)) ,
1/(2+),
0)
(Ln2 A.12 + ~ 2)/nA2 .... 1 as n~ .
n (n/(n+1)) = O(n
Hence, IZnl/[f(Vn)]1/2
= 0(1).
/Zn l/ [f(Vn)]l/2 < 1 for n>nO•
Thus, there exists an no such that
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Hence,
L
Cv)
[f(V
n>l
n
nO
)r l
f
2
2
x d F{Z < xlB ~l}
x >f(Vn)
n
n
. x 2 d F{Z <xlB -I} < 00
x >f(V )
n
n
= L [f(V )]~l f 2
1
n
•
n
The proof of the theorem now follows directly from (i)
~
(v)
and theorem 4.4 and corollary 4.5 of Strassen [11].
REMARK.
-
If we define a process T(t),
t~O
-
by T(O)
= 0, and
(2.11) ret) = (t-n) Tn+l + [l-(t-n)] Tn for n~t~n+l, n~O,
and consider a Brownian motion ~(t) for which E~(t) = 0, E[~(s) ~(t)]
= A2 s, o<s<t<oo , we have from (i) - (v) and theorem 4.4 of Strassen
that
ret) = ~(t) + o([t log log t]1/2) a.s. as t~ .
(2.12)
We start with the same set up
as in section 2, and assume F
u
(O~u<l).
E
..10 '
J (u) strictly increasing in
Consider,
(3.1)
HOI:
e = 0 against the alternatives e > 0;
(3.2)
H02 :
e=
0 against the alternatives
Sen [10] has proved that if J e: Ll , then,
where, in our notations,
e r O.
lim~
Tn
=
ne a. s. (P e) ,
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8
00
(3.3)
ne = 2
.I
- F(-x-e)]dF(x-e) - ~ .
When,
00
e > 0, ne ~ 2 J J[F(x-e) - F(-x-e)]dF(x-e) - ~
e
00
=2
00
J J[F(y)
o
F(-y-2e)]dF(y) -
~
>2
J J[F(y) - F(-y)]dF(y) o
= 0, since J(u) t in u (strict) F(x) t in
x(O~u<l,
~
x real).
Similarly, for e < 0, ne < O.
Now, to test HOI define
(3.4)
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J J[F(x-e)
o
N =
first integer n>n
- O such that Tn >- cnIn,
{ if no such n occurs,
00
where cn is some sequence positive of constants such that cn/n+O
as n~. If HOI is false then Tn+ne(>O) a. s. as n~, and hence,
c
(3.5) .
P (N=oo) = limn~ Pe (N)n) < limn~ P (Tn < -!!.)
n = O.
e
"7'JV
-
Hence, if we agree to reject HOlas soon as we observe that N <
00,
while if N=oo, we do not reject HOI' then since Pe(N<oo) = 1 for e > 0,
the test has power 1. Again, when HOI is true, by the law of iterated
logarithm in section 2, lim sUPn~
rn Tn/[A(2
log log n)1/2] = l.
Then,
(3.6)
= POCTn >
- Cn In for some n >
- nO)
c
lIlT
n
172 for some n~nO)
= p o C n 1/2 >
mCn log log n)
A(2 log log n)
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9
which can be made arbitrarily small by taking c
. . l2A(l+E)
n
(n log log n)1/2, E > 0, and nO sufficiently large.
REMARK.
The above result does not provide any explicit upper bound
for PO(N<oo).
However, if instead we take c =l2A.(l+E)(n log n)1/2,
n
we can achieve this as follows:
(3.7)
PO· {T > C In} = PO· {T > c } < tin>f E[exb{t(T -c )}]
n-n
n-nnn
o
But,
= e
-tc
n
n E[exp{t .l sgn Xi E(J(UnR .)}]
1=1
--h1
= e
-tc
n
n EE[exp{t L sgn X. E(J(U
))} Ig ]
nR. .
. 1
1
n
1=
n1
Using the fact that sgn
~n
En
are stochastically independent
x
2
under H and using the elementary inequalities (ex+e- )/2 ~ exp(x /2)
01
n
2
2
2
for x real, and HEJ (U i)] = nA < nA , one gets,
1
n
n n
E[exp{t
L
. 1
1=
and
sgn X. EJ(UnR )} Ig ]
1
n
n1.
= ~=1[1/2({exp(t EJ(UnR ))} + {exp(-t EJ(UnR ))})]
n1
n1
(3.8)
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We may remark that unlike the case of sample sum, we do not need
the assumption the J (u) is exponentially integrable; J e:: LZ suffices
the purpose.
Thus,
=
Z
Z)
exp(-cn!ZnA
= n-(l+e::).
Then,
00
PO(N < (0) ~
00
PO(Tn ~ cn) < L n
n=no
n=no
I
- (1 +e::)
<00.
One may remark that a similar problem was faced by Darling and Robbins
[Z] in connection with the derivation of Kolmogorov-Smirnov tests
with power 1, where to obtain an explicit upper bound for PO{Tn ~ en/n},
Z of
· {(l+e::) n log log n} l/Z .
cn was taken to be . {(l+e::) n log n} l /
instead
Now, for the testing problem HOZ ' define
(3.9)
.
N
=
{first in.teger n > nO such that IT
n
00
I
> C
-
n
/n
if no such n occurs,
cn defined in the same way as earlier. If HOZ is false, Tn+n e a.s.
as n+oo, where ne >«) 0 as e >«) o. Hence ITnl+lnel a.s. as n+oo.
Then,
c
Pe (N = 00)
=
limn+oo Pe (N > n) ~ limn+oo Pe (ITn I < nn) ;: O.
Noting that Tn is distributed symmetrically about 0 under HOZ and
using similar arguments as before, one reaches the conclusion that the
test is of power 1 and arbitrarily small type I error.
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REFERENCES
[1]
Bahadur,R. R. (1966).
A note on quantiles in large samples.
Ann. Math. Statist. 37, 577-580.
[2]
Darling, D. A. and Robbins, H. (1968). Some nonparametric
sequential tests with power 1. Proc. Nat. Acad. Sci. 61" 804-809.
[3]
Erdos, P. (1942).
43, 419-436.
[4]
Feller, W. (1943). The general form of the so-called law of
iterated logarithm. Trans. Amer. Math. Soc. 54, 373-402.
[5]
Ghosh, M. and Sen, P. K. (1970). On the almost sure convergence
of Von Mises' differentiable statistical functions. Cal. Stat.
Assoc. Bull. 73.
[6]
Hajek, J. and Sidak, Z. (1967).
Press, New York.
[7]
Khintchine, A. (1924).
Fund. Math. &.' 9-20.
[8]
Kolmogorov, A. (1929). Das Gesetz des iterierten Logarithmus.
Math. Ann. 101, 126-135.
[9J
Petrovski, 1. (1935). Zur ersten Randwertaufgabe der
Warmleitungsgleichung.C0!'1P0sitoMath. 1:-, 383-419.
On the law of iterated logarithm.
Theory of Rank Tests.
Ann. of Math.
Academic
..
Uber einen Satz der Wahrscheinlichkeitsrechnung.
[10]
Sen, P. K. (1970). On some convergence properties of one-sample
rank order statistics. To appear in the December issue of Ann.
Math. Statist. 41.
---
[11]
Strassen, V. (1967). Almost sure behaviour of sums of independent
random variables and martingales. Proc. Fifth. Berkelex SXIDP.
Math. Statist. and Prob. Vol. ~, Univ'. of California Press.