Kraft, Charles H.

Van Eeden, Constance

University of Minnesota Digital Conservancy

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A FLUCTU~TION THEOREM AND DISTRIBUTION FREE TEST 
by 
Charles H.- Kraft 
:-:arid 
Constance van Eeden 
Techncial Report 37 
A FLUCTUATION THEOREM AND A DISTRIBUTION-FREE TEST 
1) ' 2) Charles H. Kraft and Constance van Eeden 
University of Minnesota 
Let (~, ••• ,xn) be completely independent random variables with continuous 
and symmetr~c (about zero) distributions. If 
N'* = number of positive sums in £JS_, Xi+½, Xi+~+ x3, ••• ,lS_+ ••• :+~n) 
it has been shown by Anderson [l], [2].that N*' has, for each n, a distribution 
which is the same for the given class of sequences (~, ••• ,Xn). See Baxter [3] 
for this and a related result. 
We show here that the random variable, N, 
N = number of positive sums in ( ~ . .Xj; TC(l, 2, ••• ,n)} 
j€T 
also has a distribution which is constant for the same class of sequences 
(Xi,•••,Xn). In particular, the distribution of N is given by 
P(N = k) =_l, k = O, 1, ••• , 2n-1 • 
2n 
Proof: The conditional, given (IXi_l ,l~I , •.• ,lxnl), distribution of (Xi,•••,Xn) 
is uniform on the 2n points {(e11x11 ,e2 lx21 , ••• , enlxnl ); ei = ~1, or l}. Let 
0 = S
0 
< s1 < ••• < S be the ordered partial sums from 2n -1 
-1 
=.{ 
1 
( l~ I , .•• ' I xn I ) • Let 
jeT 
• je'T 
Then,. (see lemma), 
= number of pqsitive sums in (Si -Sk' i = o, ••• ,2n-l, ifk) • 
l) This research was supported in part by the National Science Foundation under 
Grant Number 0-19126. 
2) This research was supported in part by the National Institutes of Health under 
Grant Number 2G-43(c7+8). 
~ :'·• .. 
n Since this last expression is clearly equal to 2 -1-k and 
1 the result follows. 
Lennna: 
Proof: A one-to-one correspondence between the partial sums from 
(o1 (x11 , ••• ,onlxnl) and the elements of {Si-Sk, i=O,l, ••• ,2n-l, i:J=k) is given 
below where T = T(Sk) is defined above. 
To E o. Ix. I, jeA J J make correspond 
{
si(A) = E 1x.1 je (A-T) U (T-A) J 
i(A) = 0 
where 
A=T. 
To Si-Sk, make correspond E 8. j X. j where B is defined 
je (B-T) U (T-B) J J 
by S. = E IX. I . 
J. • B J J€ 
The number of positive sums in the two sets is the same since the corres-
pondents of the first map are equal, i.e., E o. Ix. I= s. (A)-sk . 
. A J J i ]€ 
Consistency of tests based on N (or N*). 
Let T denote one of "N*/n or N/(2n-l). A test of the null hypothesis, 
n 
x1 , .•• ,Xn are independent and symmetric about zero is to reject H0 if Tn > kn 
where kn is, for large n, approximately the solution to Bt(.\, .\) = 1-a (Bt is 
the incomplete beta-function) if T = N*/n and k = 1-a if T = N/(2n-1). 
n n n 
Since, for any random variable Z with P(O ~ Z ~ 1) = 1, 
( ) 1-EZ P Z ~ 1-€ ~ 1 - - , 
€ 1-ET 
a lower bound for the power of these tests is 1 - l-k n 
n 
will be consistent if ET ~ 1. Further, 
n 
(1) 1-ET = 
n 
-2-
Therefore the tests 
i ,.. 
0 
f 
where 
for T = N*/n 
n 
and where it is assumed that (x1 , ••• ,Xn) are independently and identically 
distributed with a distribution F(!.:.@.) for some F, and for S>O, a> O, 
a 
k 
and Y~ is a random variable distributed as ~ Xi. From (1) and the Helly-
i=l 
Bray theorem it follows that the test will be consistent if 
(2) 
Y. -ke 
P{ k < 
'1iZ a 
$e 
--} 
a 
~o. 
If, for some alternative, ET does not converge to one there will be a 
n 
size for which T will not be consistent. For example (2) is constant for the 
n 
Cauchy distribution and t is not consistent. 
n 
References. 
[l] Anderson, E. Sparre (1949). On the number of positive sums of random 
variables. Skand. Aktuarietidskr. 36 27-36. 
[2] Anderson, E. Sparre (1953). On the fluctuations of sums of random 
variables, Math. Scand. 1 263-285. 
[3] Baxter, Glen (1962). On a generalization of the finite arcsine law. 
Ann. Math. Stat. 33 909-915. 
-3-


A Fluctuation Theorem and Distribution Free Test

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A Fluctuation Theorem and Distribution Free Test

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