145
ON TESTING SYMMETRY OF
A DISTRIBUTION
YOSHIKI KUMAZAWA
I INTRODUCTION
The purpose of this paper is to give aJternative proofs of the asymptotic normaHties of test statistics defined in Kumazawa (1984) for testing symm-etry of a distribution about known or unknown location parameter. The
theory of linear combinations of order statistics is applicable for deriving the
asymptotic behaviours of statistics with known location parameter, and the asymptotic linearities with respect to shift parameter as those of linear sign-ed rank statistics are ussign-ed to prove the asymptotic normalities of statistics with estimated parameter.
For testing symmetry some distribution-free tests such as linear signed rank statistics are proposed in case of known location and those are extended
v
to deal with unknown location parameter, see Hajek and Sidak (lg67), Gupta (1967) and Antille, Kersting and Zucchini (1982). Sen (1977) considered these problem in the sequential case and gave the excellent results.
In Section II the asymptotic normalities of the statistics are proved by using the results of L-statistics. In Section III the asymptotic behaviours of the statistics with estimated parameter are given and shown to be equal to those of the linear signed rank staistics with the same score function.
146 P.scE-kK ng230e
[ ASYMPTOTIC NORMALITY WITH KNOWN LOCATION
'
Let Xi,X2,•••X. be independent and identically distributed random
variables with a distribution F. We wish to test the null hypothesis
H,:F(x)=1-F(-x) for x;lO
against the alternative
A : F(x)::lll-F(-x) for x;IO, but not synzmetric about zero.
For this testing problem Kumazawa (1984) introduced a class of test statistics
Tn : == Tn (Xi, X2, '", Xst)
: == .fee. Åë{1-Fr(e-X)}dFn(x)
nn
=n-iZ Åë{n-iX I[Xi-i-Xj>O]}
1-1 i-1
with a suitable function Åë, where F. is the sample distribution function of the Xi 's and I[B] denotes the indicator function of the event B. Note that one rejects H, in favour of A for large values of T.. For smal1-sample
n
properties of T., see Kumazawa (1984). Since n-`Z I[Xi+Xj>O] may takej' -= 1
the value O or 1 for some i, the weight function Åë must be bounded: this condition may be stringent. Here without loss of generality we may set
ip (t) = ip (1 - t) for O-<.t:$ll l .
Let IV.=nS[T.-fee..Åë{1-F(-x)}dF(x)] and g(u) = F{-F'i(1-u)}
for O:i{u;i{1, where F'i(ec) ==inf{x : F(x);llec}.
THEOREM 2.1. SuPPose that the junction g(u) has the derivative g'(u)
almost everywhere ip. Then if o\ is positive ana finite, xve have as n""'e
n
W.-n-P6Z Zi =op(nO)
'
ON TESTING SYMMETRY OF A DISTRIBUTION 147
ana
n
n-%Z Zi->N(O, a;) in law,
i1
-where
Zi = - ft [I[F(Xi) ::i{g(t)] -g(t) +g' (t) {I[F(Xi) :illt] -t}]dÅë (t)
and
ifr' = 2 fgf,! [g (t) {1 g(s) } .Ls(1 t)gr (t)gt (s) ]dip (t) dÅë (s)
- 2, .fgfg [min{t, g(s) } -tg (s) ]g' (t)dip (t) dgi (s) .
Under He, we have as n-'oo
IV.->N(O,a2) in law,
where a2=4fg Åë(u)2du.
PROOF. We have
W7. == -'nJ6fg [F.{-F.ri(1-t)}-F{-F-'(1-t) }]dgf (t)
=Nfg (Un[F{-FinJ,i(1-t)}]+Ln(t))d9f(t),
where U.(t)=:nY5{F.(F-i(t))-t} and L.(t)==n16[g{1-F(IXT.i(1-t)}-g(t)]. The results of empirical processes and L-statistics given in Shorack (1972)
yield that
SUP l Un [F{ - keti(1 - t) }]- U. {g (t) } i =op (n O)
O[.t-K.1
and
suP iL.(t)+g'(t)U.(1-t)l =op(nO)
ogtrrs"1
as nm'oo. Hence the boundess of Åë implies that W. is asymptotically
equiv-alent to
- f, [U.{g (t) } -g' (t) U. (1 - t)] dÅë (t) =n%?. i]ci Zi•
148 P.iR fuue ag 230 E)
the Central Limit Theorem yields the desired results. Q. E. D.
The asymptotic normality may be derived by the application of von Mises' statistical functional given in Fernholz (1983). Under a sequence of contiguous alternatives the asymptotic normality ls guaranteed by the result
of Behnen and Neuhaus (1975). In fact under a sequence of contiguous
alternarives F(x-bn-P5) with F symmetric and positive Fisher information and b>O, T. converges in distribution to a normal (2n>6bfgÅë(t)Åë*(t),noD as n-'oo, where Åë*(t) =-F"(F-'(t))/F'(F-i(t)). It is well known that the
signed linear rank statistic with the score function Åë also has this property.
From this fact and the asymptotic linearity with respect to shift parameter given in Section III, the statistic T. has the same asymptic behaviour as the signed linear rank statistic in the neighbourhood of the null hypothesis in large sample.
m ASYMPTOTIC NORMALITY WITH ESTIMATED PARAMETER
In this section we consider the testing problem of the composite null hypothesis
H,:F(x) ==F, (x-e), 0 unknown and F, (x) :1-F,(-x).
For this problem the statistic T. may be extended to
Ann A
T.=n'iZ ip{n"iX I[Xi+Xj>2e.]},
J-!
i--1
A
where e. is the Hodges-Lehmann estimate based on the Iinear signed rank
statistic with square integrable scores function Åë(ec). It is known that 0" . is
a robust, transformation invariant and consistent estimator of 0, so we may let 0=O.
In order to derive the asymptotic distribution of
fu.=:nY2[f.-fÅë{1-F(-x)}dF(x)], we define
ON TESTING SYMMETRY OF A DISTRIBUTION 149
and
TVn (b) =: nK{Tn (b) dSÅë{1-F(b - X) }dF (X) }
for all real b. Note that T.(O) is equal to T. defined in Section II. Then we can prove the following result along the lines of JureifkoviE (1969).
PROPOSITION 3.1. Suppose that F has a finite Fisher information. ure
have as neoo for every fixed Positiwe K
sul){ [ I'Vn (b) - I'V. (O) +2bw (gb, gb') l : ibl $K} =op (nO) ,
where v(Åë, di*) =fg Åë(t)di*• (t)dt.
' THEOREM 3.2. SuPPose thatFhasa finite Fisher information. Then we
have as n-oe
A
VV.->N(O,oZ) in law,
where
a: == o?+4v (gl , Åë*) 2v(Åë, ip)v(Åë, gf ") -2 +4v (Åë, Åë*) fi(a{g (t) } +g' (1 - t) o (t)) aÅë• (t)A
and a (t) == limn-oeE[U. (t) nJffO.].
Under H,, if F has a finite Fisher information, the asymPtotic
A
ution of I)V. is a nor"?.anl distribution with zero mean and variance
4fÅë (u) 2du + 4[Jip (u) Åë* (u) du/Jip (u) Åë* (u) du]2SÅë (u) 2du
-8SÅë(u)Åë"(u)dutÅë(u)Åë*(u)du.
PROOF. From Proposition 3.1, we have as n-'o
AA
IJV.-TLV.+nXO.2v(Åë, ipse')->O in probability.
Since a. is constructed from the signed rank test with the socres function Åë, we also have as n-.oe
nYLa.-n->6Sl] signXisb(F(iXiD)/v(sb, gb")->o in probability.
i--1
150 2ee.mbue eg230e
desired results. Q. E. D. AIn ordr to perform the test based on T. in practice, we must construct consistent estimators for v(ip, ip*) and v(Åë, Åë*). To apply the method given in Sen (1977), let
IV NN AJ
0., i :inf{a:T. (a) :.:{T., a}, e., v== suP{a:T. (a) llll -T., a}
and
A bJ ev N
Vn(Åë, Åë*) =2Tn, a/{n(0n, u'0n, l) },
Al
where T.,.is the smallest number such that for some givena:O<cr<1,
PHo{]Tnl;ill7n, a}lill-cr• Then S.(Åë, ip*) is a consistent estimator of w(Åë, Åë*). Similarly we can obtain a consistent estimator for w(Åë, ip*). Here we Amay construct an asymptotically exact test based on T. for the scoresfun-ctions Åë and Åë.
REFERENCES
Antille, A., Kersting, G. & Zucchini, W. (1982). Tesing Symmetry. J. Amer. Statist. Soc. 77, 639-646.
Fernholz, L. T. (1983). Von Mises Calculus for Statistical Functionals. Springer-Verlag,
New York.
Gupta, M. K. (1967). An Asymptotically Nonparametric Test of Symmetry. Ann.
Math. Statist. 28, 849-866.
v
Haje'k, J. and Sidak, Z. (1967). Theory of Rank Tests. Academic Press, New York. Jure6kova, J. (1969), Asymptotically Linearity of Rank Statistic in Regression ameter. Ann. Math. Statist. 40, 1889-1900.
Kurnazawa, Y. (1984). A CIass of Statistics for Testing Symmetry of a Distribution. Working Paper 9, Shiga University.
Sen, P. K. (1977). Tied-Down Wiener Process Approximations for Aligned Rank
Order Process and Some Applications. Ann. Statist. 5, 1107-1123.
