Moments and Product Moments of Bootstrap Distributions Yoko Ono
Department of Management Science, Graduate School of Engineering, Science University of Tokyo Kagurazaka, Shinjuku-ku, Tokyo 162-8601, JAPAN E-mail:
[email protected] Naoto Niki and Hiroki Hashiguchi
Department of Management Science, Science University of Tokyo Kagurazaka, Shinjuku-ku, Tokyo 162-8601, JAPAN E-mail:
[email protected],
[email protected] 1. Introduction Let X = (x1 ; : : : ; xn ) = ( [ x11 1 1 1 x1p ]; : : : ; [ xn1 1 1 1 xnp ] ) be a p-variate random sample of size n drawn from a population F . Let Y = (y 1 ; : : : ; y n ) denote a bootstrap sample of size n resampled from the empirical distribution F 3 composed from X . Distribution G3 of a statistic T (Y ) is called \the bootstrap distribution of T " and is used as an approximation to the distribution G of T (X ). The aim of this article is to express the moments of G3 in terms of the moments and product moments of F 3 for further discussions on the bootstrap method. For any p-dimensional vector
= [ 1 1 1 1 p ] 2 Np of non-negative integers,
P =
n n X X xi = xi11 1 1 1 xipp i=1
i=1
and similar notations are used hereafter.
2. Main Results The proposition below is fundamental.
Proposition 1 Let y F 3 then, for any 2 Np , it holds that EF 3 y =
1
P: n
Assumption 1 Following Bhattacharya and Ghosh(1978), we assume
T (Y ) = H (z ) = H
n 1X
n i=1
! [ f1 (y i ) : : : fk (y i ) ] ;
where H is a real-valued function on Rk and f1 ; : : : ; fk are those on Rp . For the sake of simplicity, we only consider here the case that all of f1 (y); : : : ; fk (y) are monomials in the elements of y ; that is, fj (y ) = y j (j 2 Np) as well as T (Y ) = H (z ) = H
n h 1X yi 1 : : :
n i=1
!
i yi k :
Proposition 2 If H is continuously di erentiable up to requisite order in a neighborhood of
m = EF 3 z =
1h
n
P
i
1
: : : Pk ;
then (1) G3 tends to a normal distribution as n ! 1, (2) Taylor expansions for H (z ) about z = m are symmetric polynomials in the elements of Y , (3) the cumulants of Wn = 1 n 2 (H (z ) 0 H (m)) satisfy the Cornish-Fisher assumption and formal Edgeworth expansions for the distribution of Wn are valid. Remark 1 From (2) and (3) in Proposition 2, the approximate moments of G3 can be obtained as polynomials in P 's by using Proposition 1, where the term `approximate' means `given as a formal asymptotic expansion of O(n0s ) for a xed s'. Remark 2 The moments of the (approximate) moments of G3 , as statistics based on a sample X driven from F , can be expressed as polynomials in the moments and product moments of F by using the method due to Nakagawa and Niki (1991); see also Niki, Nakagawa and Hashiguchi (1995). 3. Simple Example Let p = 1, k = 2, f1 (x) = x2 , f2 (x) = x and H ([z1 z2 ]) = z1 0 z22 , then V = T (X ) is the sample variance of F . Let U = T (Y ) denote the bootstrap version of V then 1 2 n01 n 0 1 1 P V; EF 3 U = [2] 0 2 P[1] = n n n n 1 1 2 Var(U ) = EF 3 U 2 0 (EF 3 U )2 = 2 P[4] 0 3 2 P[4] + 4 P[3] P[1] + P[2] n n 1 2 2 + 4 P[4] + 8 P[3] P[1] + 4 P[2] + 8 P[2] P[1] n 0 15 4 P[3] P[1] + 3 P[2]2 + 20 P[2] P[1]2 + 4 P[1]4 + 16 12 P[2] P[1]2 + 10 P[1]4 :
n
n
We note the bootstrap bias correction happens to work well for V towards the population variance 2 such that 1 EF (EF 3 U 0 V ) 0 (V 0 2 ) = 0 2 2 ;
n
but sometimes makes the bias larger; typically, for the unbiased estimator n=(n 0 1) V of 2 , n n n 1 2 EF ( EF 3 U0 V)0( V 0 ) = 0 2: n01 n01 n01 n
REFERENCES Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion, Ann. Statist., 6, 434{451. Nakagawa, S. and Niki, N. (1991). Change of bases of multi-system symmetric polynomials and their applications in distribution theory of multi-variate statistics (in Japanese), Bull. Comput. Statist. Jpn., 4, 35{43. Niki, N., Nakagawa, S. and Hashiguchi, H. (1995). Computer algebra application to the distribution theory of multivariate statistics, Proc. 1st Asian Tech. Conf. Math., 689{696.
RESUM E
D'autres discussions sur la \bootstrap methode" sont possibles en exprimant les moments des \bootstrap distributions" des statistiques en termes de moments des distributions empiriques.
