applications of bootstrap in two-stage shrinkage estimation

APPLICATIONS OF BOOTSTRAP IN TWO-STAGE SHRINKAGE ESTIMATION Makarand V. Ratnaparkhi,

Wright State Uliversity, Dayton, Ohio, U.S.A. mratnap@lesire. Wright.edlr Vasant B. Waikar and Frederick J. Schuurmann,

Mami Uliversity, Oxford, Ohio USA. LIntroduction Waikar and Katti (197 1) constructed the two-stage estimators of the mean vector of a multivariate normal distribution. Waikar, et al(1984) combined the two-stage and shrinkage methods for estimating the normal mean. Chang (1995) constructed the James-Stein (1961) type estimators keeping in view the two-stage approach by Waikar and Katti (1971). In this paper, we develop a new method for the estimation of a normal mean based on twostage shrinkage method of Waikar, et al(1984) and the bootstrap method [Efron and Tibshirani (1993)]. Then, using the simulations, we show that the estimators of a normal mean obtained by this new method have better effrciency as compared to the effrciency of their known counterparts.

2. Two-stage Estimation for Normal Mean (Waikar Estimation) Waikar et al. (1984) proposed the following [steps(l)-(4)] estimation scheme. Let X - N(,u , 02) where cr2 is known Let m be the initial estimate of p based on prior knowledge. Then; (1) Select two positive integers nr and n2.(2) Obtain independent random observations X ri, i =l, 2, ,. , . . ,nr (first stage sample). Let yr denote their mean(3)Test H,,:,u = ,u~ versus Hr: ,U+ ,u, at leve1 Q!using the first stage sample, and if H, is accepted the estimator of p is defined as = k kr + (1 - k) h where k = IZi/ Zcdz), Z = ( %l - ,UO)/ (c&r ), fiw and Ztd2) is the upper 100 (a/2) percentage point of a standard normal distribution. Here we note that if Ho is accepted then k < 1 and the resulting estimator has the characteristic of shrinkage toward the prior mean h. (4) If Ho is rejected, obtain a setond sample of size n2, X2; ,i =l, 2, . . * = (y s, + il2 r2 ) / (Il1 +“*) ’ * > n2 and take the estimator of p as the pooled sample mean pw The above estimator of p when compared with the usual estimator 2 based on a fixed sample of an equivalent size 11*=E(n]m) = nr( 1 + OIU) where 11=&11~ was found to be more effrcient. For a unknown, in step (3), the t-statistic is used for testing Ho with related changes. In view of improving the effrciency of the two-stage shrinkage estimators the choice of k , as to be expected, plays an important role. In the above Waikar estimator the shrinkage factor k is fixed for a given first sample and a given value of a. In our new method due to bootstrapping factor k becomes a variable quantity and allows us to select the appropriate value of k for improving the effrciency of the shrinkage estimator. 3. Bootstrapping

the Two-stage Shrinkage Estimator (Waikar - Ratnaparkhi

)

In this methodology we use the bootstrap method for generating different values of k in the first stage ( i.e. when we accept Ho) of Waikar method as follows: 1. Generate a bootstrap sample of size 111,X *r1 , X *12 , . . . , s*,,say, from the first stage sample &r , X12 , . . . , Xlnl of size tir. 2. For each bootstrap sample we calculate k*=i 2’ 1/Z(&, where the ‘*’ denotes the value of

Z- statistic for the bootstrap sample. . 3. If k’