TWO-STAGE ESTIMATION OF A LINEAR FUNCTION OF NORMAL MEANS WITH SECOND-ORDER APPROXIMATIONS

Abstract

ABSTRACT When a confidence interval tends to be too wide, its effectiveness in bolstering any inferential statements becomes limited. Hence, an experimenter may opt to construct a confidence interval with some preassigned “small” width and preassigned “large” confidence coefficient so that any inferences drawn from this can be of some value in practice. We consider k(≥2) independent normal populations with unknown means and unknown and unequal variances. We discuss the estimation problem for a linear function of the population means with a fixed-width (=2d) confidence interval having the preassigned confidence coefficient (≥1 − α), d > 0, 0 < α < 1. But, the goal of having such a confidence interval with both preassigned width and confidence coefficient is not attainable when the sample sizes are held fixed in advance [Dantzig (1940)[8], Ghosh et al. (1997, Sec. 3.7)[13]]. Chapman (1950)[4] first gave a Stein-type two-stage procedure for the problem on hand when k = 2. It is known that in a k-sample problem, the analogous two-stage procedure requires the upper percentage points of the distribution of the sum of k independent Student's t variates. First, a Cornish-Fisher expansion of such a percentage point is derived (Theorem 2.1) in general, followed by the Tables 1 and 2 of these (approximate) percentage points which are constructed by using this expansion, with the pilot sample sizes not necessarily all equal, when k = 2, α = .05, .01 and k = 3, α = .05. Next, under the limited additional assumption that each unknown population variance has a known positive lower bound, the Chapman type two-stage estimation procedure is modified along the lines of the one-sample considerations of Mukhopadhyay and Duggan (1997)[23]. For this modified two-stage procedure, various second-order expansions for both the lower and upper bounds of the average sample sizes (Theorem 2.2) and the associated confidence coefficient (Theorem 2.3) are obtained. We may remark that the second-order expansions are meant to provide faster rates of convergence for useful approximations. Then, through extensive sets of simulations we show that the extent of over-sampling experienced by the Chapman-type procedure is significantly reduced under the new modification when k = 2(1)5, α = .05. We include examples and data to illustrate usefulness of the modified k-sample two-stage estimation technique when k = 2, 3. Additionally, the importance of the asymptotic second-order terms is highlighted with the help of data analysis (Examples 1 and 2, Sec. 3).