Estimation and Testing of Hypotheses about the Quantile Function of the Normal Distribution

Abstract

Estimation (point as well as interval) and testing of the hypotheses about the quantile-function (Q(ξ). 0<ξ<1) of a two-parameter normal distribution is considered based on k(≤n) selected order statistics from a sample of size n (n large). It is shown that the asymptotic relative efficiency (ARE) expressions of the asymptotically best linear unbiased estimator (ABLUE) of Q(ξ) relative to the uniformly minimum variance unbiased estimator (UMVUE) is the same as for the corresponding 100(1–ε)% asymptotic confidence interval as well as the ε-level test of hypotheses based on the optimum ABLUE eompared with that based on the UMVUE. Hence, the optimum spacings for the ABLUE, the AMLCI (asymptotically minimum length confidence interval) and the maximum ARE-test are the same. Tables of optimum spacings, coefficients of the ABLUE and the ARE-values are provided for k=2(l)10 and ξ=.01, 05(.05) .50.