On MP Test and the MVUEs in a <Var>N(θ, cθ)</Var> Distribution with <Var>θ</Var> Unknown: Illustrations and Applications

Abstract

Consider a sequence of independent observations X1, . . . , Xn from a N(θ, cθ) distribution with 0 0) is known. We begin with the problem of testing H0 : θ = θ0 against H1 : θ = θ1 where θ0, θ1(θ0 = θ1) are specified values of θ. The most powerful (MP) level α test depends upon ∑n i=1 X 2 i , a complete and sufficient statistic for θ, which has a multiple of a non-central chi-square distribution with its non-centrality parameter involving n and the true parameter value θ under H0, H1. We first target type-I and type-II error probabilities α and β respectively, with α > 0, β > 0, α + β < 1. We set out to determine the required exact sample size which will control these error probabilities and provide two useful large-sample approximations for the sample size. The three methods provide nearly the same required sample size whether n is small, moderate or large. We also show how one may derive the minimum variance unbiased estimators (MVUEs) for a number of interesting and useful functionals of θ by combining some previous work from Mukhopadhyay and Cicconetti (2004) and Mukhopadhyay and Bhattacharjee (2010). All methodologies are illustrated with both simulated data and real data.