Abstract
Consider a sequence of independent observations X1,...,Xn from a N(θ,cθ) distribution with 0<θ <∞. We assume that θ is unknown, but c(>0) is known. We begin with the problem of testing H0: θ =θ 0 against H1: θ =θ 1 where θ 0,θ 1(θ 0≠ θ 1) are specified values of θ. The most powerful (MP) level α test depends upon ∑i=1nXi2, 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.} \keywords{Exact method, large-sample method, minimum variance unbiased estimation, monotone likelihood ratio, most powerful test, non-central chi-square distribution, one-parameter exponential family, required sample size determination, type-I error probability, type-II error probability.
