Estimation of a Common Parameter for Pooled Samples from the Uniform Distributions and the Double Exponential Distributions

Abstract

In this chapter we consider the problem to estimate an unknown real-valued parameter θ based on m samples of size n from the uniform distributions on the interval (θ — ξ_i,θ + ξ_i) (i = 1,…, m) with different nuisance parameters which is treated as a typical example in non-regular cases. In some cases the MLE and other estimators will be compared and it will be shown that the MLE based on the pooled sample is not better for both a sample of a fixed size and a large sample. We also consider an estimation problem of a common parameter θ based on m samples of each size n from the double exponential distributions with nuisance parameters τ_i, (i = 1,…, m). We obtain the asymptotic expansions of the distributions of some estimators, e.g. the MLE, the weighted median and the weighted mean, and asymptotically compare them up to the second order, i.e. the order n⁻¹/². Further we get the bound for the asymptotic distribution of the all second order asymptotically median unbiased estimators and compare it with their asymptotic distributions up to the order n⁻¹/². Related results can be found in Cohen (1976) and Bhattacharya (1981). For regular cases the reader is referred to Akahira and Takeuchi (1982) and Akahira (1986) where we have samples from m populations with densities f(x, θ, ξ_i) for i = 1,…, m with the common parameterr θ to be estimated and nuisance parameters ξ_is. It is shown that the MLE is third order asymptotically efficient but there are other estimators which are first and second order asymptotically efficient but have positive deficiency (or loss of information) compared to the MLE. However, in nonregular cases the situation becomes much more complicated and here we deal with two typical cases.