Adaptive estimation of a function of a mean vector

Abstract

A sample is taken from some r-dimensional distribution with mean vector μ and the value of θ = P(μ) is to be estimated. Here P is a polynomial. An initial sample of size n1 is taken and all components are observed. Assume that, after the initial sample, the remaining sampling budget is C and the cost of each new vector taken is c0 with an additional cost cj if component j is observed. A subset D ≠ Ø of {1,h.,r} is selected and a second sample of size n2 is taken in which only those components with index j ∈ D are observed. It is desired to select D to minimize the MSE of the estimator θ = P(X) where the components of X are the sample means of observed components.Consider the case P(μ) = ∏j−1r μj. Assume that E(∏j=1rXj2) < ∞. Except for terms which are o(1/n1) the MSE of θ depends only on the means, variances, and covariances. Without loss of generality, the MSE is computed for the case D = {1,…,m}. In the case r = 2 the inequalities are given determining when D = {1} yields smaller MSE than D = {1,2} and when D = {1} yields smaller MSE than D = {2}. In determining D the means, variances and covariances will be estimated from the initial sample.The cases of estimating ∑j=1rμj, more general polynomials in μ, and more general functions of μ are discussed briefly.