Estimating means when sampling gives probabilities as well as values or "looking a gift horse in the mouth"

Read, Robert, Thomas, Lyn and Washburn, Alan (2000) Estimating means when sampling gives probabilities as well as values or "looking a gift horse in the mouth". Statistics and Computing, 10 (3), 245-252. (doi:10.1023/A:1008995628693).

Abstract

Consider the random sampling of a discrete population. The observations, as they are collected one by one, are enhanced in that the probability mass associated with each observation is also observed. The goal is to estimate the population mean. Without this extra information about probability mass, the best general purpose estimator is the arithmetic average of the observations, XBAR. The issue is whether or not the extra information can be used to improve on XBAR. This paper examines the issues and offers four new estimators, each with its own strengths and liabilities. Some comparative performances of the four with XBAR are made.The motivating application is a Monte Carlo simulation that proceeds in two stages. The first stage independently samples n characteristics to obtain a "configuration" of some kind, together with a configuration probability p obtained, if desired, as a product of n individual probabilities. A relatively expensive calculation then determines an output X as a function of the configuration. A random sample of X could simply be averaged to estimate the mean output, but there are possibly more efficient estimators on account of the known configuration probabilities.

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