Confidence intervals for the mean of a population containing many zero values under unequal-probability sampling

Abstract

In many applications, a finite population contains a large proportion of zero values that make the population distribution severely skewed. An unequal-probability sampling plan compounds the problem, and as a result the normal approximation to the distribution of various estimators has poor precision. The central-limit-theorem-based confidence intervals for the population mean are hence unsatisfactory. Complex designs also make it hard to pin down useful likelihood functions, hence a direct likelihood approach is not an option. In this paper, we propose a pseudo-likelihood approach. The proposed pseudo-log-likelihood function is an unbiased estimator of the log-likelihood function when the entire population is sampled. Simulations have been carried out. When the inclusion probabilities are related to the unit values, the pseudo-likelihood intervals are superior to existing methods in terms of the coverage probability, the balance of non-coverage rates on the lower and upper sides, and the interval length. An application with a data set from the Canadian Labour Force Survey-2000 also shows that the pseudo-likelihood method performs more appropriately than other methods. The Canadian Journal of Statistics 38: 582–597; 2010 © 2010 Statistical Society of Canada