Comparison of distribution function estimators for forestry applications following variable probability sampling

Practical considerations often motivate employing variable probability samp ling designs when estimating characteristics of forest populations. Three d istribution function estimators, the Horvitz-Thompson estimator, a differen ce estimator, and a ratio estimator, are compared following variable probab ility sampling in which the inclusion probabilities are proportional to an auxiliary variable, X. Relative performance of the estimators is affected b y several factors, including the distribution of the inclusion probabilitie s, the correlation (\rho) between X and the response Y, and the position al ong the distribution function being estimated. Both the ratio and differenc e estimators are superior to the Horvitz-Thompson estimator. The difference estimator gains better precision than the ratio estimator toward the upper portion of the distribution function, but the ratio estimator is superior toward the lower end of the distribution function. The point along the dist ribution function at which the difference estimator becomes more precise th an the ratio estimator depends on the sampling design, as well as the coeff icient of variation of X and rho. A simple confidence interval procedure pr ovides close to nominal coverage for intervals constructed from both the di fference and ratio estimators, with the exception that coverage may be poor for the lower tail of the distribution function when using the ratio estim ator.