ASYMPTOTIC NORMALITY BEHAVIOUR OF A NON-PARAMETRIC ESTIMATOR FOR A FINITE POPULATION TOTAL USING EDGEWORTH EXPANSION
In survey sampling, the main objective is more often than not to establish information about the entire population which may include; mean, total, proportion or ratio. In this study, the nonparametric estimator of finite population total by Dorfman (1992) was adopted and the coverage probabilities using the edgeworth expansion was explored. In estimation of population parameters, three characteristics; unbiasedness, efficiency and the confidence interval are always explored. A lot of work has been done on studying unbiasedness and efficiency of the estimators and more particularly for the population total estimators. This study departs from these studies by studying the tail properties from the confidence interval instead of the unbiasness, efficiency and mean squared error. The coverage probabilities of the nonparametric estimator of the population total of three artificial populations; linear, quadratic and bump are inferred from the constructed confidence intervals. Because of its uniqueness in deriving coverage probabilities, the edgeworth expansion was used to derive the coverage probabilities. The coverage probabilities measure the extent to which the estimator is contained within the interval of the population total parameter. It was observed that the coverage probabilities from edgeworth expansion showed convergence in such a way that the estimator covered a larger interval of the population total. The edgeworth expansion gave the shortest confidence interval of the estimator and with the highest coverage probability hence optimal results. We therefore recommend the application of edgworth expansion in assessing the coverage probabilities of the other estimators of the finite population total; local polynomial and data reflection technique to assess their optimality.