Quantile Function Methods with Applications to Robust Estimation
The estimation of the population quantiles is of great interest in a broad spectrum of theories, methods and applications of parametric, robust and exploratory statistical analyses. In this dissertation, we investigate the limitations of traditional quantile function estimation methods. The lack of efficiency of sample quantile caused by the variability of individual order statistics leads us to form a weighted average of the order statistics, using an appropriate weight function, which we call the L-estimator in general. We propose a new class of quantile function estimators, namely, the semi-parametric tail extrapolated quantile estimator, which belongs to the class of L estimator and has excellent performance for estimating the extreme tails with finite sample sizes compared with other L-estimators, such as the Harrel-Davis estimator and the kernel quantile estimator. The smoothed bootstrap and direct density estimation via the characteristic function methods are developed for the estimation of the confidence intervals of quantile estimators. The new class of quantile estimators is obtained by modification of traditional quantile estimators, and therefore, should be especially appealing to researchers in estimating the extreme tails. A natural estimator of the "quantile function of a sample quantile" is defined, which is accomplished by employing a quasi-quantile estimator with bandwidth function defined as the quantile function of a specific uniform fractional order statistic. Large sample expressions for the mean and variance of the new quasi-quantile estimator are developed to order O ( n –2 ). Applications include nonparametric estimation of the moments of the u th sample quantile (0 < u < 1), new closed form bootstrap-type percentile and moment based confidence intervals for the u th quantile, two-sample confidence intervals for the difference of two medians and fractional quantile bands for quantile plots. The quick estimators of location and scale are "some useful inefficient statistics" based on the quantile estimators. When we estimate the location, the quick estimators of location are preferable since it is distribution-free and outlier-resistant compared with the well-known estimator of location, i.e., the mean. The permutation test of location shift based on the quick estimators of location is proposed. By inverting the permutation test, we obtain exact confidence intervals for the difference between two quick estimators of location. A novel confidence interval estimation method is developed by obtaining the density distribution function via the characteristic functions of the difference between quick estimators. The permutation test based on the quick estimators of location would be of great interest when the distributional assumptions of the populations are unknown. In addition, the confidence interval estimation method via the direct density approach would be preferable for small and moderate sample sizes when narrower confidence interval length is desirable. The forth part of the dissertation is to propose an empirical quantile based robust alternative measure of dependence coined quick covariance. The statistical properties of quick covariance, analogous to those of the classic moment-based covariance, lead us naturally to develop a robust nonparametric alternative to the correlation coefficient, termed the quick correlation coefficient. We examine the properties of the quick covariance, quick absolute deviation and quick correlation coefficient under bivariate normality and the asymptotic properties under the marginally symmetric assumption. An exact test that the population quick correlation coefficient is zero is developed.