Tertius de Wet

Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests

This paper analyzes the evolution of the asymptotic theory of goodness-of-fit tests. We emphasize the parallel development of this theory and the theory of empirical and quantile processes. Our study includes the analysis of the main tests of fit based on the empirical distribution function, that is, tests of the Cramér-von Mises or Kolmogorov-Smirnov type. We pay special attention to the problem of testing fit to a location scale family. We provide a new approach, based on the Wasserstein distance, to correlation and regression tests, outlining some of their properties and explaining their limitations.

Nonparametric estimation of extreme conditional quantiles

The estimation of extreme conditional quantiles is an important issue in different scientific disciplines. Up to now, the extreme value literature focused mainly on estimation procedures based on independent and identically distributed samples. Our contribution is a two-step procedure for estimating extreme conditional quantiles. In a first step nonextreme conditional quantiles are estimated nonparametrically using a local version of [Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50.] regression quantile methodology. Next, these nonparametric quantile estimates are used as analogues of univariate order statistics in procedures for extreme quantile estimation. The performance of the method is evaluated for both heavy tailed distributions and distributions with a finite right endpoint using a small sample simulation study. A bootstrap procedure is developed to guide in the selection of an optimal local bandwidth. Finally the procedure is illustrated in two case studies.

Generalized Kernel Estimators for the Weibull-Tail Coefficient

We introduce new families of estimators for the Weibull-tail coefficient, obtained from a weighted sum of a power transformation of excesses over a high random threshold. Asymptotic normality of the estimators is proven for an intermediate sequence of upper order statistics, and under classical regularity conditions for L-statistics and a second-order condition on the tail behavior of the underlying distribution. The small sample performance of two specific examples of kernel functions is evaluated in a simulation study.

Multiple Case High Leverage Diagnosis in Regression Quantiles

Regression Quantiles (RQs) (see Koenker and Bassett, 1978) can be found as optimal solutions to a Linear Programming (LP) problem. Also, these optimal solutions correspond to specific elemental regressions (ERs). On the other hand, single case ordinary least squares (OLS) leverage statistics can be expressed as weighted averages of ER ones. Using this three-tier relationship amongst RQs, ERs, and OLS leverage statistics some relationships between single case leverage statistics and ER ones are explored and deduced. We build upon these results and propose a multiple-case RQ weighted predictive leverage statistic, TJ. We do this using an ER view of the well-known leverage relationship, , by summing the ER weighted predictive leverage statistics over all ERs (RQs included) instead of over observations, i.e., . As an ad-hoc cut-off value of this statistic we make use of the analog of the Hoaglin and Welsch (1978) one, i.e., high leverage points have . So in the RQ weighted predictive leverage scenario, the cut-off value becomes , where K is the total number of ERs. We then apply this RQ high leverage diagnostic to well-known data sets in the literature. The cut-off value used generally seems too small. Some proposals of cut-off values based on some analytical bounds and a simulation study are therefore given and shown to be reasonable.

Local Estimation of the Second-Order Parameter in Extreme Value Statistics and Local Unbiased Estimation of the Tail Index

We develop and study in the framework of Pareto-type distributions a class of nonparametric kernel estimators for the conditional second order tail parameter. The estimators are obtained by local estimation of the conditional second order parameter using a moving window approach. Asymptotic normality of the proposed class of kernel estimators is proven under some suitable conditions on the kernel function and the conditional tail quantile function. The nonparametric estimators for the second order parameter are subsequently used to obtain a class of bias-corrected kernel estimators for the conditional tail index. In particular it is shown how for a given kernel function one obtains a bias-corrected kernel function, and that replacing the second order parameter in the latter with a consistent estimator does not change the limiting distribution of the bias-corrected estimator for the conditional tail index. The finite sample behavior of some specific estimators is illustrated with a simulation experiment. The developed methodology is also illustrated on fire insurance claim data.