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In this paper, we revisit the construction of confidence intervals for extreme quantiles of Pareto‐type distributions. A novel asymptotic pivotal quantity is proposed for these quantile estimators, which leads to new asymptotic confidence intervals that exhibit more accurate coverage probability. This pivotal quantity also allows for the construction of a saddle‐point approximation, from which a second set of new confidence intervals follows. The small‐sample properties and utility of these confidence intervals are studied using simulations and a case study from insurance. [ABSTRACT FROM AUTHOR]

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, whereKis 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. [ABSTRACT FROM PUBLISHER]

In the late 1960s J.H. Venter started to investigate the use of test statistics for normality based on quadratic distances between the order statistics of the sample and the corresponding hypothetical quantiles. These types of statistics are closely related to the (already known at that time) Shapiro-Wilkes'statistics, for which the limiting distribution was not yet known. The behaviour of the statistics investigated by Venter was such that the standard approach of that time, the so-called stochastic process approach, was insufficient to derive their limiting distribution. However, by writing the statistic in terms of order statistics from a uniform distribution and employing the representation of such order statistics in terms of independent, identically distributed exponential random variables, he was able to approximate the statistic by a quadratic form in independent, identically distributed random variables. This led him to the study of the limiting behaviour of the latter, for which results were not available to handle the statistics he was interested in. The results needed were derived and applied to the statistics of interest, constituting pioneering research that in later years led to the derivation of the limiting distribution of inter alia, the Shapiro-Wilk statistic and many other statistics of a quadratic type. The words of Del Barrio, Cuesta-Albertos, Matran and Rodriguez-Rodriguez in a recent paper, "All the proofs of the asymptotic behaviour of these statistics.., rely on the results in ... "', emphasize the fundamental contributions of Venter's earlier work. In the current paper the above-mentioned contribution and the work flowing from it, are discussed and placed in a historical context. In particular, it is shown that by using an expression for the distribution of uniform order statistics in terms of ratios of sums of independent, identically distributed exponential random variables, the test statistic can be shown to be asymptotically equivalent to a quadratic form in independent, identically distributed random variables. For the latter the results known at that time were insufficient and stronger results had to be developed in order to obtain a limiting distribution. This limiting distribution was obtained as that of a linear combination of independent chi-squared random variables with one degree of freedom each. The latter's characteristic function could be found quite easily and inverted numerically to obtain critical values for the test. The constants in the linear combination are closely related to the eigenvalues of the matrix whose entries are the constants in the quadratic form. It was shown how these constants can be found by transforming the required integral equation into a differential equation for which the classical orthogonal polynomials provide solutions. In this paper it is also shown how these quadratic statistics are related to degenerate U-statistics and correlation-type test statistics, leading to the same limiting distribution. A number of extensions of other researchers are also discussed, as well as more recent developments based on the original work of Venter. To be more specific, at the time of Venter's research, the general approach to deriving the limiting distribution of such quadratic statistics was by means of a so-called stochastic process approach. In that case the statistic was written as a quadratic functional of the empirical process or of the quantile process, for which the limiting distribution was known in terms of the Brownian Bridge process. However, that approach could not be applied to the type of statistics Venter considered, which led him to consider the approach based on quadratic forms discussed above. During the past 30+ years the available tools on empirical and quantile processes have developed immensely, which meant that the stochastic process approach was again a viable option.… [ABSTRACT FROM AUTHOR]