Search

This article shows how the theory of order statistics can be used to estimate confidence intervals for probabilistic risk measures, most particularly, VaR and expected shortfall. The method is very general in that it can be applied to any loss distribution, parametric or nonparametric. The method is straightforward to implement, and has a number of other advantages compared to alternatives methods of estimating confidence intervals for risk measure. [ABSTRACT FROM AUTHOR]

Let pi[sub N] be a finite population of size N whose elements have distinct X values X[sub (1)] < ... < X[sub (N)]. Let chi ([sub 1]) < ... < x(sub (n)] denote the order statistics of a simple random sample of size n taken from pi [sub N] without replacement. An outer confidence interval is formed for the quantile interval [X ([sub t]), X [sub (u)] (1

This article explores the use of t statistics based on two- or four-order statistics. The functions of the order statistics, which are used to define the t statistics, are the pivots and bipivots. The pivots are exact order statistics, while the bipivots are the means of two adjacent order statistics. Two t statistics, based on the pivots and bipivots, are examined and compared to other t statistics, including Student's t, under various sampling situations. We conclude that the proposed t statistics are not only simple to use but are robust and perform quite well in terms of the lengths of their confidence intervals. We recommend that these t statistics be used for sample sizes between 4 and 20, because they perform well and are easy to use. Tables of percentage points are provided to facilitate the use of the t statistics. [ABSTRACT FROM AUTHOR]

Some optimum nonparametric procedures for estimating and testing contrasts in two-way layouts are proposed and studied. These procedures are based on the Chernoff-Savage [4] type of rank order statistics which include the Wilcoxon and normal scores statistics among others. In the first three sections, the properties of the proposed point estimators of contrasts, such as symmetry, invariance and asymptotic normality, are studied and their asymptotic relative efficiencies with respect to the corresponding least-squares estimators are obtained. In particular, it is shown that the procedures based on the normal scores statistics are asymptotically at least as efficient as the corresponding procedures based on the method of least squares, whatever be parent cumulative distribution functions. In sections 4 and 5, the corresponding problems of testing and confidence intervals are discussed, and generalized to two-way layouts with several observations per cell. [ABSTRACT FROM AUTHOR]

Order statistics from a strongly unimodal distribution are strongly unimodal. In this note, authors focus on the strong unimodality of order statistics. Alam points out the relevance of the unimodality of order statistics for the construction of shortest confidence intervals for the largest of several location parameters. Under some regularity conditions, distribution of an order statistic is known to converge to a unimodal distribution. It is thus not surprising that some order statistics are unimodal even though the population may be far from unimodal. For instance, the bimodal density of an example, where all the order statistics, beginning with samples of size 2, are unimodal. On the other hand, order statistics from unimodal populations need not be unimodal. The main result of this note points out that order statistics from strongly unimodal distributions are necessarily strongly unimodal. As a consequence, normal order statistics are not only unimodal, they are strongly unimodal.