Inference for quantile measures of skewness

Abstract

Given a location-scale family generated by a distribution with smooth positive density, the aim is to provide distribution-free tests and confidence intervals for a skewness coefficient determined by three quantiles. It is the Bowley-Hinkley ratio S r /R r , where S r = x r + x 1−r − 2x 0.5 is the sum of two symmetric quantiles minus twice the median, and R r = x 1−r −x r is the r th interquantile range. Here, 0 < r < 0.5 is to be chosen and fixed. The sample version of this ratio depends only on three order statistics and is the basis for tests and confidence intervals. It is shown that the variance stabilized version of this statistic leads to more powerful tests than the Studentized version of the sample version of S r . Sample sizes required to obtain accurate coverage of confidence intervals with a prespecified width are provided.

Quantile versions of the Lorenz curve

“…We go one step further here, redefining the basic concept of the Lorenz curve in terms of quantiles instead of moments, and then determining what has been gained and lost in terms of conceptual clarity, inference and resistance to contamination. Examples of this approach are the standardized median in lieu of the standardized mean, and quantile measures of skewness and kurtosis, rather than the classical moment-based measures, Staudte (2013bStaudte ( , 2014Staudte ( , 2015. Ratios of quantiles based on one sample are often presented as measures of inequality, and inferential procedures for them are in Prendergast & Staudte (2015a,b).…”

Tail-Heaviness, Asymmetry, and Profitability Forecasting by Quantile Regression

“…Additionally, we have considered Bowley skewness (Staudte 2014), aka. Yule skewness (Benjamini and Krieger 1996), as another asymmetry measure because it represents a well-studied class of skewness measures in the statistics literature (Groeneveld and Meeden 1984;Groeneveld 1998 The original CoR is defined only for the right tail and uses the mean as a benchmark to contrast with the 99 th percentile.…”

A Bootstrap Test for Symmetry based on Quantiles

“…Throughout this section, we denote the asymptotic version of the test by T N A and its bootstrap one by T B . Staudte [12] has considered a test based on the statistiĉ…”

“…For more details about the statistic T ST see [12]. The asymptotic distribution of T ST under the null hypothesis is the standard normal distribution.…”

“…Since then, the symmetry tests have received a great deal of attention in the research literature. See for example, [7], [1], [6], [15] and [12].…”