Inference for quantile measures of skewness

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 Sr/Rr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_r/R_r$$\end{document}, where Sr=xr+x1-r-2x0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_r=x_r+x_{1-r}-2x_{0.5}$$\end{document} is the sum of two symmetric quantiles minus twice the median, and Rr=x1-r-xr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_r=x_{1-r}-x_r$$\end{document} is the r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document}th interquantile range. Here, 0<r<0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<r< 0.5$$\end{document} 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 Sr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_r$$\end{document}. Sample sizes required to obtain accurate coverage of confidence intervals with a prespecified width are provided.