Statistical applications of generalized quantiles. Nonparametric tolerance regions and P-P plots (Q2711188)

In his thesis the author applies the theory of generalized quantiles to build nonparametric multivariate tolerance regions and generalized PP-plots. \textit{J.H.J. Einmahl} and \textit{D.M. Mason} [Ann. Stat. 20, No. 2, 1062-1078 (1992; Zbl 0757.60012)] introduced the generalized quantile processes. These processes are based on generalized quantile functions that give a flexible manner to summarize the properties of multivariate data or probability distributions.NEWLINENEWLINENEWLINEIn Part I, the introduction of a generalized quantile function leads to the idea of minimum volume sets. This tool provides a new form of nonparametric tolerance regions. These regions are defined and studied intensely in Part I. The author defines tolerance intervals as the minimum length intervals that contain a certain number of observations. This notion can be extended to higher dimensions by defining tolerance regions as the sets of minimum volume from a general indexing class. In \(\mathbb{R}^k\) he defines, for fixed \(t_0\in[0,1]\), \(C\in R\) and \(n\in\mathbb{N}\) large enough, the tolerance region \(A_{n,t_0,C}\) as the smallest volume set from a class of sets \({\mathcal A}\), containing at least \(t_0+C/\sqrt n\) observations. The class \({\mathcal A}\) of sets can be: all closed a) ellipsoids, b) hyperrectangles with axes parallel to the coordinate hyperplanes, c) convex sets for \(k\leq 2\), that have probability between \(0\) and \(1\). The class is extended to unions of \(m\) sets of the precedent classes. The author builds also tolerance regions for spherical and circular data. He derives under weak conditions an asymptotic theory for tolerance regions. In this part the author shows that tolerance regions are asymptotically minimal with respect to the indexing class and have invariance properties. A simulation study permits to study the finite sample properties.NEWLINENEWLINENEWLINEIn Part II, the author defines graphical methods for hypothesis testing using another application of generalized quantiles. A generalized PP-plot is defined as NEWLINE\[NEWLINEm(t)=\text{sup} \{P_n(A)\;:\;P_0(A)\leq t,\;A\in {\mathcal A}\},\qquad t\in[0,1].NEWLINE\]NEWLINE The generalized PP-plot compares the empirical and the hypothetical distribution over the class \({\mathcal A}\). The generalized empirical PP-plot process can be define as NEWLINE\[NEWLINEM_n(t)=\sqrt n (\text{sup}\{P_n(A)\;:\;P_0(A)\leq t,\;A\in{\mathcal A}\}-t)\qquad t\in[0,1].NEWLINE\]NEWLINE This process is a type of inverse of the generalized quantile process. In this part the asymptotic behavior of \(M_n(t)\) is studied under the null and under contiguous alternatives. Finally, the author extends his study to consider the two-sample problem.