Minimum volume sets and generalized quantile processes (Q1275932)

Let \(X_1,X_2,\ldots \) be i.i.d.\ random vectors in \({\mathbb R}^d\) with distribution \(F\), \(\mathbb C\) a class of measurable subsets of \({\mathbb R}^d\) and \(\lambda \) a real-valued function on \(\mathbb C\). The author defines the quantile function (\(0<\alpha <1\)) \[ V(\alpha)=\inf \{\lambda (C):F(C)\geq \alpha ,C\in {\mathbb C}\} \] and the empirical quantile function \[ V_n(\alpha)=\inf \{\lambda (C):F_n(C)\geq \alpha ,C\in {\mathbb C}\} \] (\(F_n\) is the empirical distribution of the first \(n\) observations). The corresponding minimizing sets \(Q(\alpha),Q_n(\alpha)\) (if exist) are defined by \[ V(\alpha)=\lambda (Q(\alpha)),\qquad V_n(\alpha)=\lambda (Q_n(\alpha)), \] and the standardized generalized quantile process by \[ q_n(\alpha)=(V'(\alpha))^{-1}\sqrt {n} (V_n(\alpha)-V(\alpha)), \] if the derivative \(V'(\alpha)\) exists, cf. \textit{J. H. J. Einmahl} and \textit{D. M. Mason}, Ann. Stat. 20, No. 2, 1062-1078 (1992; Zbl 0757.60012). The author proves under mild assumptions consistency results for \(V_n(\alpha)\) and \(Q_n(\alpha)\). For the case of \(\lambda =\) volume, he finds the rate of convergence of \(Q_n(\alpha)\) and derives the Bahadur-Kiefer type approximation for \(q_n(\alpha)\). For particular choices of the class \(\mathbb C\), one obtains various known results. An application for multimodality testing is given and possible generalizations for non i.i.d.-random vectors are discussed.