Consistency of generalized maximum spacing estimates (Q2739869)

The author considers an i.i.d. sample from a distribution \(P_0\) with order statistics \(-\infty=\xi_{(0)}\leq \xi_{(1)}\leq\dots\xi_{(n)}\leq\xi_{(n+1)}=\infty\). It is known that \(P_0\in{\mathcal P}\). The generalized maximum spacing estimate (GMSP) \(\hat P_n\) for \(P_0\) is any \(P\in{\mathcal P}\) that maximizes NEWLINE\[NEWLINES_{\Psi,n}(P)=(n+1)^{-1}\sum_{j=0}^n \Psi((n+1)(F_P(\xi_{(j+1)})-F_P(\xi_{(j)}))),NEWLINE\]NEWLINE where \(F_P\) is the d.f. of \(P\), and \(\Psi\) is a suitable concave function on \(R_{+}\). Some results on consistency of GMSP estimates in the \(L_1\) and vague topologies are obtained. E.g., if \(\mathcal P\) is the family of probability measures with unimodal densities and \(\Psi\) is ``regular'' [e.g., \(\Psi(x)=\log x\) or \(x^\alpha{\text sgn}(1-\alpha)\)], then NEWLINE\[NEWLINE\int_R|f_{\hat P_n}(x)-f_{P_0}(x)|dx\to 0\quad \text{a.s. as}\quad n\to\infty.NEWLINE\]NEWLINE Here \(f_P\) is the density of \(P\) w.r.t. Lebesgue measure. Consistency results for GMSP estimates based on higher order spacings are also derived.