On consistency of maximum spacing estimators (Q1972641)

Let \({\mathcal P}\) be a family of continuous pseudo-distributions \(P\) on \(\mathbb R\) with \(P({\mathbb R})\leq 1\) and let \(X_j\), \(j\geq 1\), be a sequence of i.i.d.\ random variables with probability distribution \(P_0\in{\mathcal P}\). Using the order statistics \(X_{i,n}\), the authors consider the Maximum Spacing Estimator (MSE) \(P_n\in{\mathcal P}\) for an unknown distribution \(P_0\). Some general theorems on strong consistency of MSE were obtained by \textit{Y. Shao} and \textit{M.G. Hahn} [J. Hoffmann-Jørgensen et al. (eds.), Probability in Banach spaces. Prog. Probab. 35, 417-431 (1994; Zbl 0804.62028)]. However, one of the basic assumptions therein consists of the existence of densities with respect to the Lebesgue measure. In this article, the above-mentioned theorems are generalized to the case of densities with respect to any continuous positive measure on \(\mathbb R\). Some different consistency conditions for the MSE are also given.