Estimation of a monotone density in $s$-sample biased sampling models

Abstract

We study the nonparametric estimation of a decreasing density function $g_{0}$ in a general $s$-sample biased sampling model with weight (or bias) functions $w_{i}$ for $i=1,\ldots,s$. The determination of the monotone maximum likelihood estimator $\hat{g}_{n}$ and its asymptotic distribution, except for the case when $s=1$, has been long missing in the literature due to certain nonstandard structures of the likelihood function, such as nonseparability and a lack of strictly positive second order derivatives of the negative of the log-likelihood function. The existence, uniqueness, self-characterization, consistency of $\hat{g}_{n}$ and its asymptotic distribution at a fixed point are established in this article. To overcome the barriers caused by nonstandard likelihood structures, for instance, we show the tightness of $\hat{g}_{n}$ via a purely analytic argument instead of an intrinsic geometric one and propose an indirect approach to attain the $\sqrt{n}$-rate of convergence of the linear functional $\int w_{i}\hat{g}_{n}$.