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

Abstract

We study the nonparametric estimation of a decreasing density function g0 in a general s-sample biased sampling model with weight (or bias) functions wi for i = 1, …, s. The determination of the monotone maximum likelihood estimator ĝn and its asymptotic distribution, except for the case when s = 1, has been long missing in the literature due to certain non-standard structures of the likelihood function, such as non-separability and a lack of strictly positive second order derivatives of the negative of the log-likelihood function. The existence, uniqueness, self-characterization, consistency of ĝn and its asymptotic distribution at a fixed point are established in this article. To overcome the barriers caused by non-standard likelihood structures, for instance, we show the tightness of ĝn via a purely analytic argument instead of an intrinsic geometric one and propose an indirect approach to attain the n-rate of convergence of the linear functional ∫ wi ĝn,

On asymptotic equivalence of the NPMLE of a monotone density and a Grenander-type estimator in multi-sample biased sampling models

“…Large sample theory of this NPMLE was investigated in [8]. Recently, [3] established the unique existence of the decreasing NPMLÊ g n in s-sample biased sampling models and also gave its asymptotic distribution at a fixed interior point where the underlying density has a strictly negative derivative; such a problem has been open in the literature due to certain non-standard structures of the likelihood function, such as non-separability and a lack of strictly positive second order derivatives of the negative of the loglikelihood function. Formally, denote G to be the set of all decreasing densities.…”

“…It would be of both theoretical and practical interests to see if this asymptotic equivalence also holds in other models such as that in [3]. Practically, for the s-sample biased modeling models, the computation of the NPMLE is done iteratively based on a self-characterization given in [3], where an initial consistent estimator is required since the corresponding optimization problem is non-convex. In [3], we suggested to use the Grenander-type estimator as an initial estimator, where a numerically efficient method of finding G n has already been discussed in [28].…”

“…Practically, for the s-sample biased modeling models, the computation of the NPMLE is done iteratively based on a self-characterization given in [3], where an initial consistent estimator is required since the corresponding optimization problem is non-convex. In [3], we suggested to use the Grenander-type estimator as an initial estimator, where a numerically efficient method of finding G n has already been discussed in [28]. The asymptotic equivalence of the two estimators implies that that this initial estimator is already as good as the NPMLE asymptotically.…”

Censored quantile regression model with time‐varying covariates under length‐biased sampling

“…Nonparametric estimation of the survival function on (length-) biased observations include Turnbull (1976); Vardi (1982;1985); Gill et al (1988); Asgharian et al (2002) and more recently Chan et al (2018) on density function from multiple samples under monotone density constraint. To take the covariate effect into account, extensive research efforts have also been devoted to the corresponding treatments under the regression context.…”

“…(2002) and more recently Chan et al . (2018) on density function from multiple samples under monotone density constraint. To take the covariate effect into account, extensive research efforts have also been devoted to the corresponding treatments under the regression context.…”