Abstract:
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We study the nonparametric estimation of a decreasing density function in a general s-sample biased sampling model. The determination of the monotone maximum likelihood estimator (monotone MLE) 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 the monotone MLE 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 the monotone MLE via a purely analytic argument instead of an intrinsic geometric one and propose an indirect approach to attain the rate of convergence of certain linear functionals involved in the likelihood.
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