Let @math be a class of probability distributions over the discrete domain @math We show that if @math satisfies a rather general condition -- essentially, that each distribution in @math can be well-approximated by a variable-width histogram with few bins -- then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of @math unknown distributions from @math We analyze several natural types of distributions over @math , including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems.
MHR distributions: As noted above, MHR distributions appear frequently and play an important role in reliability theory and in economics (to the extent that the MHR condition is considered a standard assumption in these settings). Surprisingly, the problem of learning an unknown MHR distribution or mixture of such distributions has not been explicitly considered in the statistics literature. We note that several authors have considered the problem of estimating the hazard rate of an MHR distribution in different contexts, see e.g. @cite_48 @cite_13 @cite_14 @cite_50 .
@cite_48 @cite_14 @cite_13 @cite_50
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On construit des estimateurs non parametriques da la fonction de repartition F et de sa fonction de hasard dans la classe de toutes les distributions IFR Consider non-parametric estimation of a decreasing density function f under the random (right) censorship model. Alternatively, consider estimation of a monotone increasing (or decreasing) hazard rate i based on randomly right censored data. We show that the non-paramet- ric maximum likelihood estimator of the density f (introduced by Laslett, 1982) is asymptotically equivalent to the estimator obtained by differentiating the least concave majorant of the Kaplan-Meier estimator, the non-parametric maximum likelihood estimator of the distribution function F in the larger model without any monotonicity assumption. A similar result is shown to hold for the non-parametric maximum likelihood estimator of an increasing hazard rate A: the non-parametric maximum likelihood estimator of A (introduced in the uncensored case by Prakasa Rao, 1970) is asymptotically equivalent to the estimator obtained by differentiation of the greatest convex minorant of the Nelson-Aalen estimator, the non-parametric maximum likelihood estimator of the cumulative hazard function A in the larger model without any monotonicity assumption. In proving these asymptotic equivalences, we also establish the asymptotic distributions of the different estimators at a fixed point at which the monotonicity assumption is strictly satisfied. We propose a new method for pointwise estimation of monotone, uni- modal and U-shaped failure rates, under a right-censoring mechanism, using non- parametric likelihood ratios. The asymptotic distribution of the likelihood ratio is pivotal, though non-standard, and can therefore be used to construct asymptotic confidence intervals for the failure rate at a point of interest, via inversion. Major advantages of the new method lie in the facts that it completely avoids estima- tion of nuisance parameters, or the choice of a bandwidth tuning parameter, and is extremely easy to implement. The new method is shown to perform competi- tively in simulations, and is illustrated on a data set involving time to diagnosis of schizophrenia in the Jerusalem Perinatal Cohort Schizophrenia Study.