We study the rates of convergence of the maximum likelihood estimator (MLE) and posterior distribution in density estimation problems, where the densities are location or location-scale mixtures of normal distributions with the scale parameter lying between two positive numbers. The true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having sub-Gaussian tails. We obtain bounds for Hellinger bracketing entropies for this class, and from these bounds, we deduce the convergence rates of (sieve) MLEs in Hellinger distance. The rate turns out to be �log n � κ / √ n, where κ ≥ 1 is a constant that depends on the type of mixtures and the choice of the sieve. Next, we consider a Dirichlet mixture of normals as a prior on the unknown density. We estimate the prior probability of a certain Kullback-Leibler type neighborhood and then invoke a general theorem that computes the posterior convergence rate in terms the growth rate of the Hellinger entropy and the concentration rate of the prior. The posterior distribution is also seen to converge at the rate �log n � κ / √ n in, where κ now depends on the tail behavior of the base measure of the Dirichlet process. 1. Introduction. A

Mixture models for density estimation provide a very useful set up for the Bayesian or the maximum likelihood approach. For a density on the unit interval, mixtures of beta densities form a flexible model. The class of Bernstein densities is a muchsmaller subclass of the beta mixtures defined by Bernstein polynomials, which can approximate any continuous density. A Bernstein polynomial prior is obtained by putting a prior distribution on the class of Bernstein densities. The posterior distribution of a Bernstein polynomial prior is consistent under very general conditions. In this article, we present some results on the rate of convergence of the posterior distribution. If the underlying distribution generating the data is itself a Bernstein density, then we show that the posterior distribution converges at “nearly parametric rate ” �log n� / √ n for the Hellinger distance. If the true density is not of the Bernstein type, we show that the posterior converges at a rate n −1/3 �log n � 5/6 provided that the true density is twice differentiable and bounded away from 0. Similar rates are also obtained for sieve maximum likelihood estimates. These rates are inferior to the pointwise convergence rate of a kernel type estimator. We show that the Bayesian bootstrap method gives a proxy for the posterior distribution and has a convergence rate at par with that of the kernel estimator. 1. Introduction. Mixture models

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We show that the P Glivenko property of classes of functions F1 ; : : : ; Fk is preserved by a continuous function ' from R to R in the sense that the new class of functions x ! '(f1 (x); : : : ; fk (x)); f i 2 F i ; i = 1; : : : ; k is again a Glivenko-Cantelli class of functions if it has an integrable envelope.

For a version of the interval censoring model, case 2, in which the observation intervals are allowed to be arbitrarily small, we consider estimation of functionals that are diferentiable along Hellinger differentiable paths. The asymptotic information lower bound for such functionals can be represented as the squared L 2 -norm of the canonical gradient in the observation space. This canonical gradient has an implicit expression as a solution of an integral equation that does not belong to one of the standard types. We study an extended version of the integral equation that can also be used for discrete distribution functions like the nonparametric maximum likelihood estimator (NPMLE), and derive the asymptotic normality and efficiency of the NPMLE from properties of the solutions of the integral equations.

We study nonparametric estimation for current status data with competing risks. Our main interest is in the nonparametric maximum likelihood estimator (MLE), and for comparison we also consider the ‘naive estimator ’ of Jewell, Van der Laan and Henneman [10]. Groeneboom, Maathuis and Wellner [7] established that both estimators converge globally and locally at rate n 1/3. In this paper we use these results to derive the local limiting distributions of the estimators. The limiting distribution of the naive estimator is given by the slopes of the convex minorants of correlated Brownian motion processes with parabolic drifts. The limiting distribution of the MLE involves a new self-induced process. We prove that this process exists and is almost surely unique. Finally, we present a simulation study showing that the MLE is superior to the naive estimator in terms of mean squared error, both for small sample sizes and asymptotically.

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By a mixture density is meant a density of the form πµ(·) = � πθ (·) × µ(dθ), where(πθ)θ∈ � is a family of probability densities and µ is a probability measure on �. We consider the problem of identifying the unknown part of this model, the mixing distribution µ, from a finite sample of independent observations from πµ. Assuming that the mixing distribution has a density function, we wish to estimate this density within appropriate function classes. A general approach is proposed and its scope of application is investigated in the case of discrete distributions. Mixtures of power series distributions are more specifically studied. Standard methods for density estimation, such as kernel estimators, are available in this context, and it has been shown that these methods are rate optimal or almost rate optimal in balls of various smoothness spaces. For instance, these results apply to mixtures of the Poisson distribution parameterized by its mean. Estimators based on orthogonal polynomial sequences have also been proposed and shown to achieve similar rates. The general approach of this paper extends and simplifies such results. For instance, it allows us to prove asymptotic minimax efficiency over certain smoothness classes of the above-mentioned polynomial estimator in the Poisson case. We also study discrete location mixtures, or discrete deconvolution, and mixtures of discrete uniform distributions.

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data. 1

On nonparametric maximum likelihood for a class of stochastic inverse problems