Estimating a positive normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal population with mean μ and variance σ ². In many real life situations, specially in lifetime or reliability estimation, the parameter μ is known a priori to lie in an interval [a, ∞). This makes the usual maximum likelihood estimator (MLE) ̄ an inadmissible estimator of μ with respect to the squared error loss. This is due to the fact that it may take values outside the parameter space. Katz (1961) and Gupta and Rohatgi (1980) proposed estimators which lie completely in the given interval. In this paper we derive some new estimators for μ and present a comparative study of the risk performance of these estimators. Both the known and unknown variance cases have been explored. The new estimators are shown to have superior risk performance over the existing ones over large portions of the parameter space.     

Inequality Constrained State-Space Models

ABSTRACT

The standard Kalman filter cannot handle inequality constraints imposed on the state variables, as state truncation induces a nonlinear and non-Gaussian model. We propose a Rao-Blackwellized particle filter with the optimal importance function for forward filtering and the likelihood function evaluation. The particle filter effectively enforces the state constraints when the Kalman filter violates them. Monte Carlo experiments demonstrate excellent performance of the proposed particle filter with Rao-Blackwellization, in which the Gaussian linear sub-structure is exploited at both the cross-sectional and temporal levels.     

A new strategy for speeding Markov chain Monte Carlo algorithms

Markov chain Monte Carlo (MCMC) methods have become popular as a basis for drawing inference from complex statistical models. Two common difficulties with MCMC algorithms are slow mixing and long run-times, which are frequently closely related. Mixing over the entire state space can often be aided by careful tuning of the chain's transition kernel. In order to preserve the algorithm's stationary distribution, however, care must be taken when updating a chain's transition kernel based on that same chain's history. In this paper we introduce a technique that allows the transition kernel of the Gibbs sampler to be updated at user specified intervals, while preserving the chain's stationary distribution. This technique seems to be beneficial both in increasing efficiency of the resulting estimates (via Rao-Blackwellization) and in reducing the run-time. A reinterpretation of the modified Gibbs sampling scheme introduced in terms of auxiliary samples allows its extension to the more general Metropolis-Hastings framework. The strategies we develop are particularly helpful when calculation of the full conditional (for a Gibbs algorithm) or of the proposal distribution (for a Metropolis-Hastings algorithm) is computationally expensive. Partial financial support from FAR 2002-3, University of Insubria is gratefully acknowledged.     

Sequential importance sampling for nonparametric Bayes models: The next generation

There are two generations of Gibbs sampling methods for semiparametric models involving the Dirichlet process. The first generation suffered from a severe drawback: the locations of the clusters, or groups of parameters, could essentially become fixed, moving only rarely. Two strategies that have been proposed to create the second generation of Gibbs samplers are integration and appending a second stage to the Gibbs sampler wherein the cluster locations are moved. We show that these same strategies are easily implemented for the sequential importance sampler, and that the first strategy dramatically improves results. As in the case of Gibbs sampling, these strategies are applicable to a much wider class of models. They are shown to provide more uniform importance sampling weights and lead to additional Rao-Blackwellization of estimators.     

Correction locale de l'estimateur à noyau de la densité d'une loi de probabilité

The standard Parzen-Rosenblatt kernel density estimator is known to systematically deviate from the true value near critical points of the density curve. To overcome this difficulty, we extend the Rao-Blackwell method by using locally sufficient statistics: we define a new estimator and study its asymptotic behaviour. The interest of the method is shown by means of simulations.