The Kullback-Leibler risk of the Stein estimator and the conditional MLE

The decomposition of the Kullback-Leibler risk of the maximum likelihood estimator (MLE) is discussed in relation to the Stein estimator and the conditional MLE. A notable correspondence between the decomposition in terms of the Stein estimator and that in terms of the conditional MLE is observed. This decomposition reflects that of the expected log-likelihood ratio. Accordingly, it is concluded that these modified estimators reduce the risk by reducing the expected log-likelihood ratio. The empirical Bayes method is discussed from this point of view.     

Limit theorems for maximum likelihood estimators in the Curie-Weiss-Potts model

The Curie-Weiss-Potts model, a model in statistical mechanics, is parametrized by the inverse temperature β and the external magnetic field h. This paper studies the asymptotic behavior of the maximum likelihood estimator of the parameter β when h = 0 and the asymptotic behavior of the maximum likelihood estimator of the parameter h when β is known and the true value of h is 0. The limits of these maximum likelihood estimators reflect the phase transition in the model; i.e., different limits depending on whether β < β_c, β = β_c or β > β_c, where β_c ε (0, ∞) is the critical inverse temperature of the model.     

Geometrical expansions for the distributions of the score vector and the maximum likelihood estimator

In the present note, asymptotic expansions for conditional and unconditional distributions of the score vector are derived. Our aim is to consider these expansions in the light of differential geometry, particularly the theory of derivative strings. Expansions for the distributions of the maximum likelihood estimator are obtained from those for the score vector via transformation, with a view to interpreting from the standpoint of differential geometry the various terms entering the expansions.The present work was carried out at the Department of Theoretical Statistics, University of Aarhus, Denmark, with support from the Danish-French Cultural Exchange Programme.     

An asymptotic expansion for the distribution of asymptotic maximum likelihood estimators of vector parameters

It is shown that the probability that a suitably standardized asymptotic maximum likelihood estimator of a vector parameter (i.e., an estimator which approximates the solution of the likelihood equation in a reasonably good way) lies in a measurable convex set can be approximated by an integral involving a multidimensional normal density function and a series in $\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ with certain polynomials as coefficients.     

On large deviation expansion of distribution of maximum likelihood estimator and its application in large sample estimation

For estimating an unknown parameter , the likelihood principle yields the maximum likelihood estimator. It is often favoured especially by the applied statistician, for its good properties in the large sample case. In this paper, a large deviation expansion for the distribution of the maximum likelihood estimator is obtained. The asymptotic expansion provides a useful tool to approximate the tail probability of the maximum likelihood estimator and to make statistical inference. Theoretical and numerical examples are given. Numerical results show that the large deviation approximation performs much better than the classical normal approximation.This work is supported in part by the Natural Science and Engineering Research Council of Canada under grant NSERC A-9216.This author is also partially supported by the National Science Foundation of China.     

The maximum likelihood estimators in a multivariate normal distribution with AR(1) covariance structure for monotone data

The maximum likelihood estimators are uniquely obtained in a multivariate normal distribution with AR(1) covariance structure for monotone data. The maximum likelihood estimator of mean is unbiased.     

A note on Sheppard's corrections for grouping and maximum likelihood estimation

Sheppard's corrections for grouping can, in the case of an underlying normal distribution, be interpreted as a first step to the solution of the maximum likelihood equations which incorporate the grouping problem. This result of Lindley (for the univariate) and Haitovsky (for the bivariate) is generalized to the multivariate normal distribution, making use of recent results in matrix algebra. Also, formulae concerning the efficiency lost in grouping are generalized to the multivariate case.