Maximum likelihood signature estimation

Maximum-likelihood estimates are discussed which are based on an unlabeled sample of observations, of unknown parameters in a mixture of normal distributions. Several successive approximation procedures for obtaining such maximum-likelihood estimates are described. These procedures, which are theoretically justified by the local contractibility of certain maps, are designed to take advantage of good initial estimates of the unknown parameters. They can be applied to the signature extension problem, in which good initial estimates of the unknown parameters are obtained from segments which are geographically near the segments from which the unlabeled samples are taken. Additional problems to which these methods are applicable include: estimation of proportions and adaptive classification (estimation of mean signatures and covariances)

Maximum Lqq-likelihood estimation

In this paper, the maximum L$q$-likelihood estimator (ML$q$E), a new parameter estimator based on nonextensive entropy [Kibernetika 3 (1967) 30--35] is introduced. The properties of the ML$q$E are studied via asymptotic analysis and computer simulations. The behavior of the ML$q$E is characterized by the degree of distortion $q$ applied to the assumed model. When $q$ is properly chosen for small and moderate sample sizes, the ML$q$E can successfully trade bias for precision, resulting in a substantial reduction of the mean squared error. When the sample size is large and $q$ tends to 1, a necessary and sufficient condition to ensure a proper asymptotic normality and efficiency of ML$q$E is established.Comment: Published in at http://dx.doi.org/10.1214/09-AOS687 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model

We consider the problem of estimating the distribution function, the density and the hazard rate of the (unobservable) event time in the current status model. A well studied and natural nonparametric estimator for the distribution function in this model is the nonparametric maximum likelihood estimator (MLE). We study two alternative methods for the estimation of the distribution function, assuming some smoothness of the event time distribution. The first estimator is based on a maximum smoothed likelihood approach. The second method is based on smoothing the (discrete) MLE of the distribution function. These estimators can be used to estimate the density and hazard rate of the event time distribution based on the plug-in principle.Comment: Published in at http://dx.doi.org/10.1214/09-AOS721 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Observed Range Maximum Likelihood Estimation

The idea of maximizing the likelihood of the observed range for a set of jointly realized counts has been employed in a variety of contexts. The applicability of the MLE introduced in [1] has been extended to the general case of a multivariate sample containing interval censored outcomes. In addition, a kernel density estimator and a related score function have been proposed leading to the construction of a modified Nadaraya-Watson regression estimator. Finally, the author has treated the problems of estimating the parameters of a mutinomial distribution and the analysis of contingency tables in the presence of censoring.Comment: censored multivariate data, contingency tables with incomplete counts, nonparametric density estimation, nonparametric regressio

Maximum-likelihood method in quantum estimation

The maximum-likelihood method for quantum estimation is reviewed and applied to the reconstruction of density matrix of spin and radiation as well as to the determination of several parameters of interest in quantum optics.Comment: 12 pages, 4 figure

Symbolic Maximum Likelihood Estimation with Mathematica

Mathematica is a symbolic programming language that empowers the user to undertake complicated algebraic tasks. One such task is the derivation of maximum likelihood estimators, demonstrably an important topic in statistics at both the research and expository level. In this paper, a Mathematica package is provided that contains a function entitled SuperLog. This function utilises pattern-matching code that enhances Mathematica's ability to simplify expressions involving the natural logarithm of a product of algebraic terms. This enhancement to Mathematica's functionality can be of particular benefit for maximum likelihood estimation