Likelihood principle

About: Likelihood principle is a research topic. Over the lifetime, 1691 publications have been published within this topic receiving 101715 citations.

Information Theory and an Extension of the Maximum Likelihood Principle

Abstract: In this paper it is shown that the classical maximum likelihood principle can be considered to be a method of asymptotic realization of an optimum estimate with respect to a very general information theoretic criterion. This observation shows an extension of the principle to provide answers to many practical problems of statistical model fitting.

A reliability coefficient for maximum likelihood factor analysis

Abstract: Maximum likelihood factor analysis provides an effective method for estimation of factor matrices and a useful test statistic in the likelihood ratio for rejection of overly simple factor models. A reliability coefficient is proposed to indicate quality of representation of interrelations among attributes in a battery by a maximum likelihood factor analysis. Usually, for a large sample of individuals or objects, the likelihood ratio statistic could indicate that an otherwise acceptable factor model does not exactly represent the interrelations among the attributes for a population. The reliability coefficient could indicate a very close representation in this case and be a better indication as to whether to accept or reject the factor solution.

The behavior of maximum likelihood estimates under nonstandard conditions

Abstract: This paper proves consistency and asymptotic normality of maximum likelihood (ML) estimators under weaker conditions than usual. In particular, (i) it is not assumed that the true distribution underlying the observations belongs to the parametric family defining the ML estimator, and (ii) the regularity conditions do not involve the second and higher derivatives of the likelihood function. The need for theorems on asymptotic normality of ML estimators subject to (i) and (ii) becomes apparent in connection with robust estimation problems; for instance, if one tries to extend the author's results on robust estimation of a location parameter [4] to multivariate and other more general estimation problems. Wald's classical consistency proof [6] satisfies (ii) and can easily be modified to show that the ML estimator is consistent also in case (i), that is, it converges to the 0o characterized by the property E(logf(x, 0) log f(x, O0)) < 0 for 0 . Oo, where the expectation is taken with respect to the true underlying distribution. Asymptotic normality is more troublesome. Daniels [1] proved asymptotic normality subject to (ii), but unfortunately he overlooked that a crucial step in his proof (the use of the central limit theorem in (4.4)) is incorrect without condition (2.2) of Linnik [5]; this condition seems to be too restrictive for many purposes. In section 4 we shall prove asymptotic normality, assuming that the ML estimator is consistent. For the sake of completeness, sections 2 and 3 contain, therefore, two different sets of sufficient conditions for consistency. Otherwise, these sections are independent of each other. Section 5 presents two examples.

Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions

Abstract: Large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described. Specifically, the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived. These results generalize the work of Moran (1971), Chant (1974), and Chernoff (1954). Some of Chant's results are shown to be incorrect. The approach used in deriving these results follows from comments made by Moran and Chant. The problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate Gaussian distribution from a sample of size 1. In this representation the Gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator. Thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the Gaussian random v...

Analysis of Covariance with Qualitative Data

Abstract: In data with a group structure, incidental parameters are included to control for missing variables. Applications include longitudinal data and sibling data. In general, the joint maximum likelihood estimator of the structural parameters is not consistent as the number of groups increases, with a fixed number of observations per group. Instead a conditional likelihood function is maximized, conditional on sufficient statistics for the incidental parameters. In the logit case, a standard conditional logit program can be used. Another solution is a random effects model, in which the distribution of the incidental parameters may depend upon the exogenous variables.