Estimation theory

About: Estimation theory is a research topic. Over the lifetime, 35334 publications have been published within this topic receiving 1037566 citations.

Estimation of Regression Coefficients When Some Regressors are not Always Observed

Abstract: In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score...

The expectation-maximization algorithm

Abstract: A common task in signal processing is the estimation of the parameters of a probability distribution function Perhaps the most frequently encountered estimation problem is the estimation of the mean of a signal in noise In many parameter estimation problems the situation is more complicated because direct access to the data necessary to estimate the parameters is impossible, or some of the data are missing Such difficulties arise when an outcome is a result of an accumulation of simpler outcomes, or when outcomes are clumped together, for example, in a binning or histogram operation There may also be data dropouts or clustering in such a way that the number of underlying data points is unknown (censoring and/or truncation) The EM (expectation-maximization) algorithm is ideally suited to problems of this sort, in that it produces maximum-likelihood (ML) estimates of parameters when there is a many-to-one mapping from an underlying distribution to the distribution governing the observation The EM algorithm is presented at a level suitable for signal processing practitioners who have had some exposure to estimation theory

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...

MUSIC, maximum likelihood, and Cramer-Rao bound

Abstract: The performance of the MUSIC and ML methods is studied, and their statistical efficiency is analyzed. The Cramer-Rao bound (CRB) for the estimation problems is derived, and some useful properties of the CRB covariance matrix are established. The relationship between the MUSIC and ML estimators is investigated as well. A numerical study is reported of the statistical efficiency of the MUSIC estimator for the problem of finding the directions of two plane waves using a uniform linear array. An exact description of the results is included. >

The Robust Inference for the Cox Proportional Hazards Model

Abstract: We derive the asymptotic distribution of the maximum partial likelihood estimator β for the vector of regression coefficients β under a possibly misspecified Cox proportional hazards model. As in the parametric setting, this estimator β converges to a well-defined constant vector β*. In addition, the random vector n 1/2(β – β*) is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated. The newly proposed robust covariance matrix estimator is similar to the so-called “sandwich” variance estimators that have been extensively studied for parametric cases. For many misspecified Cox models, the asymptotic limit β* or part of it can be interpreted meaningfully. In those circumstances, valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of β and the related results for the score statistics. Extensive studies demonstrate that the proposed robust tests and interval estimation procedures...