Improved maximum-likelihood estimation for the common shape parameter of several Weibull populations

The biasness problem of the maximum-likelihood estimate (MLE) of the common shape parameter of several Weibull populations is examined in detail. A modified MLE (MMLE) approach is proposed. In the case of complete and Type II censored data, the bias of the MLE can be substantial. This is noticeable even when the sample size is large. Such a bias increases rapidly as the degree of censorship increases and as more populations are involved. The proposed MMLE, however, is nearly unbiased and much more efficient than the MLE, irrespective of the degree of censorship, the sample sizes, and the number of populations involved. Copyright © 2007 John Wiley & Sons, Ltd.     

Monte carlo estimation of the sampling distribution of nonlinear model parameter estimators

The sampling distribution of parameter estimators can be summarized by moments, fractiles or quantiles. For nonlinear models, these quantities are often approximated by power series, approximated by transformed systems, or estimated by Monte Carlo sampling. A control variate approach based on a linear approximation of the nonlinear model is introduced here to reduce the Monte Carlo sampling necessary to achieve a given accuracy. The particular linear approximation chosen has several advantages: its moments and other properties are known, it is easy to implement, and there is a correspondence to asymptotic results that permits assessment of control variate effectiveness prior to sampling via measures of nonlinearity. Empirical results for several nonlinear problems are presented.This research was supported in part by the Office of Naval Research under Contract N00014-79-C-0832.     

A new smoothing-regularization approach for a maximum-likelihood estimation problem

We consider the problem min _i=1 ^m (aⁱ,x–b_iloga ⁱ, z) subject tox 0 which occurs as a maximum-likelihood estimation problem in several areas, and particularly in positron emission tomography. After noticing that this problem is equivalent to mind(b, Ax) subject tox 0, whered is the Kullback-Leibler information divergence andA, b are the matrix and vector with rows and entriesa ⁱ,b _i, respectively, we suggest a regularized problem mind(b, Ax) + d(v, Sx), where is the regularization parameter,S is a smoothing matrix, andv is a fixed vector. We present a computationally attractive algorithm for the regularized problem, establish its convergence, and show that the regularized solutions, as goes to 0, converge to the solution of the original problem which minimizes a convex function related tod(v, Sx). We give convergence-rate results both for the regularized solutions and for their functional values.The research of A. N. Iusem was partially supported by CNPq Grant No. 301280/86-MA.     

Tumor location and parameter estimation by thermography

In non-invasive thermal diagnostics, accurate correlations between the thermal image on skin surface and interior human physiology are often desired, which require general solutions for the bioheat equation. In this study an estimation methodology is presented to determine unknown thermophysical or geometrical parameters of a tumor region using the temperature profile on the skin surface that may be obtained by infrared thermography. To solve these inverse problems a second order finite difference scheme was implemented to solve the bioheat Pennes equation with mixed boundary conditions in two and three dimensions. Then, the Pattern Search algorithm was used to estimate the different parameters by minimizing a fitness function involving the temperature profiles obtained from simulated or clinical data to those obtained by the finite different scheme.     

Recursive parameter estimation: convergence

We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general statistical model and study convergence.      

Asymptotic behavior of parameter estimators of a multivariate discontinuous density

We study the asymptotic behavior of Bayesian and maximum likelihood estimators of the parameter when the density function f(x, ) for each fixed x has discontinuities in on some surface.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 83–107, 1977.     

Recursive parameter estimation: asymptotic expansion

This paper is concerned with the asymptotic behaviour of estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. The results of the paper can be used to determine the form of the recursive procedure which is expected to have the same asymptotic properties as the corresponding non-recursive one defined as a solution of the corresponding estimating equation. Several examples are given to illustrate the theory, including an application to estimation of parameters in exponential families of Markov processes.     

On the efficiency of estimators of a spectral density multivariate parameter

Asymptotic properties of the Whittle estimator are considered. The asymptotic efficiency in the minimax sense, as well as in the Bahadur sense, are proved. The asymptotic behavior of the Whittle estimator and the maximum likelihood estimator is compared.     

On the estimation of the asymptotic variance of the rank estimators for the location parameter from a sample of random size

We obtain conditions for asymptotic normality of the estimate of asymptotic variance of rank estimators for the location parameter. The size of the sample is assumed ro be random and to depend on the sample values. We consider applications of the result obtained to sequential estimation of confidence intervals.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 97–102, 1987.     

Error structures and parameter estimation

The error calculus based on the theory of Dirichlet forms is an extension of Gauss' approach to error propagation. The aim of this paper is to derive error structures from measurements. The links with Fisher's information lay the foundations of a strong connection with experiment. Here we show that this connection behaves well towards changes of variables and is related to the theory of asymptotic statistics. Finally the study of products permits one to lay the foundation of an infinite dimensional empirical error calculus. To cite this article: N. Bouleau, C. Chorro, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A simulation based approach to the parameter estimation for the three-parameter gamma distribution

The gamma distribution is one of the commonly used statistical distribution in reliability. While maximum likelihood has traditionally been the main method for estimation of gamma parameters, Hirose has proposed a continuation method to parameter estimation for the three-parameter gamma distribution. In this paper, we propose to apply Markov chain Monte Carlo techniques to carry out a Bayesian estimation procedure using Hirose’s simulated data as well as two real data sets. The method is indeed flexible and inference for any quantity of interest is readily available.     

On stable estimation of a function of a parameter

Given the function f and the vector-statistic t_N which is a mean square consistent estimator of a parameter a, the problem is to estimate f(a). The criteria for the mean square consistency of the estimator f(t_N) are considered. In the case where the estimator f(t_N) is not mean square consistent, a class of estimators of f(a) is proposed, and it is proved that the estimators of the class are mean square consistent for all distribution of t_N. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 44–55, Perm, 1990.     

Minimax estimators for location vectors in elliptical distributions with unknown scale parameter and its application to variance reduction in simulation

In this paper, we give an ever wider and new class of minimax estimators for the location vector of an elliptical distribution (a scale mixture of normal densities) with an unknown scale parameter. The its application to variance reduction for Monte Carlo simulation when control variates are used is considered. The results obtained thus extend (i) Berger's result concerning minimax estimation of location vectors for scale mixtures of normal densities with known scale parameter and (ii) Strawderman's result on the estimation of the normal mean with common unknown variance.Research partially supported by National Science Foundation, Grant #DMS 8901922.     

The use of pontryagin estimators for on-line optimal control sequence estimation: The truck backer-upper case study

The Pontryagin optimality principle can be used in conjunction with on-line (or real-time) measurements of state data to build a local model of the control law. In this paper, we discuss and refine the use of this technique in the context of the simple truck backer-upper problem. We first compare the use of feedforward, associative and CMAC neural architectures for the local control model encoding. Algorithm implementation is then done using the CMAC architecture because of its speed of learning and local scoping. We build temporal difference state prediction models for the truck dynamics and then use these predictions to build an estimate of the best control action to take. This control action is constructed from a depth first tree search used in conjunction with optimal control information obtained by solving locally scoped control problems via the Pontryagin optimality principle. The state to control model can then be encoded into a variety of function approximation models.     

Asymptotic theory of parameter estimation by a contrast function based on interpolation error

Interpolation is an important issue for a variety fields of statistics (e.g., missing data analysis). In time series analysis, the best interpolator for missing points problem has been investigated in several ways. In this paper, the asymptotics of a contrast function estimator defined by pseudo interpolation error for stationary process are investigated. We estimate parameters of the process by minimizing the pseudo interpolation error written in terms of a fitted parametric spectral density and the periodogram based on observed stretch. The estimator has the consistency and asymptotical normality. Although the criterion for the interpolation problem is known as the best in the sense of smallest mean square error for past and future extrapolation, it is shown that the estimator is asymptotically inefficient in general parameter estimation, which leads to an unexpected result.     

Nonparametric maximum likelihood density estimation and simulation-based minimum distance estimators

Indirect inference estimators (i.e., simulation-based minimum distance estimators) in a parametric model that are based on auxiliary nonparametric maximum likelihood density estimators are shown to be asymptotically normal. If the parametricmodel is correctly specified, it is furthermore shown that the asymptotic variance-covariance matrix equals the inverse of the Fisher-information matrix. These results are based on uniform-in-parameters convergence rates and a uniform-inparameters Donsker-type theorem for nonparametric maximum likelihood density estimators.