Empirical Bayes estimation of parameters in Markov transition probability matrix with computational methods

Empirical Bayes estimator for the transition probability matrix is worked out in the cases where we have belief regarding the parameters, For example, where the states seem to be equal or not. In both cases, priors are in accordance with our beliefs. Using EM algorithm, computational methods for different hyperparameters of the empirical Bayes are described. Also, robustness of empirical Bayes procedure is investigated.     

Simultaneous robust estimation of location and scale parameters: A minimum-distance approach

Simultaneous robust estimates of location and scale parameters are derived from minimizing a minimum-distance criterion function. The criterion function measures the squared distance between the pth power (p > 0) of the empirical distribution function and the pth power of the imperfectly determined model distribution function over the real line. We show that the estimator is uniquely defined, is asymptotically bivariate normal and for p > 0.3 has positive breakdown. If the scale parameter is known, when p = 0.9 the asymptotic variance (1.0436) of the location estimator for the normal model is smaller than the asymptotic variance of the Hodges-Lehmann (HL)estimator (1.0472). Efficiencies with respect to HL and maximum-likelihood estimators (MLE) are 1.0034 and 0.9582, respectively. Similarly, if the location parameter is known, when p = 0.97 the asymptotic variance (0.6158) of the scale estimator is minimum. The efficiency with respect to the MLE is 0.8119. We show that the estimator can tolerate more corrupted observations at oo than at – for p < 1, and vice versa for p > 1.     

On the maximum likelihood estimation of parameters of a spatial discrimination model

Consideration is given here to the problem of maximum likelihood estimation of parameters in a sparial discrimination model which was proposed by switzer (1980). some moments of these estimators are derived. These results extend the work of Mardia (1984) who gave expressions for these estimators without their moments.     

Sequential estimation of a linear function of location parameters of two negative exponential distributions

The paper considers the problem of bounded risk point estimation for a linear function of location parameters of two negative exponential distributions, including the difference in a special case, when two scale parameters are unknown. Purely sequential procedures are proposed and second order expansions of the average sample sizes and risk are given. Furthermore some simulation results are provided.     

Best linear unbiased estimation of location and scale parameters of the log-logistic distribution

If an assumption, such as homoscedasticity, or some other aspect of an inference problem, such as the number of cases, is altered, our conclusions may change and different parts of the conclusions can be affected in different ways. Most diagnostic procedures measure the influence on one particular aspect of the conclusion - such as model fit or change in parameter estimates. The effect on all aspects of the conclusions can be described by the difference in two log likelihood functions and when the log likelihood functions come from an exponential family or are quasi-likelihoods, this difference can be factored into three terms: one depending only on the alteration, another depending only on the aspects of the conclusions to be considered, and a third term depending on both. The third term is interesting because it shows which aspects of the conclusions are relatively insensitive even to large alterations.     

Bounded risk estimation of a linear combination of location parameters in negative exponential distributions via three-stage sampling

The paper deals with the problem of bounded risk point estimation for a linear combination of location parameters of two negative exponential distributions. Isogai and Futschik considered the situation when the location and scale parameters are all unknown. They proposed purely sequential procedures and gave second order expansions of the average sample sizes and risks. In this paper we propose three-stage procedures and derive second order expansions of the average sample sizes and risks. Further, we compare the results with those from previous work.     

Detection of jump location curve in spatial linear regression model with two-dimensional threshold

In the context of spatial linear regression, we discuss detection of jump location curve treated as threshold curve which cannot be expressed by independent variables but indirectly determines two specific model forms. The threshold curve in this paper is described by a straight line with two location variables, longitude and latitude, and can be estimated by maximizing the coefficient difference between two one-sided linear regression models. Theoretical results show that the estimator is consistent. Our method performs well by numerical studies.     

A Bayesian model for local smoothing in kernel density estimation

A new procedure is proposed for deriving variable bandwidths in univariate kernel density estimation, based upon likelihood cross-validation and an analysis of a Bayesian graphical model. The procedure admits bandwidth selection which is flexible in terms of the amount of smoothing required. In addition, the basic model can be extended to incorporate local smoothing of the density estimate. The method is shown to perform well in both theoretical and practical situations, and we compare our method with those of Abramson (The Annals of Statistics 10: 1217–1223) and Sain and Scott (Journal of the American Statistical Association 91: 1525–1534). In particular, we note that in certain cases, the Sain and Scott method performs poorly even with relatively large sample sizes.We compare various bandwidth selection methods using standard mean integrated square error criteria to assess the quality of the density estimates. We study situations where the underlying density is assumed both known and unknown, and note that in practice, our method performs well when sample sizes are small. In addition, we also apply the methods to real data, and again we believe our methods perform at least as well as existing methods.     

Shrinkage estimation of location parameters in a multivariate skew-normal distribution

Abstract

This paper studies decision theoretic properties of Stein type shrinkage estimators in simultaneous estimation of location parameters in a multivariate skew-normal distribution with known skewness parameters under a quadratic loss. The benchmark estimator is the best location equivariant estimator which is minimax. A class of shrinkage estimators improving on the best location equivariant estimator is constructed when the dimension of the location parameters is larger than or equal to four. An empirical Bayes estimator is also derived, and motivated from the Bayesian procedure, we suggest a simple skew-adjusted shrinkage estimator and show its dominance property. The performances of these estimators are investigated by simulation.     

Maximum likelihood estimation and prediction mean square error in the spatial linear model

This paper is concerned with prediction in the spatial linear model using the maximum likelihood estimation of parameters in this model. In particular, we give some properties of predictors obtained on substituting the maximum likelihood estimators of model parameters into the form of the best-in the sense of minimum mean square prediction error-predictor. Such predictors are not optimal but we show them to be asymptotically equivalent to the optimum. We discuss practical aspects of this work and conclude by considering the connection with other areas.     

Effects of the estimation of covariance matrix parameters in the generalized multivariate linear model

We Consider the generalized multivariate linear model and assume the covariance matrix of the p x 1 vector of responses on a given individual can be represented in the general linear structure form described by Anderson (1973). The effects of the use of estimates of the parameters of the covariance matrix on the generalized least squares estimator of the regression coefficients and on the prediction of a portion of a future vector, when only the first portion of the vector has been observed, are investigated. Approximations are derived for the covariance matrix of the generalized least squares estimator and for the mean square error matrix of the usual predictor, for the practical case where estimated parameters are used.     

Robust estimation of location in samples of size three

Youden (1953) discussed the practice of averaging the two most concordant observations in sets of three measurements as a method of estimating location. Distributional results for this estimator can be found in Seth (1950) and Lieblein (1952). It follows from their work that the sample median has smaller variance for normal and uniform populations. In this paper it is shown that themedian stochastically dominates the average of the two closest observations for uniform, normal, double–exponential and Cauchy populations and thus is the superior resistant estimator in these cases for a broad class of loss functions. However, an example is given in which, for a particular contaminaion model and loss function, the mean of the closest two observations has smaller risk than the median.     

Robust estimation of covariance parameters in partial linear model for longitudinal data

For longitudinal data, the within-subject dependence structure and covariance parameters may be of practical and theoretical interests. The estimation of covariance parameters has received much attention and been studied mainly in the framework of generalized estimating equations (GEEs). The GEEs method, however, is sensitive to outliers. In this paper, an alternative set of robust generalized estimating equations for both the mean and covariance parameters are proposed in the partial linear model for longitudinal data. The asymptotic properties of the proposed estimators of regression parameters, non-parametric function and covariance parameters are obtained. Simulation studies are conducted to evaluate the performance of the proposed estimators under different contaminations. The proposed method is illustrated with a real data analysis.     

On minimum distance estimation of location based on the kolmogorov statistic

The asymptotic distribution is derived for the minimum distance estimator of a location parameter based on the Kolmogorov goodness of fit statistic. The distribution is expressed in terms of the distribution of a functional of a Brownian bridge. An upper bound is obtained for the length of the confidence interval based on the Kolmogorov statistic. A simulation study with sample sizes 10 and 20 compares the length of the interval based on the Kolmogorov statistic to the length of the interval based on the maximum likelihood estimator. Another simulation shows the effect of model misspecification on the coverage probabilities of the interval based on the Kolmogorov statistic.     

Bounded risk estimation of linear combinations of the location and scale parameters in exponential distributions under two-stage sampling

Two-stage sampling is proposed for estimating linear combinations of the location and scale parameters of exponential distributions with bounded quadratic risk functions. Exact formulae for the expected values and risks of the estimators are derived, and the performance of estimators is studied. Illustrations with real data are included.     

Robust estimation of the ratio of scale parameters for positive random variables

An estimator of the ratio of scale parameters of the distributions of two positive random variables is developed for the case where the only difference between the distributions is a difference in scale. Simulation studies demonstrate that the estimator performs much better, in terms of mean squared error, than the most popular one among those estimators currently available.     

Domains of convergence for the EM algorithm: a cautionary tale in a location estimation problem

The EM algorithm is a popular method for maximizing a likelihood in the presence of incomplete data. When the likelihood has multiple local maxima, the parameter space can be partitioned into domains of convergence, one for each local maximum. In this paper we investigate these domains for the location family generated by the t-distribution. We show that, perhaps somewhat surprisingly, these domains need not be connected sets. As an extreme case we give an example of a domain which consists of an infinite union of disjoint open intervals. Thus the convergence behaviour of the EM algorithm can be quite sensitive to the starting point.     

A robust estimation method for the linear regression model parameters with correlated error terms and outliers

Independence of error terms in a linear regression model, often not established. So a linear regression model with correlated error terms appears in many applications. According to the earlier studies, this kind of error terms, basically can affect the robustness of the linear regression model analysis. It is also shown that the robustness of the parameters estimators of a linear regression model can stay using the M-estimator. But considering that, it acquires this feature as the result of establishment of its efficiency. Whereas, it has been shown that the minimum Matusita distance estimators, has both features robustness and efficiency at the same time. On the other hand, because the Cochrane and Orcutt adjusted least squares estimators are not affected by the dependence of the error terms, so they are efficient estimators. Here we are using of a non-parametric kernel density estimation method, to give a new method of obtaining the minimum Matusita distance estimators for the linear regression model with correlated error terms in the presence of outliers. Also, simulation and real data study both are done for the introduced estimation method. In each case, the proposed method represents lower biases and mean squared errors than the other two methods.KEYWORDS: Robust estimation method, minimum Matusita distance estimation method, non-parametric kernel density estimation method, correlated error terms, outliers     

Quasi-minimax estimation in the partial linear model

This article discusses the minimax estimator in partial linear model y = Zβ + f + ε under ellipsoidal restrictions on the parameter space and quadratic loss function. The superiority of the minimax estimator over the two-step estimator is studied in the mean squared error matrix criterion.