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.     

Estimation of critical points in the mixture inverse Gaussian model

The maximum likelihood estimation for the critical points of the failure rate and the mean residual life function are presented in the case of mixture inverse Gaussian model. Several important data sets are analyzed from this point of view. For each of the data sets, Bootstrapping is used to construct confidence intervals of the critical points.     

Stochastic analysis of covariance when the error distribution is long-tailed symmetric

In this study, we consider stochastic one-way analysis of covariance model when the distribution of the error terms is long-tailed symmetric. Estimators of the unknown model parameters are obtained by using the maximum likelihood (ML) methodology. Iteratively reweighting algorithm is used to compute the ML estimates of the parameters. We also propose new test statistic based on ML estimators for testing the linear contrasts of the treatment effects. In the simulation study, we compare the efficiencies of the traditional least-squares (LS) estimators of the model parameters with the corresponding ML estimators. We also compare the power of the test statistics based on LS and ML estimators, respectively. A real-life example is given at the end of the study.     

Variable selection in elliptical linear mixed model

Variable selection in elliptical Linear Mixed Models (LMMs) with a shrinkage penalty function (SPF) is the main scope of this study. SPFs are applied for parameter estimation and variable selection simultaneously. The smoothly clipped absolute deviation penalty (SCAD) is one of the SPFs and it is adapted into the elliptical LMM in this study. The proposed idea is highly applicable to a variety of models which are set up with different distributions such as normal, student-t, Pearson VII, power exponential and so on. Simulation studies and real data example with one of the elliptical distributions show that if the variable selection is also a concern, it is worthwhile to carry on the variable selection and the parameter estimation simultaneously in the elliptical LMM.     

Robust Liu estimator for regression based on an M-estimator

Consider the regression model y = beta 0 1 + Xbeta + epsilon. Recently, the Liu estimator, which is an alternative biased estimator beta L (d) = (X'X + I) -1 (X'X + dI)beta OLS , where 0<d<1 is a parameter, has been proposed to overcome multicollinearity . The advantage of beta L (d) over the ridge estimator beta R (k) is that beta L (d) is a linear function of d. Therefore, it is easier to choose d than to choose k in the ridge estimator. However, beta L (d) is obtained by shrinking the ordinary least squares (OLS) estimator using the matrix (X'X + I) -1 (X'X + dI) so that the presence of outliers in the y direction may affect the beta L (d) estimator. To cope with this combined problem of multicollinearity and outliers, we propose an alternative class of Liu-type M-estimators (LM-estimators) obtained by shrinking an M-estimator beta M , instead of the OLS estimator using the matrix (X'X + I) -1 (X'X + dI).     

Un nouveau type d'indice de la criminalité pour le Canada*

The authors present the results of a research project in which the aim was to develop for Canada an index of criminology and delinquency based on public appraisal of the seriousness of various criminal offences. This new kind of index of crime was originally proposed in the United States by T. Sellin and M. E. Wolfgang. The possibility of using such an index in order to compensate for the shortcomings of present statistics on criminal behaviour is brought out as a research conclusion. The index has advantages as well for the development of international statistics on crime.     

Mixtures of Farlie-Gumbel-Morgenstern Copulas

Some modifications of bivariate Farlie-Gumbel-Morgenstern (FGM) copulas are going to be explained in this article. These modifications are generated by using mixtures of bivariate FGM copula functions. The main goal of this study is to determine both the ranges of association parameter and the rate of correlation, and also observe the changes in local dependence function. An application, which is related with simulated data, is conducted and results are illustrated.     

Estimation of reliability under ordered restriction on the parameters

This paper deals with the estimation of reliability for a strength-stress model under ordered restriction on the parameters. It is assumed that components have exponential distributions and are arranged in a parallel system and the failure of one component, results in increasing the failure rate of the remaining components. Results are derived when (i) the ordering of the means is taken into account and when (ii) the ordering of the means is ignored. Simulation studies are carried out to compare the results. It is noticed that, in almost all cases, in case (i) the estimates are closer to the true value with smaller mean squared error (MSE) and smaller’ standard deviation than in case (ii). Thus when the ordering of the means is present in the model, such information should be incorporated in the estimation of reliability.