ON THE LIKELIHOOD RATIO TEST FOR A SINGLE MODEL AGAINST THE MIXTURE OF TWO KNOWN DENSITIES

In this paper, we consider the likelihood ratio test for a single model against the mixture of two known densities. We research the accuracy of the limiting null distribution obtained by Titterington 1-2 Titterington, D. M. 1981. Contribution to the discussion of paper by M. Aitkin, D. Amderson and J. Hinde, J. R.. Statist. Soc. A, 144: 459 Titterington, D. M., Smith, A. F.M. and Makov, U. E. 1985. Statistical Analysis of Finite Mixture Distribution New York: Wiley.  .After that, we improve the limiting null distribution and confirm its usefulness by the simulation study.     

AN EXTENSION OF THE EM ALGORITHM FOR OPTIMIZATION OF CONSTRAINED LIKELIHOOD: AN APPLICATION IN TOXICOLOGY

Since its introduction in the mid 1970's, the EM algorithm has found a widespread popularity for solving the likelihood equations. Several investigators have used the algorithm in a variety of problems with incomplete information to obtain maximum likelihood estimates in numerous applications. The algorithm, however, becomes inappropriate when the underlying equations are subject to some constraints. Although an extension has been proposed to derive solutions when the parameters are subject to a set of linear constraints, the evaluation of likelihood equations from incomplete data when the equations are subject to a nonlinear constraint is still an open problem. Here, we consider a mixture model, a classical example of incomplete data, and discuss the problem of maximum likelihood estimation of the model parameters when the parameters have to satisfy a nonlinear constraint. An extension of the EM algorithm based on the celebrated Lagrange multiplier will be proposed to solve the equations. An application of the methodology in animal bioassay experiments for risk assessment of toxic substances will be described and data from a toxicological experiment will be used to illustrate the results.     

Nearly optimal orthogonally blocked desings for a quadratic mixture model with q components

In experiments with mixtures involving process variables, orthogonal block designs may be used to allow estimation of the parameters of the mixture components independently of estimation of the parameters of the process variables. In the class of orthogonally blocked designs based on pairs of suitably chosen Latin squares, the optimal designs consist primarily of binary blends of the mixture components, regardless of how many ingredients are available for the mixture. This paper considers ways of modifying these optimal designs so that some or all of the runs used in the experiment include a minimum proportion of each mixture ingredient. The designs considered are nearly optimal in the sense that the experimental points are chosen to follow ridges of maxima in the optimality criteria. Specific designs are discussed for mixtures involving three and four components and distinctions are identified for different designs with the same optimality properties. The ideas presented for these specific designs are readily extended to mixtures with q>4 components.     

Full Bayesian Significance Test for Zero-Inflated Distributions

ABSTRACT

In this article we propose a new Kolmogorov-Smirnov type test for testing the nonlinearity of time series, based on the one which was originally introduced by An and Cheng. Our simulation study shows that the suggested test performs better than the original test. An associated bootstrap test is also found to outperform remarkably the non-bootstrap test.     

Optimum Mixture Design Using Deficiency Criterion

In a mixture experiment the measured response is assumed to depend only on the relative proportion of ingredients or components present in the mixture. Scheffe (1958 Scheffe , H. ( 1958 ). Experiments with mixtures . J. R. Statist. Soc. B 20 : 344 – 360 . [Google Scholar], 1963 Scheffe , H. ( 1963 ). Simplex – centroid design for experiments with mixtures . J. R. Statist. Soc. B 25 : 235 – 263 . [Google Scholar]) first systematically considered this problem and introduced different models and designs suitable in such situations. Optimum designs for the estimation of parameters of different mixture models are available in the literature. The problem of estimating the optimum proportion of mixture components is of great practical importance. Pal and Mandal (2006 Pal , M. , Mandal , N. K. ( 2006 ). Optimum designs for optimum mixtures . Statist. Probab. Lett. 76 ( 13 ): 1369 – 1379 .[Crossref], [Web of Science ®] , [Google Scholar], 2007 Pal , M. , Mandal , N. K. (2007). Optimum mixture design via equivalence theory. communicated. JCISS 32:107–126. [Google Scholar]) attempted to find a solution to this problem by adopting a pseudo-Bayesian approach and using the trace criterion. Subsequently, Pal and Mandal (2008 Pal , M. , Mandal , N. K. ( 2008 ). Minimax designs for optimum mixtures . To appear in Statist. Probab. Lett.  [Google Scholar]) solved the problem using minimax criterion. In this article, the deficiency criterion due to Chatterjee and Mandal (1981 Chatterjee , S. K. , Mandal , N. K. ( 1981 ). Response surface designs for estimating the optimal point . Calcutta Statist. Assoc. Bull. . 30 : 145 – 169 .[Crossref] , [Google Scholar]) has been used as a measure for comparing the performance of competing designs.     

Estimation of Slope for Linear Regression Model with Uncertain Prior Information and Student-t Error

This article considers estimation of the slope parameter of the linear regression model with Student-t errors in the presence of uncertain prior information on the value of the unknown slope. Incorporating uncertain non sample prior information with the sample data the unrestricted, restricted, preliminary test, and shrinkage estimators are defined. The performances of the estimators are compared based on the criteria of unbiasedness and mean squared errors. Both analytical and graphical methods are explored. Although none of the estimators is uniformly superior to the others, if the non sample information is close to its true value, the shrinkage estimator over performs the rest of the estimators.     

Bayes Estimation of Change Point in Non Standard Mixture of Left-Truncated Exponential With Unknown Proportions

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,…, X _n were observed from the mixture of a degenerate and left-truncated exponential (LTE) distribution, with reliability R _1τ at time τ and minimum life length η with unknown proportion p ₁ and θ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in reliability R _2τ at time τ and unknown proportion p ₂ and θ₂. This distribution occurs in many practical situations, for instance; life of a unit may have a LTE distribution but some of the units fail instantaneously. Apart from mixture distributions, the phenomenon of change point is also observed in several situations in life testing and reliability estimation problems. It may happen that at some point of time instability in the sequence of failure times is observed. The problem of study is: When and where this change has started occurring. This is called change point inference problem. The estimators of m, R ₁(t ₀), R ₂(t ₀), p ₁, and p ₂ are derived under asymmetric loss functions namely Linex loss & general entropy loss functions. Both the non informative and informative prior are considered. The effects of prior consideration on Bayes estimates of change point are also studied.     

A Finite Mixture Three-Parameter Weibull Model for the Analysis of Wind Speed Data

This article presents a mixture three-parameter Weibull distribution to model wind speed data. The parameters are estimated by using maximum likelihood (ML) method in which the maximization problem is regarded as a nonlinear programming with only inequality constraints and is solved numerically by the interior-point method. By applying this model to four lattice-point wind speed sequences extracted from National Centers for Environmental Prediction (NCEP) reanalysis data, it is observed that the mixture three-parameter Weibull distribution model proposed in this paper provides a better fit than the existing Weibull models for the analysis of wind speed data under study.     

On the Mixture Proportional Mean Residual Life Model

In this article, we propose a new mixture model induced by the model of proportional mean residual life. Under some appropriate assumptions, it is shown that the mixing and overall variables in the model admit the positive likelihood ratio dependence structure. To see how the overall variable is affected by the stochastic variation of the mixing variable, we study some stochastic comparisons using these variables. Finally, some useful bounds for tail probability of the overall variable for large values of the mixing variable are derived.