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Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Applied Statistics Unit (ASU-Kolkata)


Sengupta, Ashis (ASU-Kolkata; ISI)

Abstract (Summary of the Work)

Mixtures of distributions are now-a-days playing very important roles in both theoretical and applied statistics. There is an abundance of real-life situations where mixture distributions are being extensively used for modelling data and drawing inference. Several books and monographs on mixtures have so far been published, e.g., Everitt and Hand (1981), Titterington et al. (1985), McLachlan and Basford (1988), McLachlan (1997) etc., which cover a wide area on different aspects of mixture distributions and their applications. The book by Titterington et al. (1985, pp. 16-21), in particular, contains a comprehensive list of references on direct applications of finite mixtures in different fields of everyday life (see also Everitt, 1985; Titterington, 1997). However, while the literature on the estimation aspect of mixture distributions is quite rich, not much work seems to have been done on optimal testing problems.One of the most important issues in finite mixtures in general and two- component mixtures in particular is that of testing for the hypothesis of ‘no mixture against a mixture alternative. The likelihood ratio test (LRT), even for a two-component mixture, is very cumbersome and regularity conditions do not hold for -2 log A to have the usual asymptotic null distribution as that of a x with degrees of freedom equal to the difference in the number of parameters in the two hypotheses. Consider, for example, the mixture densityg(x|p,θ)=∑ri=1 pifi(x,θ)where p = (P1, P2, ...Pr); pi ≥ i 0 ∀ i and ∑ri=1 Pi = 1; θ denotes a vector of unknown parameters θ1, θ2, ..., θk and fi s are the component densities. In the particular case of r 2, testing the null hypothesis of ‘no mixture’ ..... 6. can be accomplished by testing whether p1 = 0, which is on the boundary of the parameter space (Self and Liang, 1987; Feng and McCulloch, 1992), resulting in a breakdown in the standard regularity conditions.When the mixture is composed of two known but general univariate densities f1 and f2 in unknown proportions p and 1 -p respectively, Titterington et al. (1985, pp. 152-153) considered the problem of testing H0 : p = 1 against H1 p < 1 and showed that asymptotically, under H0, -2 log λ is distributed as {max(0, Y)}2 where Y is a standard normal variable.To demonstrate the failure of the likelihood asymptotics in mixtures, Hartigan (1985) considered a mixture of two univariate normals N(0, 1) and N(0,1) in unknown proportions 1-p and p and showed that for testing H0: θ = 0 against H : θ≠0, the supremum of the loglikelihood, Ln, approaches ∞ in probability as n → ∞.Ghosh and Sen (1985) derived a general form of the asymptotic distribution of the LRT in ‘strongly identifiable’ mixtures. In the example of two-component univariate normal mixtures with unknown but identifiable means µ1 and µ2, unknown mixing proportion p and common known variance, they showed that for testing H0 : p = 1, the asymptotic distribution of -2 log λ is that of a certain functional[{0,sup µ2Y(µ2)}]2


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