# Problems in the Analysis of Binary Mixture Distributions

Khan, Muhammad Farid (1989) Problems in the Analysis of Binary Mixture Distributions. PhD thesis, University of Glasgow.

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## Abstract

In Chapter 1, a finite mixture distribution is defined and background literature is briefly mentioned. A theorem is proved in Chapter 2 which states that if three density functions are proper mixtures of the same two components then one of them is a proper mixture of the other two. It is shown, with the help of a counter example, that the converse of this theorem is not necessarily true. In Chapter 3, two critical papers are reviewed. In Chapter 4, a distance function, based on the necessary condition of this theorem, is defined which 'measures' the deviation from the hypothesis that one of the densities is a proper mixture of the other two. The expressions for the mean and variance of this distance function are calculated for continuous data. Simulations are carried out to generate values of the distance function for a number of cases (null as well as non-null). For the several values of the distance function thus obtained the sample mean and sample variance were calculated and compared with the theoretical values obtained from the above mentioned expressions. The distribution of the distance function is considered and it is shown, diagrammatically, that the logarithm of this distance function is approximately normally distributed. Finally, in this chapter, a test of the null hypothesis is suggested. In Chapter 5, expressions for the mean and variance of the distance function are discussed for discrete data. A simulation study is carried out and a 'parametric' test of the null hypothesis is suggested and carried out on several data sets. Also, a non-parametric test of the null hypothesis is suggested and carried out on the same data sets. Finally, in this chapter, the various tests suggested are also applied to some fish data. In Chapter 6, a Monte Carlo test based on the above-mentioned distance function is introduced. In Chapter Via theorem is proved which determines another necessary condition for three densities to be proper mixtures of the same two components. In Chapter 8, the theory and methods developed in this thesis (which are applicable only to the case of three density functions) are generalised to the case where we have more than three density functions. Finally, finite mixtures having more than two components are discussed in some detail.

Item Type: Thesis (PhD) Doctoral Adviser: D M Titterington Statistics 1989 Enlighten Team glathesis:1989-76487 Copyright of this thesis is held by the author. 19 Nov 2019 14:16 19 Nov 2019 14:16 http://theses.gla.ac.uk/id/eprint/76487