 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 Finite mixtures of distributions with common central...
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Finite mixtures of distributions with common central moments Rennie, Robert Richard
Abstract
Let ℱ = {F} be a family of nvariate cumulative distribution functions (c.d.f.'s). If F₁...,F(k) belong to ℱ and P₁,...,P(k) are positive numbers that sum to 1, then the convex combination M(x₁ ,. . . , x(n)) = [formula omitted](x₁,...,x(n)) is called a finite mixture generated by ℱ. The F₁,…,F(k) are called the components of the mixture and the P₁,…,P(k) are called their weights, respectively. The mixture M(x₁,...,x(n)) is said to be identifiable with respect to ℱ if no other convex combination of a finite number of c.d.f.'s from ℱ will generate M(x₁,...,x(n)). We establish the identiflability of mixtures consisting of at most k components when the components belong to a family of univariate c.d.f.'s that have the following properties: (a) no two c.d.f.'s have the same mean; (b) each c.d.f. has the same r(th) central moment for r = 1,...,2k1; and (c) the first 2k1 central moments are finite. If the mixture and the 2k1 central moments are known, a solution for the weights and means of the components is given. If a random sample is taken from the mixture, then asymptotically normal estimates of the weights and means are given, providing the 2k1 central moments are known. Matrix mixtures are introduced and are found to be of use in estimating the density functions and c.d.f.'s of the components. In the .case of the above family, the estimates of the density functions are shown to have an asymptotically normal distribution. Consistent and least squares estimates are obtained for the component c.d.f.'s. We show that for multivariate mixtures identifiability of any one of the marginal mixtures implies the identifiability of the multivariate mixture, but not conversely. Finally, the univariate results are generalized to the multivariate case, and an example of the use of matrix mixtures is given.
Item Metadata
Title  Finite mixtures of distributions with common central moments 
Creator  Rennie, Robert Richard 
Publisher  University of British Columbia 
Date Issued  1968 
Description 
Let ℱ = {F} be a family of nvariate cumulative distribution functions (c.d.f.'s). If F₁...,F(k) belong to ℱ and P₁,...,P(k) are positive numbers that sum to 1, then the convex combination M(x₁ ,. . . , x(n)) = [formula omitted](x₁,...,x(n)) is called a finite mixture generated by ℱ. The F₁,…,F(k) are called the components of the mixture and the P₁,…,P(k) are called their weights, respectively. The mixture M(x₁,...,x(n)) is said to be identifiable with respect to ℱ if no other convex combination of a finite number of c.d.f.'s from ℱ will generate M(x₁,...,x(n)).
We establish the identiflability of mixtures consisting of at most k components when the components belong to a family of univariate c.d.f.'s that have the following properties: (a) no two c.d.f.'s have the same mean; (b) each c.d.f. has the same r(th) central moment for r = 1,...,2k1; and (c) the first 2k1 central moments are finite. If the mixture and the 2k1 central moments are known, a solution for the weights and means of the components is given. If a random sample is taken from the mixture, then asymptotically normal estimates of the weights and means are given, providing the 2k1 central moments are known.
Matrix mixtures are introduced and are found to be of use in estimating the density functions and c.d.f.'s of the components. In the .case of the above family, the estimates of the density functions are shown to have an asymptotically normal distribution. Consistent and least squares estimates are obtained for the component c.d.f.'s.
We show that for multivariate mixtures identifiability of any one of the marginal mixtures implies the identifiability of the multivariate mixture, but not conversely. Finally, the univariate results are generalized to the multivariate case, and an example of the use of matrix mixtures is given.

Subject  Distribution (Probability theory) 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110916 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0080609 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
Item Media
Item Citations and Data
License
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.