Mixture models for times series

• Main Author: James, S. D.
• Published: Swansea University 2001
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.637395

This thesis reviews some known results for the class of mixture models introduced by Jalali and Pemberton (1995) and presents two examples from the literature, which are based on the theory. The first has a countable number of mixture elements while the second has a finite number, <I>K</I>, and is called the Bernstein mixture model, since it involves the use of Bernstein polynomials in its construction. By including an additional parameter, λ, in the Binomial weights function, we obtain a parameterised version of the Bernstein model. The elements of the transition matrix for this model are polynomials in λ of degree <I>K</I> and the stationary distribution assumes a more complicated structure compared with its unparameterised counterpart. A series of elementary mathematical techniques is applied to reduce the elements of the transition matrix to much simpler polynomials and Cramer's Rule is adopted as a solution to obtain an explicit, analytical expression for the stationary distribution of the time series. Through maximum likelihood estimation of the parameters, λ, and <I>K</I>, in the parameterised Bernstein model, the solution developed using Cramer's Rule is compared with an alternative approach for evaluating the stationary distribution. This approach involves implementing a NAG subroutine based on Crout's factorisation method to solve the usual equations for the stationary probability row-vector. Finally, a relatively straightforward treatment of asymptotic maximum likelihood theory is given for the parameterised Bernstein model by employing regularity conditions stated in Billingsley (1961).