Digitized Theses

1993

Dissertation

Degree Name

Doctor of Philosophy

Abstract

In this thesis, I develop mean likelihood estimation (MeLE) and maximum likelihood estimation (MLE) for the parameters of a fractionally differenced autoregressive moving average (FARMA) model and its integrated form (IFARMA).;In chapter 1, I study the sampling distribution theory of MeLE by embedding the estimation problem into a Bayesian model.;In chapter 2, I apply MeLE and MLE to the estimation of the moving average parameter of a MA(1) model. The main result is that the MeLE has a smaller mean square error than the MLE when the moving average parameter is sufficiently inside the parameter space. I also demonstrate that the concentrated likelihood function of a time series with a MA(1) component has a local maximum or minimum at the boundary {dollar}\pm{dollar}1.;In chapter 3, I give the theory and methods needed to simulate FARMA series and to estimate the parameters of the FARMA model by MeLE and MLE. I explain how to calculate the exact value of the likelihood function for the parameters of a FARMA model. I show how to integrate functions over the stationary and invertible region of an ARMA process. Finally, I conduct simulation studies that compare the MLE and MeLE in combination with various algorithms to calculate them.;In the last chapter, I solve the problem of defining a likelihood function that unifies both the stationary FARMA and non-stationary IFARMA models, and I show how to select an appropriate model for a given time series by using a minimum information criteria strategy.

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