Parametric estimation for randomly censored autocorrelated data.
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1997Supervisor
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This thesis is mainly concerned with the estimation of parameters in autoregressive models with censored data. For convenience, attention is restricted to the firstorder stationary autoregressive (AR(1)) model in which the response random variables are subject to rightcensoring. In their present form, currently available methods of estimation in regression analysis with censored autocorrelated data, which includes the MLE, are applicable only if the errors of the AR component of the model are Gaussian. Use of these methods in AR processes with nonGaussian errors requires, essentially, rederivations of the estimators. Hence, in this thesis, we propose new estimators which arerobust in the sense that they can be applied with minor or no modifications to AR models with nonGaussian. We propose three estimators, two of which the form of the distribution of the errors needs to be specified. The third estimator is a distributionfree estimator. As the reference to this estimator suggests, it is free from distributional assumptions in the sense that the error distribution is calculated from the observed data. Hence, it can be used in a wide variety of applications.In the first part of the thesis, we present a summary of the various currently available estimators for the linear regression model with censored independent and identically distributed (i.i.d.) data. In our review of these estimators, we note that the linear regression model with censored i.i.d. data has been studied quite extensively. Yet, use of autoregressive models with censored data has received very little attention. Hence, the remainder of the thesis focuses on the estimation of parameters for censored autocorrelated data. First, as part of the study, we review currently available estimators in regression with censored autocorrelated data. Then we present descriptions of the new estimators for censored autocorrelated data. With the view that extensions to the AR(p), model, p > 1, and to leftcensored data can be easily achieved, all the estimators, both currently available and new, are discussed in the context of the AR(1) model. Next, we establish some asymptotic results for the estimators in which specification of the form of the error distribution is necessary. This is followed by a simulation study based on Monte Carlo experiments in which we evaluate and compare the performances of the new and currently available estimators among themselves and with the leastsquares estimator for the uncensored case. The performances of the asymptotic variance estimators of the parameter estimators are also evaluated.In summary, we establish that for each of the two new estimators for which the distribution of the errors is assumed known, under suitable conditions on the moments of the error distribution function, if the estimator is consistent, then it is also asymptotically normally distributed. For one of these estimators, if the errors are Gaussian and alternate observations are censored, then the estimator is consistent. Hence, for this special case, the estimator is consistent and asymptotically normal. The simulation results suggest that this estimator is comparable with the distributionfree estimator and a currently available pseudolikelihood (PL) estimator. All three estimators perform worse than the least squares estimator for the uncensored case. The MLE and another currently available PL estimator perform comparably not only with the least squares estimator for the uncensored case but also with estimators from the abovementioned group of three estimators, which includes the distributionfree estimator. The other new estimator for which the form of the error distribution is assumed known compares favourably with the least squares estimator for the uncensored case and better than the rest of the estimators when the true value of the autoregression parameter is 0.2. When the true value of the parameter is 0.5, this estimator performs comparably with the rest of the estimators and worse when the true value of the parameter is O.S. The simulation results of the asymptotic variance estimators suggest that for each estimator and for a fixed value of the true autoregression parameter, if the error distribution is fixed and the censoring rate is constant, the asymptotic formulas lead to values which are asymptotically insensitive to the censoring pattern. Also, the estimated asymptotic variances decrease as the sample size increases and their behaviour, with respect to changes in the true value of autoregression parameter, is consistent with the behaviour of the asymptotic variance of the leastsquares estimator for the uncensored case.Some suggestions for possible extensions conclude the thesis.
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