On maximum likelihood and sample moment estimators for the MTH(central) moment in a normal and generalized gamma population
Abbaszadehpeivasti, Hadi (2020) On maximum likelihood and sample moment estimators for the MTH(central) moment in a normal and generalized gamma population. [Thesis]
In this thesis, we consider the maximum likelihood and sample moment estimator of the mth (central) moment for a normal and generalized gamma population. We also propose using the method of moments approach new estimators for the parameters of a generalized gamma population. To introduce these maximum likelihood estimators for the mth (central) moments in a generalized gamma population we first discuss the properties of the maximum likelihood optimization problem formulated for this class and propose an efficient algorithm to solve this optimization problem. As an application of this algorithm, we show how it can be used for a given sample to estimate the maximum likelihood estimator of the mth moment of a generalized gamma distributed random variable. By means of simulation experiments, we compare in the computational section its mean squared error with the mean squared error of the mth sample moment estimator. An alternative estimator of these parameters is also proposed by applying the method of moment approach applied to the logarithmic transformation of a generalized gamma distributed random variable. Although the associated system of nonlinear equations is for small sample sizes inconsistent with a high probability, the system of nonlinear equations has a unique solution (if there is a solution) and this unique solution is easy to determine. For larger sample sizes this probability goes to zero and this is related to the accuracy of the sample moment estimator of the skewness of a generalized gamma population. Hence it is easy to determine evaluating only the sample whether the system is inconsistent. These properties are not proved for other proposals of moment estimators of the parameters of this class which appeared in the literature. Finally, we propose for any positive integer m a maximum likelihood-based estimator of the mth (central) moment in a normal population and compare the behavior of this estimator with the (classical) sample mth (central) moment estimator. In particular, we give for every computable expression for the mean and the variance of these different estimators for both the moment and the central moment estimation problem. For the mth central moment estimation problem it is shown that in a normal population one can compute a threshold value (independent of the unknown parameters) of the sample size such that beyond this sample size the mean squared error of the maximum likelihood-based estimator is smaller than the mean squared error of the sample mth central moment estimator. At the same time, this shows using the mean squared error objective that for sample sizes below a certain value the nonparametric sample moment estimator outperforms the parametric maximum likelihood-based estimator. Finally, in the computational section, we perform for these two estimation problems some simulation experiments and give some rule of thumbs for which sample sizes it is better to use the nonparametric moment estimator.
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