Contributions to the analysis of the exponential power distribution




Tumlinson, Samuel

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The exponential power distribution (EPD) is used to model firm growth rates, and rare events such as large flight path deviations in air traffic management (ATM) scenarios. This family is a three parameter generalization of the Laplace and normal distributions that contains super-Laplace alternatives. One super-Laplace case suggested for ATM scenario modeling occurs when the shape parameter of the EPD is one-half. For this case, we obtain closed form expressions for the single and product moments of the order statistics. These results are used to obtain the coefficients and variance of the best linear unbiased estimators (BLUEs) for location and scale based on complete samples of size 30 or less. We use numerical methods to obtain the BLUEs for several other cases. As part of this work, we obtain a simplification of Govindarajulu's result that allows us to express the single odd (even) moments of the order statistics from a symmetric distribution as a linear combination of the moments of the largest order statistic from the folded distribution based on even (odd) sample sizes. In addition, we show every MLE of location is a sample point that minimizes a certain function when the shape is fixed to be less than one. A simulation study is performed to investigate the performance of different estimators, and recommendations are provided. We consider Bayesian analysis of the EP sampling model to express certain beliefs regarding the shape parameter. To obtain a noninformative prior, we consider independent flat priors for location and scale, and a uniform prior for shape. These priors allow us to exercise the belief the sampling distribution is super-Laplace, Laplace, normal or close to uniform. A necessary and sufficient condition for propriety of the posterior under a certain class of priors is given. We use this prior, as well as diffuse alternatives, to perform a Bayesian analysis of economic growth rate data.


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Bayesian Inference, Best Linear Unbiased Estimation, Extended Exponential Power Distribution, Maximum Likelihood Estimation, Order Statistics



Management Science and Statistics