This paper is concerned with the maximum likelihood estimators of the Beta-Pareto distribution introduced in Akinsete et al. (2008), which comes from the mixing of two probability distributions, Beta and Pareto. Since these estimators cannot be obtained explicitly, we use nonlinear optimization methods that numerically provide these estimators. The methods we investigate are the method of Newton-Raphson, the gradient method and the conjugate gradient method. Note that for the conjugate gradient method we use the model of Fletcher-Reeves. The corresponding algorithms are developed and the performances of the methods used are confirmed by an important simulation study. In order to compare between several concurrent models, namely generalized Beta-Pareto, Beta, Pareto, Gamma and Beta-Pareto, model criteria selection are used. We firstly consider completely observed data and, secondly, the observations are assumed to be right censored and we derive the same type of results.

中文翻译：

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基于几种优化方法的贝塔-帕累托分布估计

本文与Akinsete等人引入的Beta-Pareto分布的最大似然估计有关。（2008年），它来自两个概率分布Beta和Pareto的混合。由于无法明确获得这些估计量，因此我们使用非线性优化方法以数值方式提供这些估计量。我们研究的方法是牛顿-拉夫森法，梯度法和共轭梯度法。请注意，对于共轭梯度法，我们使用Fletcher-Reeves模型。开发了相应的算法，并通过重要的仿真研究证实了所用方法的性能。为了在多个并发模型（即广义Beta-Pareto，Beta，Pareto，Gamma和Beta-Pareto）之间进行比较，使用模型标准选择。