THE TRANSMUTED HALF-NORMAL DISTRIBUTION WITH APPLICATION TO PRECIPITATION DATA

ABSTRACT The Half-Normal distribution has been intensively extended in the recent years. A review of the literature showed that at least 10 extensions of the Half-Normal distribution were introduced between 2008 and 2016. These extensions generalized the behavior of the density and hazard functions, which are restricted to monotonous decreasing and monotonically increasing, respectively. In this paper we propose a new extension called the transmuted Half-Normal distribution using the quadratic rank transmutation map, introduced by Shaw & Buckley (2009). A comprehensive account of mathematical properties of the new distribution is presented. We provide explicit expressions for the moments, moment-generating function, Shannon’s entropy, mean deviations, Bonferroni and Lorenz curves, order statistics, and reliability. The estimation of the parameters is implemented by the maximum likelihood method. The bias and accuracy of the estimators are assayed by the Monte Carlo simulations. This proposed distribution allows us to incorporate covariates directly in the mean and consequently to quantify their influences on the average of the response variable. Experiment with two real data sets show usefulness and its value as a good alternative to several extensions of the Half-Normal distribution in data modeling with and without covariates.摘要 半正态分布（Half-Normal distribution）近年来得到了广泛拓展。文献调研显示，2008年至2016年间至少已提出10种半正态分布的拓展形式。此类拓展对密度函数与风险函数的行为进行了推广，而原始半正态分布的密度函数与风险函数分别被限定为单调递减与单调递增。本文基于Shaw与Buckley（2009）提出的二次秩变换映射（quadratic rank transmutation map），提出一种新型拓展分布——变换半正态分布（transmuted Half-Normal distribution）。本文对该新型分布的数学性质展开全面阐述，给出了矩、矩生成函数、香农熵、平均偏差、邦费罗尼曲线与洛伦兹曲线、次序统计量以及可靠性的显式表达式。参数估计采用极大似然法实现。通过蒙特卡洛模拟评估了估计量的偏差与准确性。该新型分布可直接将协变量纳入均值模型，进而量化协变量对响应变量均值的影响。针对两个真实数据集的实验验证了该分布的实用价值，其可作为多种半正态分布拓展形式的优良替代方案，适用于含协变量与不含协变量的数据建模场景。