Ishaq Aliyu Ismail, Abiodun Alfred Adewole, Falgore Jamilu Yunusa
Department of Statistics, Ahmadu Bello University, Zaria, Nigeria.
Department of Statistics, University of Ilorin, Ilorin, Nigeria.
Heliyon. 2021 Oct 19;7(10):e08200. doi: 10.1016/j.heliyon.2021.e08200. eCollection 2021 Oct.
A three-parameter Maxwell-Mukherjee Islam distribution was proposed by applying Maxwell generalized family of distributions introduced by Ishaq and Abiodun [17]. The probability density and cumulative distribution functions of the proposed distribution were defined. The validity test was derived from its cumulative distribution function. The study aimed to obtain a Bayesian estimation of the scale parameter of Maxwell-Mukherjee Islam distribution by using assumptions of the Extended Jeffrey's (Uniform, Jeffrey's and Hartigan's), Inverse-Rayleigh and Inverse-Nakagami priors under the loss functions, namely, Squared Error Loss Function (SELF), Precautionary Loss Function (PLF) and Quadratic Loss Function (QLF), and their performances were compared. The posterior distribution under each prior and its corresponding loss functions was derived. The performance of the Bayesian estimation was illustrated from the basis of quantile function by using a simulation study and application to real life data set. For different sample sizes and parameter values, the QLF and SELF under Jeffrey's and Hartigan's priors produced the same estimates, bias and Mean Squared Error (MSE) just as we observed in their mathematical derivatives. Similarly, the SELF, PLF and QLF under Inverse-Rayleigh and Inverse-Nakagami priors provided the same performance when some parameter values are equal. For some parameter values, the QLF under Inverse-Nakagami and Inverse-Rayleigh priors produced the least values of MSE. In the application to real life data set, the QLF and SELF under Jeffrey's and Hartigan's priors; the SELF, PLF and QLF under Inverse-Rayleigh and Inverse-Nakagami priors provided similar results as observed in the simulation study. Therefore, the study concluded that the QLF under Inverse-Rayleigh and Inverse-Nakagami priors could effectively be used in the estimation of scale parameter of Maxwell-Mukherjee Islam distribution using Bayesian approach.
通过应用由伊沙克和阿比奥顿[17]引入的麦克斯韦广义分布族，提出了一种三参数麦克斯韦 - 穆克吉 - 伊斯兰分布。定义了所提出分布的概率密度函数和累积分布函数。有效性检验源自其累积分布函数。该研究旨在通过在损失函数（即平方误差损失函数（SELF）、预防性损失函数（PLF）和二次损失函数（QLF））下使用扩展杰弗里（均匀、杰弗里和哈蒂根）、逆瑞利和逆纳卡加米先验的假设，获得麦克斯韦 - 穆克吉 - 伊斯兰分布尺度参数的贝叶斯估计，并比较它们的性能。推导了每个先验及其相应损失函数下的后验分布。通过模拟研究和对实际生活数据集的应用，从分位数函数的角度说明了贝叶斯估计的性能。对于不同的样本大小和参数值，杰弗里和哈蒂根先验下的QLF和SELF产生了相同的估计值、偏差和均方误差（MSE），正如我们在它们的数学导数中所观察到的那样。同样，当一些参数值相等时，逆瑞利和逆纳卡加米先验下的SELF、PLF和QLF提供了相同的性能。对于一些参数值，逆纳卡加米和逆瑞利先验下的QLF产生了最小的MSE值。在对实际生活数据集的应用中，杰弗里和哈蒂根先验下的QLF和SELF；逆瑞利和逆纳卡加米先验下的SELF、PLF和QLF提供了与模拟研究中观察到的相似结果。因此，该研究得出结论，逆瑞利和逆纳卡加米先验下的QLF可以有效地用于使用贝叶斯方法估计麦克斯韦 - 穆克吉 - 伊斯兰分布的尺度参数。
