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首页> 外文期刊>British Journal of Mathematics & Computer Science >Power Transformations and Unit Mean and Constant Variance Assumptions of the Multiplicative Error Model: The Generalized Gamma Distribution
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Power Transformations and Unit Mean and Constant Variance Assumptions of the Multiplicative Error Model: The Generalized Gamma Distribution

机译：乘积误差模型的幂变换，单位均值和常数方差假设：广义伽玛分布

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Aims: To study the implications of power transformations namely; inverse-square-root, inverse, inverse-square and square transformations on the error component of the multiplicative error and determine whether the unit-mean and constant variance assumptions of the model are either retained or violated after the transformation. Methodology: We studied the distributions of the error component under the various distributional forms of the generalized gamma distribution namely; Gamma (a, b, 1), Chi-square, Exponential, Weibull, Rayleigh and Maxwell distributions. We first established the functions describing the distributional characteristics of interest for the generalized power transformed error component and secondly applied the unit-mean conditions of the untransformed distributions to the established functions. Results: We established the following important results in modeling using a multiplicative error model, where data transformation is absolutely necessary;(i) For the inverse-square-root transformation, the unit-mean and constant variance assumptions are approximately maintained for all the distributions under study except the Chi-square distribution where it was violated. (ii) For the inverse transformation, the unit-mean assumptions are violated after the transformation except for the Rayleigh and Maxwell distributions. (iii) For the inverse-square transformation, the unit-mean assumption is violated for all the distributions under study. (iv) For the square transformation, it is only the Maxwell distribution that maintained the unit-mean assumption. (v) For all the studied transformations the variances of the transformed distributions were found to be constant but greater than those of the untransformed distribution. Conclusion: The results of this study though restricted to the distributional forms of the generalized gamma distribution, however they provide a useful framework in modeling for determining where a particular power transformation is successful for a model whose error component has a particular distribution.
机译：目的：研究权力转换的含义;即：对乘性误差的误差分量进行平方根，平方，平方和逆变换，并确定变换后模型的单位均值和常数方差假设是否保留或违反。方法：我们研究了广义伽马分布的各种分布形式下误差分量的分布;即伽玛（a，b，1），卡方，指数，威布尔，瑞利和麦克斯韦分布。我们首先建立描述广义功率变换误差分量的关注分布特征的函数，然后将未变换分布的单位均值条件应用于建立的函数。结果：我们在使用乘法误差模型进行建模的过程中建立了以下重要结果，其中绝对需要数据变换;（i）对于平方根逆变换，所有分布的单位均值和常数方差假设均近似保持正在研究中，但违反卡方分布的除外。 （ii）对于逆变换，除了瑞利分布和麦克斯韦分布外，变换后均违反了均值假设。 （iii）对于平方反变换，所研究的所有分布均违反了均值假设。 （iv）对于平方变换，只有麦克斯韦分布保持单位均值假设。 （v）对于所有研究的变换，发现变换分布的方差是恒定的，但大于未变换分布的方差。结论：这项研究的结果虽然限于广义伽马分布的分布形式，但是它们为确定误差分量具有特定分布的模型的特定幂转换在何处成功提供了一个有用的建模框架。

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