Abstract
The traditional linear regression model that assumes normal residuals is applied extensively in engineering and science. However, the normality assumption of the model residuals is often ineffective. This drawback can be overcome by using a generalized normal regression model that assumes a non-normal response. In this paper, we propose regression models based on generalizations of the normal distribution. The proposed regression models can be used effectively in modeling data with a highly skewed response. Furthermore, we study in some details the structural properties of the proposed generalizations of the normal distribution. The maximum likelihood method is used for estimating the parameters of the proposed method. The performance of the maximum likelihood estimators in estimating the distributional parameters is assessed through a small simulation study. Applications to two real datasets are given to illustrate the flexibility and the usefulness of the proposed distributions and their regression models.
Highlights
Existing distributions do not always provide an adequate fit
One of the earliest works on generating distributions was done by [1] who proposed a method of differential equation as a fundamental approach to generate statistical distributions
The beta-generated (BG) family introduced by [7] has a cumulative distribution function (CDF) given by
Summary
Existing distributions do not always provide an adequate fit. generalizing distributions and studying their flexibility are of interest for researchers over recent decades. The beta-generated (BG) family introduced by [7] has a cumulative distribution function (CDF) given by. Where b(t) is the probability density function (PDF) of the beta random variable and F ( x ). [22], in turn, were interested whether other distributions with different support can be used as a generator They extended the family of BG distributions and defined the so called T-X family. [23] studied a special case of the T-X family where the link function, W (.), is a quantile function of a random variable Y. If R follows the normal distribution N (μ, σ2 ), (5) reduces to the T-normal family of distributions [24] with PDF given by h.
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