Abstract
We study multiple linear regression model under non-normally distributed random error by considering the family of generalized secant hyperbolic distributions. We derive the estimators of model parameters by using modified maximum likelihood methodology and explore the properties of the modified maximum likelihood estimators so obtained. We show that the proposed estimators are more efficient and robust than the commonly used least square estimators. We also develop the relevant test of hypothesis procedures and compared the performance of such tests vis-a-vis the classical tests that are based upon the least square approach.
Highlights
In most applications of multiple linear regression (MLR) the random errors involved are assumed to have a normal distribution
A more general and flexible family of symmetric distributions named as generalized secant hyperbolic (GSH) family is introduced by Vaughan [10] and the MMLE are derived for its parameters
In the context of MLR model, with the assumption that random errors are having a distribution in the GSH family, we used the MML method for the estimation of the model parameters
Summary
In most applications of multiple linear regression (MLR) the random errors involved are assumed to have a normal distribution. The latest contribution, related with multiple regression analysis, is that of Islam and Tiku [9] They considered three families of non-normal distributions: (a) Symmetric long-tailed distributions, (b) Symmetric short-tailed distributions, and (c) Generalized logistic distributions. A more general and flexible family of symmetric distributions named as generalized secant hyperbolic (GSH) family is introduced by Vaughan [10] and the MMLE are derived for its parameters This family consists of symmetric distributions, with kurtosis ranging from 1.8 to infinity, i.e., log-tailed (kurtosis greater than 3) and short-tailed (Kurtosis less than 3), includes the logistic as a special case, the uniform as a limiting case, and closely approximates normal and Student t with corresponding kurtosis. We will use the classical frequency method of construction of the statistical tests, that is, the T-statistic and the F-statistic
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