Abstract

Multiple linear regression is used across various fields of science to determine whether a relationship exists between the response variable and the regressors. Generally, regression model parameters are estimated using their least-squares estimators, which are equivalent to the maximum likelihood estimators when the error terms are normally distributed. This study discusses the problem of statistical inference in multiple linear regression models when the error terms are non-normally distributed, e.g., Cauchy, Student’s t, skew-normal, and skew-t, which can be commonly encountered in practice, and estimation of the regression model parameters via minimum distance estimators (MDEs) based on the Cramér-von Mises distance. Note that the MDEs of regression model parameters cannot be expressed explicitly since the corresponding score equations include nonlinear function(s) of the parameters. Therefore, the score equations are solved iteratively, which may lead to convergence issues and other problems. To overcome these problems, in this study, modified versions of the MDEs (MMDEs) of regression model parameters are derived, which have explicit algebraic forms and therefore do not require any iteration to compute. It is also shown that the MMDEs are asymptotically equivalent to their MDE counterparts. See Arslan et al. (2022) [Arslan, T., Acitas, S., and Senoglu, B. (2022). Modified minimum distance estimators: definition, properties and applications. Computational Statistics 37: 1551-1568.] for further details of the modified minimum distance estimation methodology.