New families of linear and partially linear quantile regression models under reparameterized Marshall-Olkin distributions

Abstract

In this dissertation, we propose families of linear and partially linear quantile regression models, where the response variable follows a reparameterized Marshall-Olkin distribution with support on the real line. This distribution presents great flexibility and arises from applying the Marshall- Olkin methodology to distributions of the location-scale family and then reparameterizing the location parameter as a function of the quantile. For this reason, the new distribution’s name is reparameterized Marshall-Olkin, which contains quantile, scale and skewness parameters. The first family has a structure similar to the generalized linear models that enable the use of the maximum likelihood method. Consequently, we calculate the expressions of the score vector and the observed information matrix to perform the statistical inference. The adequacy of models and outlier observations are studied through three types of residuals. In order to assess the sensitivity of the estimates, measures of global and local influence are developed. The second family is an extension of the first family by adding the description of the nonlinear relationship between the quantiles of the response variable and a continuous variable through B-splines. In this family, statistical inference tools are based on the penalized log-likelihood function. Also, analogously to the first family, the residuals and measures of global and local influence are presented. Two examples of applications are considered that illustrate the usefulness of the proposed families for data sets in the areas of health and nutrition.