Page 57 - Contributed Paper Session (CPS) - Volume 5

CPS886 Marcelo Bourguignon

A new regression model for positive random
variables
Marcelo Bourguignon
Departamento de Estat´ıstica, Universidade Federal do Rio Grande do Norte, Natal, Brazil

Abstract
In this paper, we propose a regression model where the response variable is
beta prime distributed using a new parameterization of this distribution that
is indexed by mean and precision parameters. The proposed regression model
is useful for situations where the variable of interest is continuous and
restricted to the positive real line and is related to other variables through the
mean and precision parameters. The variance function of the proposed model
has a quadratic form. In addition, the beta prime model has properties that its
competitor distributions of the exponential family do not have. Estimation is
performed by maximum likelihood. Finally, we also carry out an application to
real data that demonstrates the usefulness of the proposed model.

Keywords
Beta prime distribution; Variance function; Maximum likelihood estimator;
Regression models

1. Introduction
The concept of regression is very important in statistical data analysis
(Jørgensen, 1997). In this context, generalized linear models (Nelder &
Wedderburn, 1972) are regression models for response variables in the
exponential family.
The main aim of this paper is to propose a regression model that is tailored
for situations where the response variable is measured continuously on the
positive real line that is in several aspects, like the generalized linear models.
In particular, the proposed model is based on the assumption that the
response is beta prime (BP) distributed. We considered a new
parameterization of the BP distribution in terms of the mean and precision
parameters. Under this parameterization, we propose a regression model, and
we allow a regression structure for the mean and precision parameters by
considering the mean and precision structure separately. The variance
function of the proposed model assumes a quadratic form. The proposed
regression model is convenient for modeling asymmetric data, and it is an
alternative to the generalized linear models when the data presents skewness.
Inference, diagnostic and selection tools for the proposed class of models will
be presented.

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