A New Quantile Regression Model and Its Diagnostic Analytics for a Weibull Distributed Response with Applications

Abstract

Standard regression models focus on the mean response based on covariates. Quantile regression describes the quantile for a response conditioned to values of covariates. The relevance of quantile regression is even greater when the response follows an asymmetrical distribution. This relevance is because the mean is not a good centrality measure to resume asymmetrically distributed data. In such a scenario, the median is a better measure of the central tendency. Quantile regression, which includes median modeling, is a better alternative to describe asymmetrically distributed data. The Weibull distribution is asymmetrical, has positive support, and has been extensively studied. In this work, we propose a new approach to quantile regression based on the Weibull distribution parameterized by its quantiles. We estimate the model parameters using the maximum likelihood method, discuss their asymptotic properties, and develop hypothesis tests. Two types of residuals are presented to evaluate the model fitting to data. We conduct Monte Carlo simulations to assess the performance of the maximum likelihood estimators and residuals. Local influence techniques are also derived to analyze the impact of perturbations on the estimated parameters, allowing us to detect potentially influential observations. We apply the obtained results to a real-world data set to show how helpful this type of quantile regression model is.

An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations

“…Recently, a fourth parameterization for the Weibull distribution was introduced by Sánchez et al [41]. In this parameterization, the hazard and survival functions are given by…”

Advanced Mathematical Approaches in Psycholinguistic Data Analysis: A Methodological Insight

“…Parametric fractile regression links the response variable, a parametric component (the modeled fractile of the response), and an error component without requiring distributional assumptions for this error [18]. When assumptions are added, they are better suited to the response variable.…”

Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications

“…Then, a regression-based functional form through a link function is introduced. For example, models based on the arcsecant hyperbolic Weibull [5], Lambert-uniform [6], unit-Birnbaum-Saunders [7][8][9][10], unit-Burr-XII [11], exponentiated arcsecant hyperbolic normal [12], unit-Chen [13], logextended exponential-geometric [14], Johnson-t [15], power Johnson SB [16], L-logistic [17], unit-Weibull [18][19][20], generalized Johnson SB [21], and Kumaraswamy [22] distributions have been postulated. These formulations have been proposed to model the conditional quantiles of a bounded response variable and are well known in the literature [23].…”