Length-biased Birnbaum-Saunders quantile regression with application to water evaporation

Length-biased distributions arise naturally in environmental, reliability, and economic studies where the sampling mechanism favors larger observational units. In this paper, we propose a quantile regression model based on the length-biased Birnbaum--Saunders (QLBS) distribution. The model is constructed through a reparameterization of the length-biased Birnbaum--Saunders distribution in terms of its quantile function, thereby allowing direct interpretation of covariate effects on conditional quantiles of the response variable. We derive the log-likelihood function and the corresponding score equations, and obtain maximum likelihood estimators via numerical optimization. Asymptotic and bootstrap confidence intervals are considered. Two types of residuals are proposed for model assessment, namely the generalized Cox--Snell and randomized quantile residuals. An elaborate Monte Carlo simulation study is carried out to evaluate the performance of the maximum likelihood estimators for several sample sizes and quantile levels. The proposed methodology is illustrated with a real meteorological data set from Brazil.