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

2026-05-25

AI Summary

This study addresses the challenge of modeling response variables in environmental data subject to length-biased sampling by proposing a quantile regression model based on the length-biased Birnbaum–Saunders distribution. By reparameterizing the quantile function of this distribution, the approach directly models the effect of covariates on conditional quantiles and employs maximum likelihood estimation for parameter inference. This work represents the first integration of the length-biased Birnbaum–Saunders distribution with quantile regression, enabling intuitive interpretation of covariate effects across different quantile levels. For model diagnostics, generalized Cox–Snell residuals and randomized quantile residuals are developed. Monte Carlo simulations demonstrate that the estimators exhibit favorable finite-sample performance, and the methodology is successfully applied to model water evaporation using meteorological data from Brazil.

Abstract

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.

Problem

• length-biased
• Birnbaum-Saunders distribution
• quantile regression
• environmental data
• biased sampling

Innovation

• length-biased distribution
• Birnbaum–Saunders distribution
• quantile regression
• reparameterization
• model diagnostics