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

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.

Modeling Mortality Based on Pollution and Temperature Using a New Birnbaum-Saunders Autoregressive Moving Average Structure with Regressors and Related-Sensors Data

Environmental agencies are interested in relating mortality to pollutants and possible environmental contributors such as temperature. The Gaussianity assumption is often violated when modeling this relationship due to asymmetry and then other regression models should be considered. The class of Birnbaum–Saunders models, especially their regression formulations, has received considerable attention in the statistical literature. These models have been applied successfully in different areas with an emphasis on engineering, environment, and medicine. A common simplification of these models is that statistical dependence is often not considered. In this paper, we propose and derive a time-dependent model based on a reparameterized Birnbaum–Saunders (RBS) asymmetric distribution that allows us to analyze data in terms of a time-varying conditional mean. In particular, it is a dynamic class of autoregressive moving average (ARMA) models with regressors and a conditional RBS distribution (RBSARMAX). By means of a Monte Carlo simulation study, the statistical performance of the new methodology is assessed, showing good results. The asymmetric RBSARMAX structure is applied to the modeling of mortality as a function of pollution and temperature over time with sensor-related data. This modeling provides strong evidence that the new ARMA formulation is a good alternative for dealing with temporal data, particularly related to mortality with regressors of environmental temperature and pollution.

Optimal Sample Size for the Birnbaum–Saunders Distribution under Decision Theory with Symmetric and Asymmetric Loss Functions

The fatigue-life or Birnbaum–Saunders distribution is an asymmetrical model that has been widely applied in several areas of science and mainly in reliability. Although diverse methodologies related to this distribution have been proposed, the problem of determining the optimal sample size when estimating its mean has not yet been studied. In this paper, we derive a methodology to determine the optimal sample size under a decision-theoretic approach. In this approach, we consider symmetric and asymmetric loss functions for point and interval inference. Computational tools in the R language were implemented to use this methodology in practice. An illustrative example with real data is also provided to show potential applications.

A New Quantile Regression for Modeling Bounded Data under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics

Quantile regression provides a framework for modeling the relationship between a response variable and covariates using the quantile function. This work proposes a regression model for continuous variables bounded to the unit interval based on the unit Birnbaum–Saunders distribution as an alternative to the existing quantile regression models. By parameterizing the unit Birnbaum–Saunders distribution in terms of its quantile function allows us to model the effect of covariates across the entire response distribution, rather than only at the mean. Our proposal, especially useful for modeling quantiles using covariates, in general outperforms the other competing models available in the literature. These findings are supported by Monte Carlo simulations and applications using two real data sets. An R package, including parameter estimation, model checking as well as density, cumulative distribution, quantile and random number generating functions of the unit Birnbaum–Saunders distribution was developed and can be readily used to assess the suitability of our proposal.

Predicting PM2.5 and PM10 Levels during Critical Episodes Management in Santiago, Chile, with a Bivariate Birnbaum-Saunders Log-Linear Model

Improving air quality is an important environmental challenge of our time. Chile currently has one of the most stable and emerging economies in Latin America, where human impact on natural resources and air quality does not go unperceived. Santiago, the capital of Chile, is one of the cities in which particulate matter (PM) levels exceed national and international limits. Its location and climate cause critical conditions for human health when interaction with anthropogenic emissions is present. In this paper, we propose a predictive model based on bivariate regression to estimate PM levels, related to PM2.5 and PM10, simultaneously. Birnbaum-Saunders distributions are used in the joint modeling of real-world PM2.5 and PM10 data by considering as covariates some relevant meteorological variables employed in similar studies. The Mahalanobis distance is utilized to assess bivariate outliers and to detect suitability of the distributional assumption. In addition, we use the local influence technique for analyzing the impact of a perturbation on the overall estimation of model parameters. In the predictions, we check the categorization for the observed and predicted cases of the model according to the primary air quality regulations for PM.

A Reparameterized Weighted Lindley Distribution: Properties, Estimation and Applications

In this paper, we discuss several mathematical properties and estimation methods for a reparameterized version of the weighted Lindley (RWL) distribution. The RWL distribution can be particularly useful for modeling reliability (survival) data with bathtub-shaped or increasing hazard rate function. The inferential procedure to obtain the parameter estimates is conducted via the maximum likelihood approach considering random right-censoring. Extensive numerical simulations are carried out to investigate and evaluate the performance of the proposed estimation method. Finally, the potentiality of the RWL model is analyzed by employing two real data sets.

Extended Exponential Regression Model: Diagnostics and Application to Mineral Data

In this paper, we reparameterized the extended exponential model based on the mean in order to include covariates and facilitate the interpretation of the coefficients. The model is compared with common models defined in the positive line also reparametrized in the mean. Parameter estimation is approached based on the expectation–maximization algorithm. Furthermore, we discuss residuals and influence diagnostic tools. A simulation study for recovered parameters is presented. Finally, an application illustrating the advantages of the model in a real data set is presented.

Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data

In the present paper, a novel spatial quantile regression model based on the Birnbaum–Saunders distribution is formulated. This distribution has been widely studied and applied in many fields. To formulate such a spatial model, a parameterization of the multivariate Birnbaum–Saunders distribution, where one of its parameters is associated with the quantile of the respective marginal distribution, is established. The model parameters are estimated by the maximum likelihood method. Finally, a data set is applied for illustrating the formulated model.

Parametric modal regression with varying precision

In this paper, we propose a simple parametric modal linear regression model where the response variable is gamma distributed using a new parameterization of this distribution that is indexed by mode and precision parameters, that is, in this new regression model, the modal and precision responses are related to a linear predictor through a link function and the linear predictor involves covariates and unknown regression parameters. The main advantage of our new parameterization is the straightforward interpretation of the regression coefficients in terms of the mode of the positive response variable, as is usual in the context of generalized linear models, and direct inference in parametric mode regression based on the likelihood paradigm. Furthermore, we discuss residuals and influence diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the results. Finally, we illustrate the usefulness of the new model by two applications, to biology and demography.