Generalized Birnbaum‐Saunders distributions applied to air pollutant concentration

Abstract

SUMMARYThe generalized Birnbaum-Saunders (GBS) distribution is a new class of positively skewed models with lighter and heavier tails than the traditional Birnbaum-Saunders (BS) distribution, which is largely applied to study lifetimes. However, the theoretical argument and the interesting properties of the GBS model have made its application possible beyond the lifetime analysis. The aim of this paper is to present the GBS distribution as a useful model for describing pollution data and deriving its positive and negative moments. Based on these moments, we develop estimation and goodness-of-fit methods. Also, some properties of the proposed estimators useful for developing asymptotic inference are presented. Finally, an application with real data from Environmental Sciences is given to illustrate the methodology developed. This example shows that the empirical fit of the GBS distribution to the data is very good. Thus, the GBS model is appropriate for describing air pollutant concentration data, which produces better results than the lognormal model when the administrative target is determined for abating air pollution.

Modified slash Birnbaum-Saunders distribution

“…Following the ideas of [8] and [10], we consider the modified moment estimation method. Next, we present the modified moment estimators for the parameters of α, β and q, which are based on E(T ), E(T −1 ) and E(T 2 ) and their statistics S =…”

Revisit of relationships and models for the Birnbaum-Saunders and inverse-Gaussian distributions

“…Numerous authors have investigated different aspects related to the B-S distribution. For instance, various generalizations and extensions of the B-S distribution are proposed and discussed; see, for example, Díaz-García and Leiva-Sánchez (2005), Díaz-García and Domínguez-Molina (2006), Owen (2006), Leiva et al (2008), Sanhueza et al (2008), Gómez et al (2009), Guiraud et al (2009, Leiva et al (2009), Castillo et al (2011), Cordeiro and Lemonte (2011), Genç (2013, Cordeiro and Lemonte (2014) and Cordeiro, et al: A model with long-term survivors: Negative binomial Birnbaum-Saunders (to appear). These generalized B-S distributions have been applied to model data obtained from different disciplines including environmental sciences, reliability engineering and biomedical studies.…”

“…These generalized B-S distributions have been applied to model data obtained from different disciplines including environmental sciences, reliability engineering and biomedical studies. For example, Leiva et al (2008) used generalized B-S distributions to model air pollutant concentration, Leiva et al (2009) proposed a length-biased B-S distribution for water quality data, Guiraud et al (2009) used a non-central B-S distribution for reliability analysis and Cordeiro, et al: A model with longterm survivors: Negative binomial Birnbaum-Saunders (to appear) proposed the negative binomial B-S distribution as a cure rate survival model. Software packages written in R (R Core Team 2015) for B-S and generalized B-S distributions have been developed by Leiva et al (2006) and Barros et al (2009).…”

A note on the Fisher information matrix of the Birnbaumandash;Saunders distribution

“…Some recent applications include: modeling of hourly SO 2 concentrations at ten monitoring stations located in different zones in Santiago [20]; modeling of diameter at breast height distributions of near-natural complex structure silver fir-European beech forests [30]; modeling of hourly dissolved oxygen (DO) concentrations observed at four monitoring stations located at different areas of Santiago [33]; statistical analysis of redundant systems with one warm stand-by unit [27]. Because of the widespread study and applications of the BS distribution, it is very important to have a Fisher information matrix which works for all the values of the model parameters.…”