Influence Diagnostics in log-Birnbaum-Saunders Regression Models

Abstract

In this paper we present various diagnostic methods for a linear regression model under a logarithmic Birnbaum-Saunders distribution for the errors, which may be applied for accelerated life testing or to compare the median lives of several populations. Some influence methods, such as the local influence, total local influence of an individual and generalized leverage are derived, analysed and discussed. We also present a connection between the local influence and generalized leverage methods. A discussion of the computation of the likelihood displacement as well as the normal curvature in the local influence method are presented. Finally, an example with real data is given for illustration.Birnbaum- Saunders distribution, life distributions, sinh-normal distribution, fatigue life, log-linear models, influence diagnostic, generalized leverage, local influence, maximum likelihood estimator,

Multivariate Birnbaum-Saunders Distributions: Modelling and Applications

“…The local influence method, proposed by Cook (1987), has had an important role in regression diagnostics by assessing the effect of small perturbations in the model and/or data on the maximum likelihood (ML) estimates in the normality-based linear regression model context. Influence diagnostics have subsequently been studied for other modelling situations; see Paula (1993);Shi (1997); Galea et al (2004); Osorio et al (2007); Atkinson (2009);Santana et al (2011);Villegas et al (2011);Paula et al (2012) and . Recent works have extended influence diagnostic methods for multivariate BS regression models and BS spatial models; see and Garcia-Papani et al (2017).…”

“…The initial period contains few published papers reflecting the slow development of the methodology; see, for example, Birnbaum and Saunders (1969a); Rieck and Nedelman (1991); Johnson et al (1995); Dupuis and Mills (1998) and Owen and Padgett (1999). The second period (2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010) includes papers that discuss varied aspects of estimation, modelling and diagnostics, as well as generalizations, computational issues and novel modelling examples, but with justifications still mainly based on an argument of cumulative effects; see, for example, Owen and Padgett (2000); Volodin and Dzhungurova (2000); Tsionas (2001); Rieck (2003); Galea et al (2004); Owen (2006); Xie and Wei (2007); Lemonte et al (2008); Leiva et al (2008Leiva et al ( , 2009); Balakrishnan et al (2009) and Vilca et al (2010). The third period (2011 to the present) is characterized by a new inventiveness, breaking the link with lifetime data analysis and hence extended application in new areas such as: biology, crop yield assessment, econometrics, energy production, forestry, industry, informatics, insurance, inventory management, medicine, psychology, neurology, pollution monitoring, quality control, sociology and seismology; see, for example, Bhatti (2010); Kotz et al (2010); Balakrishnan et al (2011); Leiva et al (2010Leiva et al ( , 2011Leiva et al ( , 2012; Vilca et al (2010); Villegas et al (2011); Azevedo et al (2012); Ferreira et al (2012);...…”

“…651-63) and Leiva (2016, p. 18). In addition, a logarithmic verion of the BS distribution (log-BS) was introduced by Rieck and Nedelman (1991) allowing the extension to BS log-linear regression models; see Tsionas (2001); Galea et al (2004); Xie and Wei (2007); Desmond et al (2008); Lemonte (2011); Ferrari (2011a, 2011b).…”

On the skewed sinh-normal distribution

“…Later, Rieck [12] defined that, if δ = 2/α and g(Y; γ, σ) = sinh(y − γ/σ) in (1) then Y follow a four parameters sinh-normal (SHN) distribution with shape parameters ν ∈ ℝ , α > 0, location parameter γ ∈ ℝ, and scale parameter σ > 0, and the notation Y~SHN(α, γ, σ, ν) is used, which is reduced simply to Y~SHN(α, γ, σ) when ν = 0; for more details and applications of the SHN distribution see Rieck and Nedelman [13], Galea et al [4], and Leiva et al [9]. If Y~SHN(α, γ, 2), then T = exp(Y) follows the Birnbaum-Saunders (BS) distribution with shape parameter α > 0 and scale parameter β = exp(γ) > 0 , which is denoted by T~BS(α, β); see Birnbaum and Saunders [3], Johnson et al [8], and Sanhueza et al [14].…”

Advances on the Birnbaum-Saunders distribution

“…Among them, Rieck and Nedelman (1991) proposed a log-linear regression model based on the BS distribution. Diagnostic analysis for the BS regression model were developed by Galea et al (2004), Leiva et al (2007) and Xie and Wei (2007), while the Bayesian inference was introduced by Tsionas (2001). Barros et al (2008) proposed a class of lifetime regression models that includes the log-Birnbaum-Saunders-t (BSt) regression models as special case.…”