Robust statistical modeling using the Birnbaum‐Saunders‐t distribution applied to insurance

Abstract

In this paper, we carry out robust modeling and influence diagnostics in Birnbaum-Saunders (BS) regression models. Specifically, we present some aspects related to BS and log-BS distributions and their generalizations from the Student-t distribution, and develop BS-t regression models, including maximum likelihood estimation based on the EM algorithm and diagnostic tools. In addition, we apply the obtained results to real data from insurance, which shows the uses of the proposed model.

Influence diagnostic analysis in the possibly heteroskedastic linear model with exact restrictions

“…(ii) The vector l max , which is the eigenvector associated with the largest eigenvalue C max of F(θ ); see Cook (1986), Leiva et al (2007), Liu et al (2009) andPaula et al (2012). Large absolute values of those elements of l max , that is, those greater than 2/ √ n, indicate the corresponding observations to be influential.…”

“…The local influence method is employed in several areas of applied econometrics and statistics. For example, there are a number of applications and studies in regression modelling and time series analysis; see Cook (1986), Galea et al (1997), Liu (2000Liu ( , 2002Liu ( , 2004, Díaz-García et al (2003), Galea et al (2008) and Shi and Chen (2008) for studies in linear regression and time series models, de Castro et al (2007) and Galea and de Castro (2012) for heteroskedastic errors-in-variables models, Leiva et al (2007Leiva et al ( , 2014 for influence diagnostics with censored and uncensored data, Barros et al (2010) for a Tobit model and Paula et al (2012) for robust modelling applied to insurance data. In particular, the local influence method can play an important role in regression models involving restrictions.…”

Local influence analysis in general spatial models

“…This method is applied or extended in influence diagnostics to many regression models and multivariate statistics; see, e.g. Backman et al (1987), Lawrance (1988), Thomas and Cook (1990), Wu and Luo (1993), Shi (1997), Poon and Poon (1999), Zhu and Lee (2001), Shi and Ojeda (2004), Shi and Huang (2011) and Paula et al (2012). However, no attention has been paid to the local influence analysis for the general spatial model in the current literature.…”

Multivariate Birnbaum-Saunders Distributions: Modelling and Applications

“…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); Paula et al (2012); Santos-Neto et al (2012; Marchant et al (2013; Saulo et al (2013Saulo et al ( , 2018; Barros et al (2014); …”

“…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).…”