Semiparametric accelerated failure time models with generalized gamma and log-gamma distributions under censored data.

An accelerated failure time model is a typical tool in survival, insurance, and reliability studies. For example, (Lawless 2003) analyzes the data resulting of (Fuchs, Borowitz, Christiansen, Morris, Nash, Ramsey, Rosenstein, Smith & Wohl 1994) study. In this study, data on times to a pulmonary exacerbation for persons with cystic fibrosis was modeled. Also, he models a Leukemia survival times by two covariates: white blood cell amount (WBC) at diagnosis and binary indicator (AG) that indicates a positive or negative test related to white blood cell characteristics. In the insurance area, (Cardozo, Paula & Vanegas 2022) model the size of claims considering covariates such as operational time of claim and legal representation of a client. We can find in (Meeker, Escobar & Pascual 2003) industrial examples such as lifetime test on failure of printed circuit boards under temperature, relative humidity, and electric field. The generalized gamma family (GG) is a pretty flexible family of positive continuous distributions. This family unifies relevant distributions in reliability and survival analysis, such as Exponential, Weibull, Inverse Weibull, Gamma, Inverse Gamma and Log-normal distributions, see for instances Hager and Bain (Hager & Bain 1970), Lawless (Lawless 1980), Prentice (Prentice 1974) and Stacy and Mihram (Stacy & Mihram 1965). A quite useful subclass of the Generalized Log-gamma (GLG) family is the generalized extreme value distributions which has applications to model extremes of natural phenomena, financial markets, and biomedical data, see for example, Coles (Coles 2001) and Pinhero and Ferrari (Pinheiro & Ferrari 2016). Linear regression models with generalized log-gamma distributed errors and variants have had some activity in the last two decades. In (Ortega, Paula & Bolfarine 2003) and (Ortega, Paula & Bolfarine 2008) we found some estimation procedures, diagnostic quantities based on the local influence approach under the presence of censored observations and a three-type comparison of residuals for GLG regression models also under censoring. (Ortega, Cancho & Paula 2009) modified the generalized log-gamma regression models to include and model possible long-term observations in the data. (Fabio, Paula & Castro 2012) assumed a generalized log-gamma distribution for the random effect in random intercept Poisson models. (Hashimoto, Ortega & Cancho 2013) provides tools to estimate and make diagnostic analysis under interval censored observations. (Cox, Chu, Schneider & Munoz 2007) reported an application of GG regression to solve the limitations of Cox and Weibull regression models in the context of patterns of the hazard of death after several clinical therapies from 1984 to 2004. Lastly, (Cardozo et al. 2022) extended the works of (Ortega et al. 2003) and (Ortega et al. 2008) to the case of semiparametric systematic component and propose an iterative algorithm to calculate the parameters involved in the model under complete data. Nevertheless, no one of the preceding examples allow the possibility of a general nonlinear relationship of the covariates with the response variable under the presence of censored observations. Therefore, the purpose of this project is to offer statistical methodology when we are modelling lifetimes with semiparametric accelerated failure time models with generalized gamma distribution under the presence of censored data.