**General Abstract:** This course deals with recent advances in parametric models of interest in reliability. It covers two topics. The first one is devoted to the application of LINEX loss function in reliability models such as the generalized half-logistic and Burr type-XII models. The second one deals with advances in slash distributions and the use of these models to obtain generalisations of Birnbaum-Saunders distribution.

The course is structured in two sessions of 3 hours each, whose details are next listed.

Material (slides) will be provided to the audience in advance.

**Topics and Abstract for session:****Session 1: (June 17, 2019, 9.00-12.00)**

1 st part: Models and censoring schemes of interest in reliability.

(a) Models for reliability data.

(b) Censoring schemes.

(c) Progressive type-II right censoring.

(d) Introduction to Bayesian methods.

Summary: We introduce parametric models of interest in reliability along with common censoring mechanisms. We highlight progressive type-II right censoring scheme. We recall classical methods of estimation and Bayesian methods of inference.

2 nd part: On the use of LINEX loss function. Posterior and Bayes risk under LINEX loss function.

(a) Asymmetric loss functions in Bayesian Statistics. LINEX loss function versus quadratic loss function.

(b) Main results. Posterior risks and Bayes risks in previous setting.

(c) Applications and simulations by using R software.

Summary: To carry out estimation in Bayesian Statistics, a loss function must be specified. The most widely used loss is squared error. This function has the drawback that it is symmetric, that is, positive and negative errors of the same magnitude have the same penalty. Quite often this assumption is not realistic enough, since the loss function may not just be a measure of inaccuracy but a real loss, for example, financial. In these

cases, an asymmetric loss function may be more appropriate, and following Zellner (1986), the Linex loss may be a good choice. This loss function allows us to penalize, in a different way, positive and negative errors, and it is still mathematically manageable.

So, appealing results can be obtained from its use. In spite of this fact, there exist hardly papers dealing with this function. In this course, results will be presented for certain parametric models of interest in reliability, such as the generalized half-logistic distribution and the Burr Type-II distribution, under progressive type-II censoring.

Specifically the posterior and Bayes risks of Bayes estimators are obtained in these models, since, following Lehmann and Casella (1998), risks are crucial to assess the performance of an estimator and compare competing estimators. We obtain Bayes and posterior risks of Bayes estimators under quadratic and Linex loss function when the prior distribution is a conjugate prior, along with other intermediate results of interest (the marginal distribution that we need to obtain the risks and the relationship between the Bayes estimators). Simulations are carried out by using R, which illustrate the

behaviour of proposed estimators and their risks, along with the importance of different features involved in the progressive censoring scheme. An application to a real data set is included.

**Session 2: (June 17, 2019, 14.00-17.00)**

1 st part: Advances in slash distributions: Generalized Modified Slash (GMS) Distribution.

(a) Slash methodology.

(b) Generalized Modified Slash (GMS) Distribution.

(c) Methods of estimation in GMS model: EM-algorithm.

(d) Simulations and applications.

Summary: In real-world data, it is quite common to find symmetrical and unimodal histograms with heavy tails that do not _t well to a normal distribution. Slash models are a good option to deal with this kind of situations, in which departures of Gaussianity are a serious problem for the data analyst. This is one the main reason why slash distributions have received a great deal of attention during the last decades. In this context, we face the problem of improving slash models by introducing a generalisation able to model more kurtosis than other slash's previously proposed in literature. In slash models, the emphasis is on kurtosis, because as Moors (1988) pointed out the presence of heavy tails produces high kurtosis. In this session, we will focus on univariate symmetrical slash models. Specifically on the Generalised Modified Slash (GMS) model introduced in Reyes et al. (2019). The GMS model is defined. A closed expression for its pdf is given in terms of the confluent hypergeometric function; GMS model is expressed as a scale mixture; the convergence in law to a

normal distribution is proven; moments are obtained, with emphasis on the kurtosis coefficient; and comparisons with other slash models previously introduced in the literature are presented. As for inference, we focus on iterative and EM maximum likelihood estimation methods. A simulation study will be showed which allow to assess the performance of our results. Applications to two real datasets are included.

2 nd part: Birnbaum-Saunders model based on GMS distribution.

(a) Birnbaum-Saunders model.

(b) Birnbaum-Saunders model based on GMS distribution.

(c) Maximum likelihood based on EM-algorithm in these models.

(d) Applications and simulations by using R software.

Summary: The Birnbaum-Saunders (BS) distribution was introduced by Birnbaum and Saunders (1969). The aim of this distribution is to model the fatigue in lifetime of certain materials. Nowadays its use is being spread out to other contexts such as economic and environmental data. In these new applications, it is quite common to find real datasets in which a BS model with heavier tails would be suitable. Slash models are a good option to deal with this kind of situations. In this context, we briefly describe the BS-model and the generalisation proposed. Maximum likelihood based on EM-algorithm is carried out. Simulations and applications of interest in reliability are given.

**References**

Barranco Chamorro, Inmaculada, Luque Calvo, Pedro Luis, Jimenez Gamero, Maria Dolores, Alba Fernández, M. Virtudes. A study of risks of Bayes estimators in the generalized half-logistic distribution for progressively type-II censored samples. In: Mathematics and Computers in Simulation. 2017. Vol. 137. Pag. 130-147.10.1016/j.matcom.2016.09.003

Reyes, Jimmy, Barranco Chamorro, Inmaculada, Gallardo, Diego I., Gómez, Héctor W.: Generalized Modified Slash Birnbaum-Saunders Distribution. In: Symmetry. 2018. Vol. 10. Num. 12.10.3390/sym10120724

Reyes, Jimmy, Barranco Chamorro, Inmaculada, G_omez, H_ector W.: Generalized modified slash distribution with applications. 2019. In: Communications in Statistics – Theory and Methods. https://doi.org/10.1080/03610926.2019.1568484

and references therein.