Influence diagnostics in the capital asset pricing model under elliptical distributions

In this paper we consider the Capital Asset Pricing Model under Elliptical (symmetric) Distributions. This class of distributions, which contains the normal distribution, t, contaminated normal and power exponential, among others, offers a more flexible framework for modelling asset prices or returns. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators, the local influence method was implemented. The results are illustrated by using a set of shares from companies who trade in the Chilean Stock Market. Our main conclusion is that symmetric distributions having heavier tails than those of the normal distribution, especially the t distribution with small degrees of freedom, show a better fit and allow the reduction of the influence of atypical returns in the maximum likelihood estimators.     

Influence diagnostics for the structural sharpe model under normal/independent distributions

A method is proposed in this paper to assess the local influence of minor perturbations for the Sharpe model when the normal distribution is replaced by normal/independent (NI) distributions. The family of NI distributions is an attractive class of symmetric heavy-tailed densities that includes as special cases the normal, t-Student, slash, and the contaminated normal distributions. Since the returns of the market are not observable, the statistical analysis is carried out in the context of an errors-in-variables model. An influence analysis for detecting influential observations (atypical returns) is developed to investigate the sensitivity of the maximum likelihood estimators. Diagnostic measures are obtained based on the conditional expectation of the complete-data log-likelihood function. The results are illustrated by using a set of shares of companies traded in the Chilean stock market.     

An Application of the Local Influence Diagnostics to Ridge Regression Under Elliptical Model

In this article, we study the effect of a minor perturbation on the ridge estimator considering the elliptical distribution for the errors. The necessary matrices for assessing the local influence under the perturbation of the explanatory variables and the scale matrix are derived. The Longley data is analyzed for illustration.     

Hypotheses testing for structural calibration model

The main objective of this paper is to discuss maximum likelihood inference for the comparative structural calibration model (Barnett, in Biometrics 25:129–142, 1969), which is frequently used in the problem of assessing the relative calibrations and relative accuracies of a set of p instruments, each designed to measure the same characteristic on a common group of n experimental units. We consider asymptotic tests to answer the outlined questions. The methodology is applied to a real data set and a small simulation study is presented.     

A New Fatigue Life Model Based on the Family of Skew-Elliptical Distributions

ABSTRACT

Fatigue is structural damage produced by cyclic stress and tension. An important statistical model for fatigue life is the Birnbaum–Saunders distribution, which was developed to model ruptured lifetimes of metals that had been subjected to fatigue. This model has been previously generalized and in this article we extend it starting from a skew-elliptical distribution, the incorporation of the elliptical aspect makes the kurtosis flexible, and the skewness makes the asymmetry flexible. In this work we found the probability density, reliability, and hazard functions; as well as its moments and variation, skewness, and kurtosis coefficients. In addition, some properties of this new distribution were found.     

On a variance stabilizing model and its application to genomic data

In this paper, we propose a model based on a class of symmetric distributions, which avoids the transformation of data, stabilizes the variance of the observations, and provides robust estimation of parameters and high flexibility for modeling different types of data. Probabilistic and statistical aspects of this new model are developed throughout the article, which include mathematical properties, estimation of parameters and inference. The obtained results are illustrated by means of real genomic data.     

Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model.     

The structural Sharpe model under t-distributions

In this paper we consider Sharpe's single-index model or Sharpe's model, by assuming that the returns obtained follow a multivariate t elliptical distribution. Also, given that the returns of the market are not observable, the statistical analysis was made in the context of an errors-in-variables model. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators the local influence method [10 Cook, R. D. 1986. Assessment of local influence. J. Roy. Statist. Soc. B, 48: 133–169.  [Google Scholar]] was implemented. The results are illustrated by using a set of shares of companies belonging to the Chilean Stock Market. The main conclusion is that the t model with small degrees of freedom is able to incorporate possible outliers and influential returns in the data.     

Influence diagnostics for Grubbs’s model with asymmetric heavy-tailed distributions

Grubbs’s model (Grubbs, Encycl Stat Sci 3:42–549, 1983) is used for comparing several measuring devices, and it is common to assume that the random terms have a normal (or symmetric) distribution. In this paper, we discuss the extension of this model to the class of scale mixtures of skew-normal distributions. Our results provide a useful generalization of the symmetric Grubbs’s model (Osorio et al., Comput Stat Data Anal, 53:1249–1263, 2009) and the asymmetric skew-normal model (Montenegro et al., Stat Pap 51:701–715, 2010). We discuss the EM algorithm for parameter estimation and the local influence method (Cook, J Royal Stat Soc Ser B, 48:133–169, 1986) for assessing the robustness of these parameter estimates under some usual perturbation schemes. The results and methods developed in this paper are illustrated with a numerical example.     

Bayesian inference for sinh-normal/independent nonlinear regression models

Sinh-normal/independent distributions are a class of symmetric heavy-tailed distributions that include the sinh-normal distribution as a special case, which has been used extensively in Birnbaum–Saunders regression models. Here, we explore the use of Markov Chain Monte Carlo methods to develop a Bayesian analysis in nonlinear regression models when Sinh-normal/independent distributions are assumed for the random errors term, and it provides a robust alternative to the sinh-normal nonlinear regression model. Bayesian mechanisms for parameter estimation, residual analysis and influence diagnostics are then developed, which extend the results of Farias and Lemonte [Bayesian inference for the Birnbaum-Saunders nonlinear regression model, Stat. Methods Appl. 20 (2011), pp. 423-438] who used the Sinh-normal/independent distributions with known scale parameter. Some special cases, based on the sinh-Student-t (sinh-St), sinh-slash (sinh-SL) and sinh-contaminated normal (sinh-CN) distributions are discussed in detail. Two real datasets are finally analyzed to illustrate the developed procedures.