Risk preference and employment contract type

Summary.  We explore the possibility that a systematic relationship exists between employment within a particular type of contract and risk preference. We exploit a set of proxies for risk preference, whereby some of the proxies capture risk loving behaviour (expenditure on gambling, smoking and alcohol) whereas others capture risk averse behaviour (expenditure on life and contents insurance, and unearned income). The empirical analysis, based on pooled cross-section data from the UK Family Expenditure Survey, 1997–2000, provides evidence of a systematic relationship between employment contract type and risk preference, with, for example, self-employed workers being more or less likely to engage in the consumption of 'risky' or financial security products respectively. The results are based on the ordered generalized extreme value model, a relatively infrequently used discrete choice model, which allows for ordering and correlation in the alternatives observed.     

A comparative review of generalizations of the Gumbel extreme value distribution with an application to wind speed data

ABSTRACT

The generalized extreme value distribution and its particular case, the Gumbel extreme value distribution, are widely applied for extreme value analysis. The Gumbel distribution has certain drawbacks because it is a non-heavy-tailed distribution and is characterized by constant skewness and kurtosis. The generalized extreme value distribution is frequently used in this context because it encompasses the three possible limiting distributions for a normalized maximum of infinite samples of independent and identically distributed observations. However, the generalized extreme value distribution might not be a suitable model when each observed maximum does not come from a large number of observations. Hence, other forms of generalizations of the Gumbel distribution might be preferable. Our goal is to collect in the present literature the distributions that contain the Gumbel distribution embedded in them and to identify those that have flexible skewness and kurtosis, are heavy-tailed and could be competitive with the generalized extreme value distribution. The generalizations of the Gumbel distribution are described and compared using an application to a wind speed data set and Monte Carlo simulations. We show that some distributions suffer from overparameterization and coincide with other generalized Gumbel distributions with a smaller number of parameters, that is, are non-identifiable. Our study suggests that the generalized extreme value distribution and a mixture of two extreme value distributions should be considered in practical applications.     

A generalization of discrete Weibull distribution

In this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.     

Extreme value analysis of actuarial risks: estimation and model validation

We give an overview of several aspects arising in the statistical analysis of extreme risks with actuarial applications in view. In particular it is demonstrated that empirical process theory is a very powerful tool, both for the asymptotic analysis of extreme value estimators and to devise tools for the validation of the underlying model assumptions. While the focus of the paper is on univariate tail risk analysis, the basic ideas of the analysis of the extremal dependence between different risks are also outlined. Here we emphasize some of the limitations of classical multivariate extreme value theory and sketch how a different model proposed by Ledford and Tawn can help to avoid pitfalls. Finally, these theoretical results are used to analyze a data set of large claim sizes from health insurance.     

Spatial modeling of extreme rainfall in northeast Thailand

It is well recognized that the generalized extreme value (GEV) distribution is widely used for any extreme events. This notion is based on the study of discrete choice behavior; however, there is a limit for predicting the distribution at ungauged sites. Hence, there have been studies on spatial dependence within extreme events in continuous space using recorded observations. We model the annual maximum daily rainfall data consisting of 25 locations for the period from 1982 to 2013. The spatial GEV model that is established under observations is assumed to be mutually independent because there is no spatial dependency between the stations. Furthermore, we divide the region into two regions for a better model fit and identify the best model for each region. We show that the regional spatial GEV model reflects the spatial pattern well compared with the spatial GEV model over the entire region as the local GEV distribution. The advantage of spatial extreme modeling is that more robust return levels and some indices of extreme rainfall can be obtained for observed stations as well as for locations without observed data. Thus, the model helps to determine the effects and assessment of vulnerability due to heavy rainfall in northeast Thailand.     

A default Bayesian procedure for the generalized Pareto distribution

The generalized Pareto distribution is used to model the exceedances over a threshold in a number of fields, including the analysis of environmental extreme events and financial data analysis. We use this model in a default Bayesian framework where no prior information is available on unknown model parameters. Using a large simulation study, we compare the performance of our posterior estimations of parameters with other methods proposed in the literature. We show that our procedure also allows to make inferences in other quantities of interest in extreme value analysis without asymptotic arguments. We apply the proposed methodology to a real data set.     

A more general methodology for fitting global cross-ratio models for discrete longitudinal responses

In this paper, we focus on repeated measurement problems, comprising an interesting research area in statistics. We study longitudinal data which arise when outcomes are observed repeatedly on each experimental subject at several points. We focus on a marginal approach for this type of data with lack of independence among the observations proposed by Dale [Global cross-ratio models for bivariate, discrete, ordered responses. Biometrics. 1986;42(4):909–917] for bivariate, discrete, ordered responses. We propose an alternative estimation based on divergence measures to the full likelihood method proposed in that paper. Finally, a wide simulation study and a data example that illustrates the new methodology is provided.     

An algorithm for conditional multidimensional parameter identification with asymmetric and correlated losses of under- and overestimations

Many of today's specialized applicational tasks are obliged to consider the influence of inevitable errors in the identification of parameters appearing in a model. Favourable results can also be achieved through measuring, and then accounting for definite (e.g. current) values of factors which show a significant reaction to the values of those parameters. This paper is dedicated to the problem of the estimation of a vector of parameters, where losses resulting from their under- and overestimation are asymmetric and mutually correlated. The issue is considered from a supplementary conditional aspect, where particular coordinates of conditioning variables may be continuous, discrete, multivalued (in particular binary) or categorized (ordered and unordered). The final result is a ready-to-use algorithm for calculating the value of an estimator, optimal in the sense of minimum expectation of losses using a multidimensional asymmetric quadratic function, for practically any distributions of describing and conditioning variables.     

Bayesian modeling of dynamic extreme values: extension of generalized extreme value distributions with latent stochastic processes

This paper develops Bayesian inference of extreme value models with a flexible time-dependent latent structure. The generalized extreme value distribution is utilized to incorporate state variables that follow an autoregressive moving average (ARMA) process with Gumbel-distributed innovations. The time-dependent extreme value distribution is combined with heavy-tailed error terms. An efficient Markov chain Monte Carlo algorithm is proposed using a state-space representation with a finite mixture of normal distributions to approximate the Gumbel distribution. The methodology is illustrated by simulated data and two different sets of real data. Monthly minima of daily returns of stock price index, and monthly maxima of hourly electricity demand are fit to the proposed model and used for model comparison. Estimation results show the usefulness of the proposed model and methodology, and provide evidence that the latent autoregressive process and heavy-tailed errors play an important role to describe the monthly series of minimum stock returns and maximum electricity demand.     

The beta generalized linear exponential distribution

In this paper, a new five-parameter lifetime distribution called beta generalized linear exponential distribution (BGLED) is introduced. It includes at least 17 popular sub-models as special cases such as the beta linear exponential, the beta generalized exponential, and the exponentiated generalized linear distributions. Mathematical and statistical properties of the proposed distribution are discussed in details. In particular, explicit expression for the density function, moments, asymptotics distributions for sample extreme statistics, and other statistical measures are obtained. The estimation of the parameters by the method of maximum-likelihood is discussed and the finite sample properties of the maximum-likelihood estimators (MLEs) are investigated numerically. A real data set is used to demonstrate its superior performance fit over several existing popular lifetime models.     

Bivariate discrete generalized exponential distribution

In this paper, we develop a bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized exponential distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate distribution is a very flexible distribution and the bivariate geometric distribution can be obtained as a special case of this distribution. The proposed distribution can be seen as a natural discrete analogue of the bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally, we propose some open problems and conclude the paper.     

Bayesian latent variable models for clustered mixed outcomes

A general framework is proposed for modelling clustered mixed outcomes. A mixture of generalized linear models is used to describe the joint distribution of a set of underlying variables, and an arbitrary function relates the underlying variables to be observed outcomes. The model accommodates multilevel data structures, general covariate effects and distinct link functions and error distributions for each underlying variable. Within the framework proposed, novel models are developed for clustered multiple binary, unordered categorical and joint discrete and continuous outcomes. A Markov chain Monte Carlo sampling algorithm is described for estimating the posterior distributions of the parameters and latent variables. Because of the flexibility of the modelling framework and estimation procedure, extensions to ordered categorical outcomes and more complex data structures are straightforward. The methods are illustrated by using data from a reproductive toxicity study.     

Generalized partially linear varying coefficient models with multiple smoothing variables

This paper is concerned with semiparametric efficient estimation of a generalized partially linear varying coefficient model. The model studied in this paper is very flexible, accommodating various nonlinear relations between the response variable and a set of predictor variables. It is a structured regression model and is particularly useful in dealing with a discrete response variable. We apply the smooth backfitting technique to estimate the nonparametric part of the model and employ the profiling approach to obtain a semiparametric efficient estimator of the parametric part.     

Local likelihood smoothing of sample extremes

Trends in sample extremes are of interest in many contexts, an example being environmental statistics. Parametric models are often used to model trends in such data, but they may not be suitable for exploratory data analysis. This paper outlines a semiparametric approach to smoothing sample extremes, based on local polynomial fitting of the generalized extreme value distribution and related models. The uncertainty of fits is assessed by using resampling methods. The methods are applied to data on extreme temperatures and on record times for the women's 3000 m race.     

A semi-parametric Bayesian extreme value model using a Dirichlet process mixture of gamma densities

In this paper, we propose a model with a Dirichlet process mixture of gamma densities in the bulk part below threshold and a generalized Pareto density in the tail for extreme value estimation. The proposed model is simple and flexible for posterior density estimation and posterior inference for high quantiles. The model works well even for small sample sizes and in the absence of prior information. We evaluate the performance of the proposed model through a simulation study. Finally, the proposed model is applied to a real environmental data.     

Sparse cluster analysis of large-scale discrete variables with application to single nucleotide polymorphism data

Currently, extreme large-scale genetic data present significant challenges for cluster analysis. Most of the existing clustering methods are typically built on the Euclidean distance and geared toward analyzing continuous response. They work well for clustering, e.g. microarray gene expression data, but often perform poorly for clustering, e.g. large-scale single nucleotide polymorphism (SNP) data. In this paper, we study the penalized latent class model for clustering extremely large-scale discrete data. The penalized latent class model takes into account the discrete nature of the response using appropriate generalized linear models and adopts the lasso penalized likelihood approach for simultaneous model estimation and selection of important covariates. We develop very efficient numerical algorithms for model estimation based on the iterative coordinate descent approach and further develop the expectation–maximization algorithm to incorporate and model missing values. We use simulation studies and applications to the international HapMap SNP data to illustrate the competitive performance of the penalized latent class model.     

Estimation and testing in generalized partial linear models—A comparative study

A particular semiparametric model of interest is the generalized partial linear model (GPLM) which extends the generalized linear model (GLM) by a nonparametric component.The paper reviews different estimation procedures based on kernel methods as well as test procedures on the correct specification of this model (vs. a parametric generalized linear model). Simulations and an application to a data set on East–West German migration illustrate similarities and dissimilarities of the estimators and test statistics.     

A class of bivariate regression models for discrete and/or continuous responses

ABSTRACT

A general class of models for discrete and/or continuous responses is proposed in which joint distributions are constructed via the conditional approach. It is assumed that the distributions of one response and of the other response given the first one belong to exponential family of distributions. Furthermore, the marginal means are related to the covariates by link functions and a dependency structure between the responses is inserted into the model. Estimation methods, diagnostic analysis and a simulation study considering a Bernoulli-exponential model, a particular case of the class, are presented. Finally, this model is used in a real data set.     

Modelling small and medium enterprise loan defaults as rare events: the generalized extreme value regression model

A pivotal characteristic of credit defaults that is ignored by most credit scoring models is the rarity of the event. The most widely used model to estimate the probability of default is the logistic regression model. Since the dependent variable represents a rare event, the logistic regression model shows relevant drawbacks, for example, underestimation of the default probability, which could be very risky for banks. In order to overcome these drawbacks, we propose the generalized extreme value regression model. In particular, in a generalized linear model (GLM) with the binary-dependent variable we suggest the quantile function of the GEV distribution as link function, so our attention is focused on the tail of the response curve for values close to one. The estimation procedure used is the maximum-likelihood method. This model accommodates skewness and it presents a generalisation of GLMs with complementary log–log link function. We analyse its performance by simulation studies. Finally, we apply the proposed model to empirical data on Italian small and medium enterprises.     

Estimation of a nonlinear discriminant function from a mixture of two GEV distributions

In this paper, the identifiability of finite mixture of generalized extreme value (GEV) distributions is proved. Next, a procedure for finding maximum likelihood estimates (MLEs) of the parameters of a finite mixture of two generalized extreme value (MGEV) distributions is presented by using classified and unclassified observations. Then, a nonlinear discriminant function for a mixture of two GEV distributions is derived and the performance of the corresponding estimated discriminant function is investigated through a series of simulation experiments. Finally, the methodology is applied to real data.