A class of copulas with piecewise linear horizontal sections

We introduce the class of bivariate copulas with piecewise linear horizontal sections whose graph is composed, at most, of two segments. It is a wide class of copulas which contains some known copulas. We study several properties of the copulas in the new class concerning absolute continuity, singular components, measures of association, concordance ordering, dependence concepts and symmetry. Finally, we provide several examples.     

A new asymmetric class of bivariate copulas for modeling dependence

In this paper, we propose an asymmetric class of bivariate copulas. This class is obtained through limiting properties of the extended copula introduced by Bekrizadeh, et al. (2015 Bekrizadeh, H., Parham, G. A., Zadkarami, M. R. (2015). Extending some classes of copulas; Applications. Ph.D. Thesis, University of Shahid Chamran, Ahvaz. [Google Scholar]), and includes some of known copulas. Some general formulas for well-known association measures and concepts of dependence of the proposed model are obtained. This paper highlights the usefulness of this new bivariate copula for modeling the interested variables whose marginal distribution effect on joint distribution isn't identical. We apply some subfamilies of this new class to model a dataset of medical science to show the superiority of presented model in comparison with the known copulas. These results will be investigated using simulation.     

A general class of flexible Weibull distributions

We consider a linear combination of two logarithms of cumulative hazard functions and propose a general class of flexible Weibull distribution functions which includes some well-known modified Weibull distributions (MWDs). We suggest a very flexible Weibull distribution, which belongs to the class, and show that its hazard function is monotone, bathtub-shaped, modified bathtub-shaped, or even upside-down bathtub-shaped. We also discuss the methods of least square estimation and maximum likelihood estimation of the unknown parameters. We take two illustrated examples to compare the suggested distribution with some current MWDs, and show that the suggested distribution shows good performances.     

A recipe for bivariate copulas

ABSTRACT

We give conditions on a ? ?1, b ∈ ( ? ∞, ∞), and f and g so that C_{a, b}(x, y) = xy(1 + af(x)g(y))^b is a bivariate copula. Many well-known copulas are of this form, including the Ali–Mikhail–Haq Family, Huang–Kotz Family, Bairamov–Kotz Family, and Bekrizadeh–Parham–Zadkarmi Family. One result is that we produce an algorithm for producing such copulas. Another is a one-parameter family of copulas whose measures of concordance range from 0 to 1.     

A stochastic representation and sampling algorithm for nested Archimedean copulas

A general sampling algorithm for nested Archimedean copulas was recently suggested. It is given in two different forms, a recursive or an explicit one. The explicit form allows for a simpler version of the algorithm which is numerically more stable and faster since less function evaluations are required. The algorithm can also be given in general form, not being restricted to a particular nesting such as fully nested Archimedean copulas. Further, several examples are given.     

Supermigrative copulas and positive dependence

Recent investigations about notions of bivariate aging have underlined the need to introduce some new properties of positive dependence for a bivariate random vector. Here, by using the recent notion of supermigrativity of a bivariate copula, a?positive dependence property is introduced and investigated. Comparisons with other notions of positive dependence are also presented.     

Computationally tractable approximate and smoothed Polya trees

A discrete approximation to the Polya tree prior suitable for latent data is proposed that enjoys surprisingly simple and efficient conjugate updating. This approximation is illustrated in two applied contexts: the implementation of a nonparametric meta-analysis involving studies on the relationship between alcohol consumption and breast cancer, and random intercept Poisson regression for Ache armadillo hunting treks. The discrete approximation is then smoothed with Gaussian kernels to provide a smooth density for use with continuous data; the smoothed approximation is illustrated on a classic dataset on galaxy velocities and on recent data involving breast cancer survival in Louisiana.     

A nonparametric symmetry test for absolutely continuous bivariate copulas

Based on the works by Klement and Mesiar (Comment Math Univ Carolinae 47:141–148, 2006) and Nelsen (Stat Pap 48:329–336, 2007) on maximal asymmetry of copulas, we define and study the concept of tri-symmetry and we propose a simple statistic to test symmetry of a bivariate copula, given a random sample of an absolutely continuous bivariate random vector. We also make a power comparison against some other well known nonparametric symmetry tests.     

Derivatives and Fisher information of bivariate copulas

Data sets with complex relationships between random variables are increasingly studied in statistical applications. A popular approach to model their dependence is the use of copula functions. Our contribution is to derive expressions for the observed and expected information for several bivariate copula families, in particular for the Student’s $t$ -copula. Further likelihood derivatives which are required for numerical implementations are computed and a numerically stable implementation is provided in the R-package VineCopula. Using a real world data set of stock returns, we demonstrate the applicability of our approach for the routinely calculation of standard errors. In particular, we illustrate how this prevents overestimating the time-variation of dependence parameters in a rolling window analysis.     

Survival and lifetime data analysis with a flexible class of distributions

We introduce a general class of continuous univariate distributions with positive support obtained by transforming the class of two-piece distributions. We show that this class of distributions is very flexible, easy to implement, and contains members that can capture different tail behaviours and shapes, producing also a variety of hazard functions. The proposed distributions represent a flexible alternative to the classical choices such as the log-normal, Gamma, and Weibull distributions. We investigate empirically the inferential properties of the proposed models through an extensive simulation study. We present some applications using real data in the contexts of time-to-event and accelerated failure time models. In the second kind of applications, we explore the use of these models in the estimation of the distribution of the individual remaining life.     

The present status of confidence interval estimation for one-factor random models

This paper summarizes the present status of confidence interval estimation for one-factor random models. We provide both closed-form intervals and generalized confidence intervals for several parameters. Simple numerical examples are provided to illustrate computations. In order to limit the focus of this review, we consider only one-at-a-time (non-simultaneous) intervals with homoscedastic normal error terms, and do not include any Bayesian or bootstrap methods.     

A bivariate extension of the beta generated distribution derived from copulas

In this paper, we introduce a new class of bivariate distributions whose marginals are beta-generated distributions. Copulas are employed to construct this bivariate extension of the beta-generated distributions. It is shown that when Archimedean copulas and convex beta generators are used in generating bivariate distributions, the copulas of the resulting distributions also belong to the Archimedean family. The dependence of the proposed bivariate distributions is examined. Simulation results for beta generators and an application to financial risk management are presented.     

Bayesian inference for a flexible class of bivariate beta distributions

Several bivariate beta distributions have been proposed in the literature. In particular, Olkin and Liu [A bivariate beta distribution. Statist Probab Lett. 2003;62(4):407–412] proposed a 3 parameter bivariate beta model which Arnold and Ng [Flexible bivariate beta distributions. J Multivariate Anal. 2011;102(8):1194–1202] extend to 5 and 8 parameter models. The 3 parameter model allows for only positive correlation, while the latter models can accommodate both positive and negative correlation. However, these come at the expense of a density that is mathematically intractable. The focus of this research is on Bayesian estimation for the 5 and 8 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach. Simulation studies have been carried out for the 5 and 8 parameter cases under various priors and tolerance levels. We apply the 5 parameter model to a real data set by allowing the model to serve as a prior to correlated proportions of a bivariate beta binomial model. Results and comparisons are then discussed.     

Truncation invariant copulas and a testing procedure

ABSTRACT

The class of bivariate copulas that are invariant under truncation with respect to one variable is considered. A simulation algorithm for the members of the class and a novel construction method are presented. Moreover, inspired by a stochastic interpretation of the members of such a class, a procedure is suggested to check whether the dependence structure of a given data set is truncation invariant. The overall performance of the procedure has been illustrated on both simulated and real data.     

Idempotent and multivariate copulas with fractal support

Using special iterated function systems (IFS) Fredricks et al. (2005) constructed two-dimensional copulas with fractal supports and showed that for every s∈(1,2)$s\in (\mathrm{1,2})$ there exists a copula A whose support has Hausdorff dimension s. In the current paper we present a stronger version and prove that the same result holds for the subclass of idempotent copulas. Additionally we show that every doubly stochastic idempotent matrix N (having neither minimum nor maximum rank) induces a family of idempotent copulas such that, firstly, the corresponding Markov kernels transform according to N   and, secondly, the set of Hausdorff dimensions of the supports of elements of the family covers (1,2). Furthermore we generalize the IFS approach to arbitrary dimensions d≥2$d\ge 2$ and show that for every s∈(1,d)$s\in (1,d)$ we can find a d-dimensional copula whose support has Hausdorff dimension s.     

Nonparametric estimation of multivariate extreme-value copulas

Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estimator by its best least-squares approximation in the set of Pickands dependence functions having a discrete spectral measure supported on a sufficiently fine grid. Weak convergence of the standardized estimators is demonstrated and the finite-sample performance of the estimators is investigated by means of a simulation experiment.     

Frailty models and copulas: similarities and differences

Copulas and frailty models are important tools to model bivariate survival data. Equivalence between Archimedean copula models and shared frailty models, e.g. between the Clayton-Oakes copula model and the shared gamma frailty model, has often been claimed in the literature. In this note we show that, in both the models, there is indeed a well-known equivalence between the copula functions; the modeling of the marginal survival functions, however, is quite different. The latter fact leads to different joint survival functions.     

On some aspects of a flexible class of additive Weibull distribution

Here we consider a more flexible class of the additive Weibull distribution of Xie and Lai (Reliab. Eng. Syst. Safety, 1995) and investigate some of its important properties such as expressions for its cumulative distribution function, reliability measures, quantile function, characteristic function, raw moments, incomplete moments, etc. The distribution and moments of order statistics are obtained along with certain structural properties. The maximum-likelihood estimation of the parameters of the distribution is attempted and the usefulness of the model in certain applied areas is illustrated with the help of certain real life data sets.