Moderate deviations for the random weighted sums of END random variables with consistently varying tails

In this paper, we study moderate deviations for random weighted sums of extended negative dependent (END) random variables, which are consistently-varying tailed and not necessarily identically distributed. When these END random variables are independent of their weights, and the weights are positive random variables with two-sided bounds, the results shows END structure and the dependence between the weights have no effects on the asymptotic behavior of moderate deviations of partial sums and random sums.     

Randomly weighted sums of linearly wide quadrant-dependent random variables with heavy tails

This paper investigates tail behavior of the randomly weighted sum ∑ⁿ_{k = 1}θ_kX_k and reaches an asymptotic formula, where X_k, 1 ? k ? n, are real-valued linearly wide quadrant-dependent (LWQD) random variables with a common heavy-tailed distribution, and θ_k, 1 ? k ? n, independent of X_k, 1 ? k ? n, are n non-negative random variables without any dependence assumptions. The LWQD structure includes the linearly negative quadrant-dependent structure, the negatively associated structure, and hence the independence structure. On the other hand, it also includes some positively dependent random variables and some other random variables. The obtained result coincides with the existing ones.     

On the strong convergence of weighted sums of widely dependent random variables

ABSTRACT

For widely dependent random variables, we present some results on the strong convergence of weighted sums, including results on almost surely (a.s.) and complete convergence. To this end, we verified some Borel–Cantelli lemmas of the widely dependent random variables. The above-mentioned random variables contain common negatively dependent random variables, some positively dependent random variables, and some others; therefore, the obtained results extend and improve some existing results.     

Moderate deviations for sums of dependent claims in a size-dependent renewal risk model

In this article, we consider a non standard renewal risk model, in which pairs of claim sizes and its corresponding inter-arrival times are identically distributed, and each pair obeys a dependence structure. By assuming that the claim sizes form a sequence of extended negatively dependent random variables with consistently varying tails, moderate deviations for the aggregate amount of dependent claims are obtained.     

The effect of dependence on distribution of the functions of random variables

In this article, we study the effect of dependence on the distributional properties of functions of two random variables. Expressions for the cumulative distribution functions of the linear combinations, products, and ratios of two dependent random variables in terms of their associated copula are derived. We discuss the effect of dependence on quantities such as the variances of linear combinations of functions, the value-at-risk measure, and the stress–strength parameter. Several examples, a simulation study, and a real data analysis are provided to illustrate the result.     

Bounds for L-Statistics from Weakly Dependent Samples of Random Length

ABSTRACT

Sharp bounds on expected values of L-statistics based on a sample of possibly dependent, identically distributed random variables are given in the case when the sample size is a random variable with values in the set {0, 1, 2,…}. The dependence among observations is modeled by copulas and mixing. The bounds are attainable and provide characterizations of some non trivial distributions.     

A new measure of linear local dependence

A new local dependence function based on regression concepts is introduced. This function can characterize the dependence structure of two random variables localized at the fixed point. Some properties of the local dependence function are given. Examples of important bivariate distributions are provided.     

The strong laws of large numbers for weighted sums of extended negatively dependent random variables

In this paper, the strong laws of large numbers for maximum value of weighted sums of extended negatively dependent random variables are obtained, which improve and extend the corresponding ones for independent random variables and some dependent random variables.     

Max-Linear Competing Factor Models

Models incorporating “latent” variables have been commonplace in financial, social, and behavioral sciences. Factor model, the most popular latent model, explains the continuous observed variables in a smaller set of latent variables (factors) in a matter of linear relationship. However, complex data often simultaneously display asymmetric dependence, asymptotic dependence, and positive (negative) dependence between random variables, which linearity and Gaussian distributions and many other extant distributions are not capable of modeling. This article proposes a nonlinear factor model that can model the above-mentioned variable dependence features but still possesses a simple form of factor structure. The random variables, marginally distributed as unit Fréchet distributions, are decomposed into max linear functions of underlying Fréchet idiosyncratic risks, transformed from Gaussian copula, and independent shared external Fréchet risks. By allowing the random variables to share underlying (latent) pervasive risks with random impact parameters, various dependence structures are created. This innovates a new promising technique to generate families of distributions with simple interpretations. We dive in the multivariate extreme value properties of the proposed model and investigate maximum composite likelihood methods for the impact parameters of the latent risks. The estimates are shown to be consistent. The estimation schemes are illustrated on several sets of simulated data, where comparisons of performance are addressed. We employ a bootstrap method to obtain standard errors in real data analysis. Real application to financial data reveals inherent dependencies that previous work has not disclosed and demonstrates the model’s interpretability to real data. Supplementary materials for this article are available online.     

Positive dependence of random variables with a common marginal distribution

In this paper we review some notions of positive dependence of random variables with a common univariate marginal distribution and describe the related moment and probability inequalities. We first present a comparison between i.i.d. random variables and exchangeable random variables via an application of de Finetti's theorem, then describe some useful probability inequalities via partial orderings of the strength of their positive dependence. Finally, we state a result for random variables which are not necessarily exchangeable. Special applications to the multivariate normal distribution will be discussed, and the results involve only the correlation matrix of the distribution.     

A note on measuring the degree of dependence between two discrete random variables measured on a nominal scale

This paper discusses some of the problematic aspects of the term “dependence” and suggests a measure of a special kind of association - complete dependence - between two discrete random variables measured on a nominal scale.     

Complete convergence for weighted sums of extended negatively dependent random variables

In this article, the complete convergence for weighted sums of extended negatively dependent (END, in short) random variables without identical distribution is investigated. In addition, the complete moment convergence for weighted sums of END random variables is also obtained. As an application, the Baum–Katz type result for END random variables is established. The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.     

A class of correlated weighted Poisson processes

In this paper, we propose new classes of correlated Poisson processes and correlated weighted Poisson processes on the interval [0,1], which generalize the class of weighted Poisson processes defined by Balakrishnan and Kozubowski (2008), by incorporating a dependence structure between the standard uniform variables used in the construction. In this manner, we obtain another process that we refer to as correlated weighted Poisson process. Various properties of this process such as marginal and joint distributions, stationarity of the increments, moments, and the covariance function, are studied. The results are then illustrated through some examples, which include processes with length-biased Poisson, exponentially weighted Poisson, negative binomial, and COM-Poisson distributions.     

Some bivariate uniform distributions

Bivariate uniform distributions with dependent components are readily derived by distribution function transformations of the components of non-uniform dependent continuous bivariate random variables (X,Y). Contour plots of joint density functions show the various, and varying, forms of dependence which can arise from different distributional forms for (X,Y) and aids the choice of bivariate uniform distributions as empirical models.     

Positive dependence and monotonicity in conditional distributions

Positively dependent random variables exhibit the property that an extreme value of one of the variables tends to be accompanied by extreme values of the others. In this paper we define two notions of positive dependence which lead to monotonicity theorems for conditional distributions.     

Limiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual Cesàro alpha-integrability assumption

Asymptotically negative association is a special dependence structure. By relating such dependence condition to residual Cesàro alpha-integrability and to strongly residual Cesàro alpha-integrability, some L_p-convergence and complete convergence results of the maximum of the partial sum are derived, respectively. In addition, some of these conclusions are based on a new Rosenthal type inequality concerning asymptotically negatively associated random variables, which is of independent interest.     

Some novel results on pairwise quasi-asymptotical independence with applications to risk theory

In this article we obtain some novel results on pairwise quasi-asymptotically independent (pQAI) random variables. Concretely speaking, let X₁, …, X_n be n real-valued pQAI random variables, and W₁, …, W_n be another n non negative and arbitrarily dependent random variables, but independent of X₁, …, X_n. Under some mild conditions, we prove that W₁X₁, …, W_nX_n are still pQAI as well. Our result is in a general setting whether the primary random variables X₁, …, X_n are heavy-tailed or not. Finally, a special case of above result is applied to risk theory for investigating the finite-time ruin probability for a discrete-time risk model with a wide type of dependence structure.     

Archimedean copulas with applications to $${{\mathrm{}}}$$ estimation

Assuming absolute continuity of marginals, we give the distribution for sums of dependent random variables from some class of Archimedean copulas and the marginal distribution functions of all order statistics. We use conditional independence structure of random variables from this class of Archimedean copulas and Laplace transform. Additionally, we present an application of our results to \({{\mathrm{VaR}}}\) estimation for sums of data from Archimedean copulas.     

Dependence maps: Local dependence in practice

There is often more structure in the way two random variables are associated than a single scalar dependence measure, such as correlation, can reflect. Local dependence functions such as that of Holland and Wang (1987) are, therefore, useful. However, it can be argued that estimated local dependence functions convey information that is too detailed to be easily interpretable. We seek to remedy this difficulty, and hence make local dependence a more readily interpretable practical tool, by introducing dependence maps. Via local permutation testing, dependence maps simplify the estimated local dependence structure between two variables by identifying regions of (significant) positive, (not significant) zero and (significant) negative local dependence. When viewed in conjunction with an estimate of the joint density, a comprehensive picture of the joint behaviour of the variables is provided. A little theory, many implementational details and several examples are given.     

Randomly Weighted Sums of Pairwise Quasi Upper-Tail Independent Increments with Application to Risk Theory

This article investigates the asymptotic tail probabilities of randomly weighted sums and their maximum of pairwise quasi upper-tail independent increments with non negative and arbitrarily dependent random weights. An application of the obtained results to risk theory is proposed.