Univariate Pareto Distribution Research Articles

Bayesian Non-Parametric Inference for Multivariate Peaks-over-Threshold Models.

We consider a constructive definition of the multivariate Pareto that factorizes the random vector into a radial component and an independent angular component. The former follows a univariate Pareto distribution, and the latter is defined on the surface of the positive orthant of the infinity norm unit hypercube. We propose a method for inferring the distribution of the angular component by identifying its support as the limit of the positive orthant of the unit p-norm spheres and introduce a projected gamma family of distributions defined through the normalization of a vector of independent random gammas to the space. This serves to construct a flexible family of distributions obtained as a Dirichlet process mixture of projected gammas. For model assessment, we discuss scoring methods appropriate to distributions on the unit hypercube. In particular, working with the energy score criterion, we develop a kernel metric that produces a proper scoring rule and presents a simulation study to compare different modeling choices using the proposed metric. Using our approach, we describe the dependence structure of extreme values in the integrated vapor transport (IVT), data describing the flow of atmospheric moisture along the coast of California. We find clear but heterogeneous geographical dependence.

A Robust Estimator of R = P(X > Y ) of Heavy-tailed Distributions and its Sampling Distributions

Heavy-tailed distributions have wide ap- plications in life-time contexts, especially in reliability and risk modeling. So we consider the estimation problem of reliability, R = P (X > Y ) for small samples, when X and Y are two independent but not identically distributed random variables belonging to the family of heavy-tailed distributions, using a robust estimator, namely the harmonic moment estimator. Extensive sim- ulation studies are carried out to study the performance of this estimator. The relative efficiency of the estima- tor with the well known Hill estimator is studied. We obtain the sampling distribution of the parameters of the distribution as well as that of estimator of R which will help us to study the properties of the estimators. Also we find out the asymptotic confidence intervals of R and its performance is studied with respect to average width and the coverage probability, through simulations. especially in reliability and risk modeling. We consider the estimation problem of reliability R = P (X > Y ) for small and large samples, when X and Y are two inde- pendent but not identically distributed random variables belonging to the family of heavy-tailed distributions.The heavy-tailed distributions like the student's t and Pareto families have been used to model data with high kurto- sis see, (4). The simplest heavy tailed distribution is the Pareto distribution. It is hyperbolic over its entire range. The Pareto distribution is a power law probability dis- tribution that coincides with social, scientific, geophys- ical, actuarial and many other types of observable phe- nomena. The univariate Pareto distribution is a simple model for non-negative data with a power law probabil- ity tail, at least approximately for large values of X. It is a useful model in the analysis of income data, reliabil- ity studies, risk modeling and business failure data (14). Reference (1) gave an extensive historical survey of its uses in the context of income distribution. References (6), (16) and (19) discussed the application of the Pareto disrribution in various fields. Moreover Pareto Type I distribution has a position of importance in the field of life testing because this distribution can be considered as a distribution of failure time of a component as fol- lows.

Matrix Variate Pareto Distribution of the Second Kind

We generalize the univariate Pareto distribution of the second kind to the matrix case and give its derivation using matrix variate gamma distribution. We study several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of the Pareto matrix. Some of these results are expressed in terms of special functions of matrix arguments, zonal, and invariant polynomials.

Multivariate Pareto Distributions: Inference and Financial Applications

Univariate Pareto distributions are extensively studied. In this article, we propose a Bayesian inference methodology in the context of multivariate Pareto distributions of the second kind (Mardia's type). Computational techniques organized around Gibbs sampling with data augmentation are proposed to implement Bayesian inference in practice. The new methods are shown to work well in artificial examples involving a trivariate distribution, and to an empirical application involving daily exchange rate data for four major currencies.

A bivariate pareto model for drought

Univariate Pareto distributions have been so widely used in hydrology. It seems however that bivariate or multivariate Pareto distributions have not yet found applications in hydrology, especially with respect to drought. In this note, a drought application is described by assuming a bivariate Pareto model for the joint distribution drought durations and drought severity in the State of Nebraska. Based on this model, exact distributions are derived for the inter arrival time, magnitude and the proportion of droughts. Estimates of 2, 5, 10, 20, 50 and 100 year return periods are derived for the three variables, drought duration, drought severity and the pairwise combinations: (drought duration, drought severity), (inter arrival time of drought, proportion of drought) and (drought duration, drought magnitude). These return period estimates could have an important role in hydrology, for example, with respect to measures of vegetation water stress for plants in water-controlled ecosystems.

A multivariate pareto distribution

In this paper, we introduce a new multivariate pareto (MVP) distribution with many interesting properties. we extend the results of characterization of univariate and bivariate pareto distributions given by Krishnaji (1970) and veenus and Nair (1994) respectively. We also extend the property of dullness of univariate pareto distribution given by Talwalkar (1980) to the multivariate pareto case. We obtain the maximum likelihood estimate (MLE) of the parameters and their asymptotic multivariate normal (AMVN) distrioutions. We propose large sample studentized test for testing independence and identical marginals of the components.