Bivariate Pareto Distribution Research Articles

An efficient MEWMA chart for Gumbel's bivariate Pareto distribution

Control charts play a vital role in process monitoring to ensure the product's desired standards. Due to rapid improvements in data collection methods, multivariate charts are preferred over univariate charts. This paper proposes a bivariate exponentially weighted moving average chart for the simultaneous monitoring of the mean vector of Gumbel's bivariate Pareto type II (also known as the Lomax distribution) time-between-events data. The performance of the proposed chart is assessed through average run length, median run length, and the standard deviation of the run length criteria. To show the implementation of the chart in the real world, illustrative examples are also presented.

Decomposing China’s bilateral export growth: A firm-regional-transactions structural gravity approach

This paper undertakes a decomposition of China’s firm-transaction-level exports with an emphasis on aggregates thereof at the prefectural level. Chinese prefectures are large and, relative to some smaller countries, the larger ones are visible players in the world market. We decompose China’s prefecture-product-level bilateral exports to foreign countries into their main components. These are factor costs, market potential (including trade frictions and foreign expenditure) as well as quality and productivity. We consider the latter two as to be drawn from a bivariate Pareto distribution that is specific to prefectures, products, and countries. This strategy enables retrieving fixed market access costs and permits conducting counterfactual experiments with a focus on the distribution across Chinese prefectures.

An EM algorithm for absolutely continuous Marshall-Olkin bivariate Pareto distribution with location and scale

Block and Basu bivariate exponential distribution is one of the most popular absolutely continuous bivariate distributions. In this article, we have considered a Block–Basu type bivariate Pareto distribution. Here in the standard manner, first Marshall-Olkin type singular bivariate distribution has been constructed, and then by taking away the singular component similar to the Block and Basu model, an absolute continuous BB-BVPA model has been constructed. Further, the location and scale parameters also have been introduced. Therefore, the model has seven parameters. We drive different properties of this absolutely continuous distribution. Since the maximum likelihood estimators of the parameters cannot be expressed in a closed form, we propose to use an EM algorithm to compute the estimators of the model parameters. Some simulation experiments have been performed for illustrative purposes. We fit the model to rainfall data in the context of landslide risk estimation. The methods are implemented in the publicly available R package bvpa.

On the Bivariate Composite Gumbel–Pareto Distribution

In this paper, we propose a bivariate extension of univariate composite (two-spliced) distributions defined by a bivariate Pareto distribution for values larger than some thresholds and by a bivariate Gumbel distribution on the complementary domain. The purpose of this distribution is to capture the behavior of bivariate data consisting of mainly small and medium values but also of some extreme values. Some properties of the proposed distribution are presented. Further, two estimation procedures are discussed and illustrated on simulated data and on a real data set consisting of a bivariate sample of claims from an auto insurance portfolio. In addition, the risk of loss in this insurance portfolio is estimated by Monte Carlo simulation.

Correlations in bivariate Pareto distributions

Bivariate Pareto distributions have proven very useful in modelling lifetime data, hydrology, competing risk data, and many other datasets. In this paper, we explore Kendall tau and Gini correlations in the bivariate Pareto distributions, comparing with the popular Pearson correlation and robust quadrant correlation. It is interesting to establish the fact that zero of those correlations mutually imply independence in this family. We derive the variance of the asymptotic normality for each sample correlation via the influence function approach. When the second moment is finite, we demonstrate that the symmetric Gini correlation is asymptotically efficient as well as relatively efficient among finite samples. However, Kendall tau is more appealing in terms of compromising efficiency and robustness.

Statistical characteristics of driver acceleration behaviour and its probability model

Naturalistic driving data were applied to study driver acceleration behaviour, and a probability model of the driver was proposed. First, the question of whether the database is large enough is resolved using kernel density estimation and Kullback-Liebler divergence. Next, the convergence database is utilised to achieve the bivariate acceleration distribution pattern. Subsequently, two probability models are proposed to explain the pattern. Finally, the statistical characteristics of the acceleration behaviours are studied to verify the probability models. The longitudinal and lateral acceleration behaviours always approximate a similar Pareto distribution. The braking, accelerating, and steering manoeuvres become more intense at first and then less intense as the velocity increases. These behaviours characteristics reveal the mechanism of the quadrangle bivariate acceleration distribution pattern. The bivariate acceleration behaviour of the driver will never reach a circle-shaped pattern. The bivariate Pareto distribution model can be applied to describe the bivariate acceleration behaviour of the driver.

Exact strong laws for functions from the bivariate Pareto distribution

Exact strong laws for functions from the bivariate Pareto distribution

Non-Parametric Inference for Gini Covariance and its Variants

We obtain simple non-parametric estimators of Gini-based covariance, correlation and regression coefficients. We then establish the consistency and asymptotic normality of the proposed estimators. We provide an explicit formula for finding the asymptotic variance of the estimators. We also discuss jackknifed versions of the proposed estimators for reducing the bias of the estimators in case of small sample sizes. Finally, we evaluate the finite-sample performance of these estimators through on Monte Carlo simulations from a bivariate Pareto distribution.

A Study of Bivariate Generalized Pareto Distribution and its Dependence Structure Among Model Parameters

Several variants of the classical bivariate and multivariate generalized Pareto distributions have been discussed and studied in the literature (see Arnold (1983, Stat. Prob. Lett. 17: 361–368, 1993, 2015), Arnold and Laguna (1977), Ali and Nadarajah (2007), Rootzen and Tajvidi (2006) and the references cited therein). Ali and Nadarajah (2007) studied a truncated version of the most popular long-tailed generalized bivariate Pareto distribution (GBPD, henceforth, in short) involving six parameters. However, not much discussion exists in the current literature on the structural properties as well as on the dependence structure among the parameters in this model. In this paper we re-visit the GBPD and discuss several other new properties. In addition, we study the shape of GBPD for varying choices of the model parameters and subsequently study their interdependence. Also, we provide copula based construction of GBPD and discuss the associated local dependence measures.

Some Variations of EM Algorithms for Marshall–Olkin Bivariate Pareto Distribution with Location and Scale

Recently, Asimit et al. have used an EM algorithm to estimate the parameters of Marshall–Olkin bivariate Pareto distribution with location and scale. The distribution has seven parameters. We describe different other variations of the EM algorithm. Numerical simulation is performed to identify the best algorithm among them. A real-life data analysis is also shown for illustrative purposes.

Induced Ranked Set Sampling when Units are Inducted from Several Populations

The method of ranked set sampling when units are to be inducted from several bivariate populations is introduced in this work. The best linear unbiased estimation of a common parameter of two bivariate Pareto distributions is discussed based on the n ranked set observations, when a sample of size n 1 is drawn from a bivariate Pareto population with shape parameter a 1 and a sample of size n 2 is drawn from another bivariate Pareto with shape parameter a 2 such that n = n 1 + n 2 . The application of the results of this paper is illustrated with a real life data.

Distortion risk measures for nonnegative multivariate risks

We apply distortion functions to bivariate survival functions for nonnegative random variables. This leads to a natural extension of univariate distortion risk measures to the multivariate setting. For Gini’s principle, the proportional hazard transform distortion and the dual power transform distortion, certain families of multivariate distributions lead to a straightforward risk measure. We show that an exact analytical expression can be obtained in some cases. We consider the independence case, the bivariate Pareto distribution and the bivariate exponential distribution. An illustration of the estimation procedure and the interpretation is also included. In the case study, we consider two loss events with a single risk value and monitor the two events together over four different periods. We conclude that the dual power transform gives more weight to the observations of extreme losses, but that the distortion parameter can modulate this influence in all cases. In our example, multivariate risk clearly diminishes over time.

On the hidden truncated bivariate Pareto (IV) model and associated inferential issues

ABSTRACTIncome and wealth data are typically modelled by some variant of the classical Pareto distribution. Often, in practice, the observed data are truncated with respect to some unobserved covariate. In this paper, a hidden truncation formulation of this scenario is proposed and analysed. For this purpose, a bivariate Pareto (IV) distribution is assumed for the variable of interest and the unobserved covariate. Some important distributional properties of the resulting model as well as associated inferential methods are studied. An example is used finally to illustrate the results developed here. In this case, it is noted that hidden truncation on the left does not result in any new model, but the hidden truncation on the right does. The properties and fit of such a model pose a challenging problem and that is what is focused here in this work.

The Bivariate Pareto Model Based on Ordered Random Variables

Generalized order statistics constitute a unified model for ordered random variables that includes order statistics and record values among others. In this article, bivariate Pareto distribution is considered. Some new simple explicit expressions for single and product moments of concomitants of generalized order statistics based on a random sample drown from the considered distribution are derived. Further, applications of these results is seen in establishing some well known results given separately for order statistics and record values and obtaining some new results. Finally, the means, and variances of the concomitants of order statistics and record values are computed for various values of the parameters.

Characterizations of a family of bivariate Pareto distributions

In the present paper, we study properties of a family of bivariate Pareto distributions. The well known dullness property of the univariate Pareto model is extended to the bivariate setup. Two measures of income inequality viz. income gap ratio and mean left proportional residual income are defined in the bivariate case. We also introduce bivariate generalized failure rate useful in reliability analysis. Characterizations, using the above concepts, for various members of the family of bivariate Pareto distributions are derived.

Geometric visualization of parallel bivariate Pareto distribution surfaces

Abstract In the present paper, the differential-geometrical framework for parallel bivariate Pareto distribution surfaces ( P , P ¯ ) is given. Curvatures of a curve lying on ( P , P ¯ ) , are interpreted in terms of the parameters of P . Geometrical and statistical interpretations of some results are introduced and plotted.

Application of Concomitants of Order Statistics of Independent Non Identically Distributed Random Variables in Estimation

In this article, we obtain expressions for the pdf of a single concomitant of order statistic and the joint pdf of a pair of concomitants of order statistics of independent non identically distributed random variables. Using these expressions, we find the means, variances and covariances of order statistics arising from independent non identically distributed bivariate Pareto distributions. A method of estimation of a common parameter involved in several bivariate Pareto distributions using concomitants of order statistics is also discussed.

Approximation of the Bivariate Renewal Function

Recently, Kambo and his co-researchers (2012) proposed a method of approximation for evaluating the one-dimensional renewal function based on the first three moments. Their method is simple and elegant, which gives exact values for well-known distributions. In this article, we propose an analogous method for the evaluation of bivariate renewal function based on the first two moments of the variables and their joint moment. The proposed method yields exact results for certain widely used bivariate distributions like bivariate exponential distribution, bivariate Weibull distributions, and bivariate Pareto distributions. An illustrative example in the form of a two-dimensional warranty problem is considered and comparisons of our method are made with the results of other models.

A family of bivariate Pareto distributions

Pareto distributions have been extensively used in literature for modelling and analysis of income and lifetime data. In the present paper, we introduce a family of bivariate Pareto distributions using a generalized version of dullness property. Some important bivariate Pareto distributions are derived as special cases. Distributional properties of the family are studied. The dependency structure of the family is investigated. Finally, the family of distributions is applied to two real life data situation.

Inference for a Hidden Truncated (Both-Sided) Bivariate Pareto (II) Distribution

The Pareto distribution is a simple model for non negative data with a power law probability tail. Income and wealth data are typically modeled using some variant of the classical Pareto distribution. In practice, it is frequently likely that the observed data have been truncated with respect to some unobserved covariable. In this paper, a hidden truncation formulation of this scenario is proposed and analyzed. A bivariate Pareto (II) distribution is assumed for the variable of interest and the unobserved covariable. Distributional properties of the resulting model are investigated. A variety of parameter estimation strategies (under the classical set up) are investigated.