Regression-based orderings and measures of stochastic depence

General axiomatic approach to the so-called global dependence of a random variable xon a random vector Y= Y _t,Y _n) is proposed. natural orderings and measures of global dependence are discussed and examplified by some real and function-valued measures of dependence. orderings and measures to be introduced are referred o as regression-based as they depend only on the distributions of EX|Y X.     

Lorenz ranking of income distributions

Based on the stochastic comparison of the Lorenz curves of income distributions, five partial orderings of income distributions are obtained. Three of these orderings are the well known star shaped, stochastic and the Lorenz orderings. The other two are new and are studied in some detail. The weakest ordering which is called the Lorenz area ordering is of special importance since it enables us to compare interesting Lorenz curves. This latter ordering leads to a class of income inequality measures which are identical with the linear inequality measures considered by Mehran (1976). A discussion of these measures is presented together with an application to part of Kunzet's (1963) data.     

Modification of Pickands' Dependence Function for Ordered Bivariate Extreme Distribution

We study the characteristics of the Pickands' dependence function for bivariate extreme distribution for minima, BEVM, when considering the stochastics ordering of the two variables, X < Y. The existing Pickand's dependence function terminologies and theories are modified to suit the dependence functions of extreme minimum cases. The main result is the introduction of the restricted logistic dependence function, A _RL , and the restricted exponential function, V _RL (x, y).     

A STOCHASTIC ORDERING FOR RANDOM VARIABLES WITH APPLICATIONS

A new univariate stochastic ordering is introduced. Some characterization results for such an ordering are stated. It is proved that the ordering is an integral stochastic ordering, obtaining a maximal generator. By means of this generator, the main properties of the ordering are deduced. A method for introducing univariate stochastic orderings, suggested by the new ordering, is analysed. Relationships with other stochastic orderings are also developed. To conclude, an example of an application of the new ordering to the field of medicine is proposed.     

Uniform stochastic orderings and total positivity

Uniform stochastic orderings of random variables are expressed as total positivity (TP) of density, survival, and distribution functions. The orderings are called uniform because each is a stochastic order that persists under conditioning to a family of intervals—for example, the family consisting of all intervals of the form (-∞,x]. This paper is concerned with the preservation of uniform stochastic ordering under convolution, mixing, and the formation of coherent systems. A general TP₂ result involving preservation of total positivity under integration is presented and applied to convolutions and mixtures of distribution and survival functions. Log-concavity of distribution, survival, and density functions characterizes distributions that preserve the various orderings under convolution. Likewise, distributions that preserve orderings under mixing are characterized by TP₂ distribution and survival functions.     

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.     

On multivariate weighted distributions

The concept of weighted distributions is well-known in the literature concerning observational studies and surveys in research related to forestry, ecology, bio-medicine and many other areas (cf. Rao (1965)). This paper extends the idea of weighted distributions to multivariate case. A few multivariate orderings have been defined and some partial ordering results are presented. Some results regarding multivariate positive and negative dependence are also discussed. Multivariate weighted distributions - joint, marginal and conditional have been defined and some important results concerning them are presented along with an illustration. The Multivariate Poisson Negative Hypergeometric Distribution has been derived.     

Estimation for distributions with monotone likelihood ratio:case of vector valued parameter

An extension of a result about the estimation in Karlin and Rubin is given for the following case:The sample space, the parameter space and the decision space are subsets of a multi-dimensional Euclidean space, there is defined a suitable partial ordering in each of spaces, and a probability distribution has monotone likelihood ratio with respect to the partial orderings (see Ishii, 1976). In the special case when the loss function is quadratic a simple proof of a result in Karlin and Rubin is given. Stein's estimators are discussed as examples.     

LTD and RTI dependence orderings

The authors show how the approach of Capéra à & Genest (The Canadian Journal of Statistics, 1990) can be used to order bivariate distributions with arbitrary marginals by their degree of dependence in the LTD (left‐tail decreasing) or RTI (right‐tail increasing) sense. Some properties of these new orderings are given, along with applications to Archimedean copulas, order statistics and compound random variables.     

Importance of Components for a System

Which component is most important for a system's survival? We answer this question by ranking the information relationship between a system and its components. The mutual information (M) measures dependence between the operational states of the system and a component for a mission time as well as between their life lengths. This measure ranks each component in terms of its expected utility for predicting the system's survival. We explore some relationships between the ordering of importance of components by M and by Zellner's Maximal Data Information (MDIP) criterion. For many systems the bivariate distribution of the component and system lifetimes does not have a density with respect to the two-dimensional Lebesgue measure. For these systems, M is not defined, so we use a modification of a mutual information index to cover such situations. Our results for ordering dependence are general in terms of binary structures, sum of random variables, and order statistics.     

A coefficient of multiple association based on ranks

A generalization of Kendall's tau is formulated for describing the association between a dependent variable and a collection of independent variables. The coefficient may be defined in terms of the proportional reduction in prediction errors obtained by predicting the ordering of pairs of observations on the dependent variable based on orderings of the pairs on the independent variables. The coefficient is formulated both for continuous and discrete variables. Approximate large-sample distributions are considered for both cases. Some of the properties of this coefficient are discussed and compared with those of other multiple measures of association based on ranks.     

Tail-weighted measures of dependence

Multivariate copula models are commonly used in place of Gaussian dependence models when plots of the data suggest tail dependence and tail asymmetry. In these cases, it is useful to have simple statistics to summarize the strength of dependence in different joint tails. Measures of monotone association such as Kendall's tau and Spearman's rho are insufficient to distinguish commonly used parametric bivariate families with different tail properties. We propose lower and upper tail-weighted bivariate measures of dependence as additional scalar measures to distinguish bivariate copulas with roughly the same overall monotone dependence. These measures allow the efficient estimation of strength of dependence in the joint tails and can be used as a guide for selection of bivariate linking copulas in vine and factor models as well as for assessing the adequacy of fit of multivariate copula models. We apply the tail-weighted measures of dependence to a financial data set and show that the measures better discriminate models with different tail properties compared to commonly used risk measures – the portfolio value-at-risk and conditional tail expectation.     

Bounds on the expected maximum

Let (X₁,X₂, …,X_n) be jointly distributed random variables. Define X_n:n = max(X₁,X₂, …,X_n).Bounds on E(X_n:n), obtained by putting constraints on the distributions and/or dependence structure of the X_i's, are surveyed.     

Some new partial orderings on Sn and Sn×Sn

A new class of partial orderings on S_n, the set of permutations of {1,…,n}, is given. Each of these partial orderings is shown to be a subordering of a recently described partial ordering on S_n (Block, Chhetry, Fang and Sampson (1990)) which is related to Schriever's (1987) more associated ordering on bivariate distributions. Also given is an extension of three known partial orders on S_n to partial orders on S_n×S_n. These extensions facilitate the study of functions from S_n×S_n into , which preser these partial orderings, thereby, providing a methodology for extending the notion of arrangement increasing functions.     

Berry-Esseen bounds for kernel estimates of stationaryprocesses

In this paper we obtain nonuniform Berry-Esseen bounds of the kernel estimate of stationary process {X_i} by the method of martingale approximation. In particular, by choice of bandwidth b_n, the Berry-Esseen bounds can be n^−2/15. The dependence condition of {X_i} is expressed in terms of physical dependence measures introduced by Wu (2005).     

The one-way analysis of variance with ordered alternatives: A modification of bartholomew's Ē2 Test

Tests of homogeneity of normal means with the alternative restricted by an ordering on the means are considered. The simply ordered case, μ₁ ≤ μ₂ ≤ ··· ≤ μ_k, and the simple tree ordering, μ₁ ≤ μ_j, for; j= 2, 3,…, k, are emphasized. A modification of the likelihood-ratio test is proposed which is asymptotically equivalent to it but is more robust to violations of the hypothesized orderings. The new test has power at the points satisfying the hypothesized ordering which is similar to that of the likelihood-ratio test provided the degrees of freedom are not too small. The modified test is shown to be unbiased and consistent.     

Aspects of Dependence in Cuadras–Auge Family

In this article, we study the dependence structure of Cuadras–Auge (C–A) family of bivariate distributions. We also obtain some association measures and two local dependence functions for this family. In addition, we compare expectation of local dependence function and Pearson's rho via numerical study.     

Non-parametric Estimation of Tail Dependence

Abstract.  Dependencies between extreme events (extremal dependencies) are attracting an increasing attention in modern risk management. In practice, the concept of tail dependence represents the current standard to describe the amount of extremal dependence. In theory, multi-variate extreme-value theory turns out to be the natural choice to model the latter dependencies. The present paper embeds tail dependence into the concept of tail copulae which describes the dependence structure in the tail of multivariate distributions but works more generally. Various non-parametric estimators for tail copulae and tail dependence are discussed, and weak convergence, asymptotic normality, and strong consistency of these estimators are shown by means of a functional delta method. Further, weak convergence of a general upper-order rank-statistics for extreme events is investigated and the relationship to tail dependence is provided. A simulation study compares the introduced estimators and two financial data sets were analysed by our methods.     

Dependence function for continuous bivariate densities

The notion of cross-product ratio for discrete two-way contingency table is extended to the case of continuous bivariate densities. This results in the “local dependence function” that measues the margin-free dependence between bivariate random variables. Properties and examples of the dependence function are discussed. The bivariate normal density plays a special role since it has constant dependence. Continuous bivariate densities can be constructed by specifying the dependence function along with two marginals in analogy to the construction of two-way contingency tables given marginals and patterns of interaction. The dependence function provides a partial ordering on bivariate dependence.     

On estimation and testing arising from order restricted balanced mixed model

The randomized complete block design is one of the most widely used experimental designs to systematically control the variability arising from known nuisance sources. The balanced mixed effects model is usually appropriate for such an experiment when the blocks used in the experiment are randomly chosen. In applications with k increasing or decreasing treatment levels, there is sometimes prior knowledge about the ordering of the treatment effects. The most commonly seen orderings include simple ordering, simple tree ordering and umbrella orderings with known or unknown peaks. A natural question is how to incorporate the prior ordering information in estimating the parameters in a balanced mixed effects model so that the estimated treatment effects are consistent with the prior information and the estimated variances of the block effects and experiment errors are nonnegative. In this paper we derive the maximum likelihood estimators of the parameters in a balanced mixed model subject to any partial ordering of the treatment effects, which includes the usual maximum likelihood estimators as a special case. An example is provided to illustrate the results.