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The multivariate local dependence function

Ismihan Bayramoglu, Pelin Ersin

Abstract

The local dependence function is important in many applications of probability and statistics. We extend the bivariate local dependence function introduced by Bairamov and Kotz (2000) and further developed by Bairamov et al. (2003) to three-variate and multivariate local dependence function characterizing the dependency between three and more random variables in a given specific point. The definition and properties of the three-variate local dependence function are discussed. An example of a three-variate local dependence function for underlying three-variate normal distribution is presented. The graphs and tables with numerical values are provided. The multivariate extension of the local dependence function that can characterize the dependency between multiple random variables at a specific point is also discussed.

The multivariate local dependence function

Abstract

The local dependence function is important in many applications of probability and statistics. We extend the bivariate local dependence function introduced by Bairamov and Kotz (2000) and further developed by Bairamov et al. (2003) to three-variate and multivariate local dependence function characterizing the dependency between three and more random variables in a given specific point. The definition and properties of the three-variate local dependence function are discussed. An example of a three-variate local dependence function for underlying three-variate normal distribution is presented. The graphs and tables with numerical values are provided. The multivariate extension of the local dependence function that can characterize the dependency between multiple random variables at a specific point is also discussed.

Paper Structure

This paper contains 8 sections, 3 theorems, 78 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries. The Bivariate Local Dependence Function
  3. Properties of local dependence function H(x,y)
  4. Examples
  5. The three-variate extension of local dependence function
  6. Tables
  7. The multivariate extension of local dependence function
  8. Properties of multivariate local dependence function

Key Result

Theorem 1

(Bairamov and Kotz (2000)) The local dependence function $H(x,y)$ has the following properties: 1$^{\circ }$. If $X$ and $Y$ are independent then $H(x,y)=0$ for any $(x,y)\in N_{X,Y}.$ 2$^{\circ }$. $\left\vert H(x,y)\right\vert \leq 1,$ for alll $(x,y)\in N_{X,Y}.$ 3$^{\circ }$. If $\left\vert H(x,

Figures (13)

  • Figure 1: Graph of $H(x,y)$ given in (\ref{['bn1']}) for $\rho = 0.5$ and $\rho = -0.5$, with $-3 \leq x, y \leq 3$.
  • Figure 2: Graph of $H(x,y)$ given in (\ref{['bn1']}) for $\rho = 0.95$ and $\rho = -0.95$, with $-3 \leq x, y \leq 3$.
  • Figure 3: Graph of $H(x,y,z)$ for $x = 0$, $y,z\in (-4,4)$.
  • Figure 4: Graph of $H(x,y,z)$ for $x = 2$, $y,z\in (-4,4)$.
  • Figure 5: Graph of $H(x,y,z)$ for $x = -2$, $y,z\in (-4,4)$.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Example 1
  • Theorem 3