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We study Kendall's tau and Spearman's rho concordance measures for discrete variables. We mainly provide their best bounds using positive dependence properties. These bounds are difficult to write down explicitly in general. Here, we give the explicit formula of the best bounds in a particular Fréchet space in order to understand the behavior of the ranges of these measures. Also, based on the empirical copula which is viewed as a discrete distribution, we propose a new estimator of the copula function. Finally, we give useful dependence properties of the bivariate Poisson distribution and show the relationship between parameters of the Poisson distribution and both tau and rho.

The best known dependence property is “lack of dependence,” or what is known as stochastic independence. In many applications, independence between two random variables is assumed; this can be a strong assumption in the undertaken analysis. Taking into account the dependence structure between the variables leads to appropriate modeling approaches and correct conclusions. To study stochastic dependence, concordance concept and positive dependence are well used tools. This is because many dependence properties can be described by means of the joint distribution of the variables and these measures and properties are often margins free. In this paper we study two concordance measures, Kendall’s tau Kruskal [

Many researches have been concerned with the study of tau and rho in the case of continuous variables. Schweizer and Wolff [

In this paper, we focus on the range of the concordance measures. Aside from identifying the best bounds of tau and rho in the case of discrete random variables, we present some dependence properties of the bivariate Poisson model and discuss their relationship with the concordance measures tau and rho. The paper is organized as follows. The next section provides a method of constructing the ranges of tau and rho for discrete data. Section

Following Hoeffding [

Several results in this paper are based on the monotonicity property of Kendall's

For the remainder of the paper, we recall the property of concordance orderings, defined as follows.

Let

In the following proposition, we propose a flexible method to establish the monotonicity property given in Mesfioui and Tajar [

Let

Using Fubini's theorem, we note that

Now without loss of generality if we assume that

For any bivariate distribution function

Let

As stated earlier, the main objective in this paper is to examine the bounds of

In order to obtain the best bounds

For discrete data, the ranges of

The aim of this section is to study the effect of the marginal distributions on the range of

The best bounds for

Let

Using (

The best lower bounds of

From (

Let

In this section, we examine the symmetry of the ranges of

In space,

In space,

It is well recognized that copula provides a flexible approach to model the joint behavior of random variables. In fact, this method allows to represent a bivariate distribution as function of its univariate marginals through a linking function called a copula. Specifically, if

Suppose that the random sample

Our goal in this section is to transform the empirical copula in order to obtain a new estimator

The distribution function

For any

Now, we show the expression of

Finally, one concludes that it will be convenient to estimate the theoretical copula

Our purpose in this section is to study dependence properties of the bivariate Poisson distribution

To study further the relationships between

Now, let

Let

From (

Many statistical researches have focused on studying concepts of positive dependence for bivariate distributions, example right tail increasing, and positive quadrant dependence which are widely used in actuarial literature [

The family

Since

When

In order to appreciate the corrections of

From Table

0.2 | 0.059 | 0.075 | 0.089 | 0.094 |

0.4 | 0.120 | 0.152 | 0.180 | 0.189 |

0.6 | 0.183 | 0.231 | 0.272 | 0.286 |

0.8 | 0.248 | 0.313 | 0.365 | 0.383 |

1.0 | 0.316 | 0.398 | 0.459 | 0.482 |

1.2 | 0.388 | 0.490 | 0.554 | 0.582 |

1.4 | 0.467 | 0.589 | 0.651 | 0.684 |

1.6 | 0.556 | 0.701 | 0.749 | 0.787 |

1.8 | 0.660 | 0.832 | 0.849 | 0.892 |

The second author acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada.