1. Introduction

In multivariate studies, measures of dependence that are invariant under special transformations are too important. Also, the linear correlation has many restrictions in applications, Embrechts et al. (2003) and Mc Neil et al. (2005) considered other forms of correlations. The copula approach is a useful method for separating univariate margins and the multivariate dependence structure by using Sklar’s theorem (1959, 1996). Nelsen (2006) drew attention to copula distribution function and dependence.

The problem of copula density estimation has been studied in Biau and Wegkamp (2005), and this subject has been developed by Kallenberg (2008) by using exponential families and contamination families.

Kallenberg (2009) focused on estimating the (unknown) copula density by the selection method. In this method, the modeling step consists of an intermediate approach between a parametric family and a non-parametric approach. This is done by considering a sequence of parametric copula models and starting with a given copula density or a given family of copula densities. In order to balance between the complexity of the model and the number of parameters, the model selection techniques determine which aspects are the most important ones to capture into our model.

This paper is organized as follows. Section 2 deals with some preliminaries. In section 3 the exponential families are reviewed and the decomposition of the total error into the model error and the stochastic error is explained. In Section 4 the contamination families based on Legendre polynomials are reviewed, also this section deals with the model selection problem to choose the best dimension with fast convergence to probability 1. In section 5, for two main variables, Income and Expenditure, in Iranian Household Income and Expenditure Survey, the nearest approximation of copula density using the selection method is obtained.

2. Preliminaries

A 2-dimensional copula is a function with the following properties:

1) For every

2) For every

3) For every with

The theoretical basis of multivariate modeling by copulas is provided by a theorem due to Sklar (1959), known as Sklar’s Theorem. Let be a joint distribution function with margins which are respectively the cumulative distribution functions of the random variables and . Then there exists a copula function such that

(1)

For every where represents the extended real line. Conversely, if is a copula and are distribution functions then the function defined a joint distribution function with margins

The parametric copula approach ensures a high level of flexibility for modeling, because the dependence structure can be separated from the margins, through the function with an underlying parameter which governs the intensity of the dependence.

In the case that the bivariate distribution has a density , and this is available, we have

(2)

where is the copula density and it should be approximated in most cases.

In general, a natural and very useful way to describe a smooth function on the interval (0, 1) is to apply the orthonormal system of Legendre polynomials. This leads to a function on (0, 1) as

(3)

where is the Legendre polynomial on (0, 1) and is the Fourier coefficient, such that

(4)

For example, the Legendre polynomials are given by

(5)

In the next section, it is seen that by using Legendre Polynomials a copula density function (given a known starting copula density function) is approximated.

3. Exponential Families

The exponential families are well-known families of parametric models that are used for approximating copula density function. If is the starting copula density function, the desired copula density is then approximated by

(6)

where and are Legendre polynomials, is the vector of parameters and is a normalizing function given by

(7)

Obviously, increasing the number of parameters yields to model with more complexity, so in order to balance between complexity and the number of parameters, dimension is determined. Note that may contain unknown parameters, which should be estimated as well.

Equation (2) shows that is approximated by a linear combination of the functions minus a normalizing factor (to make its integral equal to 1). Exponential families ensure automatically that we get densities such that belongs to the natural parameter space

(8)

The criteria for choosing the best approximation might be the Kullback Leibler information, given by

(9)

It is seen that minimizing is equivalent to maximizing , which gives the asymptotic version of the maximum likelihood estimator. So, asymptotically the maximum likelihood estimator chooses that member of the exponential family which is closest to the true density in terms of Kullback Leibler information criteria.

Kallenberg (2008) showed that is the projection of into the exponential family with base , because

(10)

(11)

Where is a unique point such that

Hence as the model error, is reduced to with a reduction equal to Another extra reduction from taking a higher dimension, when going from is occurred by an amount For the exponential family, the better fit means the smaller model error and the higher dimension or the more parameters have to be estimated. Since parameters estimation in the exponential family is difficult, the idea of contamination family is developed.

4. Contamination Families

As mentioned in Kallenberg (2009), just like the exponential family, the starting point is a copula density and is approximated by a linear combination of the functions hence

(12)

where are Fourier coefficients as follows

(13)

These coefficients depend on the unknown copula density function that if it is replaced with empirical copula mass function then can be estimated as

(14)

Again when the starting copula density function belongs to a parametric family, its parameters should be estimated, then we have

(15)

Kallenberg (2009) showed that by considering the term as the model error given by

(16)

where

By equation (16) it can be seen that Total Error is decomposed by Model Error and Stochastic Error,

Total Error = Model Error + Stochastic Error

The model error expresses how good the contamination family approximates the true density and the stochastic error is due to estimation.

4.1. Model Selection

In order to obtain parameter estimations in contamination family, the best dimension should be chosen. Suppose be the largest dimension of r and s with observations, then we have

(17)

For the selection rule, taking all the coefficients for yields a large estimation error, so we consider only the largest Fourier coefficients and ignore the rest. Therefore, the estimator from (17), is replaced by restricting to the largest among with yielding

(18)

With

(19)

Random variables and depend on the data, and they are not chosen in advance. So how large should we take The optimal choice depends on but is unknown, hence a data-driven selection of the dimension is taken.

From (9), the model error for is Hence, should grow sufficiently fast in order to take a higher dimension. For that purpose a penalty is introduced, classical penalties are for example (Schwarz’s rule) or (Akaike’s criterion). It may be better to take a larger penalty, taking into account the variance of .

Kallenberg (2009), introduced a penalty as

(20)

And the selection rule as:

(21)

The estimated copula density now becomes

(22)

5. Copula Density Estimation for Iranian Household Income and Expenditure

The 2015 survey was carried out by a sample of 18839 households in urban areas and 19340 households in rural areas. The survey target population includes all private and collective settled households in urban and rural areas. three-stage cluster sampling method with strata is used in the survey. At the first stage, the census areas are classified and selected. At the second stage, the urban and rural blocks are selected and the selection of sample households is done at the third stage. The number of samples is optimized to estimate average annual income and expenditure of the sample household based on the aim of the survey. In this section, the model selection method is used to estimate copula density for Iranian Household Income and Expenditure Income and Expenditure descriptive statistics of urban and rural household are shown in Tables 1 and 2, respectively.

Table 1. Descriptive statistics for Income and Expenditure data of Urban household

Table 2. Descriptive statistics for Income and Expenditure data of Rural household

5.1. IHIE Copula Density Estimation with Contamination Families

By using empirical distributions as marginal distribution estimations for both variables as

(23)

Now the problem is to estimate the unknown copula function. In order to use a few largest Fourier coefficients, the absolute value of the Fourier coefficients are arranged from largest to smallest and compare with By using the sample size of each data set is calculated, then according to the chosen algorithm of Fourier coefficients, these coefficients are obtained. With several start copula densities (Uniform, Gaussian, Clayton and, Frank), as it is shown in Tables 3 and 4, we have several estimations of copula density for rural and urban data sets.

Table 3. Results for Urban Data

Table 4. Results for Rural Data

For Urban data the sample size is , with calculation gives that for , so and

(24)

With the Gaussian copula density as the start point, calculations give , so and

(25)

For the start with Frank copula density calculations give so and

(26)

For Rural data the sample size is with calculation gives that for so and

(27)

With the Gaussian copula density as a start point, calculations give so and

(28)

For the start with Clayton copula density calculations give , so and

(29)

It should be noted that without using this method (selection method) among known copula densities, Frank copula and Clayton copula are the appropriate copulas for Urban and Rural data respectively, here these copulas can be chosen as starting points.

5.2. Investigating Performance of the Estimated Copula Function

To check the performance of the estimated copula densities, frequency of data is compared with the estimated probabilities, based on mean absolute relative error on the same symmetric rectangles and asymmetric rectangles and also the corresponding upper tail rectangles.

It can be seen from Table 5 for Urban data copula density function with Gaussian copula as a starting point has the least Also Table 6 shows that for Rural data copula density function with Gaussian copula as the starting point has the least

Table 5. The frequencies and approximations on different rectangles for Urban data

Table 6. The frequencies and approximations on different rectangles for Rural data

For more investigating of this new method, we considered the 99% (95%) quantile based on (for both datasets) and the actual number of data points outside the rectangle with the expected number which is or was compared. The results are shown in Tables 7 and 8. In both tables, the expected numbers and the real numbers are close to each other.

Table 7. The 0.99 and 0.95 estimated and real quantiles for Urban data

Table 8. The 0.99 and 0.95 estimated and real quantiles for Rural data

6. Conclusions

In this paper in order to approximate copula density function for two variables, Income and Expenditure, of Iranian household, contamination families and selection models methods have been used. In this approach, a sequence of parametric copulas has been considered and in a few numbers of steps, accurate approximations for copula densities are obtained. By using the selection model method, the model complexity and number of model parameters have been balanced. It was shown that the best approximations for copula density function are the ones that are based on Gaussian starting copula. Also by using m.a.r.e. as a criterion, it has been shown that for both cases approximation with Gaussian copula as the starting point has the least mean absolute relative error.