Journal of Probability and Statistics

Journal of Probability and Statistics / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 5902839 | 9 pages | https://doi.org/10.1155/2018/5902839

Hybrid Clayton-Frank Convolution-Based Bivariate Archimedean Copula

Academic Editor: Ramón M. Rodríguez-Dagnino
Received05 Oct 2017
Accepted08 Feb 2018
Published14 Mar 2018

Abstract

This study exploits the closure property of the converse convolution operator to come up with a hybrid Clayton-Frank Archimedean copula for two random variables. Pairs of random variables were generated and the upper tail observation of the cumulative distribution function (CDF) was used to assess the right skew behavior of the proposed model. Various values of the converse convolution operator were used to see their effect on the proposed model. The simulation covered lengths , and 6. The proposed model was compared with about 40 other bivariate copulas (both Archimedean and elliptical). The proposed model had parameters that spanned the entire real line, thus removing restrictions on the parameters. The parameters theta and omega were varied for a selected interval and the hybrid Clayton-Frank model was, in most cases, found to outperform the other copulas under consideration.

1. Introduction

It is quite apparent that identifying and quantifying dependencies is the core of econometric modeling, especially when it comes to risk management. Pearson’s correlation factor for dependence has over time been used in this regard although it has a deficiency when it comes to issues of nonlinear dependence. Several arguments have been put forward in this regard [1] and Embrechts et al. [2]. In order to get a measure that gives more information about the dependence structure, copulas were preferred to Pearson’s correlation coefficient.

Copulas give a way of isolating the marginal behavior from the dependence structure McNeil et al. [3]. Alcock and Hatherley [4] suggested that, through copulas, the nonnormal dependence structure could be modeled by using only uniforms of the marginal distributions which allowed violation of the assumptions of normality and linear dependence. This meant that the marginals could be modeled using each type of distributions without influencing the dependence structure between them.

The use of copulas has increased greatly in all fields of study since several phenomena tend to have certain dependencies amongst or between them, no matter how small the dependencies are. Boateng et al. [5] focused on the likelihood of a pair of random variables having either an Archimedean copula or an elliptical copula. Their work involved simulating several pairs of random variables and selecting an appropriate bivariate copula family. The corresponding parameter estimates were obtained by maximum likelihood estimation. They compared AICs of the various bivariate copulas under consideration, using about forty (40) bivariate copulas, for sample sizes 30, 300, 1000, 10000, 100000, and 1000000. Their result showed that, between the Archimedean and elliptical copulas, the Archimedean copulas were the most likely to fit the simulated pairs of random variables.

There have been several applications of copulas. Some of the applications being simulation of multivariate sea storms [6]; dependence structure between the stock and foreign exchange markets [7]; operational risk management [8]; portfolio optimization in the presence of dependent financial returns with long memory [9]; risk evaluation of droughts across the Pearl River basin, China [10]; probabilistic assessment of flood risks [11]; estimation of distribution algorithms for coverage problem of wireless sensor network [12]; risk assessment of hydroclimatic variability on groundwater levels in the Manjara basin aquifer in India [13]; models of tourists’ time use and expenditure behavior with self-selection [14]; modeling wind speed dependence in system reliability assessment using copulas [15]; dependence between crude oil spot and futures markets [16]; stochastic modeling of power demand [17].

This study explores the possibility of a combination of existing bivariate Archimedean copulas performing better or just as well as the individual copulas.

2. Method

2.1. Copula

A copula is a function satisfying the following requirements:(1)Grounded: (2)Uniform marginals: (3)2-increasing: , for and .

Generator Function (Clayton)Inverse Function (Clayton)Copula (Clayton)Kendal’s Tau (Clayton)Generator Function (Frank)Inverse Function (Frank)Copula (Frank)Kendal’s Tau (Frank)where (Debye function).

2.2. -Convolution

Let and be continuous cumulative distribution functions and be a copula function. The -Convolution of and is defined by the cumulative distribution function (CDF)where .

2.3. Convolution-Based Copulas
2.3.1. Proposition

Let and be two real-valued random variables on the same probability space with a dependence structure represented by the copula function and continuous marginal distributions and . Then,

2.3.2. Proposition

Let , , and be three continuous CDFs, be a copula function, and is a copula iff .

The -Convolution operator is closed with respect to mixtures of copula functions.

Let and be bivariate copula functions.For all and for all CDFs and ,

2.4. Proposed Convolution-Based Hybrid Clayton-Frank Copula

Using the fact that the -Convolution operator is closed with respect to mixtures of copula functions, we propose a hybrid Clayton-Frank copula as follows:

2.4.1. Check for Being a Copula

(1) For It Being Grounded, That Is, . From above,Following the steps in , also gives zero .

(2) Uniform Marginals: .which is the marginal of .

Andwhich is the marginal of .

(3) 2-Increasing: , for and .The above ensures that the 2-increasing property is satisfied.

2.5. Conditional Distribution Function of Given

The conditional distribution function of given is obtained by finding the derivative of with respect to ; that is, .

2.6. Joint Distribution Function of and

The joint distribution function is obtained from .

2.7. Parameter Estimation

In order to obtain theta (), Kendal’s tau relationship with the traditional Clayton copula parameter (see (4)) is employed. Similarly, to obtain omega (), Kendal’s tau relationship with the traditional Frank copula (see (8)) is used.

3. Results and Analysis

3.1. Comparison of Clayton-Frank Copula with Other Bivariate Copulas for Two Simulated Independent Random Variables

Assuming Independence of two randomly generated variables, various starting values for the maximum likelihood estimate of the hybrid Clayton-Frank model are considered. The simulation covers lengths .

3.2. Results for ,

Standard normal bivariate random variables of length are generated and their CDF which lies in () is used for the copula analysis. The upper tail of the CDF is analysed to check whether the hybrid model works better in right skewed observations.

The interval to 4 is considered for both parameters theta and omega from the hybrid model and pairs from this interval are considered as starting values for the optimization process in the maximum likelihood estimation.

Values of lambda () are considered to see their individual effect on the hybrid Clayton-Frank copula model.

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Rotated Tawn Type 2 (180 degrees) copula with an AIC value of . This means that, compared to the other bivariate copulas, the Rotated Tawn Type 2 (180 degrees) copula had the smallest AIC value. Comparing this value with the AIC values in Table 1, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study when omega is kept constant and theta varied from to 1, when omega is kept constant and theta varied from 4 to 1, and when theta is kept constant and omega varied from 4 to 1.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Rotated Tawn Type 2 (180 degrees) copula with an AIC value of . This means that, compared to the other bivariate copulas, the Rotated Tawn Type 2 (180 degrees) copula had the smallest AIC value. Comparing this value with the AIC values in Table 2, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study when omega is kept constant and theta varied from to 1, when omega is kept constant and theta varied from 4 to 1, and when theta is kept constant and omega varied from 4 to 1.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Rotated Tawn Type 2 (180 degrees) copula with an AIC value of . This means that, compared to the other bivariate copulas, the Rotated Tawn Type 2 (180 degrees) copula had the smallest AIC value. Comparing this value with the AIC values in Table 3, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study when omega is kept constant and theta varied from to 1, when omega is kept constant and theta varied from 4 to 1, and when theta is kept constant and omega varied from 4 to 1.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

3.3. Results for ,

Standard normal bivariate random variables generated of length are generated and their CDF which lies in () is used for the copula analysis. The upper tail of the CDF is analysed to check whether the hybrid model works better in right skewed observations.

The interval to 4 is considered for both of the parameters theta and omega from the hybrid model too and pairs from this interval are considered as starting values for the optimization process in the maximum likelihood estimation.

Values of lambda are considered to see their individual effect on the overall hybrid model.

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Frank Copula with an AIC value of . This means that, compared to the other bivariate copulas, the Frank copula had the smallest AIC value. Comparing this value with the AIC values in Table 4, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Frank Copula with an AIC value of . This means that, compared to the other bivariate copulas, the Frank copula had the smallest AIC value. Comparing this value with the AIC values in Table 5, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Frank Copula with an AIC value of . This means that, compared to the other bivariate copulas, the Frank copula had the smallest AIC value. Comparing this value with the AIC values in Table 6, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned except when omega is kept constant and theta varied from to 1.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

3.4. Results for ,

Standard normal bivariate random variables of length are generated and their CDF which lies in is used for the copula analysis. The upper tail of the CDF is analysed to check whether the hybrid model works better in right skewed observations.

The interval to 4 is considered for both of the parameters theta and omega from the hybrid model and pairs from this interval are considered as starting values for the optimization process in the maximum likelihood estimation.

Values of lambda are considered to see their individual effect on the overall hybrid model.

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence Copula had the smallest AIC value. Comparing this value with the AIC values in Table 7, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence copula had the smallest AIC value. Comparing this value with the AIC values in Table 8, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence copula had the smallest AIC value. Comparing this value with the AIC values in Table 9, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

3.5. Results for ,

Standard normal bivariate random variables of length are generated and their CDF which lies in is used for the copula analysis. The upper tail of the CDF is analysed to check whether the hybrid model works better in right skewed observations.

The interval to 4 is considered for both of the parameters theta and omega from the hybrid model and pairs from this interval are considered as starting values for the optimization process in the maximum likelihood estimation.

Values of lambda are considered to see their individual effect on the overall hybrid model.

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence Copula had the smallest AIC value. Comparing this value with the AIC values in Table 10, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence copula had the smallest AIC value. Comparing this value with the AIC values in Table 11, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence copula had the smallest AIC value. Comparing this value with the AIC values in Table 12, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

3.6. Results for ,

Standard normal bivariate random variables of length are generated and their CDF which lies in is used for the copula analysis. The upper tail of the CDF is analysed to check whether the hybrid model works better in right skewed observations.

The interval to 4 is considered for both of the parameters theta and omega from the hybrid model and pairs from this interval are considered as starting values for the optimization process in the maximum likelihood estimation.

Values of lambda are considered to see their individual effect on the overall hybrid model.

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence Copula had the smallest AIC value. Comparing this value with the AIC values in Table 13, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence Copula had the smallest AIC value. Comparing this value with the AIC values in Table 14, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Using the same simulated pairs from the standard normal as used in the hybrid Clayton-Frank copula, the selected bivariate copula from a number of 40 bivariate copulas was the Independence Copula with an AIC value of 0. This means that, compared to the other bivariate copulas, the Independence Copula had the smallest AIC value. Comparing this value with the AIC values in Table 15, it can be observed that the hybrid Clayton-Frank copula outperforms all the 40 bivariate copulas under study for combinations of theta and omega earlier mentioned.


Interval descriptionNLM ML estimate (mean)AIC

Omega constant, theta varied from to 1 by 0.5
Omega constant, theta varied from 4 to 1 by 0.5
Theta constant, omega varied from to 1 by 0.5
Theta constant, omega varied from 4 to 1 by 0.5

Table 16 gives the various bivariate copulas (Archimedean and elliptical) with which the proposed hybrid Clayton-Frank copula is compared.


Copula nameNumberParameter 1Parameter 2

Gaussian-
Student
Clayton (survival)-
Rotated Clayton (90 and 270 degrees)-
Gumbel (survival)-
Rotated Gumbel (90 and 270 degrees)-
Frank-
Joe (survival)-
Rotated Joe (90 and 270 degrees)-
Clayton-Gumbel (Survival)
Rotated Clayton-Gumbel (90 and 270 degrees)
Joe-Clayton (Survival)
Rotated Joe-Clayton (90 and 270 degrees)
Joe-Frank (Survival)
Rotated Joe-Frank (90 and 270 degrees)
Tawn Type 1 (Survival)