A Comparison of Copulas Based on the Range Distribution and Its Application

This paper compares some Archimedean Copulas based on the range distribution and its application. We show that the Clayton Copula is better than the other Copula models considered in this paper. Also, Clayton Copula performed better than the regular one-way ANOVA when there were no outliers. However, when the data had outliers, the Clayton Copula performed almost at the same level (power-wise) as the regular one-way ANOVA.


Introduction
Here, we compare the performance of some Archimedean Copulas such as the Clayton Copula, Gumbel Copula, and Frank Copula in the context of the range distribution and its application.First, we use the mean and median of the range distribution for the comparison.Here, we study the effect of the sample size on the range distribution.Next, we construct a range based simultaneous test based on these Copulas in order to compare the level means of a single factor.This approach can serve as an alternative to the one-way ANOVA when the number of observations at each level are equal.As per the literature, the ANOVA test was formulated by Sir Ronald Fisher in 1918 in the context of comparing farm yields.The ANOVA test decomposes the variation among the observations into some factor variations that are mainly due to the effect of orthogonal components.In other words, this is the best application of the Pythagoras Theorem.In 1952, William Kruskal and Alan Wallis developed the nonparametric equivalent of ANOVA.But, the main issue with the nonparametric tests are that these tests are less powerful compared to the parametric tests.In this paper, we present an alternative test for simultaneous testing of the means as an application.The test that we propose uses the copulas and the range to conduct the simultaneous testing.We show that this test is more powerful than the ANOVA test.
The use of Copulas in statistical analysis became popular after Sklar's Theorem in 1959.All these different types of Archimedean Copula models were developed as a result of this theorem.Clayton, Frank, and Gumbel developed their own versions of the Archimedean Copulas.The Copulas have wide applications in Economics, Finance, Engineering, and Education.As for the research done within the last seven years, Kahadawala (2018) looked at the multivariate dependence using Clayton Copula.Hofert et al (2018) used R software to model the dependence structure by using Copulas.Munhammar and Widen (2017) used the Copula to model an autocorrelated time series.Young et al (2021) used Copula to model the quality control limits for an auto-correlated AR (1) time series.We refer the interested readers to Roger Nelsen (2006) for the literature review about the Copulas.This paper combines the concept of Copulas and Order Statistics.Order Statistics has a lot of applications.For example, reliability assessment and survival distributions involve order statistics especially in situations where there is censoring.Recently, Das and Bhattacharya (2020) looked at the application of lognormal distribution to represent the first-order statistics of wireless channels.The interested readers are referred to David and Nagaraj (2003) for the literature review about the Order Statistics.
We divide this paper as follows.We present the methodology in section 2, the numerical results in section 3, application in section 4, and conclusion and discussion in section 5.

Fx.
In other words, if X is a random observation then     F x P X x   .Moreover, let the order-statistics be In this case, the marginal distributions of the order statistics are as follows, Let the range, The density function for the range in the case of standard uniform distribution is as follows, Remark: The statistical properties are listed below.
Case 2: Exponential case where Case 3: General case where the population distribution is Gumbel Copula Density Function: Frank Copula Density Function: Remark: We will compare the performance of these Copulas based on the mean and median of the Range distribution for different sample sizes.

Numerical Study
Here, we discuss the numerical results.We study the effect of the underlying population distribution for different Copulas such as Clayton, Gumbel, and Frank based on different sample sizes.This is in the context of the median and mean of the Range distribution for the samples selected at random from the underlying distributions discussed in this section.

Numerical Results (for Range Median):
- Note: It appears based on the above numerical results, there is not much of a difference between these Archimedean Copulas.
Next, we will focus on an application.

Applications
Consider a one-way ANOVA situation where the response Where  is the overall average response, i  is the level i effect, and ij  is the error effect.Moreover, level i average response ii   

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The Copula based construction of the range distribution is very useful in the context of ANOVA type simultaneous testing of the means.We can use this method in place of ANOVA when the sample sizes are equal (balanced design).
Let us suppose that there are k number of levels.
Level 1: Remark 1: For the above data, the Levene's test for the equality of variances yielded a P-Value = 0.226 suggesting that the variability is the same for all the acids involved in this study.
Moreover, since the sample size is 48 (reasonably large), we can treat the sample means as following the normal distribution.
Note that , Under the assumption that the null hypothesis is true, we will evaluate the probability

Discussion and Conclusion
This paper presents a Copula approach based on the range distribution to see which Copula is better.In this context, the mean and the median were used.The mean and median were evaluated numerically for the Archimedean Copulas; Clayton, Gumbel, and Frank at different sample sizes.As for the mean and median of the range distribution, the Archimedean Copulas such as Clayton, Gumbel, and Frank were about the same regardless of the sample size.However in the context of the simultaneous comparison of the means, Clayton Copula seemed to perform better than Gumbel Copula and Frank Copula.This is evident from the power calculation presented above.Moreover, Clayton and Gumbel Copulas seem to possess higher power compared to the regular one-way ANOVA.This pattern was observed when there were no outliers in the data.The power computation shows that Clayton Copula is the best of the three Archimedean Copulas presented in this paper.However, in the context of outliers and negative association, the regular one-way ANOVA and the test based on the Clayton Copula performed at the same level when it comes to power.Furthermore out of the three Archimedean Copulas studied, Clayton Copula slightly performed better than Gumbel ExampleIn an analytical chemistry experiment, the finger prints measurements based on five different acids (Alanine.Glycine2, leucine, Serine, Pro-line) was obtained from 50 different individuals over a five week period.Two data points were deleted due to incomplete entries.The objective here is to compare the means obtained from the finger prints.
For the same data, regular one-way ANOVA analysis gave the following result.
Note: All the tests indicate that the means are equal at 5% significance level.Next, we compare the copula power based on a simulation study.