Testing for positive quadrant dependence

This repository contains R programs for the article, “Testing for positive quadrant dependence.” This article has been submitted for publication. Prior to using R programs on this repository, please download the main R program EL_PQD_Library.R or source it in R using the command source("https://raw.githubusercontent.com/cftang9/PQD/master/EL_PQD_Library.R") which requires installing R packages Rcpp and copula. We would like to point out that loading or executing functions in Rcpp packages may encounter some technical problems for Windows users if your R software was recently updated to the latest version. One may run these codes in Rstudio and follow what it suggests to solve the problem. After successfully loading the main R program, the function IndvsPQD will automate critical value calculations for the practitioner. To better understand the use of our R program, we start with an illustrative example.

Part 1: Illustration

1.1 A simple example

Below generates a random sample of size 10 from a Clayton copula, with a user-specified Kendall's tau, to test for independence versus positive quadrant dependence (PQD).

# Source the main R program
source("https://raw.githubusercontent.com/cftang9/PQD/master/EL_PQD_Library.R")
# Set the sample size n and the Kendall's tau
n = 10; tau = 0.2
# Generate a sample of size n
# For illustration, we set the seed to be 100
set.seed(100)
Sample = RV_CopTau(n, tau, Copula="Clayton")
# Name the sample by X and Y
X=Sample[,1];Y=Sample[,2]
# Run the test
IndvsPQD(X,Y,graph=TRUE)

A scatter plot and a plot of the corresponding pseudo-observations between X and Y will be produced.

Our proposed empirical-likelihood-based test (EL) and three distance-based tests (KS, CvM, and AD) for PQD along with the Kendall and Spearman rank tests will be performed. Results include the value of each test statistic, the corresponding p-value, reject independence (1) or not (0), and the critical value at significance level 0.05:

         test statistic p-value reject independence critical value
EL           0.39887816  0.5200                   0      1.4329523
KS           0.31884122  0.8956                   0      0.6664304
CvM          0.03267605  0.8528                   0      0.1961564
AD           1.98834920  0.7074                   0      7.8084519
spearman    -0.17575758  0.6902                   0      0.5515152
kendall     -0.20000000  0.7611                   0      0.4222222

The argument Copula="Calyton" in the function RV_CopTau above can be changed to Copula="Frank" and Copula="Gumbel" to generate a random sample from the Frank and Gumbel copulas, respectively. The Gaussian copula can also be considered. See these details in IllustrativeExamples.R.

For a quick illustration, we set n=10 above. Other sample sizes can be considered as well. However, When the sample size is large, it will take a longer time to run.

1.2 For your own data

Please use these R commands after naming the data by X and Y:

source("https://raw.githubusercontent.com/cftang9/PQD/master/EL_PQD_Library.R")
# name your data by X and Y
IndvsPQD(X,Y,graph=TRUE)

Part 2: To reproduce the simulation results

2.1 Table 1 in Section 3 of the manuscript

To reproduce Table 1, which involves four classic copulas: Clayton, Frank, Gumbel, and Gaussian, please run this R program: Clayton_Frank_Gumbel_and_Gaussian_n=100.R. But be aware of that, because the number of replications is 10,000, this program might take a long time to finish. As stated in our manuscript, our calculation of Table 1 took approximately 73 minutes on a computer with a 3.1GHz processor and 16GB of memory.

2.2 Tables C.1 and C.2 in Web Appendix C

Table 1 considers n=100. We also included the same table but with n=50 and 200 in Web Appendix C. To reproduce those two tables. Please run Clayton_Frank_Gumbel_and_Gaussian_n=50.R and Clayton_Frank_Gumbel_and_Gaussian_n=200.R, respectively.

2.3 Tables C.3-C.5 in Web Appendix C

In addition to the Clayton, Frank, Gaussian, and Gumbel copulas, we have also considered the FGM and CA copulas and a restricted bivariate t distribution family. The results are presented in Tables C.3-C.5 in Web Appendix C. To reproduce these tables, please run Restricted_t_FGM_and_CA n=50.R, Restricted_t_FGM_and_CA n=100.R, and Restricted_t_FGM_and_CA_n=200.R.

Part 3: To reproduce the real data analysis results in Section 4 of the manuscript

We applied all tests in this manuscript to three data applications. To reproduce the results of our analysis (Table 2 and Figures 2-4), please run the R program for each. The data included in the CSV file will be automatically read by the corresponding R program.

3.1 Twins Data

Data: TwinsData.csv (R program: TwinsData.R)

3.2 Education data

Data: EducationData.csv (R program: EducationData.R)

3.3 Stock Data

Data: StockData.csv (R program: StockData.R)