I have two non-parametric rank correlations matrices
sim (for example, based on Spearman’s $rho$ rank correlation coefficient):
emp <- matrix(c( 1.0000000, 0.7771328, 0.6800540, 0.2741636, 0.7771328, 1.0000000, 0.5818167, 0.2933432, 0.6800540, 0.5818167, 1.0000000, 0.3432396, 0.2741636, 0.2933432, 0.3432396, 1.0000000), ncol=4) sim <- matrix(c( 1.0000000, 0.7616454, 0.6545774, 0.3081403, 0.7616454, 1.0000000, 0.5360392, 0.3146167, 0.6545774, 0.5360392, 1.0000000, 0.3739758, 0.3081403, 0.3146167, 0.3739758, 1.0000000), ncol=4)
emp matrix is the correlation matrix that contains correlations between the emprical values (time series), the
sim matrix is the correlation matrix — the simulated values.
I need to test the null hypothesis $H_0$: matrices
sim are drawn from the same distribution.
Question. What is a test do I can use? Is is possible to use the Wishart statistic?
Follow to Stephan Kolassa‘s comment I have done a simulation.
I have tried to compare two Spearman correlations matrices
sim with the Box’s M test. The test has returned
# Chi-squared statistic = 2.6163, p-value = 0.9891
Then I have simulated 1000 times the correlations matrix
sim and plot the distribution of Chi-squared statistic $M(1-c)simchi^2(df)$.
After that I have defined the 5-% quantile of Chi-squared statistic $M(1-c)simchi^2(df)$. The defined 5-% quantile equals to
quantile(dfr$stat, probs = 0.05) # 5% # 1.505046
One can see that the 5-% quantile is less that the obtained Chi-squared statistic:
1.505046 < 2.6163 (blue line on the fugure), therefore, my
emp‘s statistic $M(1−c)$ does not fall in the left tail of the $(M(1−c))_i$.
Follow to the second Stephan Kolassa‘s comment I have calculated 95-% quantile of Chi-squared statistic $M(1-c)simchi^2(df)$ (blue line on the fugure). The defined 95-% quantile equals to
quantile(dfr$stat, probs = 0.95) # 95% # 7.362071
One can see that the
emp‘s statistic $M(1−c)$ does not fall in the right tail of the $(M(1−c))_i$.
Edit 3. I have calculated the exact $p$-value (green line on the figure) through the empirical cumulative distribution function:
ecdf(dfr$stat)(2.6163)  0.239
One can see that $p$-value=0.239 is greater than $0.05$.
Dominik Wied (2014): A Nonparametric Test for a Constant Correlation
Matrix, Econometric Reviews, DOI: 10.1080/07474938.2014.998152
Joël Bun, Jean-Philippe Bouchaud and Mark Potters (2016), Cleaning correlation matrices, Risk.net, April 2016
Li, David X., On Default Correlation: A Copula Function Approach (September 1999). Available at SSRN: https://ssrn.com/abstract=187289 or http://dx.doi.org/10.2139/ssrn.187289
G. E. P. Box, A General Distribution Theory for a Class of Likelihood Criteria. Biometrika. Vol. 36, No. 3/4 (Dec., 1949), pp. 317-346
M. S. Bartlett, Properties of Sufficiency and Statistical Tests. Proc. R. Soc. Lond. A 1937 160, 268-282
Robert I. Jennrich (1970): An Asymptotic χ2 Test for the Equality of Two
Correlation Matrices, Journal of the American Statistical Association, 65:330, 904-912.
The first founded paper that has no the assumption about normal distribution.
Reza Modarres & Robert W. Jernigan (1993) A robust test for comparing correlation matrices, Journal of Statistical Computation and Simulation, 46:3-4, 169-181