A note on positive semi-definiteness of some non-pearsonian correlation matrices

Mishra, SK (2009): A note on positive semi-definiteness of some non-pearsonian correlation matrices.

Abstract

The Pearsonian coefficient of correlation as a measure of association between two variates is highly prone to the deleterious effects of outlier observations (in data). Statisticians have proposed a number of formulas to obtain robust measures of correlation that are considered to be less affected by errors of observation, perturbation or presence of outliers. Spearman’s rho, Blomqvist’s signum, Bradley’s absolute r and Shevlyakov’s median correlation are some of such robust measures of correlation. However, in many applications, correlation matrices that satisfy the criterion of positive semi-definiteness are required. Our investigation finds that while Spearman’s rho, Blomqvist’s signum and Bradley’s absolute r make positive semi-definite correlation matrices, Shevlyakov’s median correlation very often fails to do that. The use of correlation matrices based on Shevlyakov’s formula, therefore, is problematic.

• Item Type: MPRA Paper
• Language: English
• Keywords: Robust correlation; outliers; Spearman’s rho; Blomqvist’s signum; Bradley’s absolute correlation; Shevlyakov’s median correlation; positive semi-definite matrix; fortran 77; computer program
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling

References

Blomqvist, N. (1950) "On a Measure of Dependence between Two Random Variables", Annals of Mathematical Statistics, 21(4): 593-600. Bradley, C. (1985) “The Absolute Correlation”, The Mathematical Gazette, 69(447): 12-17.

Hampel, F. R., Ronchetti, E.M., Rousseeuw, P.J. and W. A. Stahel, W.A. (1986) Robust Statistics: The Approach Based on Influence Functions, Wiley, New York.

Mishra, S.K. (2008) “The Nearest Correlation Matrix Problem: Solution by Differential Evolution Method of Global Optimization”, Journal of Quantitative Economics, New Series, 6(1&2): 240-262.

Shevlyakov, G.L. (1997) “On Robust Estimation of a Correlation Coefficient”, Journal of Mathematical Sciences, 83(3): 434-438.

Spearman, C. (1904) "The Proof and Measurement of Association between Two Things", American Journal of Psychology, 15: 88-93.