An entropy divergence, D-distribution and its selected application in multivariate financial time series

Abstract

This paper proposes an Entropy Divergence (ED) which is the difference between the two Shannon's differential entropies for multivariate discrete and continuous distributions. This is illustrated based on the multivariate version of the central limit theorem to evaluate the divergence between the multivariate scaled normal distribution (f) and scaled student's t distribution (g). This helps to firmly establish the exact distribution of ED, and its density is visualised in terms of Meijer G function. Moreover, the authors derived the distribution of the difference (D) between two EDs in terms of the sum of p-variate independent Fisher's-Z variable plus a normalising constant. Similarly, the moments of the two cases are derived which are visualised through digamma and poly-gamma functions. The percentage points of ED distribution of p-variates are computed at 5% and 1% significance level for respective degrees of freedom. Plots show the percentage points (discrepancy distance) between the multivariate distribution (normal) of financial time series and the distribution of cross-section data (student's t) graphically. The proposed ED is applied to test the sphericity of Nifty 50 stock returns in National Stock Exchange. Similarly, D-statistic is used to check the equality of two Shannon's differential entropy, where returns are grouped based on the demonetisation crisis as before and after in India.