A novel two-sample test within the space of symmetric positive definite matrix distributions and its application in finance

Abstract

This paper introduces a novel two-sample test for a broad class of orthogonally invariant positive definite symmetric matrix distributions. Our test is the first of its kind, and we derive its asymptotic distribution. To estimate the test power, we use a warp-speed bootstrap method and consider the most common matrix distributions. We provide several real data examples, including the data for main cryptocurrencies and stock data of major US companies. The real data examples demonstrate the applicability of our test in the context closely related to algorithmic trading. The popularity of matrix distributions in many applications and the need for such a test in the literature are reconciled by our findings.

Appendix

In this section, we aim to examine the behaviour of our test in a well-known theoretical scenario. Note that if $x_1, x_2, \dots , x_n \in N_p(0, \Sigma )$, then $X=(x_1, x_2, \dots , x_n)$ follows a matrix-variate normal distribution. Additionally, if $n \ge p$, then $X'X>0$ with probability 1 and $X'X \in W_p(n, \Sigma )$ (Gupta and Nagar 1999, p. 88). We have determined the test powers of the Wishart $\frac{1}{499}W_d(500, I_d)$ distribution against the $CMT_d(df, I_d)$, where we computed the sample covariance matrices on $NCov=500$ random vectors. The parameter df is selected from the set $\{1, 21, 41, \dots , 501\}$. The results for $d=2$ and $d=3$ are presented in Figs. 3 and 4. The test powers are expressed in percentage. The behaviour of our tests is as expected, since as the degrees of freedom increase, the t-distribution becomes closer to the normal distribution. Consequently, the distribution of its covariance matrix becomes closer to the appropriately scaled Wishart distribution, leading to a decrease in test powers and reaching test size for large enough degrees of freedom.

Fig. 3

The case of $2\times 2$ matrices

Fig. 4

The case of $3\times 3$ matrices