The Exact and Near-Exact Distributions for the Statistic Used to Test the Reality of Covariance Matrix in a Complex Normal Distribution

Abstract

The authors start by approximating the exact distribution of the negative logarithm of the likelihood ratio statistic, used to test the reality of the covariance matrix in a certain complex multivariate normal distribution , by an infinite mixture of Generalized Near-Integer Gamma (GNIG) distributions. Based on this representation they develop a family of near-exact distributions for the likelihood ratio statistic, which are finite mixtures of GNIG distributions and match, by construction, some of the first exact moments. Using a proximity measure based on characteristic functions the authors illustrate the excellent properties of the near-exact distributions . They are very close to the exact distribution but far more manageable and have very good asymptotic properties both for increasing sample sizes as well as for increasing number of variables. These near-exact distributions are much more accurate than the asymptotic approximation considered, namely when the sample size is small and the number of variables involved is large. Furthermore, the corresponding cumulative distribution functions allow for an easy computation of very accurate near-exact quantiles.

Appendices

Appendix A

Derivation of the Measure $\varDelta$ in (18) from the Gil–Pelaez Inversion Formula

The measure $\varDelta$ in (18) may be directly derived from the Gil–Pelaez [10] inversion formula for the c.d.f., which may be written in a number of equivalent forms, as for example

$$ F_W(w)=\frac{1}{2}-\frac{1}{2\pi }\int _{-\infty }^{+\infty }\frac{e^{-\mathrm{i}tw}\,\varPhi _W(t)}{\mathrm{i}t}\,dt\,. $$

Then, if we take $F_W(\,\cdot \,)$ and $F^+_W(\,\cdot \,)$ as the c.d.f.’s corresponding to the cf.’s $\varPhi _W(\,\cdot \,)$ and $\varPhi _W^+(\,\cdot \,)$ respectively, we have

$$ \begin{array}{rcl} \left| F_W(w)-F_W^+(w)\right| &{} = &{} \displaystyle \frac{1}{2\pi }\left| \int _{-\infty }^{+\infty }\frac{e^{-\mathrm{i}tw}}{\mathrm{i}t}\left( \varPhi _W(t)^+-\varPhi _W(t)\right) \,dt\right| \\ &{} \le &{} \displaystyle \frac{1}{2\pi }\int _{-\infty }^{+\infty }\left| \frac{e^{-\mathrm{i}tw}}{\mathrm{i}t}\left( \varPhi _W(t)^+-\varPhi _W(t)\right) \right| \,dt \end{array}$$

where, for any ${t\in \mathbb {R}}$ and any ${w\in \mathbb {R}}$,

$$ \left| \frac{e^{-\mathrm{i}tw}}{\mathrm{i}}\right| =1\,, $$

so that we may write

$$ \sup _w \left| F_W(w)-F_W^+(w)\right| \le \frac{1}{2\pi }\int _{-\infty }^{+\infty }\left| \frac{\varPhi _W(t)-\varPhi ^+_W(t)}{t}\right| dt\,. $$

The measure $\varDelta$ gives thus very sharp upper-bounds on the difference between the c.d.f.’s $F_W(\,\cdot \,)$ and $F_W^+(\,\cdot \,)$, indeed much sharper than any similar measure that would be based on the more common inversion formula for the c.d.f..

The measure $\varDelta$ clearly verifies the triangular inequality since if we take

$$ \varDelta _1=\frac{1}{2\pi }\int _{-\infty }^{+\infty }\left| \frac{\varPhi _1(t)-\varPhi _2(t)}{t}\right| dt\,,~~~~~~~~\varDelta _2=\frac{1}{2\pi }\int _{-\infty }^{+\infty }\left| \frac{\varPhi _1(t)-\varPhi _3(t)}{t}\right| dt $$

and

$$ \varDelta _3=\frac{1}{2\pi }\int _{-\infty }^{+\infty }\left| \frac{\varPhi _2(t)-\varPhi _3(t)}{t}\right| dt $$

we have ${\varDelta _1\le \varDelta _2+\varDelta _3}$ since

$$ \begin{array}{rcl} \displaystyle \left| \varPhi _1(t)-\varPhi _2(t)\right| &{} = &{} \displaystyle \left| \varPhi _1(t)-\varPhi _3(t)+\varPhi _3(t)-\varPhi _2(t)\right| \\ &{} \le &{} \displaystyle \left| \varPhi _1(t)-\varPhi _3(t)\right| +\left| \varPhi _3(t)-\varPhi _2(t)\right| \end{array}$$

and, in a similar manner, also $\varDelta _2\le \varDelta _1+\varDelta _3$ and $\varDelta _3\le \varDelta _1+\varDelta _2$.

Appendix B

Median, 0.95 and 0.99 Quantiles for the Statistic $\varLambda$

Table B.1 Median (0.5 quantile) for $\varLambda$ for ${p=6}$ ${(n=6,7,8,9,10)}$, ${p=35}$ ${(n=36,70,105)}$ and ${p=55}$ ${(n=56,110,165)}$, for the near-exact distributions that match $m^*=2,4,6$ and 10 exact moments

Table B.2 0.95 quantiles for $\varLambda$ for ${p=6}$ ${(n=6,7,8,9,10)}$, ${p=35}$ ${(n=36,70,105)}$ and ${p=55}$ ${(n=56,110,165)}$, for the near-exact distributions that match $m^*=2,4,6$ and 10 exact moments

Table B.3 0.99 quantiles for $\varLambda$ for ${p=6}$ ${(n=6,7,8,9,10)}$, ${p=35}$ ${(n=36,70,105)}$ and ${p=55}$ ${(n=56,110,165)}$, for the near-exact distributions that match $m^*=2,4,6$ and 10 exact moments