The limiting eigenvalue distribution of a multivariate F matrix. (English) Zbl 0606.62054

Silverstein, Jack W.

SIAM J. Math. Anal. 16, 641-646 (1985).

Let \(X_{ij}\), \(Y_{ij}\), \(i,j=1,2,..\). be i.i.d. N(0,1) random variables and for positive integers p,m,n, let \(\bar X{}_ p=(X_{ij})\), \(i=1,2,...,p\); \(j=1,2,...,m\), and \(\bar Y{}_ p=(Y_{ij})\), \(i=1,2,...,p\); \(j=1,2,...,n\). Suppose further that \(p/m\to y>0\) and p/n\(\to y'\in (0,)\), as \(p\to \infty\). Y. Q. Yin and P. R. Krishnaiah [J. Multivariate Anal. 13, 489-507 (1983; Zbl 0553.62018)] and Y. Q. Yin, Z. D. Bai and P. R. Krishnaiah [ibid. 13, 508-516 (1983; Zbl 0531.62018)] showed that the empirical distribution function of the eigenvalues of \[ (m^{-1}\bar X_ p\bar X^ T_ p)(n^{-1}\bar Y_ p\bar Y^ T_ p)^{-1} \] converges i.p. as \(p\to \infty\) to a nonrandom d.f. In the present paper the limiting d.f. is derived.

MSC:

62H10

Multivariate distribution of statistics

Keywords:

limiting eigenvalue distribution; multivariate F matrix; empirical distribution function; eigenvalues