Some Inference Problems Associated with the Complex Multivariate Normal Distribution

Abstract

(Security clausificeaton of title, body of abxttart and lndexing annc ratJon rnu.t be entered when the ovetall report is clahssfied) I. OnICINATING "S4. CONTRACT OR GRANT NO. Se. ORIGINATOR'S REPORT NUMBERIS)N00014-68-A-0515b. PROJECT NO. NR 042-260 C.ob. OTHER REPORT NO($) (Any other numbers Mtat may be alssiged this report) d. DISTRIDUTION ITATECENTThis document has been approved for public release and sale; its distribution is unlimited. Reproduction in whole or in part is permitted for any purpose of the United States Government. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITYOffice of Naval Research ASXIACTConsider k = q+l complex multivariate normal populations with the same variancecovariance matrix but with different means.The problem of the number of discriminant functions needed to discriminate among the k-populations is considered and shown to be equivalent to the rank of the mean-space. A test for this dimensionality being r is presented. Also developed is the statistic for testing the goodness of fit of a single hypothetical discriminant function. As in the case of the real normal populations, the test statistic for this single hypothetical function is presented as the product of two independent tactors, whose distributions are given; one measuring the direction aspect of the hypothetical function, and the other measures the collinearity aspect that is the necessity of only one function.Also included is the Bartlett decompositicn of a complex Wishart matrix and some results pertaining to the coherence of complex random variables.

On Complex Random Variables

“…Brillinger (1968), Goodman and Dubman (1968), and Young (1971) gave some practical uses of the complex analogue. Other authors, for example, Capon and Goodman (1970), Giri (1965), Kabe (1966a, b), Khatri (1964Khatri ( , 1965, Saxena (1965), Srivastava (1963), Tan (1969), and Young (1971), have discussed the complex analogues of the classical real variable multivariate analysis. For some recent references on complex random variables, see Van den Bos (1995, 1998, Dryden and Mardia (1998), Olhede (2006) and Eriksson et al (2009) …”