A note on the continuity of projection matrices with application to the asymptotic distribution of quadratic forms

G. Forchini
2005 Applicationes Mathematicae  
This paper investigates the continuity of projection matrices and illustrates an important application of this property to the derivation of the asymptotic distribution of quadratic forms. We give a new proof and an extension of a result of Stewart (1977) . An important result in statistics concerning the distribution of quadratic forms is the following: if X is a k × 1 vector having a multivariate normal distribution with mean vector µ and identity covariance matrix, and if P B is an
more » ... P B is an idempotent matrix of rank p then X P B X is χ 2 (p, δ), where δ = µ P B µ (see for instance Muirhead (1982) , Theorem 1.4.5). Sometimes we are interested in an asymptotic version of this result (examples are given below): (i) the k × 1 random vector X T indexed by T converges in distribution to a multivariate normal random variable with unknown mean µ and identity covariance matrix I k ; (ii) the k × n (n ≤ k) matrix B T converges in probability to the k × n matrix B (and we write plim T →∞ B T = B). In this application B T has rank n with probability 1 for all but a finite number of T . We want to find the asymptotic distribution of X T P B T X T as T → ∞. If the mapping B T → P B T is continuous we can conclude that X T P B T X T has an asymptotically noncentral chi-square distribution with n degrees of freedom and noncentrality parameter µ P B µ (Muirhead (1982), Theorem 1.4.5). However, if n > p = rank(B) for all but a finite number of T , then X T P B T X T often has an asymptotically noncentral chi-square distribution with (again) n degrees of freedom (rather than p) and noncentrality parameter µ (plim T →∞ P B T )µ. Two examples where such situation arises are given below. 2000 Mathematics Subject Classification: 54C05, 15A03, 62E20.
doi:10.4064/am32-1-4 fatcat:kg5yfp6f2nhzhlbb7mjdhelccm