# Linear Models

• A. K. Gupta
• T. Varga
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 240)

## Abstract

Let x1, x2,...,xn be p-dimensional vectors, such that xi ~ Ep(Bzi, Σ, ψ), where zi is a q-dimensional known vector, i = 1,...,n, and B is a p × q unknown matrix. Moreover, assume that xi, i = l,...,n are uncorrelated and their joint distribution is elliptically contoured and absolutely continuous. This model can be expressed as
$${\text{X}} \sim {\text{E}}_{{\text{p,n}}} {\text{(BZ,}}\sum { \otimes {\text{I}}_{{\text{n,}}} \psi {\text{),}}}$$
(9.1)
where X = (x1, x2,...,xn); Z = (z1, z2,...,zn) is a q × n known matrix; B (p × q) and Σ (p × p) are unknown matrices. Assume rk(Z) = q and p + q ≤ n. The joint p.d.f. of x1, x2,...,xn can be written as
$$f(X) = \frac{1} {{\left| {\sum {\left| n \right.} } \right.}}h\left( {\sum\limits_{i = 1}^n {(x_i - Bz_i )'\Sigma ^{ - 1} (x_i - Bz_i )} } \right) = \frac{1} {{\left| {\sum {\left| n \right.} } \right.}}h(tr(X - BZ)'\Sigma ^{ - 1} (X - BZ)).$$
(9.2)