3.13  Quadratic Polynomials of Joint-Normal Random Vectors

Formulas [3.27] and [3.28] provide general expressions for the mean and variance of a linear polynomial of a random vector. What about the mean and variance of a quadratic polynomial of a random vector? Unfortunately, no general formulas exist. However, if we restrict our attention to quadratic polynomials of a joint-normal random vector, there are expressions for the mean, variance, and all moments. To obtain these, we generalize an earlier result regarding chi-squared distributions.

3.13.1 Simplified Representation

In Section 3.10.3, we saw that the specific form [3.112] of quadratic polynomial of a joint standard normal random vector has a chi-squared distribution. Generalizing this, we shall demonstrate that any quadratic polynomial of any joint-normal random vector can be expressed as a linear polynomial of independent chi-squared and normal random variables. Specifically, let X ~ N_m(μ,Σ) with Σ positive definite.11 Define Y as a quadratic polynomial of X:

[3.147]

Let z be the Cholesky matrix of Σ, and define u as a square matrix whose rows comprise orthonormal eigenvectors of . By construction, u is orthogonal: u^–1 = u′. Define the change of variables

[3.148]

Then, by [3.30] and [3.31], is joint-normal with mean vector

[3.149]

[3.150]

[3.151]

[3.152]

and covariance matrix 

[3.153]

[3.154]

[3.155]

[3.156]

Accordingly,  ~ N_m(0,I). Applying our change of variables [3.148]:

[3.157]

[3.158]

[3.159]

so Y has form

[3.160]

where

[3.161]

[3.162]

[3.163]

Recall that we defined u as a matrix whose rows comprise orthonormal eigenvectors of . This means, by the spectral theorem of linear algebra, that the matrix  is diagonal with diagonal elements equal to the eigenvalues of Consequently, Y depends upon no cross terms of the form . We can write [3.159] as

[3.164]

and conclude that Y depends upon each of the variables  in one of four ways:

• No dependence:  = 0 and  = 0.
• Linear dependence: = 0 and ≠ 0, so Y depends upon a term .
• Central quadratic dependence: ≠ 0 and  = 0, so Y depends upon a term .
• Noncentral quadratic dependence: ≠ 0 and ≠ 0, so Y depends upon a term  + .

In the last case, “completing the squares” results in a dependence of the form

[3.165]

Y is a linear polynomial of independent random variables, each of which is standard normal, central chi-squared with one degree of freedom, or noncentral chi-squared with one degree of freedom and non centrality parameter .

Since a linear polynomial of independent normal random variables is itself normal, all normal terms can be combined into one. A general expression for Y is

[3.166]

where the Q_k are chi-squared with one degree of freedom, noncentrality parameters are obtainable from [3.165], and Q₀ is standard normal. The constants γ_k, β, and α can be calculated directly from the terms , , and .