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 ~ Nm(μ,Σ) with Σ positive definite.11 Define Y as a quadratic polynomial of X:
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
and covariance matrix
Accordingly, ~ Nm(0,I). Applying our change of variables [3.148]:
so Y has form
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
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
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
where the Qk are chi-squared with one degree of freedom, noncentrality parameters are obtainable from [3.165], and Q0 is standard normal. The constants γk, β, and α can be calculated directly from the terms , , and .