- Quadratic form (statistics)
If is a vector of
random variables, and is an -dimensional symmetric square matrix, then the scalarquantity is known as a quadratic form in .
It can be shown that
where and are the
expected valueand variance-covariance matrix of , respectively, and tr denotes the trace of a matrix. This result only depends on the existence of and ; in particular, normality of is not required.
In general, the variance of a quadratic form depends greatly on the distribution of . However, if does follow a multivariate normal distribution, the variance of the quadratic form becomes particularly tractable. Assume for the moment that is a symmetric matrix. Then,
In fact, this can be generalized to find the
covariancebetween two quadratic forms on the same (once again, and must both be symmetric):
Computing the variance in the non-symmetric case
Some texts incorrectly state the above variance or covariance results without enforcing to be symmetric. The case for general can be derived by noting that
But this is a quadratic form in the symmetric matrix , so the mean and variance expressions are the same, provided is replaced by therein.
Examples of quadratic forms
In the setting where one has a set of observations and an
operator matrix, then the residual sum of squarescan be written as a quadratic form in :
For procedures where the matrix is symmetric and idempotent, and the errors are Gaussian with covariance matrix , has a
chi-square distributionwith degrees of freedom and noncentrality parameter , where
may be found by matching the first two
central moments of a noncentral chi-square random variable to the expressions given in the first two sections. If estimates with no bias, then the noncentrality is zero and follows a central chi-square distribution.
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