Quadratic form (statistics)

In multivariate statistics, if is a vector of random variables, and is an -dimensional symmetric matrix, then the scalar quantity is known as a quadratic form in . It can be shown that where and are the expected value and 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. A book treatment of the topic of quadratic forms in random variables is that of Mathai and Provost. Since the quadratic form is a scalar quantity, . Next, by the cyclic property of the trace operator, Since the trace operator is a linear combination of the components of the matrix, it therefore follows from the linearity of the expectation operator that A standard property of variances then tells us that this is Applying the cyclic property of the trace operator again, we get 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 covariance between two quadratic forms on the same (once again, and must both be symmetric): In addition, a quadratic form such as this follows a generalized chi-squared distribution. Some texts incorrectly state that the above variance or covariance results hold without requiring to be symmetric. The case for general can be derived by noting that so is a quadratic form in the symmetric matrix , so the mean and variance expressions are the same, provided is replaced by therein. In the setting where one has a set of observations and an operator matrix , then the residual sum of squares can 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-squared distribution with degrees of freedom and noncentrality parameter , where may be found by matching the first two central moments of a noncentral chi-squared random variable to the expressions given in the first two sections.