Quadratic form (statistics)

Quadratic form (statistics)

If epsilon is a vector of n random variables, and Lambda is an n-dimensional symmetric square matrix, then the scalar quantity epsilon'Lambdaepsilon is known as a quadratic form in epsilon.

Expectation

It can be shown that

:operatorname{E}left [epsilon'Lambdaepsilon ight] =operatorname{tr}left [Lambda Sigma ight] + mu'Lambdamu

where mu and Sigma are the expected value and variance-covariance matrix of epsilon, respectively, and tr denotes the trace of a matrix. This result only depends on the existence of mu and Sigma; in particular, normality of epsilon is not required.

Variance

In general, the variance of a quadratic form depends greatly on the distribution of epsilon. However, if epsilon does follow a multivariate normal distribution, the variance of the quadratic form becomes particularly tractable. Assume for the moment that Lambda is a symmetric matrix. Then,

:operatorname{var}left [epsilon'Lambdaepsilon ight] =2operatorname{tr}left [Lambda SigmaLambda Sigma ight] + 4mu'LambdaSigmaLambdamu

In fact, this can be generalized to find the covariance between two quadratic forms on the same epsilon (once again, Lambda_1 and Lambda_2 must both be symmetric):

:operatorname{cov}left [epsilon'Lambda_1epsilon,epsilon'Lambda_2epsilon ight] =2operatorname{tr}left [Lambda _1SigmaLambda_2 Sigma ight] + 4mu'Lambda_1SigmaLambda_2mu

Computing the variance in the non-symmetric case

Some texts incorrectly state the above variance or covariance results without enforcing Lambda to be symmetric. The case for general Lambda can be derived by noting that

:epsilon'Lambda'epsilon=epsilon'Lambdaepsilon

so

:epsilon'Lambdaepsilon=epsilon'left(Lambda+Lambda' ight)epsilon/2

But this is a quadratic form in the symmetric matrix ilde{Lambda}=left(Lambda+Lambda' ight)/2, so the mean and variance expressions are the same, provided Lambda is replaced by ilde{Lambda} therein.

Examples of quadratic forms

In the setting where one has a set of observations y and an operator matrix H, then the residual sum of squares can be written as a quadratic form in y:

: extrm{RSS}=y'left(I-H ight)'left(I-H ight)y

For procedures where the matrix H is symmetric and idempotent, and the errors are Gaussian with covariance matrix sigma^2I, extrm{RSS}/sigma^2 has a chi-square distribution with k degrees of freedom and noncentrality parameter lambda, where

:k=operatorname{tr}left [left(I-H ight)'left(I-H ight) ight] :lambda=mu'left(I-H ight)'left(I-H ight)mu/2

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 Hy estimates mu with no bias, then the noncentrality lambda is zero and extrm{RSS}/sigma^2 follows a central chi-square distribution.

ee also

*Quadratic form


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