Quadratic form (statistics)

Expectation

It can be shown that

:\operatorname{E}\left[\varepsilon^T\Lambda\varepsilon\right]=\operatorname{tr}\left[\Lambda \Sigma\right] + \mu^T\Lambda\mu

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

A book treatment of the topic of quadratic forms in random variables is that of Mathai and Provost.

Proof

Since the quadratic form is a scalar quantity, \varepsilon^T\Lambda\varepsilon = \operatorname{tr}(\varepsilon^T\Lambda\varepsilon).

Next, by the cyclic property of the trace operator,

: \operatorname{E}[\operatorname{tr}(\varepsilon^T\Lambda\varepsilon)] = \operatorname{E}[\operatorname{tr}(\Lambda\varepsilon\varepsilon^T)].

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

: \operatorname{E}[\operatorname{tr}(\Lambda\varepsilon\varepsilon^T)] = \operatorname{tr}(\Lambda \operatorname{E}(\varepsilon\varepsilon^T)).

A standard property of variances then tells us that this is

: \operatorname{tr}(\Lambda (\Sigma + \mu \mu^T)).

Applying the cyclic property of the trace operator again, we get

: \operatorname{tr}(\Lambda\Sigma) + \operatorname{tr}(\Lambda \mu \mu^T) = \operatorname{tr}(\Lambda\Sigma) + \operatorname{tr}(\mu^T\Lambda\mu) = \operatorname{tr}(\Lambda\Sigma) + \mu^T\Lambda\mu.

Variance in the Gaussian case

In general, the variance of a quadratic form depends greatly on the distribution of \varepsilon. However, if \varepsilon 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[\varepsilon^T\Lambda\varepsilon\right] = 2\operatorname{tr}\left[\Lambda \Sigma\Lambda \Sigma\right] + 4\mu^T\Lambda\Sigma\Lambda\mu.

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

:\operatorname{cov}\left[\varepsilon^T\Lambda_1\varepsilon,\varepsilon^T\Lambda_2\varepsilon\right]=2\operatorname{tr}\left[\Lambda _1\Sigma\Lambda_2 \Sigma\right] + 4\mu^T\Lambda_1\Sigma\Lambda_2\mu.

In addition, a quadratic form such as this follows a generalized chi-squared distribution.

Computing the variance in the non-symmetric case

The case for general \Lambda can be derived by noting that

:\varepsilon^T\Lambda^T\varepsilon=\varepsilon^T\Lambda\varepsilon

so

:\varepsilon^T\tilde{\Lambda}\varepsilon=\varepsilon^T\left(\Lambda+\Lambda^T\right)\varepsilon/2

is a quadratic form in the symmetric matrix \tilde{\Lambda}=\left(\Lambda+\Lambda^T\right)/2, so the mean and variance expressions are the same, provided \Lambda is replaced by \tilde{\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:

:\textrm{RSS}=y^T(I-H)^T (I-H)y.

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

:k=\operatorname{tr}\left[(I-H)^T(I-H)\right] :\lambda=\mu^T(I-H)^T(I-H)\mu/2

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

References

References

1. Bates, Douglas. "Quadratic Forms of Random Variables". STAT 849 lectures.
2. (1992). "Quadratic Forms in Random Variables". CRC Press.
3. Rencher, Alvin C.. (2008). "Linear models in statistics". Wiley-Interscience.
4. "Matrices with applications in statistics". Belmont, Calif..