In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE)[1][2][3] was developed by C. R. Rao.
Similar to the theory of best linear unbiased estimation, MINQUE is specifically concerned with linear regression models.
[1] The method was originally conceived to estimate heteroscedastic error variance in multiple linear regression.
[1] MINQUE estimators also provide an alternative to maximum likelihood estimators or restricted maximum likelihood estimators for variance components in mixed effects models.
[3] MINQUE estimators are quadratic forms of the response variable and are used to estimate a linear function of the variances.
We are concerned with a mixed effects model for the random vector
is a design matrix for the fixed effects,
represents the unknown fixed-effect parameters,
Furthermore, any two random effect vectors are also uncorrelated (
represent the variance components of the model.
Note that this model makes no distributional assumptions about
MINQUE estimators are derived by identifying a matrix
, which represents a translation of the original fixed effect.
Under this equivalent model, the MINQUE estimator is now
Rao argued that since the underlying models are equivalent, this estimator should be equal to
Furthermore, the cyclic property of the trace is used to evaluate the expectation with respect to
To ensure that this estimator is unbiased, Rao suggested setting
, subject to the constraints In the Gauss-Markov model, the error variance
This estimator is unbiased and can be shown to minimize the Euclidean norm of the form
[1] Rao proposed a MINQUE estimator for the variance components model based on minimizing the Euclidean norm.
, the MINQUE with the Euclidean norm is obtained by identifying the matrix
, subject to the MINQUE constraints discussed above.
is the projection matrix into the column space of
represents the generalized inverse of a matrix.
That is, the vector represents the solution to the following system of equations
MINQUE estimators can be obtained without the invariance criteria, in which case the estimator is only unbiased and minimizes the norm.
[2] Such estimators have slightly different constraints on the minimization problem.
The model can be extended to estimate covariance components.
[3] In such a model, the random effects of a component are assumed to have a common covariance structure
A MINQUE estimator for a mixture of variance and covariance components was also proposed.