MINQUE

In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE) was developed by C. R. Rao. MINQUE is a theory alongside other estimation methods in estimation theory, such as the method of moments or maximum likelihood estimation. Similar to the theory of best linear unbiased estimation, MINQUE is specifically concerned with linear regression models. The method was originally conceived to estimate heteroscedastic error variance in multiple linear regression. MINQUE estimators also provide an alternative to maximum likelihood estimators or restricted maximum likelihood estimators for variance components in mixed effects models. MINQUE estimators are quadratic forms of the response variable and are used to estimate a linear function of the variances.

In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE) was developed by C. R. Rao. MINQUE is a theory alongside other estimation methods in estimation theory, such as the method of moments or maximum likelihood estimation. Similar to the theory of best linear unbiased estimation, MINQUE is specifically concerned with linear regression models. The method was originally conceived to estimate heteroscedastic error variance in multiple linear regression. MINQUE estimators also provide an alternative to maximum likelihood estimators or restricted maximum likelihood estimators for variance components in mixed effects models. MINQUE estimators are quadratic forms of the response variable and are used to estimate a linear function of the variances. \mathbf{Y} = \mathbf{X}\boldsymbol\beta + \mathbf{U}_1 \boldsymbol\xi_1 + \cdots + \mathbf{U}_k \boldsymbol\xi_k Here, \mathbf{X} \in \mathbb{R}^{n\times m} is a design matrix for the fixed effects, \boldsymbol\beta \in \mathbb{R}^m represents the unknown fixed-effect parameters, \mathbf{U}_i \in \mathbb{R}^{n\times c_i} is a design matrix for the i -th random-effect component, and \boldsymbol\xi_i\in\mathbb{R}^{c_i} is a random vector for the i -th random-effect component. The random effects are assumed to have zero mean ( \mathbb{E}[\boldsymbol\xi_i]=\mathbf{0} ) and be uncorrelated ( \mathbb{V}[\boldsymbol\xi_i]=\sigma^2_i\mathbf{I}_{c_i} ). Furthermore, any two random effect vectors are also uncorrelated ( \mathbb{V}[\boldsymbol\xi_i, \boldsymbol\xi_j]=\mathbf{0}\, \forall i\neq j ). The unknown variances \sigma^2_1,\cdots,\sigma^2_k represent the variance components of the model. This is a general model that captures commonly used linear regression models. Gauss-Markov Model : If we consider a one-component model where \mathbf{U}_1=\mathbf{I}_n , then the model is equivalent to the Gauss-Markov model \mathbf{Y}=\mathbf{X}\boldsymbol\beta + \boldsymbol\epsilon with \mathbb{E}[\boldsymbol\epsilon]=\mathbf{0} and \mathbb{V}[\boldsymbol\epsilon]=\sigma^2_1 \mathbf{I}_n . Heteroscedastic Model : Each set of random variables in \mathbf{Y} that shares a common variance can be modeled as an individual variance component with an appropriate \mathbf{U}_i . A compact representation for the model is the following, where \mathbf{U} = \left[\begin{array}{c|c|c}\mathbf{U}_1&\cdots&\mathbf{U}_k\end{array}\right] and \boldsymbol\xi^\top = \left[\begin{array}{c|c|c} \boldsymbol\xi_1^\top&\cdots&\boldsymbol\xi_k^\top\end{array}\right] . \mathbf{Y}=\mathbf{X}\boldsymbol\beta+\mathbf{U}\boldsymbol\xi Note that this model makes no distributional assumptions about \mathbf{Y} other than the first and second moments. \mathbb{E}[\mathbf{Y}] = \mathbf{X}\boldsymbol\beta \mathbb{V}[\mathbf{Y}]=\sigma^2_1\mathbf{U}_1\mathbf{U}_1^\top + \cdots + \sigma^2_k \mathbf{U}_k \mathbf{U}_k^\top \equiv \sigma^2_1\mathbf{V}_1 + \cdots + \sigma^2_k \mathbf{V}_k The goal in MINQUE is to estimate \theta = \sum_{i=1}^k p_i \sigma^2_i using a quadratic form \hat{\theta}=\mathbf{Y}^\top \mathbf{A} \mathbf{Y} . MINQUE estimators are derived by identifying a matrix \mathbf{A} such that the estimator has some desirable properties, described below. Consider a new fixed-effect parameter \boldsymbol\gamma=\boldsymbol\beta - \boldsymbol\beta_0 , which represents a translation of the original fixed effect. The new, equivalent model is now the following. \mathbf{Y} - \mathbf{X}\boldsymbol\beta_0 = \mathbf{X}\boldsymbol\gamma + \mathbf{U}\boldsymbol\xi Under this equivalent model, the MINQUE estimator is now (\mathbf{Y} - \mathbf{X}\boldsymbol\beta_0)^\top \mathbf{A} (\mathbf{Y} - \mathbf{X}\boldsymbol\beta_0) . Rao argued that since the underlying models are equivalent, this estimator should be equal to \mathbf{Y}^\top \mathbf{A} \mathbf{Y} . This can be achieved by constraining \mathbf{A} such that \mathbf{A}\mathbf{X} = \mathbf{0} , which ensures that all terms other than \mathbf{Y}^\top \mathbf{A} \mathbf{Y} in the expansion of the quadratic form are zero. Suppose that we constrain \mathbf{A}\mathbf{X} = \mathbf{0} , as argued in the section above. Then, the MINQUE estimator has the following form \begin{align} \hat{\theta} &= \mathbf{Y}^\top \mathbf{A} \mathbf{Y}\\ &= (\mathbf{X}\boldsymbol\beta + \mathbf{U}\boldsymbol\xi)^\top \mathbf{A} (\mathbf{X}\boldsymbol\beta + \mathbf{U}\boldsymbol\xi)\\ &= \boldsymbol\xi^\top\mathbf{U}^\top\mathbf{A}\mathbf{U}\boldsymbol\xi \end{align} To ensure that this estimator is unbiased, the expectation of the estimator \mathbb{E}[\hat{\theta}] must equal the parameter of interest, \theta . Below, the expectation of the estimator can be decomposed for each component since the components are uncorrelated with each other. Furthermore, the cyclic property of the trace is used to evaluate the expectation with respect to \boldsymbol\xi_i . \begin{align} \mathbb{E}[\hat{\theta}] &= \mathbb{E}[\boldsymbol\xi^\top \mathbf{U}^\top \mathbf{A} \mathbf{U} \boldsymbol\xi]\\ &= \sum_{i=1}^k \mathbb{E}[\boldsymbol\xi_i^\top\mathbf{U}_i^\top\mathbf{A}\mathbf{U}_i\boldsymbol\xi_i]\\ &= \sum_{i=1}^k \sigma_i^2 \mathrm{Tr}[\mathbf{U}_i^\top \mathbf{A} \mathbf{U}_i] \end{align} To ensure that this estimator is unbiased, Rao suggested setting \sum_{i=1}^k \sigma_i^2 \mathrm{Tr}[\mathbf{U}_i^\top \mathbf{A} \mathbf{U}_i] = \sum_{i=1}^k p_i \sigma_i^2 , which can be accomplished by constraining \mathbf{A} such that \mathrm{Tr}[\mathbf{U}_i^\top \mathbf{A} \mathbf{U}_i] = \mathrm{Tr}[\mathbf{A}\mathbf{V}_i] = p_i for all components. Rao argues that if \boldsymbol\xi were observed, a "natural" estimator for \theta would be the following since \mathbb{E}[\boldsymbol\xi_i^\top\boldsymbol\xi_i]=c_i \sigma_i^2 . Here, \boldsymbol\Delta is defined as a diagonal matrix. \frac{p_1}{c_1}\boldsymbol\xi_1^\top\boldsymbol\xi_1 + \cdots + \frac{p_k}{c_k}\boldsymbol\xi_k^\top\boldsymbol\xi_k = \boldsymbol\xi^\top\left[\mathrm{diag}\left(\frac{p_1}{c_i},\cdots,\frac{p_k}{c_k}\right)\right]\boldsymbol\xi \equiv \boldsymbol\xi^\top\boldsymbol\Delta\boldsymbol\xi The difference between the proposed estimator and the natural estimator is \boldsymbol\xi^\top (\mathbf{U}^\top \mathbf{A} \mathbf{U} - \boldsymbol\Delta)\boldsymbol\xi . This difference can be minimized by minimizing the norm of the matrix \lVert \mathbf{U}^\top\mathbf{A}\mathbf{U}-\boldsymbol\Delta \rVert . Given the constraints and optimization strategy derived from the optimal properties above, the MINQUE estimator \hat{\theta} for \theta=\sum_{i=1}^k p_i\sigma_i^2 is derived by choosing a matrix \mathbf{A} that minimizes \lVert \mathbf{U}^\top\mathbf{A}\mathbf{U}-\boldsymbol\Delta \rVert , subject to the constraints \mathbf{A}\mathbf{X}=\mathbf{0} , and \mathrm{Tr}[\mathbf{A}\mathbf{V}_i]=p_i . s^2 = \frac{1}{n-m}(\mathbf{Y}-\mathbf{X}\hat{\boldsymbol\beta})^\top(\mathbf{Y}-\mathbf{X}\hat{\boldsymbol\beta}) This estimator is unbiased and can be shown to minimize the Euclidean norm of the form \lVert \mathbf{U}^\top\mathbf{A}\mathbf{U}-\boldsymbol\Delta \rVert . Thus, the standard estimator for error variance in the Gauss-Markov model is a MINQUE estimator. For random variables Y_1,\cdots,Y_n with a common mean and different variances \sigma^2_1,\cdots,\sigma^2_n , the MINQUE estimator for \sigma^2_i is \frac{n}{n-2}(Y_i - \overline{Y})^2 - \frac{s^2}{n - 2} , where \overline{Y} = \frac{1}{n} \sum_{i=1}^n Y_i and s^2 = \frac{1}{n-1} \sum_{i=1}^n (Y_i - \overline{Y})^2 . Rao proposed a MINQUE estimator for the variance components model based on minimizing the Euclidean norm. The Euclidean norm \lVert \cdot \rVert_2 is the square root of the sum of squares of all elements in the matrix. When evaluating this norm below, \mathbf{V}=\mathbf{V}_1+\cdots+\mathbf{V}_k = \mathbf{U} \mathbf{U}^\top . Furthermore, using the cyclic property of traces, \mathrm{Tr}[\mathbf{U}^\top\mathbf{A}\mathbf{U}\boldsymbol\Delta] = \mathrm{Tr}[\mathbf{A}\mathbf{U}\boldsymbol\Delta\mathbf{U}^\top] = \mathrm{Tr}\left[\sum_{i=1}^k \frac{p_i}{c_i} \mathbf{A}\mathbf{V}_i \right] = \mathrm{Tr}[\boldsymbol\Delta\boldsymbol\Delta] . \begin{align} \lVert \mathbf{U}^\top\mathbf{A}\mathbf{U} - \boldsymbol\Delta \rVert^2_2 &= (\mathbf{U}^\top\mathbf{A}\mathbf{U} - \boldsymbol\Delta)^\top (\mathbf{U}^\top\mathbf{A}\mathbf{U} - \boldsymbol\Delta)\\ &= \mathrm{Tr}[\mathbf{U}^\top\mathbf{A}\mathbf{U}\mathbf{U}\mathbf{A}\mathbf{U}^\top] - \mathrm{Tr}[2\mathbf{U}^\top\mathbf{A}\mathbf{U}\boldsymbol\Delta] + \mathrm{Tr}[\boldsymbol\Delta\boldsymbol\Delta]\\ &= \mathrm{Tr}[\mathbf{A}\mathbf{V}\mathbf{A}\mathbf{V}] - \mathrm{Tr}[\boldsymbol\Delta\boldsymbol\Delta] \end{align} Note that since \mathrm{Tr}[\boldsymbol\Delta\boldsymbol\Delta] does not depend on \mathbf{A} , the MINQUE with the Euclidean norm is obtained by identifying the matrix \mathbf{A} that minimizes \mathrm{Tr}[\mathbf{A}\mathbf{V}\mathbf{A}\mathbf{V}] , subject to the MINQUE constraints discussed above. Rao showed that the matrix \mathbf{A} that satisfies this optimization problem is \mathbf{A}_\star=\sum_{i=1}^k \lambda_i \mathbf{R}\mathbf{V}_i\mathbf{R} , where \mathbf{R} = \mathbf{V}^{-1}(\mathbf{I}-\mathbf{P}) , \mathbf{P}=\mathbf{X}(\mathbf{X}^\top\mathbf{V}^{-1}\mathbf{X})^{-}\mathbf{X}^\top\mathbf{V}^{-1} is the projection matrix into the column space of \mathbf{X} , and (\cdot)^{-} represents the generalized inverse of a matrix. Therefore, the MINQUE estimator is the following, where the vectors \boldsymbol\lambda and \mathbf{Q} are defined based on the sum. \begin{align} \hat{\theta} &= \mathbf{Y}^\top \mathbf{A}_\star\mathbf{Y}\\ &= \sum_{i=1}^k \lambda_i \mathbf{Y}^\top\mathbf{R}\mathbf{V}_i\mathbf{R}\mathbf{Y}\\ &\equiv\sum_{i=1}^k \lambda_i Q_i\\ &\equiv \boldsymbol\lambda^\top \mathbf{Q} \end{align} The vector \boldsymbol\lambda is obtained by using the constraint \mathrm{Tr}[\mathbf{A}_\star\mathbf{V}_i]=p_i . That is, the vector represents the solution to the following system of equations \forall j\in\{1,\cdots,k\} . \begin{align} \mathrm{Tr}[\mathbf{A}_\star\mathbf{V}_j] &= p_j\\ \mathrm{Tr}\left[ \sum_{i=1}^k \lambda_i \mathbf{R}\mathbf{V}_i\mathbf{R}\mathbf{V}_j \right] &= p_j\\ \sum_{i=1}^k \lambda_i \mathrm{Tr}[\mathbf{R}\mathbf{V}_i\mathbf{R}\mathbf{V}_j] &= p_j \end{align} This can be written as a matrix product \mathbf{S}\boldsymbol\lambda=\mathbf{p} , where \mathbf{p}=[p_1\,\cdots\,p_k]^\top and \mathbf{S} is the following. \mathbf{S}=\begin{bmatrix} \mathrm{Tr}[\mathbf{R}\mathbf{V}_1\mathbf{R}\mathbf{V}_1] & \cdots & \mathrm{Tr}[\mathbf{R}\mathbf{V}_k\mathbf{R}\mathbf{V}_1]\\ \vdots & \ddots & \vdots\\ \mathrm{Tr}[\mathbf{R}\mathbf{V}_1\mathbf{R}\mathbf{V}_k] & \cdots & \mathrm{Tr}[\mathbf{R}\mathbf{V}_k\mathbf{R}\mathbf{V}_k] \end{bmatrix} Then, \boldsymbol\lambda=\mathbf{S}^{-}\mathbf{p} . This implies that the MINQUE is \hat{\theta}=\boldsymbol\lambda^\top\mathbf{Q}=\mathbf{p}^\top(\mathbf{S}^{-})^\top\mathbf{Q}=\mathbf{p}^\top\mathbf{S}^{-}\mathbf{Q} . Note that \theta=\sum_{i=1}^k p_i \sigma_i^2 = \mathbf{p}^\top\boldsymbol\sigma , where \boldsymbol\sigma = [\sigma^2_1\,\cdots\,\sigma^2_k]^\top . Therefore, the estimator for the variance components is \hat{\boldsymbol\sigma}=\mathbf{S}^{-}\mathbf{Q} . The model can be extended to estimate covariance components. In such a model, the random effects of a component are assumed to have a common covariance structure \mathbb{V}[\boldsymbol\xi_i]=\boldsymbol\Sigma . A MINQUE estimator for a mixture of variance and covariance components was also proposed. In this model, \mathbb{V}[\boldsymbol\xi_i]=\boldsymbol\Sigma for i\in \{1,\cdots,s\} and \mathbb{V}[\boldsymbol\xi_i]= \sigma_i^2\mathbf{I}_{c_i} for i\in\{s+1,\cdots,k\} .

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