MINQUE explained

In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE)[1] [2] [3] 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.

Principles

We are concerned with a mixed effects model for the random vector

Y\inRn

with the following linear structure.

Y=X\boldsymbol\beta+U1\boldsymbol\xi1++Uk\boldsymbol\xik

Here,

X\inRn x

is a design matrix for the fixed effects,

\boldsymbol\beta\inRm

represents the unknown fixed-effect parameters,

Ui\in

n x ci
R
is a design matrix for the

i

-th random-effect component, and
ci
\boldsymbol\xi
i\inR
is a random vector for the

i

-th random-effect component. The random effects are assumed to have zero mean (

E[\boldsymbol\xii]=0

) and be uncorrelated (
2
V[\boldsymbol\xi
iI
ci
). Furthermore, any two random effect vectors are also uncorrelated (

V[\boldsymbol\xii, \boldsymbol\xij]=0 \forallij

). The unknown variances
2
\sigma
k
represent the variance components of the model.

This is a general model that captures commonly used linear regression models.

  1. Gauss-Markov Model: If we consider a one-component model where

U1=In

, then the model is equivalent to the Gauss-Markov model

Y=X\boldsymbol\beta+\boldsymbol\epsilon

with

E[\boldsymbol\epsilon]=0

and
2
V[\boldsymbol\epsilon]=\sigma
1

In

.
  1. Heteroscedastic Model: Each set of random variables in

Y

that shares a common variance can be modeled as an individual variance component with an appropriate

Ui

.

A compact representation for the model is the following, where

U=\left[\begin{array}{c|c|c}U1&&Uk\end{array}\right]

and

\boldsymbol\xi\top=

\top\end{array}\right]
\left[\begin{array}{c|c|c} \boldsymbol\xi
k
.

Y=X\boldsymbol\beta+U\boldsymbol\xi

Note that this model makes no distributional assumptions about

Y

other than the first and second moments.

E[Y]=X\boldsymbol\beta

2
V[Y]=\sigma
1U

1U

\top
1

+

2
+ \sigma
k

Uk

\top \equiv
U
k
2
\sigma
1V

1++

2
\sigma
k

Vk

The goal in MINQUE is to estimate

\theta=

k
\sum
i=1

pi

2
\sigma
i
using a quadratic form

\hat{\theta}=Y\topAY

. MINQUE estimators are derived by identifying a matrix

A

such that the estimator has some desirable properties, described below.

Optimal Estimator Properties to Constrain MINQUE

Invariance to translation of the fixed effects

Consider a new fixed-effect parameter

\boldsymbol\gamma=\boldsymbol\beta-\boldsymbol\beta0

, which represents a translation of the original fixed effect. The new, equivalent model is now the following.

Y-X\boldsymbol\beta0= X\boldsymbol\gamma+U\boldsymbol\xi

Under this equivalent model, the MINQUE estimator is now

(Y-

\top
X\boldsymbol\beta
0)

A (Y-X\boldsymbol\beta0)

. Rao argued that since the underlying models are equivalent, this estimator should be equal to

Y\topAY

. This can be achieved by constraining

A

such that

AX=0

, which ensures that all terms other than

Y\topAY

in the expansion of the quadratic form are zero.

Unbiased estimation

Suppose that we constrain

AX=0

, as argued in the section above. Then, the MINQUE estimator has the following form

\begin{align} \hat{\theta}&=Y\topAY\\ &=(X\boldsymbol\beta+U\boldsymbol\xi)\topA(X\boldsymbol\beta+U\boldsymbol\xi)\\ &=\boldsymbol\xi\topU\topAU\boldsymbol\xi \end{align}

To ensure that this estimator is unbiased, the expectation of the estimator

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\xii

.

\begin{align} E[\hat{\theta}]&=E[\boldsymbol\xi\topU\topAU\boldsymbol\xi]\\ &=

k
\sum
i=1
\topAU
E[\boldsymbol\xi
i\boldsymbol\xi

i]\\ &=

k
\sum
i=1
2
\sigma
i
\top
Tr[U
i

AUi] \end{align}

To ensure that this estimator is unbiased, Rao suggested setting

k
\sum
i=1
2
\sigma
i
\top
Tr[U
i

AUi]=

k
\sum
i=1

pi

2
\sigma
i
, which can be accomplished by constraining

A

such that
\top
Tr[U
i

AUi]=Tr[AVi]=pi

for all components.

Minimum Norm

Rao argues that if

\boldsymbol\xi

were observed, a "natural" estimator for

\theta

would be the following since
\top\boldsymbol\xi
E[\boldsymbol\xi
i]=c

i

2
\sigma
i
. Here,

\boldsymbol\Delta

is defined as a diagonal matrix.
p1
c1
\top\boldsymbol\xi
\boldsymbol\xi
1

++

pk
ck
\top\boldsymbol\xi
\boldsymbol\xi
k

=

\top\left[diag\left(p1
ci
\boldsymbol\xi,,
pk
ck

\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(U\topAU-\boldsymbol\Delta)\boldsymbol\xi

. This difference can be minimized by minimizing the norm of the matrix

\lVertU\topAU-\boldsymbol\Delta\rVert

.

Procedure

Given the constraints and optimization strategy derived from the optimal properties above, the MINQUE estimator

\hat{\theta}

for
k
\theta=\sum
i=1

pi\sigma

2
i
is derived by choosing a matrix

A

that minimizes

\lVertU\topAU-\boldsymbol\Delta\rVert

, subject to the constraints

AX=0

, and

Tr[AVi]=pi

.

Examples of Estimators

Standard Estimator for Homoscedastic Error

In the Gauss-Markov model, the error variance

\sigma2

is estimated using the following.

s2=

1
n-m

(Y-X\hat{\boldsymbol\beta})\top(Y-X\hat{\boldsymbol\beta})

This estimator is unbiased and can be shown to minimize the Euclidean norm of the form

\lVertU\topAU-\boldsymbol\Delta\rVert

. Thus, the standard estimator for error variance in the Gauss-Markov model is a MINQUE estimator.

Random Variables with Common Mean and Heteroscedastic Error

For random variables

Y1,,Yn

with a common mean and different variances
2
\sigma
n
, the MINQUE estimator for
2
\sigma
i
is
n
n-2

(Yi-\overline{Y})2-

s2
n-2
, where

\overline{Y}=

1
n
n
\sum
i=1

Yi

and

s2=

1
n-1
n
\sum
i=1

(Yi-\overline{Y})2

.

Estimator for Variance Components

Rao proposed a MINQUE estimator for the variance components model based on minimizing the Euclidean norm. The Euclidean norm

\lVert\rVert2

is the square root of the sum of squares of all elements in the matrix. When evaluating this norm below,

V=V1+ … +Vk=UU\top

. Furthermore, using the cyclic property of traces,

Tr[U\topAU\boldsymbol\Delta]=Tr[AU\boldsymbol\DeltaU\top]

k
= Tr\left[\sum
i=1
pi
ci

AVi\right]= Tr[\boldsymbol\Delta\boldsymbol\Delta]

.

\begin{align} \lVertU\topAU-\boldsymbol\Delta

2
\rVert
2

&=(U\topAU-\boldsymbol\Delta)\top(U\topAU-\boldsymbol\Delta)\\ &=Tr[U\topAUUAU\top]-Tr[2U\topAU\boldsymbol\Delta]+Tr[\boldsymbol\Delta\boldsymbol\Delta]\\ &=Tr[AVAV]-Tr[\boldsymbol\Delta\boldsymbol\Delta] \end{align}

Note that since

Tr[\boldsymbol\Delta\boldsymbol\Delta]

does not depend on

A

, the MINQUE with the Euclidean norm is obtained by identifying the matrix

A

that minimizes

Tr[AVAV]

, subject to the MINQUE constraints discussed above.

Rao showed that the matrix

A

that satisfies this optimization problem is

A\star=\sum

k
i=1

λiRViR

,

where

R=V-1(I-P)

,

P=X(X\topV-1X)-X\topV-1

is the projection matrix into the column space of

X

, and

()-

represents the generalized inverse of a matrix.

Therefore, the MINQUE estimator is the following, where the vectors

\boldsymbolλ

and

Q

are defined based on the sum.

\begin{align} \hat{\theta}&=Y\topA\starY\\ &=

k
\sum
i=1

λi

\topRV
Y
iRY\\ &\equiv\sum
k
i=1

λiQi\\ &\equiv\boldsymbolλ\topQ \end{align}

The vector

\boldsymbolλ

is obtained by using the constraint

Tr[A\starVi]=pi

. That is, the vector represents the solution to the following system of equations

\forallj\in\{1,,k\}

.

\begin{align} Tr[A\starVj]&=pj\\ Tr\left[

k
\sum
i=1

λiRViRVj\right]&=pj\\ \sum

k
i=1

λiTr[RViRVj]&=pj \end{align}

This can be written as a matrix product

S\boldsymbolλ=p

, where

p=[p1 … p

\top
k]
and

S

is the following.

S=\begin{bmatrix} Tr[RV1RV1]&&Tr[RVkRV1]\\ \vdots&\ddots&\vdots\\ Tr[RV1RVk]&&Tr[RVkRVk] \end{bmatrix}

Then,

\boldsymbolλ=S-p

. This implies that the MINQUE is

\hat{\theta}=\boldsymbolλ\topQ=p\top(S-)\topQ=p\topS-Q

. Note that
k
\theta=\sum
i=1

pi

2
\sigma
i

=p\top\boldsymbol\sigma

, where

\boldsymbol\sigma=

\top
[\sigma
k]
. Therefore, the estimator for the variance components is

\hat{\boldsymbol\sigma}=S-Q

.

Extensions

MINQUE estimators can be obtained without the invariance criteria, in which case the estimator is only unbiased and minimizes the norm. Such estimators have slightly different constraints on the minimization problem.

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

V[\boldsymbol\xii]=\boldsymbol\Sigma

. A MINQUE estimator for a mixture of variance and covariance components was also proposed. In this model,

V[\boldsymbol\xii]=\boldsymbol\Sigma

for

i\in \{1,,s\}

and

V[\boldsymbol\xii]= \sigma

2I
ci
for

i\in\{s+1,,k\}

.

References

  1. Rao . C.R. . 1970 . Estimation of heteroscedastic variances in linear models . Journal of the American Statistical Association . 65 . 329 . 161 - 172 . 2283583. 10.1080/01621459.1970.10481070 .
  2. Rao . C.R. . 1971 . Estimation of variance and covariance components MINQUE theory . J Multivar Anal . 1 . 257 - 275 . 10.1016/0047-259x(71)90001-7. 10338.dmlcz/104230 . free .
  3. Rao . C.R. . 1972 . Estimation of variance and covariance components in linear models . Journal of the American Statistical Association . 67 . 337 . 112 - 115 . 2284708. 10.1080/01621459.1972.10481212 .