From Equivalent Linear Equations to Gauss-Markov Theorem

Abstract

Gauss-Markov theorem reduces linear unbiased estimation to the Least Squares Solution of inconsistent linear equations while the normal equations reduce the second one to the usual solution of consistent linear equations. It is rather surprising that the second algebraic result is usually derived in a differential way. To avoid this dissonance, we state and use an auxiliary result on equivalence of two systems of linear equations. This places us in a convenient position to attack on the main problems in the Gauss-Markov model in an easy way.

1. Introduction

The Gauss-Markov theorem is the most classical achievement in statistics. Its role in statistics is comparable with that of the Pythagorean theorem in geometry. In fact, there are close relations between both of them.

The Gauss-Markov theorem is presented in many books and derived in many ways. The most popular approaches involve

(i)geometry (cf., Kruskal [1, 2]),

(ii)differential calculus (cf., Scheffé [3], Rao [4]),

(iii)generalized inverse matrices (cf., Rao and Mitra [5], Bapat [6]),

(iv)projection operators (see Seber [7]).

We presume that such a big market has many clients. This paper is intended for some of them. Our consideration is straightforward and self-contained. Moreover, it needs only moderate prerequisites.

The main tool used in this paper is equivalence of two systems of linear equations.

2. Preliminaries

For any matrix of define the sets

(2.1)

We note that

(2.2)

It is clear that the range constitutes -dimensional linear space in spanned by the columns of , where , while constitutes -dimensional space of all vectors being orthogonal to any vector in relative to the usual inner product Thus, any vector may be presented in the form

(2.3)

Since if and only if and, hence, , we get

Denote by the linear operator from onto defined by

(2.4)

(i.e., the orthogonal projector onto ) It follows from definition (2.4) that The following lemma (see [8]) will be a key tool in the further consideration.

Lemma 2.1.

For any matrix and for any vector , the following are equivalent:

(i)

(ii)

Proof.

(i)(ii) is evident (without any condition on ).

(ii)(i). By the assumption that , we get for some . Thus, (ii) reduces to and its general solution is , where . Therefore, is a solution of (i).

Remark 2.2.

The assumption that in Lemma 2.1 is essential. To see this, let us set

(2.5)

Then, and Thus, has a solution , while is inconsistent.

3. Least Squares Solution

For any matrix of and for any vector , consider the linear equation

(3.1)

Equation (3.1) may be consistent (if ) or inconsistent (if not). In the second case we are seeking for such that the residual vector be as small as possible.

Definition 3.1.

Any vector is said to be the Least Squares Solution (LSS) of (3.1) if

(3.2)

The following theorem shows that this definition is not empty and reduces the LSS of an inconsistent equation (3.1) to the ordinary solution of a consistent one.

Theorem 3.2.

• (a) Equation (3.1) has at least one LSS.

• (b) Vector is an LSS of (3.1) if and only if

  

  (3.3)

• (c) Condition (3.3) is equivalent to

  

  (3.4)

where is the orthogonal projector onto defined by (2.4).

• (d) General solution of (3.3) may be presented in the form , where is a particular solution, while .

Remark 3.3.

In the statistical literature, (3.3) is said to be normal.

Proof.

By properties of the projector , we get

(3.5)

with the equality if and only if (3.4) holds. Moreover, by definition of , (3.4) is consistent and, by Lemma 2.1, it is equivalent to (3.3).

Statement (d) follows directly from definition of kernel.

4. Gauss-Markov Model and Gauss-Markov Theorem

Let be an arbitrary random vector in with finite second moment . Then there exist a unique vector and a unique symmetric nonnegative definite matrix of such that

(4.1)

for all vectors and all matrices and of rows. Traditionally, such and are called the expectation and the dispersion of the random vector .

As usual, we will assume that and have the representations

(4.2)

where is a given matrix of while and are unknown parameters. We will refer to the structure as to the standard Gauss-Markov model. In the context of the model we will consider unbiased estimation of the parametric vector , where is of , by estimators of the form , where is of matrix. Since is unbiased if and only if for all , is estimable if and only if

(4.3)

Without loss of generality, we may and will assume that . We note that such a matrix is uniquely determined by .

The well-known Gauss-Markov theorem provides a constructive way for estimation of the function . It is based on a solution of the normal equation which plays the role of the estimator for

Theorem 4.1.

For any estimable in the standard Gauss-Markov model , there exists a unique linear unbiased estimator with minimal dispersion. This estimator, called the Least Squares Estimator (LSE) of , may be presented in the form , where is an arbitrary LSS of or, equivalently, it is a solution of the normal equation

(4.4)

Proof.

By Theorem 3.2 the condition is equivalent to . Therefore, by (4.3), for any estimable and for any solution of (4.4), the statistic is unbiased. On the other hand,

(4.5)

Hence, any unbiased estimator of may be presented in the form , where the first component is the LSE of while . In particular the components are not correlated. Therefore, the variance of the sum is greater than the variance of the LSE , unless . Moreover, by Theorem 3.2(d) this estimator is invariant with respect to the choice of the LSS . In consequence, the LSE of the function is unique.