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From Equivalent Linear Equations to GaussMarkov Theorem
Journal of Inequalities and Applications volume 2010, Article number: 259672 (2010)
Abstract
GaussMarkov 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 GaussMarkov model in an easy way.
1. Introduction
The GaussMarkov 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 GaussMarkov 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 selfcontained. 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
We note that
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
Since if and only if and, hence, , we get
Denote by the linear operator from onto defined by
(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
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
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
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
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. GaussMarkov Model and GaussMarkov 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
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
where is a given matrix of while and are unknown parameters. We will refer to the structure as to the standard GaussMarkov 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
Without loss of generality, we may and will assume that . We note that such a matrix is uniquely determined by .
The wellknown GaussMarkov 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 GaussMarkov 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
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,
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.
References
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 [2]
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Stępniak C: Through a generalized inverse. Demonstratio Mathematica 2008, 41(2):291–296.
Acknowledgment
Thanks are due to a reviewer for his comments leading to the improvement in the presentation of this paper.
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Stępniak, C. From Equivalent Linear Equations to GaussMarkov Theorem. J Inequal Appl 2010, 259672 (2010). https://doi.org/10.1155/2010/259672
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Keywords
 Little Square Estimator
 Little Square Solution
 Pythagorean Theorem
 Inverse Matrice
 Linear Unbiased Estimation