In the mode [pic]is such that the following two conditions on the random vector [pic]are met: 1. [pic]
2. [pic]
the best (minimum variance) linear (linear functions of the [pic]) unbiased estimator of [pic]is given by least squares estimator; that is, [pic]is the best linear unbiased estimator (BLUE) of [pic]. Proof:

Let [pic]be any [pic]constant matrix and let [pic]; [pic] is a general linear function of [pic], which we shall take as an estimator of [pic]. We must specify the elements of [pic]so that [pic]will be the best unbiased estimator of [pic]. Let [pic] Since [pic] is known, we must find [pic]in order to be able to specify [pic]. For unbiasedness, we have

[pic]

But, to be unbiased, [pic] must equal [pic], and this implies that [pic]for all [pic]. Thus, unbiasedness specifies that [pic]. For property of “best” we must find the matrix [pic]that minimized [pic], where [pic], subject to the restriction [pic]. To examine this, consider the covariance

[pic]

Let [pic]Then [pic]. The diagonal elements of [pic] are respective variances of the [pic]. To minimize each [pic], we must, therefore, minimize each diagonal elements of [pic]. Since [pic]and [pic]are constants, we must find a matrix [pic]such that each diagonal element of [pic]is a minimum. But [pic] is positive semidefinite; hence [pic] Thus the diagonal elements of [pic] will attain their minimum when [pic] for [pic]. But, if [pic]then [pic] Therefore, if [pic]is to equal 0 for all [pic], it must be true that [pic]for all [pic]and [pic]. This implies that [pic]. The condition[pic]is compatible with the condition of unbiasedness, [pic]. Therefore, [pic]and [pic]. This completes the proof.

...Universidad Autónoma de Querétaro
Facultad de Ingeniería
“Iterative Methods”
“Gauss and Gauss-Seidel”
Profesor | | Nieves Fonseca Ricardo |
Mentado Camacho Félix
Navarro Escamilla Erandy
Péloquin Blancas María José
Rubio Miranda Ana Luisa
Abstract
Many real life problems give us several simultaneous linear equations to solve. And we have to find a common solution for each of them. There are several techniques to use.
Instead of using methods...

...Carl Gauss was a man who is known for making a great deal breakthroughs in the wide variety of his work in both mathematics and physics. He is responsible for immeasurable contributions to the fields of number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics, as well as many more. The concepts that he himself created have had an immense influence in many areas of the mathematic and scientific world.
Carl Gauss was born Johann...

...-------------------------------------------------
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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
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ABSTRACT / SUMMARY
The main purpose of this experiment is to investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tape red duct and to measure the flow rate and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates. The apparatus used is Bernoulli’s Theorem Demonstration Apparatus, F1-15. In this experiment, the pressure...

...BINOMIAL THEOREM :
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XI A , K V 2 , GWALIOR
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example: The coefficients appearing in the binomial...

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...pressure dynamics specified by Bernoulli’s Principle to keep their rare wheels on the ground, even while zooming off at high speed. It is successfully employed in mechanism like the carburetor and the atomizer.
The study focuses on Bernoulli’s Theorem in Fluid Application. A fluid is any substance which when acted upon by a shear force, however small, cause a continuous or unlimited deformation, but at a rate proportional to the applied force. As a matter of fact, if a fluid...

...In mathematics, the Pythagorean theorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can...

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