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 Best Linear Unbiased Estimates

Definition: The Best Linear Unbiased Estimate (BLUE) of a parameter θ based on data Y is

1. alinear function of Y. That is, the estimator can be written as b'Y,

2. unbiased (E[b'Y] = θ), and

3. has the smallest variance among all unbiased linear estimators.

Theorem 1'.1.1: For any linear combination Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is the BLUEof c'θ, where Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is the least-squaresorthogonalprojection of Y onto R(X). Proof: See lecture notes # 8 

Corollary 1'.1.2: If rank(Xn×p) = p, then, for any a, Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is the BLUE of a'β

Note: The Gauss-Markov theorem generalizes this result to the less than full rank case, for certain linear combinations a'β (the estimable functions).


Proof of Corollary 1'.1.2:

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET


1'.2. Estimable Functions

In the less than full rank case, only certain linear combinations of the components of β can be unbiasedly estimated.

Definition: A linear combination a'β is estimable if it has a linear unbiased estimate, i.e., E[b'Y] = a'β for some b for all β.

Lemma 1'.2.1: (i) a'β is estimable if and only if a ∈ R(X'). Proof: E[b'Y] = b'Xβ, which equals a'β for all β if and only if a = X'b. (ii) If a'β is estimable, there is a unique b∗ ∈ R(X) such that a = X'b∗. Proof: a'β is estimable so using (i) a = X'b. Any Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET can be uniquely decomposed as Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET, where b∗ ∈ R(X), and Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET. Then

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

Comment: Part (i) of the lemma may be a little bit surprising since all of a sudden we are talking about the row space of X, not the column space. However, the idea behind the result need not be mysterious. Every observation we have is an unbiased estimate of its expected value; the expected value of an observation is some linear combination of parameters. Such linear combinations of parameters is therefore estimable. These correspond exactly to the rows of X. Clearly, also, linear combinations of estimable functions should be estimable. These are the vectors that are spanned by the rows of X – the row space of X.


1'.3. Gauss-Markov Theorem 

Note: In the full rank case (r = p), any a'β is estimable. In particular, 

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

is a linear unbiased estimate of a'β. In this case we also know that Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is the BLUE (Corollary 1'.1.2).

Theorem 1'.3.1: (Gauss-Markov). If a'β is estimable, then
(i) a' Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is unique (i.e., the same for all solutions to the normal equations Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET).
(ii) a'Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is the BLUE of a'β.

Proof: (i) By Lemma 1'.2.1, a = X'b∗ for a unique b∗ ∈ R(X). Therefore, 

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

is unique because Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is unique. (In fact Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET since
b∗ ∈ R(X), so that Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

(ii) By Theorem 1'.1.1, Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET is the BLUE of Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NETBest linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET from part (i) and Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

1'.4. The Variance of Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

Lemma 1'.4.1: If a'β is estimable then 

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

for any generalized inverse (X'X).

Proof: If a'β is estimable, then a = X'b∗, b∗ ∈ R(X) by Lemma 1'.2.1. Then

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

regardless of the generalized inverse used.

Theorem 1'.4.2: If a'β is estimable, then 

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

Proof: Using an estimate Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

(by the Lemma) Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

Note that

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

is unique (same for all generalized inverses (X'X)).

In-class exercise: One–way ANOVA with K groups. There are K groups with J observations from each group. The model is

Best linear unbiased estimators, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

for k = 1,...,K and j = 1,...,J. As usual, E[ε] = ' and var(ε) = σ2I. In this setting we are almost never interested in the µ parameter (why not?). What are the estimable functions of the α parameters?

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FAQs on Best linear unbiased estimators, CSIR-NET Mathematical Sciences - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET

1. What are the properties of best linear unbiased estimators?
Ans. Best linear unbiased estimators (BLUE) have the following properties: - Linearity: The estimator is a linear combination of the observations. - Unbiasedness: The expected value of the estimator equals the true value of the parameter being estimated. - Minimum Variance: Among all unbiased estimators, the BLUE has the smallest variance.
2. How are best linear unbiased estimators (BLUE) different from other estimators?
Ans. Best linear unbiased estimators (BLUE) are different from other estimators in the sense that they have the minimum variance among all unbiased estimators. This means that they provide the best precision and efficiency in estimating the true value of a parameter. Other estimators may have larger variances and may not be as efficient.
3. How are best linear unbiased estimators (BLUE) calculated?
Ans. Best linear unbiased estimators (BLUE) are calculated using the method of least squares. The estimator is obtained by finding the linear combination of the observations that minimizes the sum of the squared differences between the observed values and the estimated values. This minimization problem can be solved analytically or numerically using optimization techniques.
4. What are the applications of best linear unbiased estimators (BLUE)?
Ans. Best linear unbiased estimators (BLUE) have various applications in statistical inference and decision-making. Some examples include: - Econometrics: BLUE is used in estimating parameters in economic models. - Survey Sampling: BLUE is used to estimate population parameters based on a sample. - Regression Analysis: BLUE is used to estimate the coefficients in a linear regression model. - Time Series Analysis: BLUE is used to estimate the parameters in a time series model.
5. Can best linear unbiased estimators (BLUE) be used in non-linear models?
Ans. No, best linear unbiased estimators (BLUE) are specifically designed for linear models. They assume that the relationship between the observed variables and the parameters being estimated is linear. In non-linear models, other estimation methods, such as maximum likelihood estimation or nonlinear least squares, need to be used to obtain unbiased and efficient estimators.
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