Math Problem Statement
Solution
The problem is related to a linear regression model , where . Here's the detailed interpretation and solution for both parts:
Given Information:
- : follows a multivariate normal distribution with mean and covariance matrix .
- is an design matrix with rank .
- is the unknown parameter vector.
- is an estimable parameter, where .
- is any solution of the normal equations .
Part (a): Find the distribution of
To find the distribution of , we need to recognize the following:
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Estimator for : The least squares estimator satisfies the normal equation . The general solution for is of the form: Here, is the Moore-Penrose pseudoinverse of , since is not invertible due to rank deficiency .
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Expectation of : because .
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Variance of : Since ,
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Distribution of : The quantity is a linear combination of normal variables. Therefore, is normally distributed. The mean and variance of are given by:
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Mean:
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Variance:
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Thus, the distribution of is:
Part (b): Prove that is the best linear unbiased estimator (BLUE) of
To prove that is the BLUE, we need to show the following:
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Unbiasedness: We have already shown that , so is an unbiased estimator of .
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Linearity: The estimator is a linear function of , since is a linear function of and is a linear combination of the components of .
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Best (Minimum Variance): The Gauss-Markov theorem states that the least squares estimator is the best linear unbiased estimator (BLUE) under the assumptions of the classical linear model. Specifically, the estimator has the minimum variance among all linear unbiased estimators. Hence, also has the minimum variance among all unbiased estimators of .
Thus, is the BLUE of .
Let me know if you'd like more details on any part of the derivation or if you have questions!
Here are 5 related questions to consider:
- How does the rank condition affect the estimability of ?
- What is the role of the Moore-Penrose pseudoinverse in the solution for ?
- Can be a biased estimator of ? Why or why not?
- How does the Gauss-Markov theorem guarantee the minimum variance property?
- What happens to the distribution of if has full rank?
Tip: Understanding the role of the pseudoinverse is crucial in dealing with rank-deficient regression problems, as it provides a generalized solution when the usual inverse doesn't exist.
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Multivariate Normal Distribution
Moore-Penrose Pseudoinverse
Gauss-Markov Theorem
Formulas
Y = Xβ + ϵ
c'b ∼ N(c'β, σ² c'(X'X)^+ c)
b = (X'X)^+ X'Y
Theorems
Gauss-Markov Theorem
Suitable Grade Level
Undergraduate Level
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