Math Problem Statement

Let (Y,Xβ,σ²I) be a linear regression model and let the design matrix X have rank r < k. Suppose Y ∼ N(Xβ, σ²I). Let c'β, c ∈ R^k be an estimable parameter function and let b be an arbitrary solution of the normal equations. (a) Find the distribution of c'b. (b) Prove that c'b is the best linear unbiased estimator of c'β.

Solution

The problem is related to a linear regression model Y=Xβ+ϵY = X \beta + \epsilon, where YN(Xβ,σ2I)Y \sim N(X\beta, \sigma^2 I). Here's the detailed interpretation and solution for both parts:

Given Information:

  • YN(Xβ,σ2I)Y \sim N(X \beta, \sigma^2 I): YY follows a multivariate normal distribution with mean XβX \beta and covariance matrix σ2I\sigma^2 I.
  • XX is an n×kn \times k design matrix with rank r<kr < k.
  • βRk\beta \in \mathbb{R}^k is the unknown parameter vector.
  • cβc' \beta is an estimable parameter, where cRkc \in \mathbb{R}^k.
  • bb is any solution of the normal equations XXb=XYX'X b = X'Y.

Part (a): Find the distribution of cbc'b

To find the distribution of cbc'b, we need to recognize the following:

  1. Estimator for β\beta: The least squares estimator bb satisfies the normal equation XXb=XYX'X b = X'Y. The general solution for bb is of the form: b=(XX)+XY+(some null space component).b = (X'X)^{+} X' Y + \text{(some null space component)}. Here, (XX)+(X'X)^+ is the Moore-Penrose pseudoinverse of XXX'X, since XXX'X is not invertible due to rank deficiency r<kr < k.

  2. Expectation of bb: E(b)=β,\mathbb{E}(b) = \beta, because E(Y)=Xβ\mathbb{E}(Y) = X\beta.

  3. Variance of bb: Since YN(Xβ,σ2I)Y \sim N(X \beta, \sigma^2 I), Var(b)=σ2(XX)+.\text{Var}(b) = \sigma^2 (X'X)^{+}.

  4. Distribution of cbc'b: The quantity cbc'b is a linear combination of normal variables. Therefore, cbc'b is normally distributed. The mean and variance of cbc'b are given by:

    • Mean: E(cb)=cE(b)=cβ.\mathbb{E}(c'b) = c' \mathbb{E}(b) = c' \beta.

    • Variance: Var(cb)=σ2c(XX)+c.\text{Var}(c'b) = \sigma^2 c' (X'X)^{+} c.

Thus, the distribution of cbc'b is: cbN(cβ,σ2c(XX)+c).c'b \sim N(c' \beta, \sigma^2 c' (X'X)^+ c).

Part (b): Prove that cbc'b is the best linear unbiased estimator (BLUE) of cβc'\beta

To prove that cbc'b is the BLUE, we need to show the following:

  1. Unbiasedness: We have already shown that E(cb)=cβ\mathbb{E}(c'b) = c' \beta, so cbc'b is an unbiased estimator of cβc' \beta.

  2. Linearity: The estimator cbc'b is a linear function of YY, since bb is a linear function of YY and cbc'b is a linear combination of the components of bb.

  3. 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 b=(XX)+XYb = (X'X)^+ X'Y has the minimum variance among all linear unbiased estimators. Hence, cbc'b also has the minimum variance among all unbiased estimators of cβc' \beta.

Thus, cbc'b is the BLUE of cβc' \beta.


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:

  1. How does the rank condition r<kr < k affect the estimability of β\beta?
  2. What is the role of the Moore-Penrose pseudoinverse in the solution for bb?
  3. Can bb be a biased estimator of β\beta? Why or why not?
  4. How does the Gauss-Markov theorem guarantee the minimum variance property?
  5. What happens to the distribution of cbc'b if XX 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