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

Given Information:

• $Y \sim N(X \beta, \sigma^2 I)$: $Y$ follows a multivariate normal distribution with mean $X \beta$ and covariance matrix $\sigma^2 I$.
• $X$ is an $n \times k$ design matrix with rank $r < k$.
• $\beta \in \mathbb{R}^k$ is the unknown parameter vector.
• $c' \beta$ is an estimable parameter, where $c \in \mathbb{R}^k$.
• $b$ is any solution of the normal equations $X'X b = X'Y$.

Part (a): Find the distribution of $c'b$

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

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

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

3. Variance of $b$: Since $Y \sim N(X \beta, \sigma^2 I)$, $\text{Var}(b) = \sigma^2 (X'X)^{+}.$

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

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

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

Thus, the distribution of $c'b$ is: $c'b \sim N(c' \beta, \sigma^2 c' (X'X)^+ c).$

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

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

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

2. Linearity: The estimator $c'b$ is a linear function of $Y$, since $b$ is a linear function of $Y$ and $c'b$ is a linear combination of the components of $b$.

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 $b = (X'X)^+ X'Y$ has the minimum variance among all linear unbiased estimators. Hence, $c'b$ also has the minimum variance among all unbiased estimators of $c' \beta$.

Thus, $c'b$ is the BLUE of $c' \beta$.

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