Part (c) deals with proving that $C'b$ is the Best Linear Unbiased Estimator (BLUE) of $C'B$. Let's break down the steps provided:

(c) Best Linear Unbiased Estimator (B.L.U.E.)

The goal here is to demonstrate that $C'b$ is a linear function of $Y$, which is a key property required for the estimator to be BLUE.

(i) Linearity of $C'b$

• The expression $C'b$ is written as: $C'b = C'(X'X)^{-1}X'Y$ where:

  • $b = (X'X)^{-1}X'Y$ is the least squares estimator of the coefficients.
  • $C'$ is a constant matrix.
  • $X$ is the design matrix of the linear regression.
  • $Y$ is the vector of observations.

• Then, the equation simplifies as: $C'b = U'Y \quad \text{where} \, U \in \mathbb{R}^n$ This implies that $C'b$ is a linear function of $Y$.

Thus, we have shown that $C'b$ is linear in $Y$, which is a requirement for an estimator to be the best linear unbiased estimator (BLUE).

Summary:

• The equation $C'b = C'(X'X)^{-1}X'Y$ shows that $C'b$ is a linear function of the observations $Y$, implying linearity.
• Linearity, combined with the previous parts (a) and (b), where we showed the invariance and unbiasedness, confirms that $C'b$ satisfies the criteria of being BLUE.

Let me know if you need further details or clarification!

Here are 5 related questions for further exploration:

1. What are the other conditions that an estimator must satisfy to be considered BLUE?
2. How does the Gauss-Markov theorem relate to the concept of BLUE?
3. Why is the assumption that $C'b$ is linear in $Y$ important in statistical estimation?
4. What role does the matrix $X'X$ play in the least squares estimation process?
5. How does the variance of $C'b$ influence the efficiency of the estimator?

Tip: In regression problems, BLUE estimators minimize the variance of the estimator among all linear unbiased estimators, making them optimal under the Gauss-Markov assumptions.