Consider the model $y_i = i\beta + \varepsilon_i, i = 1, 2, 3$ where $\varepsilon_1, \varepsilon_2, \varepsilon_3$ are independent with mean $0$ and variance $\sigma^2, 2\sigma^2, 3\sigma^2$ respectively. Which of the following is the best linear unbiased estimate of $\beta$?
$\frac{y_1 + y_2 + y_3}{6}$
Model Specification
The linear model is defined as:
The error terms are independent with specified variances:
Best Linear Unbiased Estimator (BLUE) Determination
The objective is to find the Best Linear Unbiased Estimator (BLUE) for . A linear estimator is represented as .
Establishing the Unbiasedness Condition
For to be unbiased, its expectation must equal :
Given , the expectation is:
The unbiasedness condition requires , leading to the constraint:
Minimizing Variance for Optimal Estimation
The variance of the linear estimator is:
Using the provided variances:
The BLUE minimizes this variance subject to the unbiasedness constraint. The Lagrange multiplier method is employed.
Applying the Lagrange Multiplier Method
Minimize \( f(a_1, a_2, a_3) = a_1^2 + 2a_2^2 + 3a_3^2 \) subject to \( g(a_1, a_2, a_3) = a_1 + 2a_2 + 3a_3 = 1 \).
Applying \( \nabla f = \lambda \nabla g \):
Substituting into the constraint:
Solving for the coefficients \( a_i \):
- \( a_1 = \frac{1/3}{2} = 1/6 \)
- \( a_2 = \frac{1/3}{2} = 1/6 \)
- \( a_3 = \frac{1/3}{2} = 1/6 \)
Final Calculation of the BLUE
The Best Linear Unbiased Estimator is constructed using these optimal coefficients:
This simplifies to:
This result corresponds to Option C.
Important Questions from Best Linear Unbiased Estimators
Consider the simple linear regression model $Y_i = \beta x_i + \epsilon_i, i = 1,2,..., n$, where $\beta > 0$, $\sum_{i=1}^n x_i^2 > 0$, and the uncorrelated errors $\epsilon_i, i = 1,2,..., n$, have zero mean and finite variance $\sigma^2 (> 0)$. Let $\tilde{\beta}_1 = \sum_{i=1}^n a_i^*Y_i$, where $a_i$'s minimize $E(\sum_{i=1}^n a_i Y_i - \beta)^2$ with respect to scalars $a_1, a_2,..., a_n$. Let $\tilde{\beta}_2$ be the ordinary least squares estimator of $\beta$. Which of the following statements are true?
Consider the linear model $Y \sim N_n(X\beta, \sigma^2 I)$, where $X$ is a $n \times (k + 1)$ matrix of rank $k + 1 < n$. Let $\hat{\beta}$ and $\hat{\sigma}^2$ be the maximum likelihood estimators of $\beta$ and $\sigma^2$ respectively. Then which of the following statements are true?
- Let $X_1, X_2, \ldots, X_n$ be a random sample from $f_\theta(x)$, a probability density function or a probability mass function. Define $s_n^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}_n)^2$, where $\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$. Then $s_n^2$ is unbiased for $\theta$ if
Let $\{X_n\}$ be a sequence of independent random variables where the distribution of $X_n$ is normal with mean $\mu$ and variance $n$ for $n = 1, 2, \ldots$ Define
$\bar{X}_n = \frac{\sum_{i=1}^n X_i}{n}$ and $S_n = \frac{\sum_{i=1}^n \frac{X_i}{i}}{\sum_{i=1}^n \frac{1}{i}}$.
Which of the following are true?
- Let $Y_1, Y_2, Y_3$ be uncorrelated observations with common variance $\sigma^2$ and expectations given by $\mathbb{E}(Y_1) = \beta_1, \mathbb{E}(Y_2) = \beta_2$ and $\mathbb{E}(Y_3) = \beta_1 + \beta_2,$ where $\beta_1, \beta_2$ are unknown parameters. The best linear unbiased estimator of $\beta_1 + \beta_2$ is
