To find the best linear unbiased estimator (BLUE) of \(\beta_1 + \beta_2\), we start by understanding the properties of the given random variables:
- \(Y_1, Y_2,\) and \(Y_3\) are uncorrelated with common variance \(\sigma^2\).
- \(\mathbb{E}(Y_1) = \beta_1\), \(\mathbb{E}(Y_2) = \beta_2\), and \(\mathbb{E}(Y_3) = \beta_1 + \beta_2\).
A linear estimator of \(\beta_1 + \beta_2\) can be expressed in the form:
\(aY_1 + bY_2 + cY_3\)
We want this to be an unbiased estimator, so:
\(\mathbb{E}(aY_1 + bY_2 + cY_3) = a\beta_1 + b\beta_2 + c(\beta_1 + \beta_2) = \beta_1 + \beta_2\)
Thus, we have the equation:
- \((a + c)\beta_1 + (b + c)\beta_2 = \beta_1 + \beta_2\)
This gives us two conditions:
\[(a + c) = 1\]\[(b + c) = 1\]Because the observations are uncorrelated, the variance of the estimator should be minimized. The variance is given by:
\(\text{Var}(aY_1 + bY_2 + cY_3) = a^2\sigma^2 + b^2\sigma^2 + c^2\sigma^2 = \sigma^2(a^2 + b^2 + c^2)\)
Substituting \(c = 1 - a\) (from equation 1) and \(c = 1 - b\) (from equation 2), we can find the values of \(a\), \(b\), and \(c\) that minimize the variance:
The values satisfying the condition with minimum variance turn out to be \(a = b = \frac{1}{3}\) and \(c = \frac{2}{3}\).
Thus, the estimator becomes:
\(\frac{1}{3}Y_1 + \frac{1}{3}Y_2 + \frac{2}{3}Y_3\)
Substituting these into the estimation formula, the required best linear unbiased estimator is:
\(\frac{1}{3}(Y_1 + Y_2 + 2Y_3)\)
Therefore, the correct answer is:
\(\frac{1}{3}(Y_1 + Y_2 + 2Y_3)\)
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 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 $\beta$?
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?
