The problem at hand involves finding the best linear unbiased estimator (BLUE) for a common mean \(\theta\) with a given variance-covariance structure.
- The random variables \(Y_1, Y_2, \ldots, Y_n\) have a common unknown mean \(\theta\). This implies that we are dealing with a setting where we assume each \(Y_i\) is distributed with mean \(\theta\).
- The variance-covariance matrix, denoted as \(V\), has its inverse such that all diagonal elements are equal to \(c\) and all off-diagonal elements equal to \(d\).
- We need to determine the best linear unbiased estimator (BLUE), \(T_1\), and the ordinary least squares (OLS) estimator, \(T_2\), for \(\theta\).
Using the properties of linear models and the given structure of \(V^{-1}\), it is known that the BLUE of \(\theta\) in this context (when variance-covariance matrix has this specific form) is simply the sample mean: - Since the ordinary least squares estimator for the common mean in a simple estimation problem of this type is also the sample mean,
we can conclude: - Thus, \(T_1 = T_2 = \bar{Y}\), where \(\bar{Y}\) is the mean of the observations.
- Regarding the linear functions of \(Y_1, Y_2, \ldots, Y_n\), if each function has a zero expectation, it implies that all these functions are centered around zero. Given \(n\) observations and looking at affine transformations on the dataset, we can construct exactly \((n - 1)\) linearly independent linear functions since one degree of freedom is consumed in estimating \(\theta\).
Therefore, the truth is represented in the options:
- \(T_1 = \frac{1}{n} \sum_{i=1}^n Y_i = T_2\).
There are exactly \((n - 1)\) linearly independent linear functions of \(Y_1, Y_2, \ldots, Y_n\) each with zero expectation.
These explanations confirm that both these statements are true given the problem's setting. The first option is true since the OLS estimator (which in this case, is the same as the BLUE because of homogeneity) is the arithmetic mean. The second option is true since we can construct \(n - 1\) orthogonally projecting linear functions.
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?
