The problem requires identifying an estimable parametric function from a given linear model.
Linear Model Definition
The linear model is defined as:
- $Y_t = \theta_1 + \theta_2 + \epsilon_t$ for $t = 1, 2$
- $Y_t = \theta_1 - \theta_3 + \epsilon_t$ for $t = 3, 4$
The error terms $\epsilon_t$ are independent with $E(\epsilon_t) = 0$ and $Var(\epsilon_t) = \sigma^2 > 0$. The parameters are $\theta_1, \theta_2, \theta_3 \in \mathbb{R}$.
Estimability Criterion
A linear parametric function $c^T \theta$ is estimable if it can be expressed as the expected value of a linear combination of the observed variables $Y$. Formally, in the model $Y = X\theta + \epsilon$, a function $c^T \theta$ is estimable if and only if the vector $c$ belongs to the column space of $X^T$. An alternative and often simpler condition is that $c^T \theta$ is estimable if $c$ is orthogonal to the null space of the design matrix $X$, meaning $c^T z = 0$ for all $z \in N(X)$.
Model Matrix and Null Space Calculation
The model can be written in matrix form $Y = X\theta + \epsilon$, where $\theta = (\theta_1, \theta_2, \theta_3)^T$. The design matrix $X$ is:
| $\theta_1$ | $\theta_2$ | $\theta_3$ | |
| $Y_1$ | 1 | 1 | 0 |
| $Y_2$ | 1 | 1 | 0 |
| $Y_3$ | 1 | 0 | -1 |
| $Y_4$ | 1 | 0 | -1 |
We find the null space $N(X)$ by solving $Xz = 0$ for $z = (z_1, z_2, z_3)^T$:
$Xz = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & -1 \\ 1 & 0 & -1 \end{pmatrix} \begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$
This leads to the equations:
- $z_1 + z_2 = 0 \implies z_2 = -z_1$
- $z_1 - z_3 = 0 \implies z_3 = z_1$
The null space $N(X)$ is spanned by the vector $(1, -1, 1)^T$. Any vector $z \in N(X)$ is of the form $k(1, -1, 1)^T$.
Applying the Estimability Condition
A parametric function $c^T \theta = c_1 \theta_1 + c_2 \theta_2 + c_3 \theta_3$ is estimable if $c = (c_1, c_2, c_3)^T$ satisfies $c^T z = 0$ for all $z \in N(X)$. Using the basis vector $(1, -1, 1)^T$ for $N(X)$:
$c^T (1, -1, 1)^T = c_1(1) + c_2(-1) + c_3(1) = c_1 - c_2 + c_3 = 0$.
Thus, the condition for a function $c_1 \theta_1 + c_2 \theta_2 + c_3 \theta_3$ to be estimable is $c_1 - c_2 + c_3 = 0$.
Evaluation of Options
We test each option using the condition $c_1 - c_2 + c_3 = 0$:
- Option 1: $\theta_1 + \theta_3$
- Here, $c = (1, 0, 1)^T$.
- Checking the condition: $1 - 0 + 1 = 2$. Since $2 \neq 0$, this function is not estimable.
- Option 2: $\theta_2 - \theta_3$
- Here, $c = (0, 1, -1)^T$.
- Checking the condition: $0 - 1 + (-1) = -2$. Since $-2 \neq 0$, this function is not estimable.
- Option 3: $\theta_2 + \theta_3$
- Here, $c = (0, 1, 1)^T$.
- Checking the condition: $0 - 1 + 1 = 0$. Since $0 = 0$, this function is estimable.
- Option 4: $\theta_1 + \theta_2 + \theta_3$
- Here, $c = (1, 1, 1)^T$.
- Checking the condition: $1 - 1 + 1 = 1$. Since $1 \neq 0$, this function is not estimable.
Final Conclusion
Only the parametric function $\theta_2 + \theta_3$ satisfies the derived estimability condition ($c_1 - c_2 + c_3 = 0$).
Important Questions from Gauss-Markov Models
- Let $X_1, X_2,..., X_n$ ($n > 5$) be independent random variables such that
$X_t = \alpha + \alpha^2 t + \epsilon_t$, for $t = 1,..., n$,
where, $\epsilon_1, \epsilon_2,..., \epsilon_n$ are independent and identically distributed $N(0, \sigma^2)$ random variables. Here $\alpha \in R$ and $\sigma > 0$ are unknown parameters. Which of the following statements are true? Consider the linear statistical model
$y_{ij} = \mu + \tau_i + \varepsilon_{ij}; \quad i = 1, 2, \cdots, a; \quad j = 1, 2, \cdots, n$
where $\mu$ is unknown, $\tau_i$ are independently and identically distributed as $N(0, \sigma_\tau^2)$, $\varepsilon_{ij}$ are independently and identically distributed as $N(0, \sigma^2)$; $\tau_i$ and $\varepsilon_{ij}$ are independent for all $i$ and $j$. Note that $\tau_i$ is the $i^{th}$ treatment effect. Suppose $SS_{total}, SS_{treatment}, SS_{error}$ are total sum of squares, total treatment sum of squares and error sum of squares, respectively. To test :$H_0 : \sigma_\tau^2 = 0$ vs $H_A : \sigma_\tau^2 > 0$ which of the following statements is not true?
Consider the model $Y = X\beta + \epsilon$, where $Y = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}, X = \left( (x_{ij}) \right)_{n \times p}$ and $\beta = \begin{bmatrix} \beta_1 \\ \vdots \\ \beta_p \end{bmatrix}, \epsilon = \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_n \end{bmatrix}$. $E(\epsilon) = 0$ and $D(\epsilon) = \sigma^2 I_n, p < n$. Let $\hat{\beta}$ be the solution of $X^T X \beta = X^T Y$. Which of the following are true?
Let $X_1, X_2, \ldots$ be a random sample from uniform $(0, 3\theta), \theta > 0$. Define
$T_n = \frac{1}{3} \max\{X_1, X_2, \ldots, X_n\}$. Which of the following is NOT true?
- Consider the following regression problem $y_i = \beta_1 e^i + \beta_2 e^{-i} + \epsilon_i; \ i = 1, \ldots, n$. Here $\epsilon_1, \ldots, \epsilon_n$ are i.i.d. $N(0, \sigma^2)$ random variables. If $\hat{\beta}_1$ and $\hat{\beta}_2$ are the least square estimators of $\beta_1$ and $\beta_2$, respectively, then which of the following statements are correct?
