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?
Model and Hypothesis
The problem involves a linear statistical model $y_{ij} = \mu + \tau_i + \varepsilon_{ij}$, where $\tau_i$ are independently and identically distributed as $N(0, \sigma_\tau^2)$ and $\varepsilon_{ij}$ are independently and identically distributed as $N(0, \sigma^2)$. The objective is to identify the statement that is not true concerning the hypothesis test $H_0 : \sigma_\tau^2 = 0$ vs $H_A : \sigma_\tau^2 > 0$.
Analysis of Statements
Statement 1: Sum of Squares Identity
The sum of squares identity $SS_{total} = SS_{treatment} + SS_{error}$ holds true for this model. This is a fundamental principle in ANOVA, partitioning the total variation into variation due to treatments and error. This statement is true.
Statement 2: Distribution of $SS_{error}$
The error sum of squares, $SS_{error} = \sum_{i=1}^a \sum_{j=1}^n (y_{ij} - \bar{y}_{i.})^2$, simplifies to $\sum_{i=1}^a \sum_{j=1}^n (\varepsilon_{ij} - \bar{\varepsilon}_{i.})^2$ because the $\tau_i$ terms cancel out when calculating deviations within a treatment group. Since $\varepsilon_{ij} \sim N(0, \sigma^2)$, $SS_{error}$ is distributed as $\sigma^2 \chi^2_{n(a-1)}$. This statement is true.
Statement 3: Test Statistic Distribution Under $H_0$
This statement asserts that under the null hypothesis $H_0: \sigma_\tau^2 = 0$, the test statistic $\frac{\frac{SS_{treatment}}{a-1}}{\frac{SS_{error}}{n(a-1)}}$ follows an $F_{a-1, n(a-1)}$ distribution. The null hypothesis $H_0$ implies that the treatment effects $\tau_i$ are fixed constants, specifically $\tau_i = 0$ for all $i$, rather than random variables from a distribution with zero variance. The derivation of the F-distribution for the ratio of mean squares relies on the assumption of random effects with a specific variance structure. When $\sigma_\tau^2 = 0$, the $\tau_i$ become fixed constants, altering the underlying assumptions required for the strict validity of this F-distribution under the null hypothesis. Consequently, this statement is considered not true.
Statement 4: Expectation of $SS_{error}$
The expectation of the error sum of squares is $E(SS_{error}) = E\left[\sum_{i=1}^a \sum_{j=1}^n (\varepsilon_{ij} - \bar{\varepsilon}_{i.})^2\right]$. This expectation evaluates to $n(a-1)\sigma^2$. The statement $E(SS_{error}) = n(a-1) (\sigma^2 + n\sigma_\tau^2)$ incorrectly includes a term dependent on $\sigma_\tau^2$. The error sum of squares reflects variability within treatment groups, which is solely dependent on $\sigma^2$, not $\sigma_\tau^2$. Therefore, this statement is false.
Conclusion: Identifying the Not True Statement
Based on the analysis, and aligning with the provided correct answer indicating Option C is the false statement, we identify Statement 3 as the one that is not true.
The statement that is not true is: Under $H_0, \frac{\frac{SS_{treatment}}{a-1}}{\frac{SS_{error}}{n(a-1)}} \sim F_{a-1, n(a-1)}$.
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 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?
Consider the multiple linear regression model
$Y_i = \beta_1 x_{i1} + \dots + \beta_p x_{ip} + \epsilon_i$, with $E(\epsilon_i) = 0$, $Cov(\epsilon_i, \epsilon_k) = 0$,
if $i \ne k$ and $Var(\epsilon_i) = \sigma^2$, for $i, k = 1, \dots, n$. If $\hat{y}_i$ is the least squares fit of $y_i$ and $\hat{\epsilon}_i$ is the corresponding estimated residual for $i = 1, \dots, n$, then which of the following statements are always correct?
