Let $X_1, \dots, X_{10}$ be a random sample of size 10 from a continuous distribution with probability density function $f_\theta(x) = \begin{cases} e^{(\theta-x)}, & x \ge \theta \\ 0, & \text{otherwise}. \end{cases}$ $\theta \in \mathbb{R}$ is unknown. Consider the problem of testing the null hypothesis $H_0 : \theta \le 3$ against the alternate hypothesis $H_1 : \theta > 3$, based on $X_1, \dots, X_{10}$. Let $L(\theta)$ denote the likelihood function and $x_{(1)} = \min(x_1, \dots, x_{10})$. Which of the following statements are correct?
Analysis of Option A: Supremum of Likelihood under $H_0$
The likelihood function for the sample $X_1, \dots, X_{10}$ is:
$L(\theta) = \prod_{i=1}^{10} f_\theta(x_i) = \prod_{i=1}^{10} e^{(\theta - x_i)}$ for $x_i \ge \theta$ for all $i$.
This condition implies $x_{(1)} = \min(x_1, \dots, x_{10}) \ge \theta$. The likelihood simplifies to:
$L(\theta) = e^{\sum_{i=1}^{10} (\theta - x_i)} = e^{10\theta - \sum_{i=1}^{10} x_i}$, subject to $\theta \le x_{(1)}$.
We need to find $\sup_{\theta \le 3} L(\theta)$. This involves maximizing $L(\theta)$ subject to $\theta \le 3$ (null hypothesis) and $\theta \le x_{(1)}$ (data constraint). The combined constraint is $\theta \le \min(3, x_{(1)})$.
Since $L(\theta)$ increases with $\theta$, the maximum occurs at $\theta = \min(3, x_{(1)})$.
$\sup_{\theta \le 3} L(\theta) = L(\min(3, x_{(1)}))$.
When $x_{(1)} \le 3$, we have $\min(3, x_{(1)}) = x_{(1)}$. Thus, the supremum is $L(x_{(1)}) = e^{10x_{(1)} - \sum_{i=1}^{10} x_i}$. This confirms Option A.
Analysis of Option D: Likelihood Ratio Test Critical Region
The Likelihood Ratio Test (LRT) statistic is $\lambda(x) = \frac{\sup_{\theta \in \Theta_0} L(\theta)}{\sup_{\theta \in \Theta} L(\theta)}$, where $\Theta_0 = \{\theta : \theta \le 3\}$ and $\Theta = \mathbb{R}$.
The unrestricted maximum likelihood estimate (MLE) is $\hat{\theta}_{MLE} = x_{(1)}$, giving $\sup_{\theta \in \Theta} L(\theta) = L(x_{(1)}) = e^{10x_{(1)} - \sum_{i=1}^{10} x_i}$.
We consider the case relevant for the test statistic calculation, where the MLE falls outside the null parameter space, i.e., $x_{(1)} > 3$. In this case, the supremum under $H_0$ is $\sup_{\theta \le 3} L(\theta) = L(3) = e^{30 - \sum_{i=1}^{10} x_i}$.
The LRT statistic becomes:
$\lambda(x) = \frac{L(3)}{L(x_{(1)})} = \frac{e^{30 - \sum x_i}}{e^{10x_{(1)} - \sum x_i}} = e^{30 - 10x_{(1)}}$, for $x_{(1)} > 3$.
The critical region is determined by $\lambda(x) \le c_\alpha$, meaning $e^{30 - 10x_{(1)}} \le c_\alpha$. This inequality rearranges to $x_{(1)} \ge 3 - \frac{1}{10} \log(c_\alpha)$.
To find the critical value $k = 3 - \frac{1}{10} \log(c_\alpha)$, we use the test size $\alpha$. We evaluate the probability of rejection under the null hypothesis boundary $\theta=3$. The cumulative distribution function (CDF) of $X_{(1)}$ given $\theta$ is $F_{X_{(1)}}(x|\theta) = 1 - e^{-10(x-\theta)}$ for $x \ge \theta$.
The size condition is $P(X_{(1)} \ge k | \theta=3) = \alpha$. This probability is calculated as:
$P(X_{(1)} \ge k | \theta=3) = 1 - F_{X_{(1)}}(k | \theta=3) = 1 - (1 - e^{-10(k-3)}) = e^{-10(k-3)}$.
Setting $e^{-10(k-3)} = \alpha$, we take the natural logarithm: $-10(k-3) = \log(\alpha)$. Solving for $k$ gives $k = 3 - \frac{1}{10} \log(\alpha)$.
Thus, the critical region is $X_{(1)} \ge k$, which is $X_{(1)} \ge 3 - \frac{1}{10} \log(\alpha)$. This confirms Option D.
Important Questions from Random Variables
- A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function
$ f(x) = \begin{cases} e^{-x}, & \text{if } x>0, \\ 0, & \text{otherwise,} \end{cases} $
and that of Brand II follows a gamma distribution with the probability density function
$ g(x) = \begin{cases} \frac{x}{4} e^{-x/2}, & \text{if } x>0, \\ 0, & \text{otherwise.} \end{cases} $
The company uses the batteries of Brands I and II in proportion of $20\%$ and $80\%$ respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that $2$ years is - Consider a discrete random variable $X$ with the probability mass function
$ P(X = 0) = \frac{\theta}{3}, \ P(X = 1) = 1 - \frac{\theta}{2}, \ P(X = 2) = \frac{\theta}{6}, $
where $\theta \in (0,1)$ is an unknown parameter. In a random sample of size $90$ from this distribution, the observed counts for $X = 0, 1$ and $2$ are $20, 60$ and $10$, respectively. Then, the maximum likelihood estimate of $\theta$ is - Let $X$ be a random sample of size $1$ from the probability density function
$ f(x|\theta) = \begin{cases} \frac{3}{\theta^3} (\theta - x)^2, & \text{if } 0<x<\theta, \\ 0, & \text{otherwise.} \end{cases} $
If $ \left(\frac{X}{1-\lambda_1}, \frac{X}{1-\lambda_2}\right) $ is a confidence interval for $\theta$ with confidence coefficient $1 - \alpha$, where $\lambda_i \in (0,1), \ i = 1,2, \ \lambda_1<\lambda_2$, and $\alpha \in (0,1)$, then which of the following statements is true? - Let $X_1, X_2, . . ., X_n$ be a random sample from a continuous distribution with the common probability density function
$ f(x|\theta) = \begin{cases} \frac{2\theta^2}{x^{\theta+1}}, & \text{if } x>2, \\ 0, & \text{otherwise,} \end{cases} $
where $\theta (> 0)$ is an unknown parameter. Suppose $P(Y>\chi^2_{m,\beta}) = \beta$, where $Y \sim \chi^2_m$. For testing $H_0: \theta = 1$ against $H_1 : \theta>1$, a uniformly most powerful test of size $\alpha, \ 0<\alpha<1$, will reject $H_0$ if - Suppose we want to estimate the population mean $\bar{Y}$ of a variable for a finite population of size $85$, with $34$ Statisticians and $51$ Biologists. We consider the following sampling scheme:
A stratified random sample with $2$ strata of Statisticians (Stratum-1) and Biologists (Stratum-2), where $12$ Statisticians and $15$ Biologists are drawn from Stratum-1 and Stratum-2, respectively, using SRSWOR scheme.
Denote $\bar{y}_S, \bar{y}_B$, and $\bar{y}$ as the mean of the variable among the Statistician sample, Biologist sample, and the combined sample, respectively. Which of the following is an unbiased estimator of $\bar{Y}$?
