Question:
Let \(X\) be a single observation from a distribution having a probability density function \(f_\theta\), \(\theta \in \Theta\). For testing \(H_0: \theta = 1\) against \(H_1: \theta \neq 1\), under which of the following options a uniformly most powerful test of level \(0.05\) exists?
Let \(X\) be a single observation from a distribution having a probability density function \(f_\theta\), \(\theta \in \Theta\). For testing \(H_0: \theta = 1\) against \(H_1: \theta \neq 1\), under which of the following options a uniformly most powerful test of level \(0.05\) exists?
Show Hint
A UMP test for a two sided hypothesis can exist only when the parameter also fixes the boundary of the support of \(X\), because then one side of the alternative is detected for free.
Updated On: Aug 3, 2026
- \[ f_\theta(x) = \begin{cases} \dfrac{1}{\theta} & \text{if } 0 < x < \theta \\ 0 & \text{otherwise} \end{cases} ; \quad \Theta = (0, \infty) \]
- \[ f_\theta(x) = \begin{cases} e^{-(x-\theta)} & \text{if } x > \theta \\ 0 & \text{otherwise} \end{cases} ; \quad \Theta = (-\infty, \infty) \]
- \[ f_\theta(x) = \begin{cases} \dfrac{1}{\theta} e^{-x/\theta} & \text{if } x > 0 \\ 0 & \text{otherwise} \end{cases} ; \quad \Theta = (0, \infty) \]
- \[ f_\theta(x) = \begin{cases} \dfrac{1}{\theta} e^{-(x-1)/\theta} & \text{if } x > 1 \\ 0 & \text{otherwise} \end{cases} ; \quad \Theta = (0, \infty) \]
Show Solution
Verified By Collegedunia
The Correct Option is A, B
Solution and Explanation
Step 1: When does a UMP two-sided test exist?
Only when \(\theta\) controls the boundary of the support, so one direction is caught for free.
Step 2: (A) Uniform(0,\(\theta\)).
\(x>1\) proves \(\theta>1\) for free; add \(x<0.05\) for the other side. UMP exists. TRUE.
Step 3: (B) shifted exponential support \((\theta,\infty)\).
\(x<1\) proves \(\theta<1\) for free; add \(x>c\) tail. UMP exists. TRUE.
Step 4: (C) Exponential(\(\theta\)), support \((0,\infty)\) fixed.
No free direction, no UMP. FALSE.
Step 5: (D) support \((1,\infty)\) fixed regardless of \(\theta\).
Same issue as (C), no UMP. FALSE.
Final Answer: \[ \boxed{\text{(A) and (B)}} \]
Only when \(\theta\) controls the boundary of the support, so one direction is caught for free.
Step 2: (A) Uniform(0,\(\theta\)).
\(x>1\) proves \(\theta>1\) for free; add \(x<0.05\) for the other side. UMP exists. TRUE.
Step 3: (B) shifted exponential support \((\theta,\infty)\).
\(x<1\) proves \(\theta<1\) for free; add \(x>c\) tail. UMP exists. TRUE.
Step 4: (C) Exponential(\(\theta\)), support \((0,\infty)\) fixed.
No free direction, no UMP. FALSE.
Step 5: (D) support \((1,\infty)\) fixed regardless of \(\theta\).
Same issue as (C), no UMP. FALSE.
Final Answer: \[ \boxed{\text{(A) and (B)}} \]
Was this answer helpful?
0
0
Top GATE ST Statistics Questions
- Let \( X \) be a random variable having the Poisson distribution with mean \( \log_e 2 \). Then \( E\left( e^{(\log_e 3)X} \right) \) equals:
Let \( (X_1, X_2, X_3) \) follow the multinomial distribution with the number of trials being 100 and the probability vector \( \left( \frac{3}{10}, \frac{1}{10}, \frac{3}{5} \right) \).
Then \( E(X_2 | X_3 = 40) \) equals:- Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables with the common probability density function \[ f(x) = \frac{1}{\pi(1 + x^2)}, \quad -\infty<x<\infty. \] Define \[ Y_n = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(X_n) { for } n = 1, 2, \dots. \] Then which one of the following options is correct?
- Let \( \{ N(t): t \geq 0 \} \) be a homogeneous Poisson process with the intensity/rate \( \lambda = 2 \). Let} \[ X = N(6) - N(1), \quad Y = N(5) - N(3), \quad W = N(6) - N(5), \quad Z = N(3) - N(1). \] Then which one of the following options is correct?
- Let \( T \) be a complete and sufficient statistic for a family \( \mathcal{P} \) of distributions and let \( U \) be a sufficient statistic for \( \mathcal{P} \). If \( P_f(T \geq 0) = 1 \) for all \( f \in \mathcal{P} \), then which one of the following options is NOT necessarily correct?
View More Questions
Top GATE ST Testing of Hypotheses Questions
- For testing a null hypothesis \(H_0\) against an alternative hypothesis \(H_1\) at level of significance \(\alpha \in (0,1)\), which of the following statements is correct?
- Let \(X\) be a single observation from a distribution having the probability density function
\[f_\theta(x)=\begin{cases}1 & \text{if } \theta<x<\theta+1\\0 & \text{otherwise,}\end{cases}\]
where \(\theta\in(-\infty,\infty)\). For testing \(H_0:\theta\le0\) against \(H_1:\theta>1\), let \(\beta\) be the power of the uniformly most powerful test of level \(0.05\). Then \(225\beta\) equals ______ (answer in integer). - Let \(X_1,X_2,\ldots,X_n\ (n\geq2)\) be a random sample from the probability density function \(f(x)\). Consider the following hypotheses:
\[ H_0: f(x)=\frac{1}{\sqrt{2\pi}}\,e^{-\frac{x^2}{2}};\quad -\infty<x<\infty \]
\[ H_1: f(x)=\frac{1}{2}\,e^{-|x|};\quad -\infty<x<\infty. \]
For testing \(H_0\) against \(H_1\), let \(R\) denote the critical region based on the likelihood ratio test having level \(0.05\). Then, for some constant \(c\), the region \(R\) is - Let \(X_1,X_2,\ldots,X_n\ (n>1)\) be a random sample from the following probability density function
\[ f_{\beta}(x)=\begin{cases}\beta e^{-x}(1-e^{-x})^{\beta-1} & \text{if } x>0\\ 0 & \text{otherwise},\end{cases} \]
where \(\beta>0\) is an unknown parameter. For testing the following hypotheses,
\[ H_0:\beta=1 \quad \text{against} \quad H_1:\beta>1, \]
at level \(\alpha\in(0,1)\), which of the following statements is correct? - Let \(X_1, X_2, \ldots, X_5\) be random observations from a continuous distribution. Let \(\theta_p\) be the \(p\)-th population quantile. Consider the following hypotheses
\[ H_0:\theta_{1/2}=2.5 \quad \text{against} \quad H_1:\theta_{1/2}>2.5. \]
Let \(X_{(r)}\) denote the \(r\)-th order statistic of the given observations. Then which of the following is a critical region of a level \(0.05\) test?
View More Questions
Top GATE ST Questions
- Even though I had planned to go skiing with my friends, I had to ............. at the last moment because of an injury.
Select the most appropriate option to complete the above sentence. - The President, along with the Council of Ministers, ............. to visit India next week.
Select the most appropriate option to complete the above sentence. An electricity utility company charges ₹7 per kWh. If a 40-watt desk light is left on for 10 hours each night for 180 days, what would be the cost of energy consumption? If the desk light is on for 2 more hours each night for the 180 days, what would be the percentage-increase in the cost of energy consumption?
In the context of the given figure, which one of the following options correctly represents the entries in the blocks labelled (i), (ii), (iii), and (iv), respectively?
A bag contains Violet (V), Yellow (Y), Red (R), and Green (G) balls. On counting them, the following results are obtained:
(i) The sum of Yellow balls and twice the number of Violet balls is 50.
(ii) The sum of Violet and Green balls is 50.
(iii) The sum of Yellow and Red balls is 50.
(iv) The sum of Violet and twice the number of Red balls is 50.
Which one of the following Pie charts correctly represents the balls in the bag?
View More Questions