Let \(X_1,X_2,X_3\) be a random sample from a distribution having probability mass function
\[ f_{\theta}(x)=\begin{cases}\theta & \text{if } x=1\\ 1-\theta & \text{if } x=2\\ 0, & \text{otherwise},\end{cases} \]
where \(\theta\in\Theta=(0,1)\). Let \(\underline{X}=(X_1,X_2,X_3)\). Then which of the following is NOT a sufficient statistic for \(\theta\)?
\[ f_{\theta}(x)=\begin{cases}\theta & \text{if } x=1\\ 1-\theta & \text{if } x=2\\ 0, & \text{otherwise},\end{cases} \]
where \(\theta\in\Theta=(0,1)\). Let \(\underline{X}=(X_1,X_2,X_3)\). Then which of the following is NOT a sufficient statistic for \(\theta\)?
Show Hint
- \(T_1(\underline{X})=(X_1-X_2,\ X_1+X_2,\ X_1+X_3)\)
- \(T_2(\underline{X})=(X_1+X_2,\ X_1-X_3,\ X_2+X_3)\)
- \(T_3(\underline{X})=(X_1-X_2,\ X_2-X_3,\ X_3)\)
- \(T_4(\underline{X})=(X_1+X_2,\ X_3)\)
The Correct Option is B
Solution and Explanation
With \(n_1\) = count of \(X_i=1\), joint pmf = \(\theta^{n_1}(1-\theta)^{3-n_1}\). So \(n_1\) (equivalently \(\sum X_i\)) is sufficient.
Step 2: Check (A).
$T_1$ recovers \(X_1,X_2,X_3\) fully (invertible linear system), so sufficient.
Step 3: Check (C).
$T_3$ also recovers \(X_1,X_2,X_3\) fully, sufficient.
Step 4: Check (D).
$T_4$'s sum gives \(X_1+X_2+X_3\), determines \(n_1\), sufficient.
Step 5: Check (B) with two samples.
$(1,2,1)$: \(n_1=2\), \(T_2=(3,0,3)\). $(2,1,2)$: \(n_1=1\), \(T_2=(3,0,3)\). Same \(T_2\) but different \(n_1\) and different likelihoods, so NOT sufficient.
Final Answer:
\[ \boxed{T_2(\underline{X})=(X_1+X_2,\ X_1-X_3,\ X_2+X_3)} \]
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?
Top GATE ST Estimation Questions
- Let \( X_1, \dots, X_n \) be a random sample from a uniform distribution over the interval \( \left( -\frac{\theta}{2}, \frac{\theta}{2} \right) \), where \( \theta>0 \) is an unknown parameter. Then which of the following options is/are correct?
Let \( X_1, X_2, \dots, X_7 \) be a random sample from a population having the probability density function \[ f(x) = \frac{1}{2} \lambda^3 x^2 e^{-\lambda x}, \quad x>0, \] where \( \lambda>0 \) is an unknown parameter. Let \( \hat{\lambda} \) be the maximum likelihood estimator of \( \lambda \), and \( E(\hat{\lambda} - \lambda) = \alpha \lambda \) be the corresponding bias, where \( \alpha \) is a real constant. Then the value of \( \frac{1}{\alpha} \) equals __________ (answer in integer).
- Let \( X_1, X_2, \dots, X_7 \) be a random sample from a population having the probability density function \[ f(x) = \frac{1}{2} \lambda^3 x^2 e^{-\lambda x}, \quad x>0, \] where \( \lambda>0 \) is an unknown parameter. Let \( \hat{\lambda} \) be the maximum likelihood estimator of \( \lambda \), and \( E(\hat{\lambda} - \lambda) = \alpha \lambda \) be the corresponding bias, where \( \alpha \) is a real constant. Then the value of \( \frac{1}{\alpha} \) equals __________ (answer in integer).
- Let \(X_1,X_2\) be a random sample from the following probability density function
\[ f_\alpha(x)= \begin{cases} \alpha x^{\alpha-1}e^{-x^\alpha} & \text{if } x>0 \\ 0 & \text{otherwise,} \end{cases} \]
where \(\alpha\in(0,\infty)\) is an unknown parameter. Let
\[ X_{(1)}=\min\{X_1,X_2\} \quad\text{and}\quad X_{(2)}=\max\{X_1,X_2\}. \]
Then which of the following statements is correct? - Let \(X_1\) and \(X_2\) be independent and identically distributed random variables following normal distribution with mean \(\theta\in(-\infty,\infty)\) and variance \(1\). Then which of the following estimators of their expected values attains the Cramer-Rao lower bound?
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?