Sufficiency, Completeness and Ancillarity - Estimation - Statistics Previous Year Questions

1

2020 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2020

Let \(X_1, \ldots, X_n\) be a random sample of size \(n \ (\geq 2)\) from a uniform distribution on the interval \([-\theta, \theta]\), where \(\theta \in (0, \infty)\). A minimal sufficient statistic for \(\theta\) is

A

\(\max_{1 \leq i \leq n} X_i\)

B

\(\left( \min_{1 \leq i \leq n} X_i, \max_{1 \leq i \leq n} X_i \right)\)

C

\(\max_{1 \leq i \leq n} |X_i|\)

D

\(\min_{1 \leq i \leq n} |X_i|\)

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2

2020 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2020

Let \( X_1, ..., X_n \) be a random sample of size \( n \; (n \geq 2) \) from an exponential distribution with the probability density function
\[ f(x; \theta) = \begin{cases} e^{-(x - 2\theta)}, & x > 2\theta \\ 0, & \text{otherwise} \end{cases} \]
where \( \theta \in (0, \infty) \). If \( X_{(1)} = \min\{X_1, ..., X_n\} \) then the conditional expectation
\[ E\left[ \frac{1}{\theta} \left( X_{(1)} - \frac{1}{n} \right) \mid X_1 - X_2 = 2 \right] = \) ___________

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3

2020 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2020

Let \( X_1, ..., X_n \) be a random sample of size \( n \) (\( n \ge 2 \)) from an exponential distribution with the probability density function \( f(x; \theta) = \begin{cases} \frac{1}{\theta} e^{-(x - \theta)/\theta}, & x > \theta \\ 0, & \text{otherwise} \end{cases} \), where \( \theta \in (0, \infty) \). Which of the following statements is TRUE?

A

\(\min_{1 \le i \le n} X_i \) is a minimal sufficient statistic

B

\(\sum_{i=1}^n X_i \) is a minimal sufficient statistic

C

Any minimal sufficient statistic is complete

D

\(\left( \min_{1 \le i \le n} X_i, \frac{1}{n} \sum_{i=1}^n X_i \right) \) is minimal sufficient statistic

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4

2021 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2021

Let X_1, X_2, ..., X_n be a random sample of size n (n ≥ 2) from a distribution having the probability density function
f(x; θ) = (1/θ) e^{-x/θ}, x > 0,
0, otherwise,
where θ ∈ (0, ∞). Let X_(1) = min{X_1, X_2, ..., X_n} and T = ∑_{i=1}^n X_i. Then E(X_(1) | T) equals

A

T / n^2

B

T / n

C

(n + 1)T / (2n)

D

(n + 1)^2 T / (4n^2)

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5

2021 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2021

Let \( X_1, X_2, ..., X_n \) be a random sample of size \( n \ (\ge 2) \) from a uniform distribution on \( [-\theta, \theta] \), where \( \theta \in (0, \infty) \). Let \( X_{(1)} = \min\{X_1, X_2, ..., X_n\} \) and \( X_{(n)} = \max\{X_1, X_2, ..., X_n\} \). Then which of the following statements is/are true?
P : \( (X_{(1)}, X_{(n)}) \) is a complete statistic.
Q : \( X_{(n)} - X_{(1)} \) is an ancillary statistic.

A

P only

B

Q only

C

Both P and Q

D

Neither P nor Q

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6

2022 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2022

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population $f(x; \theta)$, where $\theta$ is a parameter. Then which one of the following statements is NOT true?

A

$\sum_{i=1}^n X_i$ is a complete and sufficient statistic for $\theta$, if $f(x; \theta) = \frac{e^{-\theta} \theta^x}{x!}$ , $x = 0,1,2, \ldots$, and $\theta > 0$

B

$(\sum_{i=1}^n X_i , \sum_{i=1}^n X_i^2)$ is a complete and sufficient statistic for $\theta$, if $f(x; \theta) = \frac{1}{\sqrt{2\pi}\theta} e^{-\frac{1}{2\theta^2}(x-\theta)^2} , -\infty < x < \infty, \theta > 0$

C

$f(x; \theta) = \theta x^{\theta-1}, 0 < x < 1, \theta > 0$ has monotone likelihood ratio property in $\prod_{i=1}^n X_i$

D

$X_{(n)} - X_{(1)}$ is ancillary statistic for $\theta$ if $f(x; \theta) = 1$, $0 < \theta < x < \theta + 1$, where $X_{(1)} = \min\{X_1, X_2, \ldots, X_n\}$ and $X_{(n)} = \max\{X_1, X_2, \ldots, X_n\}$

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7

2024 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2024

Let \( X_1, X_2, X_3 \) be three independent and identically distributed binomial random variables with number of trials \( n = 100 \) and success probability \( p \) (0 < p < 1), which is an unknown parameter. Let \( T_1 = (X_1 + X_2, X_3) \) and \( T_2 = X_1 + X_2 + X_3 \). Consider the following statements:
(I) The distribution of \( T_2 \) given \( T_1 = t_1 \) is independent of \( p \).
(II) The distribution of \( T_1 \) given \( T_2 = t_2 \) is independent of \( p \).
Which of the above statements is/are true?

A

Only (I)

B

Only (II)

C

Both (I) and (II)

D

Neither (I) nor (II)

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8

2025 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2025

Let T be a complete and sufficient statistic for a family P of distributions and let U be a sufficient statistic for P. If P_T(T ≥ 0) = 1 for all f ∈ P, then which one of the following options is NOT necessarily correct?

A

T^2 is a complete statistic for P

B

T^2 is a minimal sufficient statistic for P

C

T is a function of U

D

U is a function of T

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9

2025 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2025

Let $X_1, \ldots, X_n, n \geq 2$, be a random sample from a $N(-\theta, \theta)$ distribution, where $\theta > 0$ is an unknown parameter. Then which one of the following options is correct?

A

$\sum_{i=1}^n X_i$ is a minimal sufficient statistic

B

$\sum_{i=1}^n X_i^2$ is a minimal sufficient statistic

C

$\left(\frac{1}{n} \sum_{i=1}^n X_i, \frac{1}{n-1} \sum_{i=1}^n \left(X_i - \frac{1}{n} \sum_{i=1}^n X_i\right)^2\right)$ is a complete statistic

D

$-\frac{1}{n} \sum_{i=1}^n X_i$ is a uniformly minimum variance unbiased estimator of $\theta$

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10

2025 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2025

Let \(X_1, ..., X_n\) 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?

A

\(2 \max\{X_1, ..., X_n\}\) is the maximum likelihood estimator of \(\theta\)

B

\((\min\{X_1, ..., X_n\}, \max\{X_1, ..., X_n\})\) is a sufficient statistic

C

\((\min\{X_1, ..., X_n\}, \max\{X_1, ..., X_n\})\) is a complete statistic

D

\(\frac{2(n+1)}{n} \max\{|X_1|, ..., |X_n|\}\) is a uniformly minimum variance unbiased estimator of \(\theta\)

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11

2026 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2026

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?

A

\((X_1 + X_2, X_{(1)})\) is a minimal sufficient statistic

B

\((X_1, X_{(2)})\) is a minimal sufficient statistic

C

\((X_1 - X_2, X_{(1)})\) is a sufficient statistic but not a minimal sufficient statistic

D

\((X_1 - X_2, X_{(2)})\) is a complete sufficient statistic

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12

2026 · Statistics · Estimation · Sufficiency, Completeness and Ancillarity

Statistics (ST) 2026

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 \( \theta \)?

A

\(T_1(\underline{X}) = (X_1 - X_2, X_1 + X_2, X_1 + X_3) \)

B

\(T_2(\underline{X}) = (X_1 + X_2, X_1 - X_3, X_2 + X_3) \)

C

\(T_3(\underline{X}) = (X_1 - X_2, X_2 - X_3, X_3) \)

D

\(T_4(\underline{X}) = (X_1 + X_2, X_3) \)

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