Moment and Maximum-likelihood Estimation - Estimation - Statistics Previous Year Questions

1

2019 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2019

Let \(X_1, ..., X_n\) be a random sample drawn from a population with probability density function \(f(x; \theta) = \theta x^{\theta-1}, 0 \le x \le 1, \theta > 0\). Then the maximum likelihood estimator of \(\theta\) is

A

\(\dfrac{-n}{\sum_{i=1}^n \log_e X_i}\)

B

\(\dfrac{-\sum_{i=1}^n \log_e X_i}{n}\)

C

\((\prod_{i=1}^n X_i)^{1/n}\)

D

\(\dfrac{\prod_{i=1}^n X_i}{n}\)

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2

2019 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2019

Let \(X_1, \ldots, X_n\) be a random sample from a population having probability density function \(f_X(x; \theta) = \frac{2x}{\theta^2}, 0 < x < \theta\). Then the method of moments estimator of \(\theta\) is

A

\(\frac{3 \sum_{i=1}^n X_i}{2n}\)

B

\(\frac{3 \sum_{i=1}^n X_i^2}{2n}\)

C

\(\frac{\sum_{i=1}^n X_i}{n}\)

D

\(\frac{3 \sum_{i=1}^n X_i(X_i-1)}{2n}\)

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3

2019 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2019

Let \(X_1, ..., X_{10}\) be a random sample from a population with probability density function \[ f(x; \theta) = \frac{e^{-|x-\theta|}}{2}, -\infty < x < \infty, -\infty < \theta < \infty. \] Then the maximum likelihood estimator of \(\theta\)

A

does not exist.

B

is not unique.

C

is the sample mean.

D

is the smallest observation.

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4

2019 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2019

Consider a fixed effects one-way analysis of variance model \(Y_{ij} = \mu + \tau_i + \epsilon_{ij}\), for \(i = 1, ..., a\), \(j = 1, ..., r\), and \(\epsilon_{ij}\)'s are independent and identically distributed normal random variables with mean zero and variance \(\sigma^2\). Here, \(r\) and \(a\) are positive integers. Let \(\bar{Y}_{i.} = \frac{\sum_{j=1}^{r} Y_{ij}}{r}\). Then \(\bar{Y}_{i.}\) is the least squares estimator for

A

\(\mu + \frac{\tau_i}{2}\)

B

\(\tau_i\)

C

\(\mu + \tau_i\)

D

\(\mu\)

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5

2020 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2020

Let {0,1,2,3} be an observed sample of size 4 from \( N(\theta, 5) \) distribution, where \( \theta \in [2, \infty) \). Then the maximum likelihood estimate of \( \theta \) based on the observed sample is ___________

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6

2021 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2021

Let \(X_1, X_2, \ldots, X_n\) be a random sample of size \(n\) (≥ 2) from a distribution having the probability density function \[f(x; \theta) = \begin{cases} \frac{1}{\theta} e^{-\frac{x-\theta}{\theta}}, & x > \theta, \\ 0, & \text{otherwise}, \end{cases}\] where \(\theta \in (0, \infty)\). Then the method of moments estimator of \(\theta\) equals

A

\(\frac{1}{2n} \sum_{i=1}^n X_i\)

B

\(\frac{2}{n} \sum_{i=1}^n X_i\)

C

\(\frac{1}{n} \sum_{i=1}^n X_i\)

D

\(\frac{n}{\sum_{i=1}^n X_i}\)

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7

2021 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2021

Let \(\{0.90, 0.50, 0.01, 0.95\}\) be a realization of a random sample of size 4 from the probability density function
\(f(x) = \begin{cases} \frac{\theta}{1 - \theta} x^{(2\theta - 1)/(1 - \theta)}, & 0 < x < 1, \\ 0, & \text{otherwise}, \end{cases}\).
where \(0.5 \leq \theta < 1\). Then the maximum likelihood estimate of \(\theta\) based on the observed sample equals __________ (round off to 2 decimal places).

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8

2022 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2022

A random sample of size 4 is taken from the distribution with the probability density function \( f(x; \theta) = \begin{cases} \frac{2(\theta - x)}{\theta^2}, & 0 < x < \theta, \\ 0, & \text{elsewhere}. \end{cases} \) If the observed sample values are 6, 5, 3, 6, then the method of moments estimate (in integer) of the parameter \(\theta\), based on these observations, is ________

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9

2022 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2022

A random sample of size 5 is taken from a distribution with the probability density function
\[ f(x; \theta) = \begin{cases} \frac{3x^2}{\theta^3}, & 0 < x < \theta, \\ 0, & \text{otherwise}, \end{cases} \]
where \(\theta\) is an unknown parameter. If the observed values of the random sample are \(3, 6, 4, 7, 5\), then the maximum likelihood estimate of the \(\frac{1}{8}\)th quantile of the distribution (rounded off to one decimal place) is ______

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10

2023 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2023

Let \( X \) be a random variable with probability density function
\( f(x; \lambda) = \begin{cases} \frac{1}{\lambda} e^{-\frac{x}{\lambda}} & \text{if } x > 0 \\ 0 & \text{otherwise}, \end{cases} \)
where \( \lambda > 0 \) is an unknown parameter. Let \( Y_1, Y_2, \ldots, Y_n \) be a random sample of size \( n \) from a population having the same distribution as \( X^2 \).
If \( \bar{Y} = \frac{1}{n} \sum_{i=1}^n Y_i \), then which one of the following statements is true?

A

\(\sqrt{\frac{\bar{Y}}{2}} \) is a method of moments estimator of \(\lambda \)

B

\(\sqrt{\bar{Y}} \) is a method of moments estimator of \(\lambda \)

C

\(\frac{1}{2} \sqrt{\bar{Y}} \) is a method of moments estimator of \(\lambda \)

D

\(2\sqrt{\bar{Y}} \) is a method of moments estimator of \(\lambda \)

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11

2023 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2023

Let \( X_1, X_2, \ldots, X_n \) be a random sample of size \( n \ (n \geq 2) \) from a population having probability density function
\( f(x; \theta) = \begin{cases} \frac{2}{\theta x} (-\log_e x) e^{-\frac{(\log_e x)^2}{\theta}} & \text{if } 0 < x < 1 \\ 0 & \text{otherwise}, \end{cases} \)
where \( \theta > 0 \) is an unknown parameter. Then which one of the following statements is true?

A

\(\frac{1}{n} \sum_{i=1}^n (\log_e X_i)^2 \) is the maximum likelihood estimator of \(\theta \)

B

\(\frac{1}{n-1} \sum_{i=1}^n (\log_e X_i)^2 \) is the maximum likelihood estimator of \(\theta \)

C

\(\frac{1}{n} \sum_{i=1}^n \log_e X_i \) is the maximum likelihood estimator of \(\theta \)

D

\(\frac{1}{n-1} \sum_{i=1}^n \log_e X_i \) is the maximum likelihood estimator of \(\theta \)

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12

2023 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2023

Let \( X_1, X_2, \ldots, X_n \) be a random sample of size \( n \) from a population having probability density function
\[ f(x; \mu) = \begin{cases} e^{-(x-\mu)} & \text{if } \mu \le x < \infty \\ 0 & \text{otherwise}, \end{cases} \]
where \( \mu \in \mathbb{R} \) is an unknown parameter. If \( \widehat{M} \) is the maximum likelihood estimator of the median of \( X_1 \), then which one of the following statements is true?

A

\(P(\widehat{M} \le 2) = 1 - e^{-n(1-\log_e 2)} \) if \(\mu = 1 \)

B

\(P(\widehat{M} \le 1) = 1 - e^{-n \log_e 2} \) if \(\mu = 1 \)

C

\(P(\widehat{M} \le 3) = 1 - e^{-n(1-\log_e 2)} \) if \(\mu = 1 \)

D

\(P(\widehat{M} \le 4) = 1 - e^{-n(2\log_e 2 - 1)} \) if \(\mu = 1 \)

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13

2024 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2024

Let \( X_1, X_2, ..., X_n \) be a random sample of size \( n \) (n ≥ 2) from a population having probability density function
\[ f(x;\theta) = \begin{cases} \theta(2x)^{\theta-1} & \text{if } 0 < x \leq \frac{1}{2} \\ \theta(2-2x)^{\theta-1} & \text{if } \frac{1}{2} < x \leq 1 \\ 0 & \text{otherwise}, \end{cases} \]
where \( \theta > 0 \) is an unknown parameter. Then which one of the following is a maximum likelihood estimator of \( \theta \)?

A

\(n \left[ \sum_{\{i: X_i \leq \frac{1}{2}\}} \log_e 2X_i + \sum_{\{i: X_i > \frac{1}{2}\}} \log_e (2-2X_i) \right]^{-1} \)

B

\(-n \left[ \sum_{\{i: X_i \leq \frac{1}{2}\}} \log_e 2X_i + \sum_{\{i: X_i > \frac{1}{2}\}} \log_e (2-2X_i) \right]^{-1} \)

C

\(n \left[ \sum_{\{1 \leq i \leq n\}} \log_e 2X_i + \sum_{\{1 \leq i \leq n\}} \log_e (2-2X_i) \right]^{-1} \)

D

\(-n \left[ \sum_{\{1 \leq i \leq n\}} \log_e 2X_i + \sum_{\{1 \leq i \leq n\}} \log_e (2-2X_i) \right]^{-1} \)

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14

2025 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2025

Let \( x_1 = 0, x_2 = 1, x_3 = 1, x_4 = 1 \) and \( x_5 = 0 \) be observed values of a random sample of size 5 from \( \text{Bin}(1, \theta) \) distribution, where \( \theta \in (0, 0.7] \). Then the maximum likelihood estimate of \( \theta \) based on the above sample is ______________ (rounded off to two decimal places).

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15

2025 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2025

Let \( X_1, X_2, ..., X_7 \) be a random sample from a population having the probability density function
\( f(x) = \frac{1}{2} \lambda^3 x^2 e^{-\lambda x} \), \( x > 0 \),
where \( \lambda > 0 \) is an unknown parameter. Let \( \hat{\lambda} \) be the maximum likelihood estimator of \( \lambda \), and \( E(\hat{\lambda} - \lambda) = a\lambda \) be the corresponding bias, where \( a \) is a real constant. Then the value of \( \frac{1}{a} \) equals ______________ (answer in integer).

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16

2026 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2026

Let x₁, x₂, ..., xₙ (n ≥ 2) be the observed values of a random sample from the following probability density function
f(x) = { (λ^α / Γ(α)) x^{α-1} e^{-λx} if x > 0,
0 otherwise,
where α ∈ (0, ∞) and λ ∈ (0, ∞) are unknown parameters. If
(x₁ + x₂ + ⋯ + xₙ)/n = 2 and (x₁² + x₂² + ⋯ + xₙ²)/n = 5,
then the method of moments estimate of α equals __________ (answer in integer).

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17

2026 · Statistics · Estimation · Moment and Maximum-likelihood Estimation

Statistics (ST) 2026

Let \(X_1, X_2, \ldots, X_n\) (\(n \geq 2\)) be a random sample from the following probability density function
\[f(x) = \frac{1}{2} e^{-|x-\mu|}, \quad -\infty < x < \infty,\]
where \(\mu \in (-\infty, \infty)\) is an unknown parameter. Let \(\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i\) and \(\hat{\mu}\) denote the maximum likelihood estimator of \(\mu\), whenever it exists. Then which of the following statements is/are correct?

A

\(\hat{\mu}\) will always exist and \(\hat{\mu} = \bar{X}\)

B

\(\hat{\mu}\) may not always exist

C

\(\hat{\mu}\) always exists but it may not be unique

D

\(\hat{\mu}\) is a consistent estimator of \(\mu\)

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