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Probability, Statistics & Linear Programming

Estimation

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1. Sufficiency, completeness, UMVUE, Cramér–Rao

Read this as a chapter →

Lec-12 Sufficiency

NPTEL · Statistical Inference

The factorisation theorem.

Lec-14 Minimal Sufficiency, Completeness

NPTEL · Statistical Inference

Lec-15 UMVU Estimation, Ancillarity

NPTEL · Statistical Inference

Rao–Blackwell and Lehmann–Scheffé — the UMVUE machine.

Lec-09 Lower Bounds for Variance - II

NPTEL · Statistical Inference

Cramér–Rao, including when the regularity conditions fail.

Sufficiency, completeness and the UMVUE machine free

In plain words

A sufficient statistic holds everything the data says about the parameter. Complete and sufficient, plus unbiased, hands you the best estimator outright with no variance calculation needed.

Sufficiency

Factorisation theorem: T is sufficient ⇔ f(x|θ)=g(T(x),θ)h(x)\theta) = g(T(x), \theta)h(x).

Family Sufficient statistic
N(μ,σ2)N(\mu, \sigma^{2}), both unknown (∑Xi,∑Xi2)(\sum{}X_{i}, \sum{}X_{i}^{2})
Bernoulli/Binomial/Poisson ∑Xi\sum{}X_{i}
Uniform(0,θ)(0, \theta) X(n)X_{(n)}
Uniform(θ,θ+1)(\theta, \theta+1) (X(1),X(n))(X_{(1)}, X_{(n)})
Shifted exponential eθ−xe^{\theta-x}, x≥θx \ge \theta X(1)X_{(1)}
Exponential family the natural statistic

Minimal sufficient: use the ratio criterion — f(x|θ)/f(y\theta)/f(y|θ)\theta) is θ−\theta-free ⇔ T(x) = T(y).

Completeness

T is complete if Eθ[g(T)]=0E_\theta[g(T)] = 0 for all θ⇒g≡0\theta \Rightarrow g \equiv 0. Full-rank exponential families and the order statistics of Uniform(0,θ)(0,\theta) are complete. Uniform(−θ,θ)(-\theta, \theta) is not: X(n)X_{(n)} is sufficient but ∑Xi\sum{}X_{i} has zero expectation without being zero.

The UMVUE machine

  • Rao–Blackwell: conditioning any unbiased estimator on a sufficient T improves it.
  • Lehmann–Scheffé: if T is complete sufficient and g(T) is unbiased, then g(T) is the unique UMVUE.

Standard results:

Model Parameter UMVUE
Uniform(0,θ)(0,\theta) θ\theta (n+1)X(n)/n(n+1)X_{(n)}/n
Shifted exp eθ−xe^{\theta-x} θ\theta X(1)−1/nX_{(1)} - 1/n
N(μ,σ2)N(\mu,\sigma^{2}) μ\mu X̄
θxθ−1\theta{}x^{\theta-1} on (0,1) 1/θ1/\theta −(1/n)∑ln⁡Xi-(1/n)\sum{}\ln X_{i}
Bernoulli(p) p X̄

Cramér–Rao

Var(T)≥[ψ′(θ)]2/(Var(T) \ge [\psi'(\theta)]^{2}/(nI(θ))(\theta)) for unbiased T of ψ(θ)\psi(\theta), under regularity. Equality ⇔ the model is exponential family with T the natural statistic.

Regularity fails when the support depends on θ(\theta (uniform, shifted exponential) — there the UMVUE can beat the "bound", which is why those examples appear so often.

Key takeaways

  • Factorisation ⇒ sufficient; ratio criterion ⇒ minimal.
  • Complete + sufficient + unbiased = UMVUE (Lehmann–Scheffé).
  • Cramér–Rao does not apply when the support moves with θ\theta.

The Lehmann–Scheffé pipelineinteractive

This visual is part of Trap Analysis

Sufficient, then complete, then unbiased — the three steps that produce a UMVUE.

Unlock interactive visuals

Exam focus: Factorisation gives sufficiency; Lehmann–Scheffé turns a complete sufficient statistic plus unbiasedness into the UMVUE. Cramér–Rao gives a bound that is often not attained.

Practice

Part C2025 Decembershow ▾
Let X1,X2,…,Xn(n≥2)X_{1}, X_{2}, \dots, X_{n} (n\ge2) be a random sample from an Exponential distribution with the probability density function f(x∣μ,σ)=(1/σ)exp⁡((μ−x)/σ)\mu,\sigma) = (1/\sigma)\exp((\mu-x)/\sigma) if x>μ,0x>\mu, 0 otherwise, where parameters μ\mu and σ\sigma are unknown and positive. Let Xˉn,Sn2{\bar{X}}_{n}, S_{n}^{2} and X1:nX_{1}:_{n} denote the sample mean, the sample variance and the sample smallest order statistic, respectively, and let θ=σ/μ\theta = \sigma/\mu. Then which of the following statements are true?
  1. A.Sn(X1:n)S_{n}(X_{1}:_{n})⁻1^{1} is a consistent estimator of θ\theta✓ Correct
  2. B.(Xˉn−X1:n)(X1:n)({\bar{X}}_{n} - X_{1}:_{n})(X_{1}:_{n})⁻1^{1} is a consistent estimator of θ\theta✓ Correct
  3. C.(Xˉn−X1:n)(Xˉn−Sn)({\bar{X}}_{n} - X_{1}:_{n})({\bar{X}}_{n} - S_{n})⁻1^{1} is a consistent estimator of θ\theta✓ Correct
  4. D.Sn(Xˉn)S_{n}({\bar{X}}_{n})⁻1^{1} is a consistent estimator of θ\theta

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾
Let X1,X2,…,Xn(n>5)X_{1},X_{2},\dots,X_{n} (n>5) be independent random variables such that Xt=α+α2t+εtX_{t}=\alpha+\alpha^{2}t+\varepsilon_{t}, for t=1,…,n, where ε1,ε2,…,εn\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{n} are independent and identically distributed N(0,σ2)N(0,\sigma^{2}) random variables. Here α∈R\alpha\in\mathbb{R} and σ>0\sigma>0 are unknown parameters. Which of the following statements are true?
  1. A.(X1,X2,…,Xn)(X_{1},X_{2},\dots,X_{n}) is a sufficient statistic for (α,σ)(\alpha,\sigma)✓ Correct
  2. B.(∑tXt,∑t(\sum{}t X_{t}, \sum{}t tXt,∑tt2Xt2)_{t}, \sum{}t t^{2}X_{t}^{2}) is a jointly minimal sufficient statistic for (α,σ)(\alpha,\sigma)
  3. C.X4−X3−X2+X1X_{4}-X_{3}-X_{2}+X_{1} is an ancillary statistic
  4. D.(X4+X1−X2−X3)/(X5+X2−X3−X4)(X_{4}+X_{1}-X_{2}-X_{3})/(X_{5}+X_{2}-X_{3}-X_{4}) is an ancillary statistic✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
Let X1,X2,…,Xn(n≥3)X_{1}, X_{2}, \dots, X_{n} (n \ge 3) be a random sample from the uniform distribution on the interval (θ1−θ2,θ1+θ2)(\theta_{1} - \theta_{2}, \theta_{1} + \theta_{2}), where θ1∈R\theta_{1} \in \mathbb{R} and θ2>0\theta_{2} > 0 are unknown parameters. Let X(j)X_{(j)} be the jᵗʰ order statistic, j = 1, 2, …, n, and let Xˉ=(1/n)∑i{\bar{X}} = (1/n)\sum_{i}₌1nXi_{1}^{n} X_{i}. Here, (X(1),X(n))(X_{(1)}, X_{(n)}) is a complete and sufficient statistic for (θ1,θ2)(\theta_{1}, \theta_{2}). Then, which of the following statements are true?
  1. A.X̄ is an unbiased estimator of θ1\theta_{1}✓ Correct
  2. B.(Xˉ−X(1))({\bar{X}} - X_{(1)}) is an unbiased estimator of θ2\theta_{2}
  3. C.(X(1)+X(n))/2(X_{(1)} + X_{(n)})/2 is the uniformly minimum variance unbiased estimator of θ1\theta_{1}✓ Correct
  4. D.(n+1)(X(n)−X(1))/(2(n−1))(n + 1)(X_{(n)} - X_{(1)})/(2(n - 1)) is the uniformly minimum variance unbiased estimator of θ2\theta_{2}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
Let X1,X2,…,Xn(n≥2)X_{1}, X_{2}, \dots, X_{n} (n \ge 2) be a random sample from a continuous distribution with the probability density function f(x|θ)=\theta) = (1/2θ)e−∣x−θ∣/θ(1/2\theta)e^{-|x-\theta|/\theta}, x∈Rx \in \mathbb{R}, where θ(>0)\theta (> 0) is an unknown parameter. Let Un=(1/n)∑iU_{n} = (1/n)\sum_{i}₌1nXi,Vn=(1/n)∑i_{1}^{n} X_{i}, V_{n} = (1/n)\sum_{i}₌1nXi2_{1}^{n} X_{i}^{2}, and Sn2=(1/(n−1))∑iS_{n}^{2} = (1/(n-1))\sum_{i}₌1n(Xi−Un)2_{1}^{n}(X_{i} - U_{n})^{2}. Then, which of the following statements are true?
  1. A.UnU_{n} is an unbiased estimator of θ\theta✓ Correct
  2. B.Sn2S_{n}^{2} is an unbiased estimator of θ2\theta^{2}
  3. C.(1/3)Vn(1/3)V_{n} is a consistent estimator of θ2\theta^{2}✓ Correct
  4. D.The statistic (Un,Vn)(U_{n}, V_{n}) is complete

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾
Let X be a single sample from an absolutely continuous distribution with probability density function f(x∣θ)=(2/θ2)(θ−x)\theta) = (2/\theta^{2})(\theta-x) if 0<x<θ0 < x < \theta, and 0 otherwise, where θ>0\theta > 0 is unknown. Which of the following intervals is a 95% confidence interval for θ\theta?
  1. A.(X,X/(1−0.95))(X, X/(1-\sqrt{0.95}))✓ Correct
  2. B.(X, X/0.95)
  3. C.(X,X/0.95)(X, X/\sqrt{0.95})
  4. D.(0.95X, X)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1,X2,…,X13X_{1}, X_{2}, \dots, X_{13} be independent and identically distributed (i.i.d.) Poisson(θ)(\theta) random variables, where θ>0\theta > 0. Then, which of the following statements are true?
  1. A.(∑(i=1..8)Xi)(∑(i=8..13)Xi)(\sum(i=1..8)X_{i})(\sum(i=8..13)X_{i}) is an unbiased estimator of 48θ248\theta^{2}.
  2. B.(1/13)∑(i=1..13)Xi(1/13)\sum(i=1..13)X_{i} is method of moments estimator of θ\theta.✓ Correct
  3. C.There does not exist any unbiased estimator of e(−7θ)e^(-7\theta).
  4. D.(e(X7),∑(i=1..6)Xi,∑(i=8..13)Xi)(e^(X_{7}), \sum(i=1..6)X_{i}, \sum(i=8..13)X_{i}) is a sufficient statistic for θ\theta.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X, Y, Z be independent and identically distributed (i.i.d.) random variables, each following Bernoulli(θ),0<θ<1(\theta), 0 < \theta < 1. Then, which of the following statements are true?
  1. A.(79X+2024Y,23Z2)(79X + 2024Y, 23Z^{2}) is a sufficient statistic for θ\theta.✓ Correct
  2. B.13(X+Y+Z)313(X + Y + Z)^{3} is a minimal sufficient statistic for θ\theta.✓ Correct
  3. C.2(X + Y + Z) is a complete sufficient statistic for θ\theta.✓ Correct
  4. D.(X − 2Y + Z) is an ancillary statistic.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let X1,…,Xn(n≥3)X_{1}, \dots, X_{n} (n \ge 3) be a random sample from a distribution having probability density function f(x | θ)=θe(−θx)\theta) = \theta{}e^(-\theta{}x) if x > 0, and 0 otherwise, where θ>0\theta > 0 is an unknown parameter. Let Tn=(1/n)∑iT_{n} = (1/n)\sum_{i}₌1nXi_{1}^{n}X_{i}. Which of the following statements are true?
  1. A.Uniformly minimum variance unbiased estimator of θ\theta is (n − 1)/(nTn)_{n})✓ Correct
  2. B.Cramer-Rao lower bound for the variance of any unbiased estimator of θ\theta is θ2/n\theta^{2}/n✓ Correct
  3. C.Uniformly minimum variance unbiased estimator of θ\theta attains the Cramer-Rao lower bound
  4. D.(1−e(−1/Tn))(1 - e^(-1/T_{n})) is a consistent estimator of Pθ(X1≤1)P_\theta(X_{1} \le 1)✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let X1,…,XnX_{1}, \dots, X_{n} be independent and identically distributed U(0,θ),θ>0U(0, \theta), \theta > 0 random variables. Define X(n)=max⁡X_{(n)} = \max{X1,…,XnX_{1}, \dots, X_{n}} and X(1)=min⁡X_{(1)} = \min{X1,…,XnX_{1}, \dots, X_{n}}. Which of the following statements are true?
  1. A.Cov(X(n)/X(1),X(n))=0Cov(X_{(n)}/X_{(1)}, X_{(n)}) = 0✓ Correct
  2. B.E(X(1)/X(n))=E(X(1))/E(X(n))E(X_{(1)}/X_{(n)}) = E(X_{(1)})/E(X_{(n)})✓ Correct
  3. C.Cov(X(1)/X(n),X(n))=0Cov(X_{(1)}/X_{(n)}, X_{(n)}) = 0✓ Correct
  4. D.Cov(ln⁡(X(1))−ln⁡(X(1)+X(n)),X(n))<0Cov(\ln(X_{(1)}) - \ln(X_{(1)} + X_{(n)}), X_{(n)}) < 0

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Juneshow ▾
Let {Xi:1≤i≤2nX_{i} : 1 \le i \le 2n} be i.i.d. normal with mean μ\mu and variance 1, independent of a standard Cauchy random variable W. Which of the following statistics are consistent for μ\mu?
  1. A.n⁻1∑i=1n^{1} \sum_{i=1}^{n} XiX_{i}✓ Correct
  2. B.n⁻1∑i=12n^{1} \sum_{i=1}^{2n} XiX_{i}
  3. C.n⁻1∑i=1n^{1} \sum_{i=1}^{n} X2iX_{2i}₋1_{1}✓ Correct
  4. D.n⁻1(∑i=1n^{1}(\sum_{i=1}^{n} Xi+W)X_{i} + W)✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾
For n ≥ 2, let X1,…,XnX_{1}, \dots, X_{n} be a random sample with density f(x|θ)=\theta) = θxθ−1\theta{}x^{\theta-1} on 0<x<1,θ>00 < x < 1, \theta > 0 unknown. Which of the following is the UMVUE for 1/θ1/\theta?
  1. A.−(1/n)∑ln⁡Xi-(1/n) \sum \ln X_{i}✓ Correct
  2. B.−n/∑ln⁡Xi-n/\sum \ln X_{i}
  3. C.−(n−1)/∑ln⁡Xi-(n - 1)/\sum \ln X_{i}
  4. D.−(2/n)∑ln⁡Xi-(2/n) \sum \ln X_{i}

Solution

−ln⁡Xi-\ln X_{i} ~ Exp(θ)(\theta) with mean 1/θ1/\theta, and ∑ln⁡Xi\sum \ln X_{i} is complete sufficient; the sample mean of −ln⁡Xi-\ln X_{i} is unbiased for 1/θ1/\theta.

Save for spaced revision (free account)
Part B2023 Juneshow ▾
Let X be a Poisson random variable with mean λ\lambda. Which of the following parametric functions is not estimable?
  1. A.λ\lambda⁻1^{1}✓ Correct
  2. B.λ\lambda
  3. C.λ2\lambda^{2}
  4. D.e−λe^{-\lambda}

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾
For n ≥ 2, let ε1,…,εn\varepsilon_{1}, \dots, \varepsilon_{n} be i.i.d.N(0,σ2)i.i.d. N(0, \sigma^{2}) and Yi=iα+i2α2+εiY_{i} = i\alpha + i^{2}\alpha^{2} + \varepsilon_{i}, where σ>0\sigma > 0 and α∈R\alpha \in \mathbb{R} are unknown. Which of the following is a jointly minimal sufficient statistic for (α,σ)(\alpha, \sigma)?
  1. A.(∑Yi2,∑(\sum{}Y_{i}^{2}, \sum iYi,∑i2Yi)_{i}, \sum i^{2}Y_{i})✓ Correct
  2. B.(∑Yi2,∑(\sum{}Y_{i}^{2}, \sum iYi,∑i2Yi2)_{i}, \sum i^{2}Y_{i}^{2})
  3. C.(∑(\sum iYi,∑i2Yi2)_{i}, \sum i^{2}Y_{i}^{2})
  4. D.(∑Yi,∑(\sum{}Y_{i}, \sum iYi)_{i})

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let X1,…,XnX_{1}, \dots, X_{n} be i.i.d. Uniform(0,θ)(0, \theta). Which are true?
  1. A.X(n)X_{(n)} is sufficient for θ\theta✓ Correct
  2. B.X(n)X_{(n)} is complete✓ Correct
  3. C.(n+1)X(n)/n(n+1)X_{(n)}/n is the UMVUE of θ\theta✓ Correct
  4. D.2X̄ is the UMVUE of θ\theta

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let X1,…,XnX_{1}, \dots, X_{n} be i.i.d.N(μ,1)i.i.d. N(\mu, 1). Which of the following are true?
  1. A.X̄ is sufficient for μ\mu.✓ Correct
  2. B.X̄ is complete for the family {N(μ,1):μ∈RN(\mu, 1) : \mu \in \mathbb{R}}.✓ Correct
  3. C.X̄ is the UMVUE of μ\mu.✓ Correct
  4. D.The sample median is also a sufficient statistic for μ\mu.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Which of the following statements about point estimation are true?
  1. A.If a UMVUE exists, it is almost surely unique.✓ Correct
  2. B.Conditioning an unbiased estimator on a sufficient statistic yields an unbiased estimator with variance no larger than the original.✓ Correct
  3. C.Every unbiased estimator attains the Cramér–Rao lower bound.
  4. D.Completeness of a statistic implies its sufficiency.

The worked solution and the per-option analysis are in Trap Analysis.

2. MLE and method of moments

Read this as a chapter →

Lec-04 Finding Estimators - I

NPTEL · Statistical Inference

Method of moments and maximum likelihood side by side.

Lec-07 Properties of MLEs

NPTEL · Statistical Inference

Invariance, consistency, asymptotic normality — and the failures.

Lec-18 Bayes and Minimax Estimation - I

NPTEL · Statistical Inference

Bayes estimators; posterior mean under squared error loss.

Maximum likelihood and the method of moments free

In plain words

Maximum likelihood picks the parameter making the data you saw most probable, and it transforms correctly under any relabelling. It is often biased, and sometimes sits on a boundary.

MLE

Maximise L(θ)=∏f(xiL(\theta) = \prod{}f(x_{i}|θ)\theta), usually via ℓ=log⁡L\ell = \log L. Check the boundary when the support depends on θ\theta — differentiation will not find it.

Model MLE
N(μ,σ2)N(\mu, \sigma^{2}) μ\mû =Xˉ,σ= {\bar{X}}, \sigmâ2=(1/n)∑(Xi−Xˉ)2(^{2} = (1/n)\sum(X_{i} - {\bar{X}})^{2} (biased)
Bernoulli(p) X̄
Poisson(λ)(\lambda) X̄
Exponential(λ)(\lambda) 1/X̄
Uniform(0,θ)(0,\theta) X(n)(X_{(n)} (boundary, not a derivative)
Uniform(θ,θ+1)(\theta, \theta+1) any θ∈[X(n)−1,X(1)]\theta \in [X_{(n)} - 1, X_{(1)}] — not unique
Shifted exp eθ−xe^{\theta-x} X(1)X_{(1)}
θxθ−1\theta{}x^{\theta-1} −n/∑ln⁡Xi-n/\sum{}\ln X_{i}

Properties. Invariance: the MLE of g(θ)g(\theta) is g(θg(\thetâ). Consistent and asymptotically N(θ,1/(N(\theta, 1/(nI(θ)))(\theta))) under regularity. But: may be biased (σ(\sigmâ2,X(n))^{2}, X_{(n)}), may not be unique, may not exist.

Method of moments

Equate sample and population moments. Easy but generally inefficient and sometimes absurd (it can give estimates outside the parameter space).

For f(x|θ)=\theta) = eθ−xe^{\theta-x}, x≥θ:E[X]=θ+1x \ge \theta: E[X] = \theta + 1, so the moment estimator is X̄ − 1, while the MLE is X(1)X_{(1)} and the UMVUE is X(1)−1/nX_{(1)} - 1/n.

Bayes estimators

Posterior ∝ prior × likelihood; under squared error loss the Bayes estimate is the posterior mean.

Conjugate pairs: Beta–Binomial, Gamma–Poisson, Normal–Normal. With Beta(a,b) prior and x heads in n tosses, the posterior is Beta(a+x, b+n−x) with mean (a+x)/(a+b+n).

Key takeaways

  • Support depends on θ⇒\theta \Rightarrow maximise by inspection, not calculus.
  • MLE is invariant; UMVUE is not.
  • Bayes under squared error = posterior mean; use conjugacy to write it down.

When the MLE is not found by differentiatinginteractive

This visual is part of Trap Analysis

The uniform case, where calculus finds nothing and the answer sits on the boundary.

Unlock interactive visuals

Exam focus: MLE is invariant and asymptotically efficient but can be biased and need not be unique; moment estimators are easy but usually inefficient.

Practice

Part B2025 Juneshow ▾
Consider a discrete random variable X with the probability mass function P(X=0)=θ/3,P(X=1)=1−θ/2,P(X=2)=θ/6P(X = 0) = \theta/3, P(X = 1) = 1 - \theta/2, P(X = 2) = \theta/6, where θ∈(0,1)\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 θ\theta is
  1. A.1/3
  2. B.1/2
  3. C.2/3✓ Correct
  4. D.3/4

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾
Let X1,X2,…,Xn(n≥2)X_{1}, X_{2}, \dots, X_{n} (n\ge2) be a random sample from Uniform[−3θ/2,θ/2][-3\theta/2, \theta/2] distribution, where θ>0\theta>0 is an unknown parameter. Let Xˉ,X(1){\bar{X}}, X_{(1)} and X(n)X_{(n)} denote respectively the mean, the smallest order statistic and the largest order statistic of the sample. Then which of the following statements are true?
  1. A.The method of moments estimator of θ\theta is −2X̄✓ Correct
  2. B.The method of moments estimator of θ\theta is 2X̄
  3. C.The maximum likelihood estimator of θ\theta is min{−2X(1)/3,2X(n)-2X_{(1)}/3, 2X_{(n)}}
  4. D.The maximum likelihood estimator of θ\theta is max{−2X(1)/3,2X(n)-2X_{(1)}/3, 2X_{(n)}}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾
Let X be a Binomial(n, p) random variable, where n ∈ {5, 6} and p ∈ {1/4, 3/4}. If X = 3 is observed, then the maximum likelihood estimate of (n, p) is
  1. A.(5, 1/4)
  2. B.(5, 3/4)✓ Correct
  3. C.(6, 3/4)
  4. D.(6, 1/4)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1,X2,…,X10X_{1}, X_{2}, \dots, X_{10} be a random sample of size 10 from a Bernoulli distribution with success probability 1/(1+eθ)1/(1 + e^\theta), where θ∈R\theta \in \mathbb{R} is an unknown parameter. If M=∑(i=1..10)XiM = \sum(i=1..10)X_{i}, then which of the following statements are true?
  1. A.For M = 0, the maximum likelihood estimate of θ\theta is equal to 0.
  2. B.For M = 10, the maximum likelihood estimate of θ\theta does not exist.✓ Correct
  3. C.The maximum likelihood estimator of θ\theta exists for M ∈ {1, 2, …, 9} and is equal to ln(10/M − 1).✓ Correct
  4. D.The method of moments estimator of θ\theta exists and is equal to M/10.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let X1,…,Xn(n≥2)X_{1}, \dots, X_{n} (n \ge 2) be a random sample from aU(−θ,2θ)a U(-\theta, 2\theta) distribution, where θ>0\theta > 0 is an unknown parameter. Let Xˉ=(1/n)∑i{\bar{X}} = (1/n)\sum_{i}₌1nXi,X(1)=min⁡_{1}^{n}X_{i}, X_{(1)} = \min{X1,…,XnX_{1}, \dots, X_{n}} and X(n)=max⁡X_{(n)} = \max{X1,…,XnX_{1}, \dots, X_{n}}. Which of the following statements are true?
  1. A.Maximum likelihood estimator of θ\theta is min{X(1),X(n)/2X_{(1)}, X_{(n)}/2}
  2. B.Maximum likelihood estimator of θ\theta is max{−X(1),X(n)/2-X_{(1)}, X_{(n)}/2}✓ Correct
  3. C.Method of moments estimator of θ\theta is 2X̄✓ Correct
  4. D.Method of moments estimator of θ\theta is 2X̄/3

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n} be a random sample of size n from U(θ,θ+1)U(\theta, \theta + 1) distribution, where θ∈R\theta \in \mathbb{R} is an unknown parameter. If Tn=max⁡T_{n} = \max{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}, n = 1, 2, …, then which of the following statements are true?
  1. A.TnT_{n} is a consistent estimator of θ\theta.
  2. B.TnT_{n} is an unbiased estimator of θ\theta.
  3. C.Tn−1T_{n} - 1 is a consistent estimator of θ\theta.✓ Correct
  4. D.lim⁡(n→∞)Eθ(Tn)=θ+1,∀θ∈R\lim(n\to\infty) E_\theta(T_{n}) = \theta + 1, \forall\theta \in \mathbb{R}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Consider a manufacturing unit producing three-component series systems, where component's lifetimes are independent and identically distributed random variables with hazard function h(x)=λh(x) = \lambda if x > 0, and 0 otherwise. Here, λ>0\lambda > 0 is an unknown parameter. Let X1,X2,X3X_{1}, X_{2}, X_{3} be a random sample of size 3 on lifetimes of systems produced by the manufacturing unit. Let θ=Pλ(X1>1/2)\theta = P_\lambda(X_{1} > 1/2). If θ\thetâ is the maximum likelihood estimate of θ\theta based on the realization x1=1/2,x2=1/6,x3=1/3x_{1} = 1/2, x_{2} = 1/6, x_{3} = 1/3 of the random sample, then which of the following statements are true?
  1. A.θ\thetâ ∈ (0.0, 0.2)
  2. B.θ\thetâ ∈ (0.1, 0.3)✓ Correct
  3. C.θ\thetâ ∈ (0.2, 0.4)✓ Correct
  4. D.θ\thetâ ∈ (0.3, 0.5)

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Let X and Y be independent random variables such that X follows U(0, 1) distribution and Y follows Bernoulli distribution with success probability p ∈ (0, 1). Define Z = X + Y. Let z1=0.5,z2=1.2,z3=1.3,z4=0.9,z5=0.1,z6=0.7z_{1} = 0.5, z_{2} = 1.2, z_{3} = 1.3, z_{4} = 0.9, z_{5} = 0.1, z_{6} = 0.7 be the observed values from the distribution of Z. Then the maximum likelihood estimate of p equals
  1. A.1/4
  2. B.1/2
  3. C.1/3✓ Correct
  4. D.1/6

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾
Consider a distribution with probability mass function f(x | θ)=(1−θ)/2\theta) = (1 - \theta)/2 if x = 0; 1/2 if x=1;θ/2x = 1; \theta/2 if x = 2; and 0 otherwise, where θ∈(0,1)\theta \in (0, 1) is an unknown parameter. In a random sample of size 100 from the above distribution, the observed counts of 0, 1 and 2 are 20, 30 and 50 respectively. Then, the maximum likelihood estimate of θ\theta based on the observed data is
  1. A.1
  2. B.5/7✓ Correct
  3. C.1/2
  4. D.2/7

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Let X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n} be a random sample from distribution Poisson(θ),θ>0(\theta), \theta > 0. Let π(θ)=e(−θ),θ>0\pi(\theta) = e^(-\theta), \theta > 0, be the prior distribution of θ\theta. Under the squared error loss function, which of the following is the Bayes estimator of e(−2θ)e^(-2\theta)?
  1. A.((n+1)/(n+3))(∑(i=1..n)Xi+1)((n+1)/(n+3))^(\sum(i=1..n)X_{i} + 1)✓ Correct
  2. B.((n+1)/(n+3))e(−(2/n)∑(i=1..n)Xi)((n+1)/(n+3))e^(-(2/n)\sum(i=1..n)X_{i})
  3. C.((n+1)/(n+3))(∑(i=1..n)Xi)((n+1)/(n+3))^(\sum(i=1..n)X_{i})
  4. D.((n+1)/(n+3))e(−(2/n)(∑(i=1..n)Xi+1))((n+1)/(n+3))e^(-(2/n)(\sum(i=1..n)X_{i} + 1))

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n} be a random sample of size n from U(θ,θ+1)U(\theta, \theta + 1) distribution, where θ∈R\theta \in \mathbb{R} is the unknown parameter. If Tn=min⁡T_{n} = \min{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}, n = 1, 2, …, then which of the following statements are true?
  1. A.TnT_{n} is an unbiased estimator of θ\theta.
  2. B.lim⁡(n→∞)Eθ(Tn)=θ\lim(n\to\infty) E_\theta(T_{n}) = \theta for all θ∈R\theta \in \mathbb{R}.✓ Correct
  3. C.TnT_{n} is a consistent estimator of θ\theta.✓ Correct
  4. D.max{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}} is a consistent estimator of θ\theta.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾
Let X1,…,X10X_{1}, \dots, X_{10} be a random sample from a distribution with the probability density function f(x|θ)=θx(θ−1)\theta) = \theta{}x^(\theta-1) if 0 < x < 1, and 0 otherwise, where θ>0\theta > 0 is an unknown parameter. The prior distribution of θ\theta is given by π(θ)=θe(−θ)\pi(\theta) = \theta{}e^(-\theta) if θ>0\theta > 0, and 0 otherwise. The Bayes estimator of θ\theta under squared error loss is
  1. A.12/(1−∑i12 / (1 - \sum_{i}₌110ln⁡Xi)_{1}^{10} \ln X_{i})✓ Correct
  2. B.11/(2−∑i11 / (2 - \sum_{i}₌110ln⁡Xi)_{1}^{10} \ln X_{i})
  3. C.(3+∑i(3 + \sum_{i}₌110ln⁡Xi)/13_{1}^{10} \ln X_{i}) / 13
  4. D.(2+∑i(2 + \sum_{i}₌110ln⁡Xi)/11_{1}^{10} \ln X_{i}) / 11

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n} be a random sample from N(θ,1)N(\theta, 1) distribution, where θ∈R\theta \in \mathbb{R} is unknown. Let δn\delta_{n} be the Bayes estimator of θ\theta, under the squared error loss function L(θ,a)=(a−θ)2,a,θ∈RL(\theta, a) = (a - \theta)^{2}, a, \theta \in \mathbb{R} and the prior distribution N(1, 2). If (1/n)[(2n+1)δn−1−2nθ](1/\sqrt{n})[(2n + 1)\delta_{n} - 1 - 2n\theta] converges in distribution to a random variable Z, as n→∞n \to \infty, then which of the following statements are true?
  1. A.δn\delta_{n} converges in probability to θ\theta, as n→∞n \to \infty, for all θ∈R\theta \in \mathbb{R}✓ Correct
  2. B.Z follows normal distribution.✓ Correct
  3. C.E(Z4)=48E(Z^{4}) = 48✓ Correct
  4. D.E(Z2)=1E(Z^{2}) = 1

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Suppose that the probability density function of the random variable X is f(x)=2(θ−x)/θ2f(x) = 2(\theta - x)/\theta^{2} if 0≤x≤θ0 \le x \le \theta, and 0 otherwise, where θ>0\theta > 0 is an unknown parameter. Based on a single observation X, the confidence coefficient of the confidence interval [(2/5)X, (5/2)X] for θ\theta is
  1. A.0.36✓ Correct
  2. B.0.55
  3. C.0.76
  4. D.0.95

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Let X1,X2,…,Xn(n≥2)X_{1}, X_{2}, \dots, X_{n} (n \ge 2) be a random sample from Uniform(θ,1−θ)(\theta, 1 - \theta) distribution, where −∞<θ<1/2-\infty < \theta < 1/2. The maximum likelihood estimator of θ\theta is
  1. A.min{1 − min{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}, max{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}}
  2. B.max{1 − min{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}, max{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}}
  3. C.min{min{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}, 1 − max{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}}✓ Correct
  4. D.max{min{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}, 1 − max{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}}

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n} be a random sample from an exponential distribution with probability density function f(x|θ)=e(−(x−θ))\theta) = e^(-(x-\theta)) if x≥θx \ge \theta, and 0 otherwise, where θ∈R\theta \in \mathbb{R} is an unknown parameter. If X(1)=min⁡X_{(1)} = \min{X1,X2,…,XnX_{1}, X_{2}, \dots, X_{n}}, then which of the following statements are true?
  1. A.(X(1)−(2/n)ln⁡5,X(1))(X_{(1)} - (2/n)\ln 5, X_{(1)}) is a 97% confidence interval for θ\theta.
  2. B.(X(1)−(2/n)ln⁡5,X(1))(X_{(1)} - (2/n)\ln 5, X_{(1)}) is a 96% confidence interval for θ\theta.✓ Correct
  3. C.(X(1)−(1/n)ln⁡20,X(1))(X_{(1)} - (1/n)\ln 20, X_{(1)}) is a 95% confidence interval for θ\theta.✓ Correct
  4. D.(X(1)−(1/n)ln⁡20,X(1))(X_{(1)} - (1/n)\ln 20, X_{(1)}) is a 96% confidence interval for θ\theta.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let Y1,…,Yn(n≥2)Y_{1}, \dots, Y_{n} (n \ge 2) be independent observations; YiY_{i} ~ N(βxi,σ2),i=1,…,nN(\beta{}x_{i}, \sigma^{2}), i = 1, \dots, n; where x1,…,xnx_{1}, \dots, x_{n} and σ2(>0)\sigma^{2}(> 0) are known constants and β∈R\beta \in \mathbb{R} is an unknown parameter. Consider N(β0,τ2)N(\beta_{0}, \tau^{2}) prior for the parameter β\beta, where β0\beta_{0} and τ2(>0)\tau^{2}(> 0) are known constants, and N(μ,λ2)N(\mu, \lambda^{2}) denotes a normal distribution with mean μ\mu and variance λ2\lambda^{2}. Suppose ȳ =(1/n)∑i= (1/n)\sum_{i}₌1nyi_{1}^{n}y_{i} and xˉ=(1/n)∑i{\bar{x}} = (1/n)\sum_{i}₌1nxi_{1}^{n}x_{i} are observed sample means. Under squared error loss function, which of the following statements are true?
  1. A.Bayes estimate of β\beta tends to β0\beta_{0} as τ2→0\tau^{2} \to 0✓ Correct
  2. B.Bayes estimate of β\beta tends to ȳ/x̄ as τ2→0\tau^{2} \to 0
  3. C.Bayes estimate of β\beta tends to the BLUE of β\beta as τ2→∞\tau^{2} \to \infty✓ Correct
  4. D.Bayes estimate of β\beta tends to MLE of β\beta as τ2→∞\tau^{2} \to \infty✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾
The probability of a head in tossing a coin is p ∈ (0, 1). The coin is independently tossed 25 times and heads appear 10 times. The Bayes estimate of p with respect to the prior Beta(5, 5) and squared error loss is
  1. A.3/7✓ Correct
  2. B.3/5
  3. C.1/2
  4. D.2/5

Solution

Posterior is Beta(5 + 10, 5 + 15) = Beta(15, 20) with mean 15/35 = 3/7.

Save for spaced revision (free account)
Part C2023 Decembershow ▾
Let X1,…,XnX_{1}, \dots, X_{n} be a random sample with density f(x|θ)=\theta) = eθ−xe^{\theta-x} for x≥θx \ge \theta, where θ∈R\theta \in \mathbb{R} is unknown. Define X̄ and X(1)=min⁡XiX_{(1)} = \min X_{i}. Which of the following statements are true?
  1. A.X̄ is the method of moments estimator of θ\theta
  2. B.X(1)X_{(1)} is the maximum likelihood estimator of θ\theta✓ Correct
  3. C.X(1)−1/nX_{(1)} - 1/n is the uniformly minimum variance unbiased estimator of θ\theta✓ Correct
  4. D.X(1)X_{(1)} is a sufficient statistic for θ\theta✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾
Let X1,…,XnX_{1}, \dots, X_{n} be i.i.d. with mean μ\mu and finite variance. As an estimator of μ\mu, the sample mean X̄ is
  1. A.unbiased and consistent✓ Correct
  2. B.unbiased but not consistent
  3. C.biased but consistent
  4. D.biased and inconsistent

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let X1,…,XnX_{1}, \dots, X_{n} be i.i.d. Uniform(0,θ),θ>0(0, \theta), \theta > 0 unknown. Which of the following are true?
  1. A.The MLE of θ\theta is X(n)=max⁡XiX_{(n)} = \max X_{i}.✓ Correct
  2. B.The method-of-moments estimator of θ\theta is 2X̄.✓ Correct
  3. C.The MLE is unbiased.
  4. D.E[X(n)]=nθ/(n+1)E[X_{(n)}] = n\theta/(n + 1), so ((n+1)/n)⋅X(n)((n + 1)/n)\cdot{}X_{(n)} is unbiased.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.