Indian Statistical Institute admission tests (BStat, MStat, JRF in Statistics) · India · Indian Statistical Institute
Original practice questions written to the published syllabus · not past papersHow to estimate well and know it: sufficiency and completeness, unbiasedness and minimum variance, the Cramer-Rao bound, the Rao-Blackwell and Lehmann-Scheffe theorems, the methods of moments, maximum likelihood and minimum chi-square, and confidence intervals.
In the exams: AICE-JRF Ag Stat Unit 1 (heavy: each theorem named); GATE ST 'Estimation' (heavy); IIT JAM MS (heavy); UPSC ISS Statistics I (heavy); ISI and CSIR-NET (present)
How our questions for Indian Statistical Institute admission tests spread across the units of Statistical Inference I: Estimation. Difficulty: 32 easy, 28 medium, 5 hard.
| Minimum chi-square and other methods | 8 | |
| Sufficiency, the factorisation theorem and minimal sufficiency | 7 | |
| Maximum likelihood: properties, invariance and asymptotic normality | 7 | |
| Iterative maximum likelihood: Newton-Raphson and Fisher scoring | 7 | |
| Point estimation: unbiasedness, consistency and efficiency | 6 | |
| Method of moments | 6 | |
| Interval estimation: pivotal quantities and coverage | 6 | |
| Cramer-Rao inequality and Fisher information | 5 | |
| Large-sample confidence intervals | 5 | |
| Completeness, ancillarity and Basu's theorem | 4 | |
| Minimum variance unbiased estimation; Rao-Blackwell and Lehmann-Scheffe theorems | 4 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Four observations from uniform (0, θ) have maximum 9.0. What is the UMVUE estimate of θ?
D. 11.25
(n + 1)X₍ₙ₎/n = 5 × 9/4 = 11.25, unbiased and a function of the complete sufficient X₍ₙ₎. 9.00 is the MLE, biased downward; 7.20 corrects in the wrong direction.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
What does the Rao-Blackwell theorem guarantee for an unbiased estimator U and a sufficient statistic T?
B. E(U | T) is unbiased with variance ≤ Var(U) for every θ
Conditioning on a sufficient statistic keeps unbiasedness and cannot raise variance (law of total variance). Uniqueness and UMVUE status need completeness as well (Lehmann-Scheffé); attaining the CRLB is not guaranteed.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
Counts are iid Poisson(λ). What is the Cramér-Rao lower bound for an unbiased estimator of ψ(λ) = e^(−λ), at λ = 2 with n = 10?
D. 0.0037
Bound = [ψ′(λ)]²/[nI(λ)] = e^(−2λ)/(n/λ) = λe^(−2λ)/n = 2e^(−4)/10 = 0.0037. 0.2000 is the bound for λ itself; omitting the square of ψ′ gives 0.0271; dividing by λ instead of multiplying gives 0.0009.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
Five observations from uniform (0, θ) have mean 3.1 and maximum 6.5. What is the method-of-moments estimate, and what is wrong with it?
B. 6.2, which is impossible because it is below the observed maximum of 6.5
E(X) = θ/2 gives θ̂ = 2X̄ = 6.2, yet θ must be at least 6.5. Moment estimates can fall outside the parameter space; the MLE X₍ₙ₎ = 6.5 or the UMVUE 7.8 respect it.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
In practice, what is the most common modern use of method-of-moments estimates in models fitted by maximum likelihood?
C. As starting values for maximum-likelihood iterations
Moment estimates are cheap, consistent and usually close enough for Newton-Raphson or scoring to converge from them. They are not more efficient, and they do not give standard errors directly.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
For multinomial data, minimising the likelihood-ratio statistic G² = 2ΣOⱼ log(Oⱼ/Eⱼ(θ)) over θ is equivalent to which estimation method?
C. Maximum likelihood
G² = 2[ΣOⱼ log Oⱼ − ΣOⱼ log Eⱼ(θ)], and the first term does not involve θ, so minimising G² maximises ΣOⱼ log pⱼ(θ), the multinomial log-likelihood.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
Ten plots of a new wheat variety give mean 4.2 t/ha and s = 0.5 t/ha. With t₀.₀₂₅,₉ = 2.262, what is the 95% confidence interval for the mean yield?
D. 3.84 to 4.56 t/ha
x̄ ± t s/√n = 4.2 ± 2.262 × 0.5/√10 = 4.2 ± 0.358. Using z = 1.96 ignores that σ is estimated; omitting √n gives the range for single plots, not for the mean.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
Why is the Wilson score interval preferred to the Wald interval for a binomial proportion?
D. It inverts the score test, keeps coverage near nominal for small n and stays inside [0, 1]
The Wilson interval uses the null standard error √(p(1 − p)/n) and solves for p, which corrects the Wald interval's undercoverage near 0 and 1. It is still an approximation and is not always narrower; exact (Clopper-Pearson) intervals guarantee coverage.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
What is the Newton-Raphson update for maximising a log-likelihood ℓ(θ) of one parameter?
C. θ₍ₖ₊₁₎ = θ₍ₖ₎ − ℓ′(θ₍ₖ₎) / ℓ″(θ₍ₖ₎)
Expanding ℓ′(θ) ≈ ℓ′(θₖ) + (θ − θₖ)ℓ″(θₖ) = 0 gives the update with a minus sign; near a maximum ℓ″ < 0, so the step moves in the direction of the score. θ − ℓ/ℓ′ solves ℓ = 0, the wrong equation.
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
For a generalised linear model with canonical link, such as logistic regression for germination data, which statement is correct?
C. Fisher scoring and Newton-Raphson coincide, each step being a weighted least-squares fit
With the canonical link the observed and expected information are equal, so the two algorithms are the same and take the iteratively reweighted least squares form. Starting values are still needed (usually from the data themselves).
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
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