ASRB NET - Agricultural Statistics (and Computer Application / Bioinformatics) · India · Agricultural Scientists Recruitment Board (ICAR)
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)
ASRB NET / ARS Agricultural Statistics
Applied estimation: standard errors of means, proportions and ratios, sample size for a given precision, iterative estimation in bioassay and genetics.
How our questions for ASRB NET spread across the units of Statistical Inference I: Estimation. Difficulty: 14 easy, 12 medium, 2 hard.
| Minimum variance unbiased estimation; Rao-Blackwell and Lehmann-Scheffe theorems | 4 | |
| Method of moments | 4 | |
| Large-sample confidence intervals | 4 | |
| Maximum likelihood: properties, invariance and asymptotic normality | 3 | |
| Minimum chi-square and other methods | 3 | |
| Interval estimation: pivotal quantities and coverage | 3 | |
| Point estimation: unbiasedness, consistency and efficiency | 2 | |
| Sufficiency, the factorisation theorem and minimal sufficiency | 2 | |
| Iterative maximum likelihood: Newton-Raphson and Fisher scoring | 2 | |
| Cramer-Rao inequality and Fisher information | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
X₁, ..., Xₙ are iid with density θx^(θ−1) on (0, 1), θ > 0. Which statistic is sufficient for θ?
A. Σ log Xᵢ
L = θⁿ(ΠXᵢ)^(θ−1) = θⁿ exp[(θ − 1)Σ log Xᵢ], so ΠXᵢ, equivalently Σ log Xᵢ, is sufficient. This is a beta(θ, 1) exponential family with T(x) = log x.
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
What does the Lehmann-Scheffé theorem state?
B. An unbiased estimator that is a function of a complete sufficient statistic is the unique UMVUE
Completeness ensures uniqueness of the unbiased function of T, and Rao-Blackwell ensures it beats every other unbiased estimator. Sufficiency without completeness is not enough, and the MLE is often biased (uniform maximum, normal σ̂²).
Further reading: Casella and Berger, Statistical Inference; Lehmann and Casella, Theory of Point Estimation; Rohatgi and Saleh, An Introduction to Probability and Statistics
Aphid counts per leaf have mean 2 and variance 6. What is the method-of-moments estimate of the negative binomial k?
A. 1.0
From variance = m + m²/k: k̂ = X̄²/(S² − X̄) = 4/4 = 1.0. 3.0 is the dispersion index S²/X̄.
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
How many farms must be surveyed to estimate an adoption proportion within ±0.05 with 95% confidence, when nothing is known about p?
A. 385
n = 1.96² p(1 − p)/d² with the worst case p = 0.5: 3.8416 × 0.25/0.0025 = 384.2, so 385. 2000 is the Chebyshev requirement; 20 forgets to square; 1537 omits the factor p(1 − p) = 0.25.
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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