ICAR AICE-JRF/SRF - Agricultural Statistics, Computer Application, Bioinformatics (PhD admission and fellowship) · India · National Testing Agency for ICAR
Original practice questions written to the published syllabus · not past papersThe algorithms inside modern statistics: random-number generation, Monte Carlo integration, the bootstrap and jackknife, permutation tests, the EM algorithm, MCMC and numerical linear algebra, with attention to accuracy and reproducibility.
In the exams: AICE-JRF Computer Application Unit 10 (present); GATE DA (present); PhD course in most programmes; light in national exams
How our questions for ICAR AICE-JRF/SRF spread across the units of Statistical Computing: Simulation, Resampling & Algorithms. Difficulty: 7 easy, 29 medium, 11 hard.
| Monte Carlo integration and variance reduction | 5 | |
| EM algorithm and its applications (missing data, mixtures) | 5 | |
| MCMC algorithms in practice | 5 | |
| Numerical linear algebra for statistics: QR and Cholesky | 5 | |
| Pseudo-random number generation and tests of randomness | 4 | |
| Generating random variates: inversion, acceptance-rejection and transformation methods | 4 | |
| Bootstrap: percentile and BCa intervals | 4 | |
| Jackknife | 4 | |
| Permutation and randomisation tests | 4 | |
| Parallel and high-performance computing for statistics | 4 | |
| Simulation studies: design and reporting | 3 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
In acceptance–rejection, the proposal density g must satisfy:
A. f(x) ≤ c g(x) for all x
The envelope must cover the target everywhere; lighter tails would violate it.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
The probability that a given observation is absent from a bootstrap sample of size n = 100 is about:
D. 0.37
(1 − 1/100)¹⁰⁰ ≈ 0.366, close to e⁻¹; 0.63 is the chance it appears.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
The jackknife estimate of bias is:
B. (n − 1)(θ̂₍.₎ − θ̂)
θ̂₍.₎ is the mean of the leave-one-out estimates.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
The jackknife variance estimate is inconsistent for:
C. The sample median
Leave-one-out medians change in jumps; use the bootstrap or delete-d jackknife.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
Jackknife pseudo-values are defined as:
D. nθ̂ − (n − 1)θ̂₍ᵢ₎
Their mean gives the bias-corrected estimate.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
EM convergence is slow when:
D. Much information is missing
The rate is linear, governed by the missing information.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
Theory suggests a target acceptance rate for random-walk Metropolis in high dimensions of about:
B. 0.23
About 0.44 in one dimension.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
Column pivoting in QR is used in regression software to:
C. Detect aliased columns
Aliased columns are reported as not estimable.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
How many replicates are needed for a Monte Carlo SE of 0.005 on 95% coverage?
B. 1,900
0.95 × 0.05/0.005² = 1,900.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
A useful check of a parallel simulation is that, with the same random streams, its results:
B. Equal those of a serial run
Reproducibility tests catch stream-assignment errors.
Further reading: Robert and Casella, Monte Carlo Statistical Methods (2004); Rizzo, Statistical Computing with R (2019); Efron and Tibshirani, An Introduction to the Bootstrap (1993); Givens and Hoeting, Computational Statistics (2013)
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