Indian Statistical Institute admission tests (BStat, MStat, JRF in Statistics) · India · Indian Statistical Institute
Original practice questions written to the published syllabus · not past papersMethods that hold without a normal model: the empirical distribution, goodness-of-fit, sign and rank tests for one, two and many samples, rank correlation, and tests used for scored or ranked agricultural data such as disease ratings and sensory panels.
In the exams: AICE-JRF Ag Stat Unit 2 (heavy: run, sign, rank, Wilcoxon, Kruskal-Wallis, Mann-Whitney, Cochran, Friedman named); GATE ST 'Non-parametric Statistics' (heavy); UPSC ISS Statistics II (present)
How our questions for Indian Statistical Institute admission tests spread across the units of Nonparametric & Rank-Based Methods. Difficulty: 20 easy, 27 medium, 9 hard.
| Mann-Whitney U test and two-sample Kolmogorov-Smirnov test | 9 | |
| Asymptotic relative efficiency | 7 | |
| Empirical distribution function and its properties | 6 | |
| Goodness of fit: chi-square and Kolmogorov-Smirnov tests | 6 | |
| Friedman's test for randomised blocks | 6 | |
| Sign test and Wilcoxon signed-rank test | 5 | |
| Spearman's and Kendall's rank correlations | 5 | |
| Kruskal-Wallis test | 4 | |
| Run test for randomness | 3 | |
| Analysing ordinal scores: disease ratings and sensory panels | 3 | |
| Cochran's Q test | 2 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What are E[Fₙ(x)] and Var[Fₙ(x)] for the empirical distribution function at a fixed x?
B. F(x) and F(x)[1 − F(x)]/n
Since nFₙ(x) ~ binomial(n, F(x)), Fₙ(x) is unbiased for F(x) with variance F(x)[1 − F(x)]/n. 1/(4n) is only the maximum of this variance, reached at the median.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Which statement compares Pearson's chi-square and the Kolmogorov-Smirnov goodness-of-fit tests correctly?
B. KS uses ungrouped continuous data; chi-square suits counts and grouped data
KS is exact and distribution-free for continuous, fully specified F₀ and avoids grouping. Chi-square handles discrete data and estimated parameters (with df adjustment) but loses information by grouping and ignores the order of classes.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
What does the Wilcoxon signed-rank test assume about the distribution of the paired differences that the sign test does not?
A. Symmetry of the differences
Both need independence and continuity; the signed-rank test also assumes symmetry, which is what lets each rank carry either sign with probability 1/2. Neither assumes normality.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Two independent samples have sizes 6 and 6, and sample 1's rank sum is W₁ = 25. What is the Mann-Whitney statistic U₁?
A. 4
U₁ = W₁ − n₁(n₁ + 1)/2 = 25 − 21 = 4. U₂ = n₁n₂ − U₁ = 32. 14 = 25 − 11 omits the correct offset.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Six treated and six untreated plants are scored for disease; the Mann-Whitney U for the treated group is 4. What is the normal-approximation Z, ignoring ties and continuity correction?
C. −2.24
E(U) = 36/2 = 18, Var(U) = 36 × 13/12 = 39, Z = (4 − 18)/√39 = −2.24. Using 36/4 as the variance gives −4.67.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
For k = 2 groups, how is the Kruskal-Wallis H related to the Mann-Whitney test?
D. H equals the square of the Mann-Whitney Z
With two groups H is the squared standardised rank-sum statistic, Z², and χ²₁ is the distribution of Z² for normal Z.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Under H₀ in Friedman's test with b blocks and k treatments, what are the mean and variance of a single treatment's rank sum Rⱼ?
D. Mean b(k + 1)/2, variance b(k² − 1)/12
In each block a treatment's rank is uniform on 1, ..., k, with mean (k + 1)/2 and variance (k² − 1)/12; summing over b independent blocks multiplies both by b.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Under independence, what is the variance of Kendall's tau for n = 10?
B. 0.0617
Var(τ) = 2(2n + 5)/[9n(n − 1)] = 50/810 = 0.0617. 1/(n − 1) = 0.1111 is the null variance of Spearman's ρ_s.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
How is the Mann-Whitney test related to the proportional-odds model for comparing two groups on an ordinal scale?
A. It is closely related to the score test for the group effect in that model
For two groups, the score test of β = 0 in the cumulative logit model is essentially the Wilcoxon rank-sum test with mid-ranks, which is why the two usually agree; the model adds odds-ratio estimates and covariates.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
A blocked trial has many treatments but only three blocks. Which rank test weights blocks by their range of responses and is often more powerful than Friedman's test here?
D. Quade's test
Quade's test ranks the blocks by their ranges and weights within-block ranks accordingly; with few blocks and many treatments it tends to beat Friedman. Page's test targets ordered alternatives.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
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