Indian Statistical Service Examination (with Indian Economic Service) · India · Union Public Service Commission
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 Service Examination spread across the units of Nonparametric & Rank-Based Methods. Difficulty: 6 easy, 11 medium, 8 hard.
| Asymptotic relative efficiency | 5 | |
| Run test for randomness | 3 | |
| Kruskal-Wallis test | 3 | |
| Empirical distribution function and its properties | 2 | |
| Goodness of fit: chi-square and Kolmogorov-Smirnov tests | 2 | |
| Mann-Whitney U test and two-sample Kolmogorov-Smirnov test | 2 | |
| Friedman's test for randomised blocks | 2 | |
| Cochran's Q test | 2 | |
| Spearman's and Kendall's rank correlations | 2 | |
| Analysing ordinal scores: disease ratings and sensory panels | 1 | |
| Sign test and Wilcoxon signed-rank test | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
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
In a runs test for randomness, what do far too many runs suggest?
D. Alternation or negative dependence
Many short runs mean the two symbols alternate more than chance allows, as with regular spacing or negative serial correlation. Too few runs indicate clustering or trend.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
What is a weakness of the Wald-Wolfowitz two-sample runs test compared with the Mann-Whitney test?
D. Low power against a pure location shift
The runs test reacts to any difference in distribution, but for a location shift Mann-Whitney is much more powerful. Neither needs normality, and the runs test does detect spread differences.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Under randomness, how does the number of runs R behave for large n₁ and n₂?
C. Approximately normal with mean 2n₁n₂/N + 1
R is asymptotically normal (Wald and Wolfowitz), with the stated mean and variance 2n₁n₂(2n₁n₂ − N)/[N²(N − 1)], which justifies the Z approximation.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
What is the null hypothesis of Cochran's Q test?
D. The success probability is the same for all k treatments
Q tests equality of the treatment marginal success proportions, allowing blocks to differ. It does not assume p = 0.5 or test blocks.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Hodges and Lehmann proved a lower bound for the ARE of the Wilcoxon (or Mann-Whitney) test relative to the t-test, over all continuous symmetric distributions. What is it?
A. 0.864
108/125 = 0.864; it can never be lower, while it can be arbitrarily large for heavy-tailed distributions. 0.955 is the value at the normal, not the minimum.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
In the Wilcoxon signed-rank test, how is the null variance of T⁺ adjusted when some |dᵢ| are tied and given mid-ranks?
A. It is reduced by Σ(t³ − t)/48 over tie groups of size t
Mid-ranks keep E(T⁺) but lower the variance of the ranks, so Var(T⁺) = n(n + 1)(2n + 1)/24 − Σ(t³ − t)/48.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
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