IIT JAM - Mathematical Statistics (MS) · India · IITs (JAM 2026 organised by IIT Bombay)
Original practice questions written to the published syllabus · not past papersAnalysing many correlated variables together: the multivariate normal and its estimation, Wishart and Hotelling's T-squared, MANOVA, and the reduction and classification tools agricultural scientists use daily (principal components, factor analysis, discriminant and cluster analysis, Mahalanobis D-squared).
In the exams: AICE-JRF Ag Stat Unit 2 (heavy: MVN, Wishart, Hotelling, MANOVA, PCA, FA, discriminant, cluster named); GATE ST 'Multivariate Analysis' (heavy); UPSC ISS Statistics II (heavy); GATE DA PCA (present)
How our questions for IIT JAM spread across the units of Multivariate Analysis. Difficulty: 20 easy, 44 medium, 9 hard.
| Multivariate normal distribution: properties, marginals and conditionals | 7 | |
| Estimation of the mean vector and dispersion matrix | 7 | |
| Discriminant analysis and classification | 7 | |
| Cluster analysis: hierarchical and k-means | 7 | |
| Hotelling's T-squared and its applications | 6 | |
| Multivariate analysis of variance (MANOVA) | 6 | |
| Principal component analysis | 6 | |
| Factor analysis | 6 | |
| Canonical correlation analysis | 6 | |
| Multidimensional scaling and correspondence analysis | 6 | |
| Wishart distribution | 5 | |
| Mahalanobis D-squared and genetic divergence | 4 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Which condition is needed for the sample dispersion matrix S of p variables to be non-singular (with probability one, under normality)?
A. n > p
S has rank at most n − 1, so n − 1 ≥ p, i.e. n > p, is necessary; under normality it is then non-singular with probability one. In practice n should be several times p.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Two traits keep the same variances, but their correlation rises from 0.5 to 0.9. What happens to the generalised variance |S|?
D. It decreases
|S| = s₁²s₂²(1 − r²), which shrinks as |r| grows: the scatter ellipse flattens. The total variance tr(S) would be unchanged. |S| cannot be negative for a valid dispersion matrix.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
W₁ ~ Wₚ(m₁, Σ) and W₂ ~ Wₚ(m₂, Σ) are independent. What is the distribution of W₁ + W₂?
D. Wₚ(m₁ + m₂, Σ)
Additivity: pooling independent Wishart matrices with the same Σ adds degrees of freedom, which is why pooled within-group SSCP matrices are Wishart.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
For p = 1, how is Hotelling's T² related to Student's t?
A. T² = t²
With one variable, S⁻¹ = 1/s², so T² = n(x̄ − μ₀)²/s² = t², and the F(1, n − 1) reference distribution is that of t².
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
A standardised variable has loadings 0.8 and 0.3 on two orthogonal factors. What is its communality?
C. 0.73
h² = 0.8² + 0.3² = 0.64 + 0.09 = 0.73; its uniqueness is 1 − 0.73 = 0.27. Adding loadings without squaring gives 1.10.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Canonical correlations 0.6 and 0.3 are found between 3 soil and 2 crop variables on 50 fields. What is Bartlett's chi-square for H₀: all ρᵢ = 0, and its df?
B. 24.87 with 6 df
−[n − 1 − (p + q + 1)/2]Σ ln(1 − rᵢ²) = −46 × [ln 0.64 + ln 0.91] = 24.87, with pq = 6 df. Omitting the correction gives 26.49; using ln(1 − r) instead of ln(1 − r²) is a slip.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
What does the cophenetic correlation measure?
C. Dendrogram fidelity to original distances
Values close to 1 mean the tree represents the original dissimilarities faithfully; low values warn that the dendrogram distorts the data.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
For uncorrelated traits with standard deviations σⱼ, what does D² between two population means reduce to?
C. Σ(δⱼ/σⱼ)²
With diagonal Σ, Σ⁻¹ has entries 1/σⱼ², so D² = Σδⱼ²/σⱼ²: the sum of squared standardised differences.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Classical multidimensional scaling of Euclidean distances computed from a data matrix gives the same configuration as which method?
A. Principal component analysis
Principal coordinate analysis of Euclidean distances reproduces the PCA scores (up to sign and rotation).
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Farm-household survey data contain many categorical variables (crop type, irrigation source, tenure, credit access). Which method maps all their categories together?
B. Multiple correspondence analysis
MCA extends correspondence analysis to several categorical variables via the indicator or Burt matrix; PCA on arbitrary category codes treats nominal codes as numbers.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
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