Counterexample bank for CSIR NET Maths Part C

Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Probability & Statistics › Estimation

“A sufficient statistic is complete” — false

Counterexample: X₍ₙ₎ (or (X₍₁₎, X₍ₙ₎)) for Uniform(−θ, θ)

Sufficient but not complete: symmetry gives non-zero functions with zero expectation.

statistics

#2 · Probability & Statistics › Estimation

“The Cramér–Rao bound is attained by the UMVUE” — false

Counterexample locked — unlock with Notes + PYQ

statistics

#3 · Probability & Statistics › Estimation

“The MLE is unbiased” — false

Counterexample: σ̂² = (1/n)Σ(Xᵢ − X̄)² for N(μ, σ²), or X₍ₙ₎ for Uniform(0,θ)

Both underestimate systematically; $E [X_{(n)}] = n θ / (n + 1)$ .

statistics

#4 · Probability & Statistics › Estimation

“The MLE is unique” — false

Counterexample locked — unlock with Notes + PYQ

statistics

#5 · Probability & Statistics › Hypothesis Testing

“A UMP test exists for every testing problem” — false

Counterexample: H₀: μ = 0 vs H₁: μ ≠ 0 for N(μ, 1)

The MP test for $μ > 0$ rejects for large X̄, for $μ < 0$ for small X̄; no single test is best against both.

statisticstesting

#6 · Probability & Statistics › Hypothesis Testing

“Any interval of the form [X̄ − (S/√n)t, ∞) with a 90% quantile t is a 90% confidence interval” — false

Counterexample: t = t_{n−1,0.9} under the convention t_{m,α} = (1−α)-th quantile

That t is the 10th percentile, so the interval has coverage 10%, not 90%.

statistics

#7 · Probability & Statistics › Linear Models and Multivariate

“OLS is the BLUE in every linear model” — false

Counterexample: Yᵢ = βxᵢ + εᵢ with Var(εᵢ) = σ²xᵢ²

Gauss–Markov assumes constant variance; here weighted least squares (the mean of $Y_{i} / x_{i})$ has smaller variance.

statisticsregression

#8 · Probability & Statistics › Linear Models and Multivariate

“Every parameter in the one-way ANOVA model is estimable” — false

Counterexample locked — unlock with Notes + PYQ

statisticsanova

#9 · Probability & Statistics › Linear Models and Multivariate

“If every marginal is normal then the vector is multivariate normal” — false

Counterexample: X ~ N(0,1) and Y = εX with ε = ±1 independent

Both marginals are N(0,1) but X + Y is 0 half the time — not normal, so the pair is not jointly normal.

statisticsmultivariate

#10 · Probability & Statistics › Sampling and Design of Experiments

“Systematic sampling is always at least as efficient as SRS” — false

Counterexample: A population with a periodic pattern of period k, sampled every k-th unit

Every sampled unit falls at the same phase, so the sample can be maximally unrepresentative.

statisticssampling