Competitive Stats

Statistical Inference

133 questions across 6 topics. Also examined in CNET, ISS, GATE ST, GATE DA, JAM MS.

Practise Statistical Inference Filter in the bank

Methods of Estimation, Properties of Estimators

Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio Tests

106Jump points of a distribution functionMSET June 2025Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
133Existence of UMP tests: one-sided versus two-sidedMSET June 2025Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
143Pearson's chi-square for a fully specified multinomialMSET June 2025Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
133Size of a critical region — summing a binomial tailMSET April 2024Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
135Hypothesis testing — the one statement that survives scrutinyMSET April 2024Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
141Testing a Bernoulli proportion — the Wald statisticMSET April 2024Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
133Size of a test with a critical region unreachable under H0MSET March 2023Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
150A strongly-supported null hypothesis means a large p-valueMSET March 2023Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
115Power and size of a min-based test for a uniform location parameterMSET September 2021Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
150Nested rejection regions across significance levelsMSET September 2021Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
115Matching the chi^2 degrees of freedom to the estimated meanMSET December 2020Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
118The F-test for equality of variances needs both independence and normalityMSET December 2020Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
120The standard one-sample proportion test statisticMSET December 2020Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
121Degrees of freedom for testing homogeneity of r multinomial populationsMSET December 2020Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
150Monotone likelihood ratio in bar X gives the UMP upper-tail Poisson testMSET December 2020Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
115Computing test size directly under the discrete nullMSET June 2019Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
118The chi^2n-1 null distribution for the sample-variance test statistic requires normalityMSET June 2019Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
150A randomised Neyman–Pearson test: reject at x=4 with probability 0.2MSET June 2019Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
66One Bernoulli observation: the size-0.05 MP test must randomise at X=1MSET January 2018 Paper IITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
68For a shifted exponential the minimum carries all the information about the locationMSET January 2018 Paper IITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
71K=mu0+z0.05,sigma/sqrt n with sigma=2MSET January 2018 Paper IITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
100For simple hypotheses the NP test is a likelihood-ratio testMSET January 2018 Paper IIITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
101Poisson MLR in T=sum Xi: reject for large totalsMSET January 2018 Paper IIITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
102Small totals favour a smaller exponential mean; 2T/theta0simchi^22nMSET January 2018 Paper IIITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
103Unbiased: rejection probability at most alpha under H0, at least alpha under H1MSET January 2018 Paper IIITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
107Wilks's theorem: -2loglambdaoversetdlongrightarrowchi^21 under H0MSET January 2018 Paper IIITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
64MLR makes the MP test for one theta1theta0 UMP for all of themMSET April 2017 Paper IITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
67Beta(3,1) against uniform: the likelihood ratio 3x^2 increases in xMSET April 2017 Paper IITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
69Small sum Xi^2 is evidence of a small sigmaMSET April 2017 Paper IITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
70alpha=P2(Xge1)=e^-2 and beta=P1(X<1)=1-e^-1MSET April 2017 Paper IITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
98Uniform scale family: a UMP two-sided test exists (misprinted hypotheses)MSET April 2017 Paper IIITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
99The UMPU normal test is the symmetric two-sided z testMSET April 2017 Paper IIITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
100LMP Cauchy test: sum Yi has null variance n/8, so the cut-off is sqrtn/8,zalphaMSET April 2017 Paper IIITests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard

Elementary Bayesian Inference

Confidence Intervals

Analysis of Discrete Data and Chi-Square Test of Goodness of Fit

Nonparametric Tests, Rank Correlation, Test for Independence