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ICAR AICE-JRF/SRF Statistical Inference II: Testing of Hypotheses & Sequential Analysis: practice questions

ICAR AICE-JRF/SRF - Agricultural Statistics, Computer Application, Bioinformatics (PhD admission and fellowship) · India · National Testing Agency for ICAR

Original practice questions written to the published syllabus · not past papers
10worked here, free
44written to this exam in this subject
11syllabus units

What this subject covers

The logic and optimality of tests: errors and power, the Neyman-Pearson lemma, uniformly most powerful and unbiased tests, likelihood-ratio tests and their large-sample forms, and Wald's sequential probability ratio test.

In the exams: AICE-JRF Ag Stat Units 1-2 incl. SPRT (heavy); GATE ST 'Testing of Hypotheses' (heavy); IIT JAM MS (heavy); UPSC ISS Statistics I-II (heavy)

What ICAR AICE-JRF/SRF asks from it

ICAR AICE-JRF/SRF Agricultural Statistics (Units 1 and 2)

Neyman-Pearson lemma, UMP and likelihood-ratio tests, the standard tests, and Wald's SPRT with OC and ASN functions; numerical boundaries and statements of conditions.

Where the 44 questions fall

How our questions for ICAR AICE-JRF/SRF spread across the units of Statistical Inference II: Testing of Hypotheses & Sequential Analysis. Difficulty: 8 easy, 18 medium, 18 hard.

Sequential probability ratio test: OC and ASN functions10
Unbiased tests and UMPU tests for exponential families7
Monotone likelihood ratio and uniformly most powerful tests6
Wald and score (Rao) tests5
Relation between tests and confidence intervals5
Likelihood ratio tests and Wilks' theorem3
Hypotheses, the two types of error, size and power2
Neyman-Pearson lemma and most powerful tests2
Multiple testing: family-wise error rate and false discovery rate2
Tests based on Z, t, chi-square and F distributions1
Power and sample-size calculation for common tests1

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Hypotheses, the two types of error, size and power · hard

Plot yields are N(μ, 16). From 16 plots, H₀: μ = 50 is rejected in favour of H₁: μ = 52 when x̄ > 51.645. What is the power of the test?

  1. 0.639
  2. 0.361
  3. 0.050
  4. 0.535
Answer and working

A. 0.639

SE = 4/√16 = 1. Power = P(X̄ > 51.645 | μ = 52) = P(Z > −0.355) = 0.639. 0.050 is the size; 0.361 is β; using σ = 4 instead of the standard error gives 0.535.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

2 · Neyman-Pearson lemma and most powerful tests · hard

For H₀: p = 0.5 against H₁: p = 0.8 with 10 Bernoulli trials, the MP test rejects for large X. P(X ≥ 9 | H₀) = 11/1024 and P(X = 8 | H₀) = 45/1024. For an exact size of 0.05, with what probability should H₀ be rejected when X = 8?

  1. 1.138
  2. 0.893
  3. 0.050
  4. 0.107
Answer and working

B. 0.893

Size = P(X ≥ 9) + γP(X = 8) = 0.05 gives γ = (0.05 − 11/1024)/(45/1024) = 0.893. Dividing 0.05 by P(X = 8) alone ignores the rejections already made at X ≥ 9.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

3 · Monotone likelihood ratio and uniformly most powerful tests · medium

Under a monotone likelihood ratio in T, what does the Karlin-Rubin theorem give for H₀: θ ≤ θ₀ against H₁: θ > θ₀?

  1. A UMP level-α test that rejects for small T
  2. An LRT that is UMP only as n becomes large
  3. A UMPU test, though no UMP test exists here
  4. A UMP level-α test that rejects for large T
Answer and working

D. A UMP level-α test that rejects for large T

Karlin-Rubin: rejecting when T > c with P_θ₀(T > c) = α is UMP for the one-sided problem, and its power function is non-decreasing in θ.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

4 · Monotone likelihood ratio and uniformly most powerful tests · hard

Why is there no UMP test of a one-sided hypothesis about the location θ of a Cauchy distribution based on one observation?

  1. The Cauchy family has a parameter-dependent support (unlike the normal), which rules out all MP tests
  2. The Cauchy distribution has no mean, so no test of its location can be constructed
  3. The likelihood ratio f(x; θ₁)/f(x; θ₀) is not monotone in x, so the MP region depends on θ₁
  4. Its Fisher information is zero, so every test has power equal to its size
Answer and working

C. The likelihood ratio f(x; θ₁)/f(x; θ₀) is not monotone in x, so the MP region depends on θ₁

f(x; θ₁)/f(x; θ₀) rises and then falls back to 1 as x → ±∞, so the MP region against θ₁ is a bounded interval whose location depends on θ₁; no single region is best for all θ₁ > θ₀. Tests do exist; the support is the whole line; information is positive.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

5 · Unbiased tests and UMPU tests for exponential families · hard

In what sense is Fisher's exact test for a 2 × 2 table optimal?

  1. It has exactly size α for every table, with no randomisation needed
  2. Its randomised version is UMPU for the odds ratio, by conditioning on the margins
  3. It is the likelihood-ratio test, so Wilks' theorem gives its exact size
  4. It is UMP among all tests of independence, conditionally and unconditionally, for any table size
Answer and working

B. Its randomised version is UMPU for the odds ratio, by conditioning on the margins

The margins are sufficient for the nuisance parameters; conditioning on them gives a hypergeometric null distribution, and the randomised conditional test is UMPU. The non-randomised version is conservative.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

6 · Likelihood ratio tests and Wilks' theorem · easy

The generalised likelihood-ratio statistic for H₀: θ ∈ Θ₀ is Λ = sup_Θ₀ L(θ)/sup_Θ L(θ). When is H₀ rejected?

  1. When Λ exceeds 1, so the null fits better
  2. When Λ equals 0.5, the midpoint of its range
  3. When Λ is small, i.e. when −2 log Λ is large
  4. When Λ is large, close to its maximum of 1
Answer and working

C. When Λ is small, i.e. when −2 log Λ is large

Λ lies in [0, 1]; values near 1 mean the restricted model fits nearly as well as the full one, small values mean H₀ fits poorly. Λ cannot exceed 1 because the full maximum is at least the restricted one.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

7 · Wald and score (Rao) tests · hard

Which drawback is specific to the Wald test among the three asymptotic likelihood tests?

  1. It needs the model to be refitted under H₀ every time
  2. It has a different asymptotic null distribution from the others
  3. It cannot be used for more than one parameter at a time
  4. It is not invariant to reparameterisation of θ
Answer and working

D. It is not invariant to reparameterisation of θ

Testing θ = 1 and log θ = 0 by Wald can give different statistics, because the standard error is computed on the chosen scale. The LRT is invariant; all three are asymptotically χ² under H₀ and all extend to several parameters.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

8 · Sequential probability ratio test: OC and ASN functions · medium

In Wald's SPRT with boundaries B < 1 < A on the likelihood ratio Λₙ = L₁/L₀, what is the rule after n observations?

  1. Reject H₀ if Λₙ ≤ B, accept H₀ if Λₙ ≥ A, otherwise take another observation
  2. Accept H₀ if Λₙ ≤ A, otherwise reject H₀, after a fixed number of steps
  3. Accept H₀ if Λₙ ≤ B, reject H₀ if Λₙ ≥ A, otherwise take another observation
  4. Reject H₀ if B < Λₙ < A, otherwise stop sampling and accept H₀
Answer and working

C. Accept H₀ if Λₙ ≤ B, reject H₀ if Λₙ ≥ A, otherwise take another observation

Λₙ large favours H₁ (reject H₀); small favours H₀. Sampling continues while the ratio stays between the boundaries.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

9 · Sequential probability ratio test: OC and ASN functions · medium

Where is the average sample number (ASN) of an SPRT largest?

  1. It is the same at every θ, because the boundaries are fixed
  2. At θ₀, because the test is designed to accept H₀ quickly there
  3. Far beyond θ₁, because large effects need more data to confirm
  4. At parameter values between θ₀ and θ₁, where evidence accumulates slowly
Answer and working

D. At parameter values between θ₀ and θ₁, where evidence accumulates slowly

Between the two hypotheses E_θ(Z) is near zero, so the random walk drifts slowly and takes longest to hit a boundary. Far from both values decisions are quick.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

10 · Sequential probability ratio test: OC and ASN functions · hard

For the SPRT of μ₀ = 0 against μ₁ = 1 with N(μ, 1) data, α = 0.05 and β = 0.10, E₀(Z) = −1/2. Using Wald's approximation, what is the average sample number under H₀?

  1. 3.99
  2. 9
  3. 1.99
  4. 4.50
Answer and working

A. 3.99

E₀(N) ≈ [(1 − α)log B + α log A]/E₀(Z) = [0.95(−2.251) + 0.05(2.890)]/(−0.5) = 3.99. The fixed-sample one-sided test with the same α and β needs (1.645 + 1.282)² = 8.56, i.e. 9 observations; 4.50 ignores the chance of rejecting.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

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