Statistics 2022 Paper I 50 marks Construct

Q2

(a) Let a random variable X have exponential distribution with mean 1/θ, θ > 0. To test H₀ : θ = 3 against H₁ : θ = 2, construct sequential probability ratio test. Show that probability of terminating the test at the first stage when null hypothesis is true is 1 – 8/27 ((A–B)/AB), where B and A, B < A, are stopping bounds. (20 marks) (b) Each Sunday a fisherman visits one of three possible locations near his home : he goes to the sea with probability 1/2, to a river with probability 1/4, or to a lake with probability 1/4. If he goes to the sea there is an 80% chance that he will catch fish; corresponding figures for the river and the lake are 40% and 60% respectively. (i) Find the probability that, on a given Sunday, he catches fish. (ii) If, on a particular Sunday, he comes home without catching anything, determine the most likely place that he has been to. (5+10=15 marks) (c) Let X₁ < X₂ < X₃ be the order statistics from uniform population having probability density function f(x; θ) = 1/θ, 0 < x < θ. Show that 4X₁ is an unbiased estimator of θ. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) मान लीजिए कि एक यादृच्छिक चर X का बंटन चरघातांकी है जिसका माध्य 1/θ, θ > 0 है। H₀ : θ = 3 का H₁ : θ = 2 के विरुद्ध परीक्षण करने के लिए अनुक्रमिक प्रायिकता अनुपात परीक्षण की रचना कीजिए। यदि निराकरणीय परिकल्पना सत्य है, तो दर्शाइए कि प्रथम चरण में परीक्षण निरस्त होने की प्रायिकता 1 – 8/27 ((A–B)/AB) है, जहाँ B और A, B < A, समाप्ति सीमाएँ हैं। (20 अंक) (b) प्रत्येक रविवार को एक मछुआरा अपने घर के पास तीन संभावित स्थानों में से किसी एक स्थान पर जाता है : वह प्रायिकता 1/2 के साथ समुद्र को, प्रायिकता 1/4 के साथ एक नदी को, या प्रायिकता 1/4 के साथ एक सरोवर को जाता है। यदि वह समुद्र को जाता है, तो उसके मछली पकड़ने का संयोग 80% है; अनुरूपी संख्याएँ नदी और सरोवर के लिए क्रमशः 40% और 60% हैं। (i) दिए गए एक रविवार के दिन वह मछली पकड़े, इस बात की प्रायिकता ज्ञात कीजिए। (ii) यदि किसी दिए गए रविवार के दिन वह बिना मछली पकड़े घर वापस आता है, तो वह जहाँ से वापस आया, उस अधिकतम संभावित स्थान का निर्धारण कीजिए। (5+10=15 अंक) (c) मान लीजिए कि एकसमान समष्टि जिसका प्रायिकता घनत्व फलन f(x; θ) = 1/θ, 0 < x < θ है, से X₁ < X₂ < X₃ क्रम प्रतिदर्शज लिए गए हैं। दर्शाइए कि 4X₁, θ का एक अनभिनत आकलक है। (15 अंक)

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How this answer will be evaluated

Approach

Construct the sequential probability ratio test for part (a) by deriving the likelihood ratio and identifying stopping bounds, allocating approximately 40% of effort given its 20 marks. For part (b), apply Bayes' theorem to solve the probability and posterior location problem, spending ~30% of time. For part (c), derive the distribution of the first order statistic and verify unbiasedness, using the remaining ~30%. Present derivations step-by-step with clear probabilistic reasoning throughout.

Key points expected

  • Part (a): Derive likelihood ratio Λₙ = (3/2)ⁿ exp(-∑Xᵢ/6) for SPRT with stopping bounds A and B, and show termination probability at first stage under H₀ equals 1 − P(B < (3/2)exp(−X₁/6) < A)
  • Part (a): Evaluate P(termination at stage 1 | H₀) = 1 − [exp(−6ln(2A/3)) − exp(−6ln(2B/3))] and simplify to 1 − (8/27)((A−B)/AB) using exponential CDF
  • Part (b)(i): Apply total probability theorem: P(catch) = (1/2)(0.8) + (1/4)(0.4) + (1/4)(0.6) = 0.65
  • Part (b)(ii): Use Bayes' theorem to find P(sea|no catch) = 0.1/0.35, P(river|no catch) = 0.15/0.35, P(lake|no catch) = 0.1/0.35; identify river as most likely location
  • Part (c): Derive PDF of X₍₁₎ as f₍₁₎(x) = 3(θ−x)²/θ³ for 0 < x < θ, compute E(X₍₁₎) = θ/4, and conclude E(4X₍₁₎) = θ proving unbiasedness

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly specifies exponential likelihoods under H₀ and H₁ for (a); properly defines sample space and events for fisherman's locations in (b); accurately states uniform order statistics framework for (c) with correct supportIdentifies distributions but makes minor errors in parameter specification or support; partially correct event definitions in (b); order statistics setup has notation errorsWrong distributions assumed (e.g., confuses mean 1/θ with rate θ); fundamental misunderstanding of SPRT setup or Bayes' structure; incorrect order statistics framework
Method choice20%10Selects Wald's SPRT with proper likelihood ratio construction for (a); applies total probability and Bayes' theorem correctly in (b); uses PDF transformation/jacobian method for order statistics in (c)Correct general approach but inefficient or partially correct methods; attempts SPRT but with wrong inequality direction; uses Bayes' but with calculation shortcutsUses fixed-sample test instead of SPRT for (a); attempts naive probability without conditioning in (b); tries direct integration without order statistics theory in (c)
Computation accuracy20%10Precise algebraic manipulation yielding exact form 1 − (8/27)((A−B)/AB) for (a); correct arithmetic 0.65 and posterior probabilities 2/7, 3/7, 2/7 for (b); exact integration showing E(X₍₁₎) = θ/4 for (c)Correct approach with minor arithmetic slips; right structure but simplified incorrectly at final step; integration correct but evaluation errorMajor computational errors in likelihood ratio simplification; wrong probability values in (b); failed integration or wrong moments for order statistics
Interpretation20%10Explains why termination at stage 1 relates to likelihood ratio falling outside continuation region for (a); interprets 'most likely' as MAP estimate and justifies river choice numerically for (b); explains why 4X₍₁₎ specifically provides unbiasedness versus other estimators for (c)States conclusions without full justification; identifies correct location in (b) but without comparing all three posterior probabilities; mentions unbiasedness without insightNo interpretation of results; fails to identify most likely location; no explanation of why estimator is unbiased or what it means for inference
Final answer & units20%10All three parts answered completely: exact symbolic expression for (a), numerical probability 0.65 and identified location 'river' for (b), proven statement '4X₍₁₎ is unbiased estimator of θ' for (c); proper mathematical notation throughoutMost answers present but incomplete or with notation issues; missing one sub-part answer; correct final forms but with presentation problemsMissing multiple final answers; incomplete derivations with no boxed/conclusive statements; major notational confusion between parameters and estimators

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