Statistics 2023 Paper I 50 marks Explain

Q4

(a) What is the role of properties of completeness and sufficiency in Statistical Inference ? Explain. In U (0, θ), find out Uniformly Minimum Variance Unbiased Estimator (UMVUE) of θ. (20 marks) (b) A survey of 400 families with four children each have the following distribution : | Number of boys | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | Number of families | 16 | 89 | 145 | 118 | 32 | Is this result consistent with the hypothesis that male and female births are equally probable at 5% level of significance ? It is given that χ²_(.05) for 4 degrees of freedom = 9·488 and χ²_(.05) for 5 degrees of freedom = 11·070. (c) Define Likelihood Ratio Test. In N(θ, σ²), where σ² is unknown, find out LR test for testing H₀ : θ = θ₀ against H₁ : θ ∈ (Ω – θ₀), where Ω is the parametric space for θ. α is the size of the test.

हिंदी में प्रश्न पढ़ें

(a) सांख्यिकी निष्कर्ष में पूर्णता एवं पर्याप्तता के गुणों की क्या भूमिका है ? स्पष्ट कीजिए । U (0, θ) में, θ का एकसमान न्यूनतम प्रसरण अनभिनत आकलक (UMVUE) ज्ञात कीजिए । (20 अंक) (b) 400 परिवारों, जिनमें प्रत्येक में चार बच्चे हैं, के सर्वेक्षण का बंटन निम्नलिखित है : | लड़कों की संख्या | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | परिवारों की संख्या | 16 | 89 | 145 | 118 | 32 | क्या यह परिणाम 5% सार्थकता स्तर पर इस परिकल्पना से संगत है कि लड़कों एवं लड़कियों के जन्म होने की संभावना बराबर है ? यह दिया गया है कि χ²_(.०५) 4 स्वतंत्र कोटि के लिए = 9·488 एवं χ²_(.०५) 5 स्वतंत्र कोटि के लिए = 11·070. (c) संभाव्यता अनुपात परीक्षण को परिभाषित कीजिए । N(θ, σ²), जहाँ σ² अज्ञात है, में H₀ : θ = θ₀ विरुद्ध H₁ : θ ∈ (Ω – θ₀), जहाँ Ω, θ के लिए प्राचलिक अंतराल है, के परीक्षण के लिए संभाव्यता अनुपात (LR) परीक्षण ज्ञात कीजिए । α परीक्षण का आकार है ।

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Directive word: Explain

This question asks you to explain. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

Explain the theoretical foundations and derive the required estimators and tests across all three parts. Spend approximately 40% of effort on part (a) covering completeness, sufficiency and UMVUE derivation; 30% on part (b) for chi-square goodness-of-fit test with correct degrees of freedom and conclusion; and 30% on part (c) for likelihood ratio test definition and derivation in normal distribution with unknown variance. Structure with clear headings for each sub-part, stating definitions first, then derivations, and ending with explicit final answers.

Key points expected

  • Part (a): Define completeness and sufficiency; explain their roles in reducing data without loss of information and enabling unbiased estimation via Rao-Blackwell and Lehmann-Scheffé theorems
  • Part (a): For U(0,θ), identify sufficient statistic T = X_(n), prove completeness, and derive UMVUE of θ as ((n+1)/n)X_(n) with proper justification
  • Part (b): Set up H₀: p = 0.5 (equal probability), calculate expected frequencies under Binomial(4, 0.5), compute chi-square statistic correctly
  • Part (b): Use correct degrees of freedom = 4 (not 5), compare with critical value 9.488, and state conclusion about hypothesis
  • Part (c): Define Likelihood Ratio Test (LRT) as λ(x) = sup_{θ∈Θ₀}L(θ)/sup_{θ∈Θ}L(θ)
  • Part (c): Derive LRT statistic for N(θ,σ²) with unknown σ², showing it reduces to t-test with rejection region |t| > t_{α/2,n-1}

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%2Correctly identifies sufficient statistic T=X_(n) for U(0,θ) in (a); sets up proper binomial null hypothesis with p=0.5 in (b); correctly specifies parameter spaces Θ₀={θ₀} and Θ=ℝ in (c)Identifies some correct elements but confuses parameter spaces or uses incorrect null hypothesis; may use p≠0.5 or wrong sufficient statisticWrong setup for all parts; confuses uniform distribution bounds, uses normal approximation incorrectly, or misidentifies parameter spaces
Method choice20%2Applies Lehmann-Scheffé theorem correctly for UMVUE in (a); uses chi-square goodness-of-fit test with correct df calculation in (b); derives LRT leading to t-test in (c)Uses correct general methods but with errors in application; may use MLE instead of UMVUE or wrong degrees of freedomWrong methods throughout; uses method of moments for UMVUE, z-test instead of chi-square, or Wald test instead of LRT
Computation accuracy20%2Correct UMVUE ((n+1)/n)X_(n) with proper expectation calculation; accurate expected frequencies (25, 100, 150, 100, 25) and χ²≈9.344 in (b); correct LRT statistic -2logλ = n(x̄-θ₀)²/s² or equivalent t-form in (c)Correct approach with minor calculation errors; wrong arithmetic in χ² components or algebraic slips in LRT derivationMajor computational errors; wrong UMVUE formula, completely wrong χ² value, or incorrect final test statistic form
Interpretation20%2Correctly interprets completeness as enabling unique unbiased estimation; concludes data is consistent with equal probability hypothesis since 9.344<9.488 in (b); explains LRT rejection criterion and connection to t-distribution in (c)Partial interpretation; states conclusions without proper justification or misinterprets significance levelNo meaningful interpretation; fails to conclude hypothesis test or explain why methods work
Final answer & units20%2Explicit final answers: UMVUE stated clearly, hypothesis test conclusion with significance level, LRT rejection region |t|>t_{α/2,n-1}; all parts clearly demarcatedFinal answers present but incomplete or unclear; missing explicit rejection region or conclusion statementNo clear final answers; derivations without conclusions or missing answers for one or more parts

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