Statistics 2023 Paper I 50 marks Compulsory Solve

Q5

(a) (i) If **X** = (X₁ X₂ X₃)' is distributed as N₃ (μ, Σ), find the distribution of [(X₁ – X₂) (X₂ – X₃)]'. (5 marks) (ii) Suppose that **X** = (X₁ X₂ X₃)' ~ N₃ (**0**, Σ), where Σ = $\begin{pmatrix} 1 & \rho & 0 \\ \rho & 1 & \rho \\ 0 & \rho & 1 \end{pmatrix}$. Is there a value of ρ for which (X₁ + X₂ + X₃) and (X₁ – X₂ – X₃) are independent ? (5 marks) (b) Show that **X** = (X₁, X₂, ..., Xₚ)' has p-variate normal distribution if and only if every linear combination (l₁X₁ + l₂X₂ + ... + lₚXₚ) of **X** follows a univariate normal distribution. (10 marks) (c) Let x₁, x₂, ..., xₙ be n given observations, and suppose that Yᵢ = β₀ + β₁xᵢ + eᵢ; i = 1, 2, ..., n, where β₀, β₁ are unknown parameters and eᵢ are mutually independent normal random variables with E(eᵢ) = 0 and V(eᵢ) = σ², i = 1, 2, ..., n. Also, σ² is assumed to be unknown. Test the null hypothesis H₀ : β₀ = β₁ = 0. (10 marks) (d) Complete the following analysis of variance table of a design and examine whether there is a significant difference between the treatments at 5% level of significance: | Source of Variation | Degrees of Freedom | Sum of Squares | Mean Sum of Squares | Variance Ratio | |---------------------|-------------------|----------------|---------------------|----------------| | Blocks | — | 21 | 4·2 | — | | Treatments | — | — | 5·0 | — | | Error | 15 | 12 | — | | | Total | — | — | | | Given that F_{·05}(3, 15) = 8·70, F_{·05}(5, 15) = 4·62 (10 marks) (e) Define regression estimator used for the estimation of population mean. Obtain its bias and Mean Square Error (MSE) to the first order of approximation. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) यदि **X** = (X₁ X₂ X₃)' का बंटन N₃ (μ, Σ) है, तब [(X₁ – X₂) (X₂ – X₃)]' का बंटन ज्ञात कीजिए । (5 अंक) (ii) माना कि **X** = (X₁ X₂ X₃)' ~ N₃ (**0**, Σ) है, जहाँ Σ = $\begin{pmatrix} 1 & \rho & 0 \\ \rho & 1 & \rho \\ 0 & \rho & 1 \end{pmatrix}$ है । क्या ρ का ऐसा कोई मान है जिसके लिए (X₁ + X₂ + X₃) एवं (X₁ – X₂ – X₃) स्वतंत्र हैं ? (5 अंक) (b) दिखाइए कि **X** = (X₁, X₂, ..., Xₚ)' का बंटन p-चरिय प्रसामान्य बंटन है, यदि और केवल यदि **X** के प्रत्येक रैखीय युग्म (l₁X₁ + l₂X₂ + ... + lₚXₚ) का बंटन एकचरिय (एकविचर) प्रसामान्य बंटन है । (10 अंक) (c) माना x₁, x₂, ..., xₙ दिए हुए n प्रेक्षण हैं तथा Yᵢ = β₀ + β₁xᵢ + eᵢ; i = 1, 2, ..., n, जहाँ β₀, β₁ अज्ञात प्राचल हैं तथा सभी eᵢ E(eᵢ) = 0 एवं V(eᵢ) = σ², i = 1, 2, ..., n के साथ परस्पर स्वतंत्र प्रसामान्य यादृच्छिक चर हैं । σ² को अज्ञात माना गया है । निराकरणीय परिकल्पना H₀ : β₀ = β₁ = 0 का परीक्षण कीजिए । (10 अंक) (d) एक अभिकल्पना की निम्नलिखित प्रसरण विल्लेखन सारणी को पूर्ण कीजिए एवं 5% सार्थकता स्तर पर बताइए कि क्या व्यवहारों के मध्य सार्थक अंतर है : | विचरण स्रोत | स्वतंत्र कोटि | वर्गों का योग | माध्य वर्गों का योग | प्रसरण अनुपात | |------------|-------------|-------------|------------------|-------------| | खंड | — | 21 | 4·2 | — | | व्यवहार | — | — | 5·0 | — | | त्रुटि | 15 | 12 | — | | | योग | — | — | | | दिया गया है F_{·05}(3, 15) = 8·70, F_{·05}(5, 15) = 4·62 (10 अंक) (e) समष्टि माध्य के आकलन के लिए प्रयुक्त समाश्रयण आकलक को परिभाषित कीजिए । इसकी अभिनति (बायस) एवं माध्य वर्ग त्रुटि (एम.एस.ई.) को प्रथम सन्निकटन क्रम तक प्राप्त कीजिए । (10 अंक)

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Approach

Solve this multi-part numerical problem by allocating time proportionally to marks: spend ~20% on (a)(i)-(ii) combined, ~20% on (b), ~20% on (c), ~20% on (d), and ~20% on (e). Begin each sub-part by stating the relevant theorem or formula, show complete derivation/calculation steps, and conclude with precise final answers. For (d), complete the ANOVA table systematically before hypothesis testing. For (e), clearly define the estimator before deriving bias and MSE.

Key points expected

  • (a)(i) Apply linear transformation theorem: if Y = AX, then Y ~ N₂(Aμ, AΣA') with correct matrix A = [[1,-1,0],[0,1,-1]]
  • (a)(ii) Use independence condition Cov(X₁+X₂+X₃, X₁-X₂-X₃) = 0; solve for ρ = -1/2 and verify validity
  • (b) Prove both directions: (⇒) by definition of MVN, (⇒) using characteristic functions or Cramér-Wold theorem
  • (c) Set up F-test for H₀: β₀=β₁=0 using extra sum of squares; compute F = [(SSR/2)]/[SSE/(n-2)] with correct df
  • (d) Complete ANOVA table: Blocks df=5, Treatments df=3, Total df=23, Total SS=33, Error MS=0.8; compute F_Treatments=6.25 and compare with critical value
  • (e) Define regression estimator Ŷ_reg = ȳ + b(X̄ - x̄); derive bias ≈ 0 and MSE ≈ (1-f)S²_y(1-ρ²)/n to first order

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies distributions, states all assumptions, and sets up proper null hypotheses for all sub-parts; for (a) correctly specifies transformation matrices and for (c) properly defines the linear model with error structureIdentifies most distributions and hypotheses correctly but misses some assumptions or has minor errors in transformation matrix setup; incomplete specification of model conditionsWrong distributional assumptions, incorrect hypothesis formulation, or fundamental misunderstanding of linear transformation properties; fails to specify the linear model correctly
Method choice20%10Selects optimal methods: linear transformation theorem for (a), Cramér-Wold/characteristic functions for (b), extra sum of squares F-test for (c), standard ANOVA procedure for (d), and large-sample approximation for (e); justifies choices appropriatelyUses correct but suboptimal methods (e.g., direct covariance calculation instead of matrix approach for (a)); or applies correct methods without clear justification; minor errors in method selection for one sub-partWrong methods chosen (e.g., t-tests instead of F-test for (c), or ignores Cramér-Wold for (b)); applies ANOVA incorrectly or uses wrong approximation order for (e)
Computation accuracy20%10All matrix multiplications, covariance calculations, F-statistics, and ANOVA table entries computed flawlessly; correct algebraic manipulation in bias/MSE derivation; no arithmetic errorsMost calculations correct with minor arithmetic slips (e.g., sign error in ρ solution, small df miscalculation, or rounding errors in final ANOVA entries); correct approach but execution lapsesMajor computational errors: wrong matrix products, incorrect covariance values, wrong F-statistic formula, or fundamentally flawed ANOVA table completion; incorrect bias/MSE derivation
Interpretation20%10Correctly interprets independence condition in (a)(ii), explains significance of Cramér-Wold characterization, draws proper conclusion about H₀ in (c) with clear decision rule, correctly interprets F-test result in (d) with context, and explains when regression estimator is preferredBasic interpretation present but lacks depth; states conclusions without clear connection to computed values; misses nuanced interpretation of why methods workNo interpretation of results, wrong conclusion from test statistics, or fails to state final decision; confuses independence with zero correlation; misinterprets ANOVA output
Final answer & units20%10All final answers explicitly stated: complete distribution parameters for (a), verified ρ value, clear statement of iff proof, explicit F-statistic with rejection decision, completed ANOVA table with conclusion, and final bias/MSE expressions; proper notation throughoutMost final answers present but some incomplete or buried in working; missing explicit statement of distribution in (a)(i) or final decision in (c); ANOVA table filled but conclusion not clearly statedMissing final answers, incomplete ANOVA table, no conclusion on hypothesis tests, or answers without proper statistical notation; fails to box or highlight key results

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