Q3
(a) Let X and Y be two independent random variables following exponential distribution with mean $\frac{1}{\lambda}$ and $\frac{1}{\mu}$ respectively, $\lambda > 0$, $\mu > 0$. Suppose that $(X_1, X_2, ..., X_n)$ and $(Y_1, Y_2, ..., Y_n)$ are sequences of observations on X and Y respectively. A random variable $U_i$ is defined as $$U_i = \begin{cases} 1, & \text{if} \quad X_i \geq Y_i, \quad i = 1, 2, ..., n \\ 0, & \text{otherwise} \end{cases}$$ Construct Wald's SPRT procedure based on $U_i$'s for testing H : $\lambda = \mu$ versus K : $\lambda = 2\mu$ with strength $(\alpha, \beta)$. (20 marks) (b) Let $Y_i$, $i \geq 1$ be independent and identical $U(-1, 1)$ random variables. Determine if the following sequences converge in probability : (i) $\left\{\frac{Y_i}{i}\right\}$ (ii) $\left\{(Y_i)^i\right\}$ (5+10 marks) (c) Let X₁, X₂, ..., Xₙ be a random sample from uniform distribution U(− θ, θ), θ > 0. Find the complete sufficient statistic for θ. Hence, obtain the best unbiased estimator of θ. (15 marks)
हिंदी में प्रश्न पढ़ें
(a) माना X और Y चर्यातांकी बंटन से लिए गए दो स्वतंत्र यादृच्छिक चर हैं जिनका माध्य क्रमशः $\frac{1}{\lambda}$ और $\frac{1}{\mu}$, $\lambda > 0$, $\mu > 0$ है । माना $(X_1, X_2, ..., X_n)$ और $(Y_1, Y_2, ..., Y_n)$ क्रमशः X और Y से लिए गए प्रेक्षणों के अनुक्रम हैं । एक यादृच्छिक चर $U_i$ इस प्रकार से परिभाषित है $$U_i = \begin{cases} 1, & \text{यदि} \quad X_i \geq Y_i, \quad i = 1, 2, ..., n \\ 0, & \text{अन्यथा} \end{cases}$$ $U_i$ पर आधारित H : $\lambda = \mu$ विरुद्ध K : $\lambda = 2\mu$ के परीक्षण के लिए वाल्ड SPRT विधि की रचना कीजिए जिसकी शक्ति $(\alpha, \beta)$ है । (20 अंक) (b) माना $Y_i$, $i \geq 1$, स्वतंत्र और सर्वसम $U(-1, 1)$ यादृच्छिक चर हैं । ज्ञात कीजिए कि क्या निम्न अनुक्रम प्रायिकता में अभिसरित हैं : (i) $\left\{\frac{Y_i}{i}\right\}$ (ii) $\left\{(Y_i)^i\right\}$ (5+10 अंक) (c) माना X₁, X₂, ..., Xₙ एकसमान बंटन U(− θ, θ), θ > 0 से लिया गया एक यादृच्छिक प्रतिदर्श है । θ का पूर्ण पर्याप्त प्रतिदर्शज्ञात कीजिए । इससे θ का सर्वोत्तम अनभिनत आकलक प्राप्त कीजिए । (15 अंक)
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How this answer will be evaluated
Approach
Construct the Wald's SPRT procedure for part (a) by deriving the likelihood ratio for Bernoulli outcomes, then determine convergence properties for sequences in part (b) using appropriate limit theorems, and finally derive the complete sufficient statistic and MVUE for part (c). Allocate approximately 40% time to part (a) given its 20 marks, 30% to part (c) for its 15 marks, and 30% to part (b) for its 10 marks. Structure with clear headings for each sub-part, showing derivations step-by-step and concluding with explicit final answers.
Key points expected
- For (a): Derive P(X_i ≥ Y_i) = μ/(λ+μ) under H and K, showing U_i ~ Bernoulli with p = 1/2 under H and p = 1/3 under K
- For (a): Construct Wald's SPRT with likelihood ratio Λ_n = (2/3)^T_n × (4/3)^(n-T_n) where T_n = ΣU_i, and specify continuation region with bounds A ≈ (1-β)/α and B ≈ β/(1-α)
- For (b)(i): Show Y_i/i → 0 in probability using Chebyshev's inequality or direct calculation of P(|Y_i/i| > ε)
- For (b)(ii): Analyze (Y_i)^i convergence by considering cases Y_i ∈ (-1,1), showing convergence to 0 in probability
- For (c): Identify T = max(|X_(1)|, |X_(n)|) or equivalently max(-X_(1), X_(n)) as complete sufficient statistic using factorization theorem and completeness of uniform family
- For (c): Derive E[T] = nθ/(n+1) and construct unbiased estimator θ̂ = (n+1)T/n, verifying it is the UMVUE via Lehmann-Scheffé theorem
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies distributions: for (a) establishes U_i are i.i.d. Bernoulli with correct probabilities under H and K; for (b) properly sets up convergence definitions; for (c) correctly identifies the parameter space and support dependence on θ | Identifies most distributions correctly but makes minor errors in probability calculations under hypotheses or misstates support conditions | Fundamental errors in distributional assumptions, such as treating U_i as continuous or failing to recognize parameter-dependent support in (c) |
| Method choice | 20% | 10 | Selects optimal methods: Wald's SPRT formulation with correct likelihood ratio structure for (a); appropriate convergence criteria (Chebyshev/Markov or direct) for (b); factorization theorem plus completeness argument for (c) | Uses correct general approaches but with suboptimal technique choices, such as using weak law instead of direct probability calculation for (b)(i) | Incorrect method selection, such as using Neyman-Pearson lemma instead of SPRT for (a), or attempting MLE when MVUE is asked for in (c) |
| Computation accuracy | 20% | 10 | Precise calculations: correct integration P(X≥Y) = ∫∫_{x≥y} λμe^{-λx-μy}dxdy = μ/(λ+μ); accurate derivation of continuation boundaries; correct evaluation of E[max|X_i|] using order statistics | Correct approach with minor computational slips, such as arithmetic errors in boundary constants or incomplete evaluation of expectations | Major computational errors including wrong integration limits, incorrect likelihood ratio simplification, or failure to compute normalizing constants |
| Interpretation | 20% | 10 | Clear interpretation: explains SPRT's sequential nature and expected sample size advantage; interprets convergence results probabilistically; explains why T is minimal sufficient and completeness holds | Provides correct answers with limited interpretation, such as stating convergence without explaining rate or mechanism | Missing or incorrect interpretation, such as confusing almost sure convergence with convergence in probability, or failing to explain sufficiency completeness link |
| Final answer & units | 20% | 10 | Explicit final answers: SPRT decision rule with clear A, B bounds and stopping condition; definitive convergence statements with limits for (b); closed-form UMVUE θ̂ = (n+1)max(|X|_(n), |X|_(1))/n with verification | Correct final forms but incomplete presentation, such as implicit stopping boundaries or unverified unbiasedness | Missing final answers, incomplete decision rules, or answers without proper mathematical closure |
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