(a) Let Y₁, Y₂, Y₃, ... be independent and identical Poisson random variables with parameter 1. Use central limit theorem to establish
$$n! \simeq \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$$
for large value of positive integer n. (20 marks)

(b) Let X₁

Question

(a) Let Y₁, Y₂, Y₃, ... be independent and identical Poisson random variables with parameter 1. Use central limit theorem to establish
$$n! \simeq \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$$
for large value of positive integer n. (20 marks)

(b) Let X₁, X₂, ..., Xₙ be a random sample such that log Xᵢ ~ N(θ, θ) distribution with θ > 0 unknown. Show that one of the solutions of the likelihood equation is the unique MLE of θ. Obtain asymptotic distribution of MLE of θ. (15 marks)

(c) (i) State the sufficient conditions for a function φ(t) to be a characteristic function.
(ii) Investigate if the following functions are characteristic functions :
1. e⁻ᵗ⁴
2. [1 + |t|]⁻¹
Justify your answer. (5+10 marks)

UPSC Answer Check · Accepted Answer

Derive the Stirling approximation in part (a) by applying CLT to Poisson sums and carefully manipulating the resulting normal approximation. For part (b), derive the likelihood equation, verify the MLE solution, and obtain its asymptotic normality via Fisher information. In part (c), state Bochner's theorem precisely, then investigate the two functions using properties of positive definiteness and Polya's criteria. Allocate approximately 40% time to (a), 30% to (b), and 30% to (c), ensuring rigorous justification at each step.
- Part (a): Define Sₙ = Y₁ + ... + Yₙ ~ Poisson(n), apply CLT to (Sₙ - n)/√n → N(0,1), and use P(Sₙ = n) with Stirling's manipulation
- Part (a): Equate Poisson pmf at n to normal density approximation and solve for n! to obtain √2πn(n/e)ⁿ
- Part (b): Construct log-likelihood l(θ) = -n/2 log(2πθ) - 1/(2θ)Σ(log Xᵢ - θ)², derive score function and likelihood equation
- Part (b): Verify second-order condition (negative Fisher information) to confirm unique MLE, then apply standard asymptotic theory: √n(θ̂ - θ) → N(0, I(θ)⁻¹)
- Part (c)(i): State Bochner's theorem: φ(0)=1, continuous at 0, positive definite (non-negative definite matrices from φ(tᵢ-tⱼ))
- Part (c)(ii): Show e⁻ᵗ⁴ fails (fourth derivative at 0 gives E[X⁴]=0, contradiction) or check positive definiteness failure
- Part (c)(ii): Verify [1+|t|]⁻¹ satisfies Polya's criteria (convex on t>0, φ(0)=1, even, continuous, φ(∞)=0) hence is characteristic function

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies Sₙ ~ Poisson(n) in (a), proper log-normal specification in (b), and accurately states Bochner's theorem with all three conditions in (c)(i); all random variables and parameters properly defined	Minor errors in parameter specification or missing one condition in Bochner's theorem; correct identification of distributions but sloppy notation	Fundamental errors like treating Yᵢ as normal, wrong likelihood construction, or stating incorrect conditions for characteristic functions
Method choice	20%	10	Selects CLT with continuity correction insight for (a), standard likelihood theory with Fisher information for (b), and appropriate tests (derivatives/Polya criteria) for (c)(ii); methods match mark allocation	Correct general methods but misses optimal approach (e.g., direct Stirling formula without CLT justification, or missing Polya's criteria for second function)	Inappropriate methods like Taylor expansion without probabilistic basis, or attempting to find explicit distributions when asymptotic theory suffices
Computation accuracy	20%	10	Flawless algebra in likelihood differentiation, correct Fisher information calculation I(θ) = (2θ+1)/(2θ²), accurate derivative checks for e⁻ᵗ⁴; all limits and approximations properly justified	Minor computational slips in coefficients or signs, correct final expressions but missing intermediate steps, or incomplete verification of convexity for Polya's criteria	Major errors in differentiation, wrong asymptotic variance, incorrect conclusion about characteristic function status, or algebraic mistakes leading to wrong Stirling constant
Interpretation	20%	10	Explains why CLT applies to discrete pmf via local limit theorem intuition, interprets MLE uniqueness through strict concavity, and distinguishes between necessary vs sufficient conditions for characteristic functions with counterexample insight	Correct conclusions without deeper insight; states results without explaining why methods work or what assumptions are crucial	Misinterprets asymptotic results as exact, confuses characteristic function conditions, or fails to recognize when approximation quality matters
Final answer & units	20%	10	Precise Stirling formula with correct constant √2π, explicit MLE formula θ̂ and its asymptotic distribution N(θ, 2θ²/(2θ+1)/n), clear verdicts on both functions with complete justification; proper mathematical notation throughout	Correct final forms but missing explicit constants, or correct conclusions with incomplete justification; minor notational inconsistencies	Missing final expressions, wrong conclusions on characteristic functions, or failure to state asymptotic variance explicitly

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