Question 1 :

Let X₁,…, Xm be a random sample from Poisson distribution with parameters λ, 0 < λ. Which of the following is an unbiased estimator of λ?

Correct Answer : Option B: X̅, S^2, aX̅ + (1-a)S^2; 0<=a<=1

Question 2 :

Let three random samples of sizes n₁ = 20, n₂ = 10 and n3 = 8 be taken from a population with mean μ and variance σ2. Then which of the following is an unbiased estimator of σ².

Correct Answer : Option D: 20S²1+10S²2+8S²3/38

Question 3 :

Let X₁,…, Xn be a random sample from a gamma population with parameters r and λ. Find the moment estimators of λ and r.

Correct Answer : Option C: X̅/(1/n)Σni=1X2i-X̅2, X̅2/(1/n)Σni=1X2i-X̅2

Question 4 :

Let X₁,…, Xn be a random sample from N(0, 1), where θ > 0. Find the MLE of θ.

Correct Answer : Option A: X̅

Question 5 :

Let X₁,…, Xn be a random sample from exponential distribution with mean μ. Find the MSE of an estimator T = 1/n+1Σni=Xi of μ.

Correct Answer : Option B: μ²/n+1

Question 6 :

Let X be a Bernoulli random variable with P(X = 1) = p and P(X = 0) = 1-p, 0 < p < 1. If μn denotes the nth moment about mean and μ2n+1 = 0 iff

Correct Answer : Option C: p = 1/2

Question 7 :

Let X₁,…, Xn be iid random variables with EXi = μ and E|Xi|2 < ∞. Which among the following is a consistent estimator for μ? 

(NOTE*: Solve this by your own)

Answer : Option A: (Πni=1/Xi)¹/n

Question 8 :

Let -2,-6,5,9,-5,-9 be the observed values of a random sample of size 6 from population having density function given by fθ(x) = e-(x-θ), x > θ Then MLE of θ is

Correct Answer : Option D: -4/3

Question 9 :