Let 𝑋1, 𝑋2, …, 𝑋𝑛 be a random sample has from a continuous distribution with probability density function

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1:

Let 𝑋1, 𝑋2, …, 𝑋𝑛 be a random sample has from a continuous distribution with probability density function given by

Find:

𝑓(𝑥, 𝜇) = 𝑒−2√𝜋(𝑥−𝜇) 4

1. The method of moments estimator of 𝜇.

2. T: The maximum likelihood estimator of 𝜇.

3. Prove that T is consistent.

4. The ML estimate of 𝜇, when 𝑥1 = 3.4, 𝑥2 = 2.5, 𝑥3 = 3.1, 𝑥4 = 3.2, 𝑥5 = 2.2,𝑥6 = 2, 𝑥7 = 2.6, 𝑥8 = 2.

Question 2:

Suppos e X1, X2, • • • , Xn are i.i.d. random variables with density function

Question 3:

We consider a sample X1, X2, .., XN of i .i .d. discrete random variables, where Xi has a geometric distribution with a pmf given by:

fX (x, θ) = Pr (X = x) = θ × (1 — θ)x —1 6x ∈ (1, 2, 3, ..}

where the success probability θ satisfies 0 < θ < 1 and is unknown. We assume that: E (X )=1 θ V (X )=1 — θ θ2 a) Write the log-likelihood function of the sample (x1, x2, ..xN } b) Determine the maximum likelihood estimator of the success probability θ

If 𝑋1, 𝑋2, … , 𝑋𝑛 be independent identical random variables with pdf f(x, 𝜃)=(2x)/ 𝜃2, 0 < 𝑥 < 𝜃. 1) Find: 1. The method of moments estimator of 𝜃. 2. The maximum likelihood estimator of 𝜃. 3. The mean of 𝑇=𝑋̅. 4. The MSE of T. 2) Let S=3T/2. Prove that 1. S is an unbiased estimator of 𝜃 ; 2. S is consistent estimator of 𝜃.

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