Posted: April 10th, 2022

Statistical Inference: Maximum Likelihood Estimation

Answers to all questions should be handed in to the postbox in the statistics corridor by 2pm on
Thursday 25st January. I need help writing my essay – research paper remember to fill the assignment form in all the required
parts. You can find the form in the Statistics corridor.
• Suppose that X¯ = (X1, . . . , Xn) are i.i.d. continuous random variables with p.d.f. given by
f1,θ(x) = (θ + 1)x
θ1(0,1)(x)
for θ > −1.
(i) Describe the statistical model. [2 marks]
(ii) Construct the likelihood and plot it for values n = 3 and x = (.2, .6, .1). [4 marks]
(iii) Construct the Maximum Likelihood Estimate ˆθ(x1, . . . , xn). [4 marks]
(iv) Compute the mean of the Maximum Likelihood Estimator (MLE). What happens? [2 marks]
(v) Suppose that we do not want to estimate θ but a functiont of θ given by
ζ(θ) = θ + 1
θ + 2
.
We use the estimator
ˆζ(X1, . . . , Xn) = 1
n
Xn
x=1
Xi
.
Is this an unbiased estimator? Compute the variance and Mean Square Error. [4 marks]
(vi) From Theorem 2 of section 9.2 of the textbook, we can immediately derive that the MLE for
parameter ζ is given by
ˆζMLE(X1, . . . , Xn) =
ˆθ(X1, . . . , Xn) + 1
ˆθ(X1, . . . , Xn) + 2
= 1 −
1
ˆθ(X1, . . . , Xn) + 2
.
Compare the two estimators ˆζ and ˆζMLE, for n = 1. [4 marks]

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