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. Please 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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