Posted: April 10th, 2022

Statistical Inference: Maximum Likelihood Estimation

Solutions to all questions needs to be handed in to the postbox within the statistics hall by 2pm on
Thursday 25st January. Please bear in mind to fill the task kind in all of the required
elements. Yow will discover the shape within the Statistics hall.
• Suppose that X¯ = (X1, . . . , Xn) are i.i.d. steady random variables with p.d.f. given by
f1,θ(x) = (θ + 1)x
θ1(zero,1)(x)
for θ > −1.
(i) Describe the statistical mannequin. [2 marks]
(ii) Assemble the chance and plot it for values n = three and x = (.2, .6, .1). [4 marks]
(iii) Assemble the Maximum Likelihood Estimate ˆθ(x1, . . . , xn). [4 marks]
(iv) Compute the imply of the Maximum Likelihood Estimator (MLE). What occurs? [2 marks]
(v) Suppose that we don’t need to estimate θ however 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 that this an unbiased estimator? Compute the variance and Imply Sq. Error. [4 marks]
(vi) From Theorem 2 of part 9.2 of the textbook, we will instantly 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
.
Evaluate the 2 estimators ˆζ and ˆζMLE, for n = 1. [4 marks]

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