MATH38001 Statistical Inference Coursework
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MATH38001 Statistical Inference Coursework
Q1. Suppose X1 , . . . , Xn is a random sample from Uniform [0, θ].
i) Write down the likelihood function of θ ,
ii) Plot the likelihood function of θ versus θ and hence show that = max (X1 , . . . , Xn ) is the maximum likelihood estimator of θ ,
iii) Show that = max (X1 , . . . , Xn ) is a biased estimator of θ ,
iv) Show that = max (X1 , . . . , Xn ) is a consistent estimator of θ .
[A total of 10 marks for Q1]
Q2. Let X1 , . . . , Xn be a random sample from a distribution with probability density function given by
for x > 0, where α > 0 and β > 0 are unknown parameters.
i) Calculate the score function of (α, β),
ii) Calculate the observed information matrix of (α, β),
iii) Calculate the Fisher information matrix of (α, β).
[A total of 10 marks for Q2]
[The total marks for the two questions is 20]
2023-11-09