Math330 Exercises Week 3
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WS3.1
Let be independent with for known constants .
Calculate the log-likelihood .
Hence calculate the MLE, .
Calculate also the Fisher’s information matrix .
Give a condition on which leads to the orthogonality of and .
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WS3.2
Suppose are independent identically distributed random variables from pdf:
where .
For this model
where , and
Suppose now we have data from this model, . Show that the likelihood function can be written
By first maximising in or otherwise, find expressions for as functions of .
Is unbiased for ? Is unbiased for ? Briefly justify your answers.
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QZ3.3
This question will be completed as the quiz on Moodle.
Figure 1 (First Link, Second Link) shows a simulated dataset from the Gamma distribution, with , . Four points are labelled ‘1’ to ‘4’ on the contour plot; corresponding density functions ‘a’ to ‘d’ are superimposed onto the histogram. This week’s Moodle quiz will ask you to match the density functions with the points ‘1’ to ‘4’.
Figure 1: First Link, Second Link, Caption: Simulated dataset for Gamma distribution, with , . -
WS3.4
(Invariance principle) Let be some parameter space and be a one-to-one function ( if ). Also, let be a density function. We want to use to reparameterise our model class. What is a natural choice for our new parameter space ? Also define the new likelihood function for all . Show that if achieves the highest likelihood, i.e. for all and some sample then also for all . Finally, show that this also holds in the opposite direction, i.e. if for all then also for all .
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CW3.5
In this exercise, you may use that an Exponential distribution, with has density function
for , and expected value and variance . Let be observations assumed to be independently generated from
where and . Here is a known covariate for observation and are unknown parameters.
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(a)
Show that expected information matrix is given by
HINT: Recall that the s are treated as known and the s as random.
(3 marks) -
(b)
Use the expected information matrix to give a condition on the under which parameter orthogonality holds. Explain the relevance of parameter orthogonality for inference. (2 marks)
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(c)
Find the asymptotic distribution of the maximum likelihood estimator , quoting the general result used. Note: You may use the following result
(2 marks)
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(d)
Find the standard error of in the following two cases: (1) when is also unknown, and (2) when is known. Under what circumstances are both standard errors identical? (Justify your answer.) (3 marks)
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(a)