Unit 7

Question 1

Note

Answer 1

Worth reading pages 1-4 for understanding

Question 2

If α is an unknown parameter, how is the likelihood of the data denoted?

Answer 2

f(X1,…,Xn|α) (line shows likelihood depends on α)
Then define

L(α) = f(X1,…,Xn|α)

Question 3

How can we develop L(α) once we assume that Xi are independent?

Answer 3

We can then see the likelihood is the product of the individual densities

L(α) = f(X1|α)…f(Xn|α)

Question 4

What is the idea of MLE?

Answer 4

To choose the estimate of α to make the likelihood L(α) as large as possible!

Ie. To make the observed data as likely as possible!

Question 5

How to optimise L(α)?

Answer 5

L=objective function
α=choice variable

Differentiate L wrt α, equate to 0 then obtain α estimate, & check with 2nd order conditions it is a maximum point

Question 6

What can we do to avoid algebraic mess?

Answer 6

Since log(x) increases with x, L(α) will take its highest value of α at same point as logL(α) tf can take logs of both sides - log-likelihood function

Question 7

What can we say about MLE estimators in large samples?

Answer 7

They are normally distributed with mean θ (true value of parameter) and variance CRLB (Cramer-Rao Lower Bound) (don’t need to know how to calculate)

Question 8

How do we know MLE are asymptotically unbiased?

Answer 8

As n->infinity, E(θestimate) -> θ

Question 9

See optional material pp.11-14 and summary and other examples 15->

Answer 9

Now