Likelihood

Question 1

The probability of d given θ is f(d | θ), what is the likelihood function for observed data values d = d٭?

Answer 1

𝑙(θ) = f(d٭ | θ)

Question 2

Given a likelihood function what is the log likelihood?

Answer 2

L(θ) = log(𝑙(θ))

Question 3

What is the maximum likelihood estimator?

Answer 3

The value θ^{^} which maximises the likelihood 𝑙(θ) for θ given observed data d٭

Question 4

Give three reasons why it is good to use maximum likelihood estimators.

Answer 4

1. Simple, intuitive way to find an estimator for any parametric probability model
2. For large samples, MLE has very good properites
3. Strong links with other types of estimators

Question 5

How do you find the maximum likelihood estimator?

Answer 5

1. Log the likelihood function
2. Differentiate the log likelihood
3. Set equal to zero and make θ^{^} the subject
4. Check max by differentiating again and ensuring it is less than 0

Question 6

When can we call y = (y_1, … ,y_n) an independent and identically distributed (iid) sample?

Answer 6

If each y₁, … , y_n are independent, each with distribution f(y | θ)

Question 7

If y = (y₁, … ,y_n) is an iid sample of size n from f(y | θ) what is the likelihood function and therefore log likelihood function?

Answer 7

Question 8

Define a sufficient statistic.

Answer 8

In general, a statistic, T is sufficent for parameter θ if we can factorise the probability function as

Question 9

What is another name for

Answer 9

Factorisation Criterion

Question 10

In a sufficent statsitic what does the MLE only depends on the sample through?

Answer 10

The value of the sufficient statistic T