Week 3

Question 1

CRLB theorem

Answer 1

Equality holds for switching deriv (not expected to prove why)

Question 2

Efficiency of an unbiased estimator

Answer 2

Question 3

Def likelihood function

Answer 3

In effect likelihood function is reversed role of the argument of join PDF or PMF, I.e:

l is a function of params for given sample whereas pdf/pmf is a function of sample for given param

Question 4

Def strong likelihood principle

Answer 4

Question 5

Sufficiency principle

Answer 5

For any 2 sufficient statistics

Likelihood is proportional

Wrt likelihood principle, these are equally good estimators

Question 6

Likelihood function (discrete case)

Answer 6

Question 7

P of observing x_i for continuous dist

Answer 7

As P(X = x) = 0

For a sample x = (x₁, x₂, …, x_n)^T

And each observation has an associated measurement error ε > 0 (precision)

Question 8

Use mid value theorem to form L estimator for continuous dist

Answer 8

Question 9

Def Plausibility

Answer 9

Question 10

Score function

Answer 10

Question 11

Observed information matrix

Answer 11

Looks like fisher information without expectation

Question 12

MLEθ^ for binomial

Answer 12

Crucially removing binomial coefficient as this doesn’t depend on θ

Question 13

Jensen’s for MLE

Answer 13

Unbiased therefor can estimate fxs

Question 14

Multivariate MLE is

Answer 14

NOT EXAMINABLE :)

Question 15

For Gaussian Rv (x-μ)² = ?

Answer 15

σ²