6. MLE

Question 1

Likelihood function

Answer 1

the pdf seen as a function of @ for a given sample x, the ratio of likelihood functions gives us which @ is more likely to be the correct estimator
- if L(x) =c(x,x)L(x) where c depends solely on x and x*, then inference coincides

Question 2

MLE and how to find it

Answer 2

it the estimator that maximises the likelihood function:

• look at the shape of the function
• maximize the function using first and second derivative on the log-likelihood
• cannot compute it

Question 3

Invariance property

Answer 3

if A is the MLE for @ then g(A) is the MLE for g(@)

Question 4

MLE and SUFF

Answer 4

By the factorization th, the MLE is always a function of the SUFF stat

Question 5

Newton-Raphson algorithm

Answer 5

it’s an all-purpose algorithm to find the solutions of non linear equations: knowing that A is a root of l’, then using the Taylor expansion we use a guess (a) and, by iteration, we get increasingly closer to A:
A~= a - l’(a)/l’‘(a)

Question 6

Asymptotic properties

Answer 6

Under regularity conditions, the MLE is asymptotically normal and efficient and consistent in probability
- the asymptotic variance (Mann-Wald + CR) is (g’(@))^2/I_X