9. Hypothesis testing

Question 1

Types of errors

Answer 1

First: rejecting H0 when true > probability alpha
Second: not rejecting H0 when false > probability beta
inversely correlated

Question 2

Power function

Answer 2

Q is the probability of rejecting H0

Question 3

Dimension and level of a test

Answer 3

A test is said to have dimension a when the sup over Theta0 of Q is a, the test is of level a if the sup over Theta0 of Q is less or equal to a

Question 4

UMP test: definition

Answer 4

The power function of the test is greater than the PF of any other test for @ in Theta1
If H1 is simple, we can identify the MP test, the one with power function greater for @1
! remember the graph

Question 5

Neyman-Pearson Lemma

Answer 5

MP test for simple hyp using the likelihood function:

we reject H0 when L(@1,x) => kL(@0,x), with k is such that Q(@0)= a, then this is the MP test of level a

Question 6

Proof of the Neyman-Pearson lemma

Answer 6

a= int(R) f0 = int (R+R) f0 +int (R+A) f0
a => int ((R) then int(R+A) f0 <= int (R+A) f0
Q(@1) = int(R) f1 = …split
=> int(R+A) kf0 + int(R+R) f1
=> k int(A+R) f0 + …
=> k int (A+R) 1/k f1+ int(A+R) f1 = (int(R) f1 = Q(@1)

Question 7

UMP test: how to find it

Answer 7

1) if a test is MP (H0: @=@0 and H1: @=@1 (with @1<> @0) and the critical regions do not depend on the value of @1, the test is also UMP for h1: @<>@0
2) the Monotone Likelihood Ratio and Karlin Rubin

Question 8

MLR property

Answer 8

the ratio of likehood functions with different @s is nondecreasing or nonincreasing in T
For the EXPO FAM it’s enough to check b(@)

Question 9

Karlin Rubin theorem

Answer 9

For LR non decreasing and H1: @>@0, we reject for T>k

Question 10

Likelihood Ratio test

Answer 10

it’s based on the statistic Lambda: the ratio of supL over Theta0 and supL all over Theta (i.e. with the MLE), we reject H0 for Lambda < k for k st Pr{X in R}= a

Question 11

LRT for large samples

Answer 11

-2lnLambda(@,x) ~ (under H0) as a Chi^2_m

Question 12

When to use what:

1. LRT
2. UMP
3. MP test with @=@0 and @=@1 (@1<>@0)
4. @=@0 and @ not @0
5. @< a and @>a

Answer 12

1. LRT
2. Karlin Rubin
3. Neyman Pearson Lemma
4. LRT
5. Karlin Rubin