Week 8 Flashcards
A hypothesis is simple if? If not?
Θ0 or equivalently Θ1 is a singleton set (a single point)
If not, then it is a composite hypothesis
A hypothesis test is one sided if
Summarise hypothesis formation for testing
H0 is the opposite of what we want to show
We can only get enough data to reject null or not reject it (we don’t accept it)
H1 is the complement of null and the statement we are interested in ‘proving’
Steps in hypothesis testing
Table of test error
Power function of a hypothesis test
Test and level
Define a uniformly most power class test
Define test function
A test function φ(x) for a hypothesis testing procedure is a function of the sample space
It takes value 1 if x is in the rejection region and 0 of it is in the acceptance region
It’s an indicator function
Neyman Pearson lemma
Any (simple hypothesis) test that satisfies (1) and (2) is a UMP level α test
Prove Neyman Pearson lemma
Size of type 2 error
Generalisation of NP lemma
What is point of UMP
A good test is one that minimises P of type 1 error (α) and minimises P of type 2 error (γ)
Minimising one maximises the other therefore
UMP test: we fix α and then minimise γ given the fixed α
