Hypothesis testing and confidence intervals

Question 1

Define a power function

Answer 1

binary function: 
πΦ(theta) = P( X in RΦ) = E[ Φ(X) ] = P (reject H0)
- if 1 reject H0
- if 0 do not reject H0
where Φ is the test

Question 2

Define a UMP at level alpha

Answer 2

test Φ at level alpha satisfying:

• sup_θ πΦ(θ) = P( X in RΦ) = E[ Φ(X) ] = P (reject H0)
• for all θ, πΦ(θ) <= πΦ(θ) for any other level alpha test Φ

Question 3

State the Neyman-Pearson lemma for simple hypothesis
H0: θ = θ0
H1: θ = θ1

Answer 3

UMP is Φ(x) =
1 if f_theta1(x)/f_theta0(x) >= k
0 if f_theta1(x)/f_theta0(x) < k

compute k : P(f_theta1(x)/f_theta0(x) > k) = alpha or P(x > c) for non-decreasing likelihood function

Question 4

State the Karlin-Rubin theorem for one-sided hypothsis
H0: theta <= theta_0 (or theta = theta_0)
H1: theta > theta_0

Answer 4

if there exists a T(X) such that f has monotone likelihood (ie. f_theta2/f_theta1 is non-decreasing wrt T(X) for any theta2 > theta1) then UMP is Φ(x) =
1 if T(x) >= k
0 if T(x) < k

compute k : P(T(X) >= k) = alpha

Question 5

State the Wilk’s theorem for “two-sided” hypothesis
H0 : theta = theta_0
H1 : theta =/= theta_0

Answer 5

if f_theta satisfies the model assumptions 3.1, then

2log Λ(X) ->d X2_p (chi-squared where p is dimension of theta)

Question 6

Define a confidence interval

Answer 6

C(x) such that P( theta in C(X)) = 1-alpha for all theta

Question 7

Strategies to find CI

Answer 7

• using pivotal quantities (whose quantity doesn’t depend on theta) and rearranging
• using (1-alpha) quantiles

cf. revision handout