You design a statistical test of some hypothesis H0 which has asymptotic size 5% but you are unsure of the approximation in finite samples. You run a simulation experiment on your computer to check if the asymptotic distribution is a good approximation. You generate data which satisfies H0.

On each simulated sample, you compute the test. Out of B Æ 50 independent trials you find 5 rejections and 45 acceptances.

(a) Based on the B Æ 50 simulation trials, what is your estimate b p of p, the probability of rejection?

(b) Find the asymptotic distribution for p

B

¡

b p ¡p

¢

as B !1.

(c) Test the hypothesis that p Æ 0.05 against p 6Æ 0.05. Does the simulation evidence support or reject the hypothesis that the size is 5%?

Hint: P[jN(0,1)j ¸ 1.96] Æ 0.05.

Hint: There may be more than one method to implement a test. That is okay. It is sufficient to describe one method.