korona zoom teaching, Wed 15.4 and Mon 20.4

There'll be zoom teaching Wed 15.4 and Mon 20.4 from 13:00, but not very long sessions. Check also Hjelpetr?d 3, and use it actively. Main themes now are

-- consolidation of likelihood things;

-- confidence curves;

-- basics of empirical processes.

-- Extra 1: let x be binomial (n,\theta), with n = 50 and x = 18. First do simple classical things for estimating \theta, with a 90 % c.i. Then make *two* confidence curves for \theta, and draw them in the same diagram: the simple normal approximation; and the one using the deviance function, ccval = pchisq(devval, 1), where you compute the deviance D(\theta) first. Check with Exercise 86.

-- Extra 2: suppose y_1, ..., y_n come from the Cauchy with unknown centre parameter \theta, i.e. with density

f = (1/\pi) / ( 1 + (y-\theta)^2 ). 

Note that the mean is infinite, so the \bar y is hopeless. But there is a ML. Find the limit distribution for \rootn ( \hat\theta_\ml - \theta ). If n = 100, and I tell you that y_{(40)} = 3.33 and y_{(60)} = 5.55, estimate \theta, and find a c.i. 

-- Extra 3: Consider Brownian motion, W(t) for t \ge 0. Simulate say 10 paths on create them on your screen. With M = \max_{0\le t \le 1} W(t), try to find p = Pr(M \le 1.234) by simulation. Explain why this p is also the limit of p_n = Pr(\max (S_1,S_2,\ldots,S_n) \le 1.234 \sqrt{n}), where S_1,S_2,... are partial sums of U_1,U_2,..., where these are i.i.d. with zero mean and variance one.