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Suppose that n = 100 people were asked to throw a ball at a target. Let < yi < 2m be the angle (in radians) of the ith participant's shot from the target, for i = 1, 100. For example, yi will be a little larger than 0 if the ith participant's shot was slightly to the right of the target, a little less than 2m if the shot was slightly to the left of the target and close to a if the shot was in entirely the wrong direction. We transform the data to Z; = cos(y), so that - -15451 The more accurate the shot, the closer Z; will be to 1. We will say Z has cosine-transformed von Mises distribution with scale parameter N. and write Z ~ if Z has probability density function - - 1 < Z 1, N Here h(s) is chosen to make sure that = 1 for every K. Plots of h(.) and its first two derivatives are shown on the final page. (a) Initially, we model the Z, as independent and identically distributed random variables, with each Z; n CVM(x), for a single unknown scale parameter K 0. We would like to estimate is from the observed data Z1 Zn. i. [5 marks] Find the loglikelihood for N, and hence show that the score is u(s)=nz-nh'(s). Find the observed and expected information. Are the Newton-Raphson and the Fisher scoring methods identical for this problem? Justify your answer. ii. [1 mark] Assuming zu > 0, write down an equation, the solution of which will give the maximum likelihood estimate i of N. iii. [2 marks] Suppose that z = 0.70. By using your answer to (ii), and the plots of k(.) and its derivatives on the final page, find the approximate value of R. iv. [2 marks] If xl < 0, what is the value of A? Justify your answer. v. [4 marks] Again assuming zu = 0.70, find an approximate standard error of your estimate R. Use this standard error to find an approximate 95% confidence interval for N. 1

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