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An insurer has arranged a reinsurance treaty for the office's non-life insurance portfolio. Under this treaty the reinsurer pays the excess of each individual claim that exceeds the retention limit M. So, for each claim reported to the insurer, the reinsurer pays the greater of X - M and zero, where X denotes the gross amount of the claim before reinsurance. The reinsurer assumes that the random variable X has a Weibull distribution with distribution function (where 0 is a positive parameter): F(x) =1-exp(-ex ), 120 (i) In the past year M was set at 250,000. Fifty claims exceeded this amount and, for these claims. you are given that Zay = 2,600, where x, is the gross amount of the ith claim before reinsurance. Use this information to calculate . the maximum likelihood estimate of O, and estimate the standard error of A [9] (ii) Now suppose you are given the extra information that in the past year the insurer's portfolio also produced 600 claims that did not exceed (50,000. (a) Use this additional information, together with the information in part (i), to write down the equation satisfied by O', the revised maximum likelihood estimate of 8. (b] By substituting @ into the equation for O' or otherwise, determine whether ' is greater or less than d. (C) Without doing any further calculations, explain briefly why or is not equal to o and also explain briefly which of the two estimates should generally be preferred. [10] [Total 19]