Macroeconomics questions model.give answers

The following data refers to a certain group of 150 cancer patients. They were diagnosed on 1 January 1990 and followed up until 1 January 1998. Time, r No. of patients No. of cancer No. of withdrawals (measured from present at the deaths during during the year 1.1.1990) beginning of the the year rtor+1, year a to r + 1, artor + 1, Er 150 39 107 19 86 73 66 60 54 49 The "withdrawals" are those lives with whom contact was lost, or who died from a cause other than cancer. Define qp as the independent probability that a patient who has been under observation for r years will die of cancer within a year. For a = 0, 1,2, ...,7, calculate the independent q-type exposed to risk, Ey, and estimate the independent probability that such a patient will die from cancer within a year, (8, by the "actuarial method".Example IF'.'E:.1. The management of a large life ofce wishes to conduct an investigation into the mortality rates of group life assurance business in the period 1981;] to 1983. The ofce conducts group life assurance business on a year to year basis1 although lives may withdraw from group life assurance cover during the \"scheme year" (Le. the year between two scheme renewal dates] if they leave service. You have been given the following data in respect of each scheme year beginning in the period 1981;] to 1932: I the number of lives covered at the start of the scheme year; I the number of lives dying during the scheme year; and I the number of lives leaving service during the scheme year; classied in each case according to age next birthday at the preceding (or present) scheme renewal date. There are no new entrants during the scheme year. Obtain practical formulae for the calculation of the dependent qtype exposed to risk and the corresponding crude death rates during the period 1981;] to 1933. At which average ages do these rates apply? A discrete random variable has a probability function given by: ll Give the range of possible values for the unknown parameter or. [1] A random sample of 313! observations gave respective frequencies of T, IS and 1?. (ii) [iii] (iv) Calculate the method of moments estimator of tr. [3] Write down an expression for the likelihood of these data and hence show that the maximum likelihood estimate ti\" satises the quadratic equation: lso\" +?o=o [5] Hence determine the maximum likelihood estimate and explain why the second root is rejected as a possible estimate of or. [3] [Total 12]