Answer the following questions.,,,
A group of six lives was observed over a period of time as part of a mortality investigation. Each of the lives was under observation at all ages from age 55 until they died or were censored. The table below shows the sex, age at exit and reason for exit from the investigation. Life Sex Age at exit Reason for exit M 56 death F 62 censored F 63 death OutWN- M 66 death M 67 censored M 67 censored The following model has been suggested for the force of mortality: #(x |Z = =) = /(x)e#= where: x denotes age Mo (x) is the baseline hazard 2 = 0 for males and z = 1 for females. Write down the partial likelihood for these observations using the model above.A mortality investigation covered the period 1 January 2007 to 1 January 2008. Time is measured in years from 1 January 2007 and P. () denotes the number of lives at time / aged x last birthday. The following data were recorded for each x : d, = number of deaths aged x next birthday P. (0) and P.(1) (i) Obtain an expression for the initial exposed to risk in terms of the available census data that may be used to estimate the initial rate of mortality q,+. stating your assumptions. (ii) Determine the value of / , stating any assumptions you make.Claims arising from a particular class of general insurance business can be classified into three mutually exclusive types: S. M and L. The proportions of claims in each category are 80%, 15% and 59%. The distribution of the amounts of individual claims in each category can be modelled by the probability density function fo(x)=20-/ x*, (x > #) where X represents the size of an individual claim. The parameters for the three categories are 0, = $100, By = 21,000, and 01 = 12,500. Given only that the amount of an individual claim was f5,000, find the probability that it belongs to each of the three categories.You bought a box of light bulbs from a market stall a few months ago. You know that the bulbs are all either short-life bulbs with a mean life of 500 hours or long-life bulbs with a mean life of 2,500) hours, but you cannot tell which because there was no label on the box. As you have not shopped at this stall before, you initially have no opinion as to whether you have been sold long-life bulbs or the cheaper alternative. After approximately 300 hours, the 5 bulbs you have been using are all still going. Assuming that the life of an individual light bulb has an exponential distribution, how would you now assess the probability that you bought long-life bulbs?The following model for the force of mortality for a life insurance company's annuitants has been proposed: an, a = (ours o.ooo1r).exp[e[x, 71]] + y,- + 9:3,] where: pair, 1'} = force of mortality for the ith life, in calendar year 200D +t ; x,- = age of the ith life; y,- = 1 if the iih life is a smoker, or y,- = 1} if a non-smoker; 2,- =1 ifthe ith life is male, or z,- =|I if female; and tr, y are the parameters of the model. The following data have been observed over the calendar year 2003: Risk characteristics Number dying Male non-smoker average age 55 Male smoker average age El] , 5 Female non-smoker, average age Ti] 2 Female smoker, average age I55 151] 1 You can assume that the numbers of armuitants in each class remained constant throughout the investigation period, and that the average age for each class can be treated as representing the value of x,- for each individual in that class. {i} Explain why this model is a proportional hazards model. [2] {ii} Explain the importance of subdividing the data by age, x and smoking status, and explain whether you think each of the parameters or, ,5 and ywould be likely to be positive or negative. [3] {iii} Calculate the force of mortality for female non-smokers with average age Tl] in 20-07, according to this model. [1] {iv} {a} Obtain an expression for the partial likelihood based on the given data, expressing your answer in terms of tr, ,5 and y only. {b} State how you would estimate the parameters of the model using the partial likelihood. [a] [Total 12]
