I would like to ask for the solution of the attached assignments, thanks.

1 AMA531 Assignment 3 Due: Tuesday, 20 Nov. 2018 1. [40 marks] A random sample of size 100 has been taken from a gamma distribution with P known to be 2 but unknown. For this sample, 100 j=1 xj = 30, 000. The prior distribution for is inverse gamma with taking the role of and taking the role of . sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m (a) Determine the exact posterior distribution of . At this point the values of and have yet to be specified. (b) The population mean is 2. Determine the posterior mean of 2 using the prior distribution first with = = 0 [this is equivalent to () = 1 ] and then with = 2 and = 250 (which is a prior mean of 250). Then, in each case, determine a 95% credibility interval with 2.5% probability on each side. (c) Determine the posterior variance of 2 and use the Bayesian central limit theorem to construct a 95% credibility interval for 2 using each of the two prior distributions given in part (b). (d) Determine the maximum likelihood estimate of and then use the estimated variance to construct a 95% confidence interval for 2. 2. [20 marks] Fifteen individuals were observed from birth to death. Their death ages are recorded as follows: Data for Question 2 3 6 1 2 15 1 20 1 23 3 28 1 34 1 42 2 46 1 50 2 Th Age Number of Deaths (a) Fit the data with an exponential distribution and draw a p-p plot of the fitted values versus the empirical values. Comment on the goodness of fit. (b) Use Kolmogorov-Smirnov test to determine whether the exponential distribution is a plausible model for the data. https://www.coursehero.com/file/35452584/Ass3-18pdf/ 2 3. [20 marks] The following table gives the numbers of policy holders under a health insurance plan who had 0, 1, . . . , 6 claims during a one-year period: Data for Question 3 0 152 1 2 120 94 3 43 4 22 5 6 5 1 sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m Number of claims Number of Policy Holders (a) Assume a negative binomial distribution for the data. Estimate the parameters r and by matching the population mean and variance with the sample mean and variance respectively (i.e., setting E(X) = x and V ar(X) = s2 ). (b) Use the chi-square test to determine (at 5% level of significance) whether the negative binomial distribution from part (a) is suitable to analyze the data. (If necessary, combine groups to ensure Ej 5 for each group). 4. [20 marks] In searching for an appropriate model to analyze a set of survival data of size 80, six models were attempted and the log-likelihood of each fitted model was obtained as follows: (i) Exponential distribution, with log L = 152.28. (ii) Gamma distribution, with log L = 149.52. (iii) Log-normal distribution, with log L = 150.51. Th (iv) Burr model with density f (x) = (x/) ; x[1 + (x/) ]+1 log L = 147.05. (v) Piecewise Weibull distribution with hazard rate ( 1 t1 1 /11 t