Actuarial Statistics Assignment on Probability Mass Function and

School: University of New South Wales - Course: MATH ZZSC5905 - Subject: Accounting

ACST3061 Actuarial Statistics Assignment 2 - 2021 Session 1 Answer the following questions by providing atypedreport inPDF. You should include detailed explanations to the answers, R codes used for the computations and the numeri- cal results. Your report should not exceed 8 pages and you must submit your assignment to iLearn before 12pm on 27 May 2021 (Wednesday). 1. Consider the following probability mass function forY f(y;γ, ρ) =r ρ 2πy3 e - ρ(y-γ) 2 2γ2 y whereγ >0, ρ >0 and the random variableYcan take any positive values. (a) Show that this distribution can be written in the form of exponential family and write down the relevant expressions in terms ofθ, b(θ) andc(y, φ) given thatφ=1 ρ andw= 1.(6 marks) (b) Determine the mean and variance ofYin terms ofθ.(3 marks) (c) Following from part (b), write down the canonical link function and express the variance function, both as function of the mean parameterμ.(2 marks) (d) Following from above, suppose that we use a link function ofμi=h(ηi ) = 1 √ η i to perform the parameter estimatation for the model, whereηi=∑p j=1 βjx ij is the linear predictor for observationi. Write down and simplify the set of score equations that should be solved to obtain the regression coefficientsβj , j= 1,2, . . . , p. Would the estimated regression coefficients be different if we use the canonical link function instead? Explain.(6 marks) (e) Derive (i) the scaled residual deviance for the model; (ii) the deviance residual for observationi. Show your steps clearly.(8 marks) 1

2. A real insurance data set of private passenger bodily injury insurance average claims and the respective number of claims for 5 states in the United States over 12 quarters from July 1970 to June 1973 is considered. The data set can be obtained from the R package "actuar" using the command "data(hachemeister)". In the package, the data set is presented in a matrix form with 5 rows and 25 columns.The indexes of the five individual states are presented in the first col- umn. Columns 2-13 contain the claim averages. Columns 14-25 contain the claim numbers. Here we are interested in modelling the claim averages for each of the 5 states. Specifically, it is supposed that for eachi= 1,2,· · ·,5, the claim averages from the i-th individual state are independent and identically distributed random variables from an exponential distribution with the unknown parameterαi representingthe reciprocal ofthe mean claim averages for thei-th individual risk. Furthermore, it is assumed thatαi , i= 1,2,· · ·,5 are independent and identically distributed random variables from a Gamma distribution with the shape parameter γ= 26.62653 and the rate parameterθ= 42284.25. For each of the following parts, you should perform the calculations and display the results using R. You should also explain in words the mathematical reasonings behind the computations briefly. (a) State the distribution of the Bayesian posterior densities of the unknown means 1 α i and compute the relevant parameters for all 5 individual risks.(5 marks) (b) Compute the 95% Bayesian confidence intervals forαi .(3 marks) (c) Assuming the zero-one loss function, that is, assigning a zero loss if decision rulea(x) is sufficiently close to the unknown parameterα, and a loss of 1 otherwise, compute the Bayesian estimates ofαi .(3 marks) (d) Compute the Bayesian credibility estimates and the respective credibility fac- tors of1 α i . Verify numerically that the Bayesian credibility estimates coincide with the Bayesian posterior mean.(4 marks)