This assignment focuses on estimating model parameters for Weibull, Pareto, and Lognormal distribution using the maximum likelihood method. It also involves calculating the 95th and 99th percentiles of the claims and commenting on the appropriateness of the three models for the claims.
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Mathematical Theory of Risk assignment 11 Mathematical Theory of Risk Assignment 1 Student Name Course Code: ACTST861 Associate Professor Jackie Li
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Mathematical Theory of Risk assignment 12 (a)Using the maximum likelihood method to estimate the model parameters for Weibull, Pareto and Lognormal distribution. (i)Weibull distribution Maximum Likelihood estimate. The weibull probability density function f(x)=Υ α¿)x≥0,Υ>0,α>0 The probability density function above is chosen for purposes of simplifying the maximum likelihood function L(x1,…,xn;Υ,α)=∏ i=1 n ¿¿ We take the logarithm ofL(x1,…,xn;Υ,α)and differentiating with respect toΥ∧αin turns and equating it to zero to obtain the estimators of the equation. ∂lnL ∂Υ=n Υ+∑ i=1 n lnxi−1 α∑ i=1 n xi Υlnxi=0 ∂lnL ∂Υ=−n α+1 α2∑ i=1 n xi Υ We eliminateαfrom the equation ∑ i=1 n xi Υlnxi ∑ i=1 n xi Υ −1 Υ=1 n∑ i=1 n lnxi The estimate of parameterΥis thus ^Υ=1 n∑ i=1 n lnxi The α estimator is thus ^α=1 n∑ i=1 n xi ^Υ From the given data we can estimate the parameters using excel ^Υ=1 n∑ i=1 n lnxi=393.4389 35=0.705207301 ^α=1 n∑ i=1 n xi ^Υ= 151947.9155
Mathematical Theory of Risk assignment 13 (ii)Log normal distribution MLE The parameter estimates for the maximum likelihood. L(μ,σ2׀X)=∏ i=1 n [f(xi∨¿μ,σ2)]¿ Deriving the parameter from the above we will have ^μ= ∑ i=1 n lnxi n From the given data, using excel we find the values of∑ i=1 n lnxi=393.4389 The value of n=35 ^μ= ∑ i=1 n lnxi n =393.4389 35 ^μ=11.24111 The Maximum Likelihood estimate ofσ2for a lognormal distribution is: ^σ2=∑ i=1 n ¿¿¿ ^σ2=4422.049 35=126.3442 (iii)The pareto distribution MLE for parameters given the data f(x|α,k)=αkα xα+1k≤x≤∞,α,k>0 The MLE ofαis^α= n ∑ i=1 n log(xi k) The MLE of k=min{xi}Thus k=6766
Mathematical Theory of Risk assignment 14 From the given claims data, we have n=35,∑ i=1 n log(xi k)=36.80672 ^α= n ∑ i=1 n log(xi k)=35 36.80672= 0.950913 (b)Calculate the 95th and 99th percentiles of the claims and also plot the probability density function. (i)Lognormal distribution Pdf function plot 95thpercentile=8.16656E+12 99thpercentile=1.73401E+16 (ii)Weibull distribution
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Mathematical Theory of Risk assignment 15 The probability distribution was beyond the scope of the claims data provided. 95thpercentile= 720090.10 99thpercentile=1324927.82 (iii)Pareto distribution 95thpercentile=157951 99thpercentile=858171 (c)Comment on the appropriateness of the three models for the claims. From the data provided and the modeling subjected to the claims data using the three probability distributions, it is clear that the lognormal distribution and the Weibull distributions are not
Mathematical Theory of Risk assignment 16 appropriate in the modeling of the data. The parameters produced by the data are outliers and this makes it hard for the two probability distributions to fit the data. The pareto distribution indicates a precision as far as the estimation of the percentiles are concerned and thus the most appropriate in the modeling of the claims data. From the data obtained and the estimates calculated, we can use the pareto distribution for modeling the claims.