KlugmanB - 2.3. ESTIMATION FOR PARAMETRIC MODELS 23 This...

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2.3. ESTIMATION FOR PARAMETRIC MODELS 23 This formula is equivalent to assuming that every observation in a group is equal to the group’s midpoint. For a limit of 125,000 the expected cost is 1 227 [3 , 750(99) + 12 , 500(42) + 25 , 000(29) + 50 , 000(28) + 96 , 250(17) + 125 , 000(12)] = 27 , 125 . 55 . For a limit of 300,000, the expected cost is 1 227 [3 , 750(99) + 12 , 500(42) + 25 , 000(29) + 50 , 000(28) + 96 , 250(17) + 212 , 500(9) + 300 , 000(3)] =32 , 907 . 49 . The ratio is 32 , 907 . 49 / 27 , 125 . 55 = 1 . 213 , or a 21.3% increase. Note that if the last group has a boundary of in f nity, the f nal integral will be unde f ned, but the contribution to the sum in the last two lines is still correct. ¤ Exercise 16 Estimate the variance of the number of accidents using Data Set A assuming that all seven drivers with f ve or more accidents had exactly f ve accidents. Exercise 17 Estimate the expected cost per loss and per payment for a deductible of 25,000 and a maximum payment of 275,000 using Data Set C. Exercise 18 (*) You are studying the length of time attorneys are involved in settling bodily injury lawsuits. T represents the number of months from the time an attorney is assigned such a case to the time the case is settled. Nine cases were observed during the study period, two of which were not settled at the conclusion of the study. For those two cases, the time spent up to the conclusion of the study, 4 months and 6 months, was recorded instead. The observed values of T for the other sevencasesareasfo l lows—1 ,3 ,5 ,8 ,9 . Es t ima te Pr(3 T 5) using the Kaplan-Meier product-limit estimate. 2.3 Estimation for parametric models 2.3.1 Introduction If a phenomenon is to be modeled using a parametric model, it is necessary to assign values to the parameters. This could be done arbitrarily, but it would seem to be more reasonable to base the assignment on observations from that phenomenon. In particular, we will assume that n independent observations have been collected. For some of the techniques used in this Section it will be further assumed that all the observations are from the same random variable. For others, that restriction will be relaxed. The methods introduced in the next Subsection are relatively easy to implement, but tend to give poor results. The following Subsection covers maximum likelihood estimation. This method is more di cult to use, but has superior statistical properties and is considerably more F exible. 2.3.2 Method of moments and percentile matching For these methods we assume that all n observations are from the same parametric distribution. In particular, let the distribution function be given by F ( x | θ ) , θ T =( θ 1 , θ 2 ,..., θ p ) .
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24 CHAPTER 2. MODEL ESTIMATION That is, θ is a vector containing the p parameters to be estimated. Furthermore, let μ 0 k ( θ )= E ( X k | θ ) be the k th raw moment, and let π g ( θ ) be the 100 g th percentile of the random variable. That is, F [ π g ( θ ) | θ ]= g .F o rasamp leo f n independent observations from this random variable, let ˆ μ 0 k = 1 n n X j =1 x k j be the empirical estimate of the k th moment, and let ˆ π g be the empirical estimate of the 100 g th percentile.
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This note was uploaded on 03/26/2012 for the course ACTUARIAL 306 taught by Professor Arthuringham during the Spring '12 term at Birmingham City University.

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KlugmanB - 2.3. ESTIMATION FOR PARAMETRIC MODELS 23 This...

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