Estimating Parameters for Incomplete Data. William White

Transcription

1 Estimating Parameters for Incomplete Data William White

2 Insurance Agent Auto Insurance Agency Task Claims in a week Boss, Is this a good representation of the population?

3 Insurance Agent Things to think of. How should it look? The distribution should be skewed right. Frequency of Claims $ per Claim

4 Insurance Agent Exponential Distribution, If is 1 If is ,000 80,000

5 Insurance Agent How can we estimate the value of? Find an estimator What is an estimator? Uses sample data to find approximations of actual parameters

6 Estimator What do we need to look for? Consistent The estimator value converges to the population value. Estimate Error True Parameter Sample Size

7 Estimator What do we need to look for? Efficient For a fixed sample size, there is less variability in the estimator. Sample Mean Sample Median Sample means have less variability than sample medians.

8 Estimator What do we need to look for? Unbiased True Parameter Estimate Sample Size True Parameter Sample Size As people take more samples, the expected value of the parameter will become the population parameter.

9 Maximum Likelihood Estimator Sir Ronald A. Fisher ( ) Maximum Likelihood Estimator (MLE) Solve the problems of estimation Written in 1912 Completed in 1922

10 Maximum Likelihood Estimator Characteristics of the MLE Very versatile Applies to most types of data Simplistic Can be very efficient with little calculations

11 Maximum Likelihood Estimator Uses the likelihood function Finds the probability of obtaining the sample results that were obtained Product of probability density functions (pdf) with independent random variables

12 Probability Maximum Likelihood Estimator

13 Maximum Likelihood Estimator Likelihood function Sample Data- Claims What parameter is most likely for our sample? If we knew is the probability density not the probability

14 Maximum Likelihood Estimator Likelihood function Probability density function Our samples are identically distributed Restate: If we had a value for the parameter, what is the likelihood we would get the sample set? Because the events are independent of each other is the probability density not the probability

15 Maximum Likelihood Estimator Likelihood function is the probability density not the probability

16 Maximum Likelihood Estimator What makes our product maximized? Probability

17 Maximum Likelihood Estimator Loglikelihood Function Taking the product can be cumbersome Often easier due to properties of Logarithms Do logarithms change up our evaluation? No, because logarithms are increasing, we are still looking for the maximum value.

18 Maximum Likelihood Estimator Example using the Exponential Distribution

19 Maximum Likelihood Estimator

20 Maximum Likelihood Estimator With calculus we can find the MLE by taking the derivative, setting it equal to 0, and solving for the parameter. (We can use the 2 nd derivative to check maximum.) Because this is are estimate for the population parameter we are also concluding that the sample mean is an estimate for the population mean.

21 Let s use our claims with the Exponential Distribution, sample mean= What Do We Think?

22 What Do We Think? Why are there no claims below 294? Probability of Claim $ per Claim

23 Deductible We forgot there is a $250 deductible! No one is going to file a claim if the damage is not worth $250. Incomplete data- Truncated

24 Incomplete Data The MLE also works with incomplete data. Incomplete data occurs when specific observations are either lost or are not recorded exactly. Two Types Truncated data When data is excluded. Censored When the number of observations is known, but the values of the observations are unknown.

25 Incomplete Data Truncated Data Vehicle insurance with a Deductible of $250 Claims are filed when greater than $

26 Incomplete Data This is an example of data that is truncated from below, or the left, since the data below the set value, $250, is truncated. Truncated from above, the right, is when data is truncated above a set value. Probability of Claim =undefined $250 $5,000 $ per Claim

27 Incomplete Data Censored data Policy Limit All values above $1,000, are set equal to $1,

28 Incomplete Data This example would be considered censored from above, or the right, since the data above the set value, 1000, is censored. Censored from below, or the left, would be the case when data is censored below a set value. Probability of Claim =$1,000 $500 $1,000 $ per Claim

29 Incomplete Data Estimate with deductible and policy limit What are we estimating for? We want to estimate for our entire sample using truncated and censored data We want our estimate to be unbiased.

30 Incomplete Data Estimating with incomplete data Group X- modified value, claim amount Group Y- modified values, amount paid Group X Group Y

31 Incomplete Data Probability y 750 y Group Y

32 Incomplete Data Estimating with incomplete data Probability y 750 y Group Y

33 Incomplete Data Solving with incomplete Probability 750 y Group Y

34 Incomplete Data Group Y

35 What s Our Result? Boss, Is this a good representation of the population? Excel File What do we need to tell the boss? Estimated mean is $ If we compare this too what our complete data set mean, $565.05, we observe that our estimate is too high. This may mean that we have a considerably high amount of accidents below the deductible.

36 What s Our Result? The results show that it is a good representation of our received claims, but it is not a good representation for our population.

37 Incomplete Data Why should we use the MLE? One of the major attractions of this estimator is that it is almost always available. That is, if you can write an expression for the desired probabilities, you can execute this method. If you cannot write and evaluate an expression for probabilities using your model, there is no point in postulating that model in the first place because you will not be able to use it to solve your problem. (Klugman, Panjer, and Willmot)

38 Thanks! Dr. Troy Riggs- Project Advisor Dr. Matt Lunsford, Seminar Instructor

39 References Klugman, Stuart A., Harry H. Panjer, and Gordon E. Willmot. Loss Models: From Data to Decisions. New York: John Wiley and Sons, Inc, Loss Models: From Data to Decisions. 2nd ed. New York: John Wiley and Sons, Inc, Myung, In Jae. "Tutorial on Maximum Likelihood Estimation." Journal of Mathematical Psychology. 47 (2003): 93.