AY1516 T2 Team Hew - Prediction

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Claim Amount Distribution Fitting

Analyze Distribution

Vaughn (1996) and a number of actuaries have mentioned that ClaimPaid amounts have to factor in inflation for greater accuracy. However, as the highest inflation rate attained by Indonesia for the time period of study is about 7%, this is rather negligible. Using JMP Pro 12, we first excluded rows where ClaimPaid = 0 and derived the distribution statistics of:

N	Min	Max	Median	Mean	Standard Deviation
49,066	2,570	3,295,056,701	4,297,418	21,090,896	90,927,485

From figure above, we can see that there are many instances of small claim amounts, with few instances of extreme values, and the distribution shape is right skewed. Similar distribution shapes for Claim amounts have also been derived from insurance companies based in other regions, as seen in figure below taken from Eling’s research (2011).

Fit Distribution

We then attempt to fit several distributions on the Claim amounts column, using JMP Pro’s Fit Distribution function. The most suitable distribution selected will be the one which minimizes the -2Log(Likelihood), taken to be a measure of variation or uncertainty in the sample.

We can see that -2Log(Likelihood) is minimized with the LogNormal distribution. According to JMP Pro documentation, maximum likelihood estimation is used in determining the parameters for the LogNormal distribution. Naturally, we can see why the LogNormal distribution derives the lowest -2Log(Likelihood) as the problem of maximizing the likelihood function is reformulated to become a minimization of the negative of the natural logarithm of the likelihood function.

We did another Goodness-of-Fit test using Kolmogrov’s D test with the null hypothesis as “H0: The data is from the LogNormal Distribution”, thus obtaining a D-statistic of 0.048569 and a p-value of 0.01. As p-values are the probability of getting an even more extreme statistic given the true value being tested is at the hypothesized value (usually at zero), a small p-value means that the statistic is unlikely to be that extreme by coincidence. In this case, the p-value of 0.01 is fairly small, indicating significance that the fitted distribution is LogNormal.

Using JMP Pro’s Diagnostic Plot function, we can visually affirm the goodness-of-fit, as seen in the figure below. The actual Claim amount is plotted against its corresponding estimated LogNormal probability on the Y-axis. We can see that most of the data points fall on the Line of Fit (red line) for Claim amounts within the mid-range, thus depicting a good fit.

Linear Regression

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Research & Methodology

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References