We thank the Office of Research and Sponsored Programs for supporting this research, and Learning & Technology Services for printing this poster. Fitting Alpha-Skew Distributions to Insurance Data Aaron Leinwander Mathematics Department University of Wisconsin -Eau Claire Faculty Mentor: Mohammad Aziz In this work, we investigated application of alpha-skew distributions to insurance claims data and compared their performance to standard models such as the normal and skew-normal distributions. I NTRODUCTION A LPHA - SKEW - NORMAL The alpha-skew-logistic is similar to the alpha-skew-normal distribution, but it is an altered logistic distribution. The continuous probability density function is given by: We consider two types of alpha-skew distributions in our research: Alpha-skew- normal and Alpha-skew-logistic A LPHA - SKEW - LOGISTIC Alpha-skew-logistic distribution For some selected values of alpha We found the MLE’s of the parameters of Alpha-skew normal distributions and alpha-skew-logistic distributions using GenSA package in R. In addition, all the graphs in our work were produced using statistical software R. A PPLICATION TO R EAL D ATA F ITTING D ISTRIBUTIONS TO SIMULATED BIMODAL DATA As a precursor to empirical data, we simulated a bimodal data set by including two normal distributions with different means. The alpha-skew distributions clearly have the distinct advantage in modeling this data, while the normal, skew-normal, and skew-t show their limitations. Alpha-skew-normal distribution for Selected values of alpha W HAT ARE ALPHA - SKEW DISTRIBUTIONS ? A continuous random variable X is said to follow alpha-skew-normal distribution if its pdf is given by : Alpha-skew-normal can have up to two modes as seen from the following density plot. We collected 2013 dental claims data from New Hampshire Public Health Department. For our analysis we used a random sample of 5,000 claims. The log of the data is used in conjunction with the original. From the histograms below, we observe high skewness in the original data and slight bimodal characteristic in the log data. The table shows that skew-t was the best distribution for this data, but alpha-skew-logistic appears also to be a competitive model with values fairly close to that of skew-t model. While looking at the histograms of the data it is apparent that both of these models are fairly competitive and are capable of taking into account the sharp skewness of the data. Models fitted to original dental data Comparing the log of dental data C ONCLUSION From both the table and the histograms, it can be seen that the skew-t is the best model for the given dental data set. However, Alpha-skew- logistic is competitive in the original data and alpha-skew-normal is competitive in the log data and even models the slight second mode that can be seen. Because the data is not strongly bimodal In our case, the skew-t becomes a better fit, but alpha- skew distributions are still competitive because the data is skewed. Bimodal data can be found in various other applications In which case alpha-skew distributions can be suitably applied along with other existing models. R EFERENCES Elal-Olivero, David. "Alpha-Skew-Normal Distribution." Proyecciones Journal of Mathematics 29.3 (2010): Web. Eling, Martin. "Fitting Insurance Claims to Skewed Distributions: Are the Skew-normal and Skew-student Good Models?" Insurance: Mathematics and Economics 51.2 (2012): Web. A CKNOWLEDGEMENTS We thank the Office of Research and Sponsored Programs for supporting this research and learning and Technology Services for printing this poster. Alpha-skew-logistic distribution is also bimodal yet skewed as can be seen from the density plot. The following table shows model fitting results. In both cases, log-likelihood and AIC the best model is the one with the lowest absolute value. Dental data original (left) and log (right)