Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?

This paper deals with the issue of the likelihood inference for nonlinear models with a flexible skew-t-normal (FSTN) distribution, which is proposed within a general framework of flexible skew-symmetric (FSS) distributions by combining with skew-t-normal (STN) distribution. In comparison with the common skewed distributions such as skew normal (SN), and skew-t (ST) as well as scale mixtures of skew normal (SMSN), the FSTN distribution can accommodate more flexibility and robustness in the presence of skewed, heavy-tailed, especially multimodal outcomes. However, for this distribution, a usual approach of maximum likelihood estimates based on EM algorithm becomes unavailable and an alternative way is to return to the original Newton-Raphson type method. In order to improve the estimation as well as the way for confidence estimation and hypothesis test for the parameters of interest, a modified Newton-Raphson iterative algorithm is presented in this paper, based on profile likelihood for nonlinear regression models with FSTN distribution, and, then, the confidence interval and hypothesis test are also developed. Furthermore, a real example and simulation are conducted to demonstrate the usefulness and the superiority of our approach.

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Skewed and Flexible Skewed Distributions: A Modern Look at the Distribution of BMI

American Journal of Undergraduate Research ◽

10.33697/ajur.2017.013 ◽

2017 ◽

Vol 14 (2) ◽

Author(s):

Thao Tran ◽

Cara Wiskow ◽

Mohammad Aziz

Keyword(s):

Body Mass Index ◽

Normal Distribution ◽

Body Mass ◽

Model Parameters ◽

Skew Normal Distribution ◽

Scale Mixture ◽

Skewed Distributions ◽

Goodness Of Fit Test ◽

T Distribution ◽

Skew Normal

The purpose of this study is to find distributions that best model body mass index (BMI) data. BMI has become a standard health indicator and numerous studies have been done to examine the distribution of BMI. Due to the skew and bimodal nature, we focus on modeling BMI with flexible skewed distributions. The distributions are fitted to University of Wisconsin–Eau Claire (UWEC) BMI data and to a data obtained from National Health and Nutrition Survey (NHANES). The model parameters are obtained using maximum likelihood estimation method. We compare flexible models to more conventional distributions, such as skew-normal, and skew-t distributions using AIC and BIC and Kolmogorov-Smirnov (K-S) goodness-of-fit test. Our results indicate that the skew-t and Alpha-Skew-Laplace distributions are reasonably competitive when describing unimodal BMI data whereas Alpha-Skew-Laplace and finite mixture of scale mixture of skew-normal and skew-t distributions are better alternatives to both unimodal and bimodal conventional distributions. The results we obtained are useful because we believe the models discussed in ours study will offer a framework for testing features such as bimodality, asymmetry, and robustness of the BMI data, thus providing a more detailed and accurate understanding of the distribution of BMI. KEYWORDS: Body Mass Index; Skew-normal distribution; Skew-t distribution; Flexible skewed distributions; Mixture distributions; Scale mixture of skew-normal distribution; K-S test

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