A Thought on Exotic Statistical Distributions

Authors: R K Sinha

Abstract:

The statistical distributions are modeled in explaining nature of various types of data sets. Although these distributions are mostly uni-modal, it is quite common to see multiple modes in the observed distribution of the underlying variables, which make the precise modeling unrealistic. The observed data do not exhibit smoothness not necessarily due to randomness, but could also be due to non-randomness resulting in zigzag curves, oscillations, humps etc. The present paper argues that trigonometric functions, which have not been used in probability functions of distributions so far, have the potential to take care of this, if incorporated in the distribution appropriately. A simple distribution (named as, Sinoform Distribution), involving trigonometric functions, is illustrated in the paper with a data set. The importance of trigonometric functions is demonstrated in the paper, which have the characteristics to make statistical distributions exotic. It is possible to have multiple modes, oscillations and zigzag curves in the density, which could be suitable to explain the underlying nature of select data set.

Keywords: Exotic Statistical Distributions, Kurtosis, Mixture Distributions, Multi-modal

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1334219

References:

[1] R. Ciumara, "An actuarial model based on composite Weibull-Pareto distribution," Mathematical Reports, vol. 8 (58), no. 4, 2006.
[2] K. Cooray and M. M. A. Ananda "Modeling actuarial data with a composite lognormal-Pareto model," Scandinavian Actuarial Journal, vol. 5, pp. 321-334, 2005.
[3] A. Frigessi, O. Haug and A. Rue "Dynamic mixture model for unsupervised tail estimation without threshold selection," Extremes, vol. 5, pp. 219-235, 2002.
[4] E. Kolker, B. C. Tjaden, R. Hubley, E. N. Trifonov and A. F. Siegel "Spectral analysis of distributions: finding periodic components in eukaryotic enzyme length data," OMICS, Journal of Integrated Biology, vol. 6, no.1, pp. 123-130, 2002.
[5] A. McNeil "estimating the tails of loss severity distributions using extreme value theory," ASTIN Bulletin, vol. 27, no. 1, pp. 117-137, 1997.
[6] S. Resnick "Discussion of the Danish data on large fire insurance losses," ASTIN Bulletin, vol.27, no.1, 139-151, 1997.
[7] D. V. S. Sastry and R. K. Sinha "A revisit to Danish fire loss data," Conference Proceedings, 12th Global Conference of Actuaries (GCA), Mumbai, India, 2010.
[8] D. V. S. Sastry and R. K. Sinha "Length of stay - a data analytic approach," Journal of Quantitative Economics, The Indian Econometric Society, vol. 8, no. 2, pp. 42-60, 2010.
[9] D. P. M. Scollnik "On composite lognormal-Pareto model," Scandinavian Actuarial Journal, vol. 7, no. 1, pp. 20-33, 2007.