Alpha Normal Distribution and Its Application in China’s Logistics Prosperity Index

Alpha Normal Distribution and Its Application in China’s Logistics Prosperity Index

Xiaheng ZhangRonggang Zhang Lingling Li 

Business School, Northwest University of Political Science and Law, Xi’an 710000, China

Corresponding Author Email:
9 May 2017
| |
23 May 2017
| | Citation



Despite the extensive use in data processing, the symmetric distribution fails to yield satisfactory results. Based on the traditional normal distribution, the asymmetric distribution is often adopted to handle the features of asymmetric-tailed data. In view of the above, this paper proposes a new normal distribution, alpha normal distribution, to deal with skewed and heavy-tailed data. The validity of the proposed distribution was verified by fitting the data of China’s logistics prosperity index (LPI).


Skewed Data, Heavy-tailed data, Alpha normal distribution, Logistics Prosperity Index (LPI).

1. Introduction
2. Alpha Normal Distribution
3. Properties of Alpha Normal Distribution
4. Maximum Likelihood Estimation
5. Data Analysis
6. Conclusion

We acknowledge the financial support from the “Young Academic Innovation Team of Northwest University of Political Science and Law”, the Special research program of Shaanxi Provincial Department of Education of the “Operational Mechanism and Implementation Path of E-commerce in the Precision Poverty Reduction Strategy” (17JK0795).


1. S.J. Fan, Study on concentration distribution of urban air pollutants, 1993, Studies in Logic, no. 1, pp. 163-167.

2. W.R. Ott, A physical explanation of the lognormality of pollutant concentrations, 1990, Journal of the Air & Waste Management Association, vol. 40, no. 10, pp. 1378.

3. L.G. Blackwood, The lognormal distribution, environmental data, and radiological monitoring, 1992, Environmental Monitoring & Assessment, vol. 21, no. 3, pp. 193.

4. J. Han, Z.G. Dai, W.T. Li, Analysis of PM2.5 concentration and meteorological conditions in haze weather of Xi’an city, 2014, Environmental Pollution & Control, vol. 36, no. 2, pp. 52-56.

5. J. Chen, Statistical characteristics of air pollutant concentration and its meteorological impact in Yulin, 2015, Chang'an University Press, Xi’an.

6. H.D. Kan, B.H. Chen, Statistical distributions of ambient air pollutants in Shanghai, China, 2004, Biomedical and Environmental Sciences, vol. 17, no. 3, pp. 366-372.

7. G.L. Xu, Y.J. Lv, Approach of multi-attribute decision-making based on normal distribution internal number, 2011, Systems Engineering, vol. 29, no. 9, pp. 120-123.

8. Z.F. Zhao, H. Yan, L. Yao, Application of normal distribution to estimate the quality of regional geochemical survey samples, 2013, Rock and Mineral Analysis, vol. 32, no. 1, pp. 96-100.

9. A. Azzalini, A class of distributions which includes the normal ones, 1985, Scandinavian Journal of Statistics, vol. 12, no. 2, pp. 171-178.

10. H. Norbert, A probabilistic representation of the “skew-normal” distribution, 1986, Scandinavian Journal of Statistics, vol. 13, no. 4, pp. 271-275.

11. A. Azzalini, A. Capitanio, Statistical applications of the multivariate skew normal distribution, 1999, Journal of the Royal Statistical Society, vol. 61, no. 3, pp. 579-602.

12. W.J. Huang, Y.H. Chen, Generalized skew-Cauchy distribution, 2007, Statistics & Probability Letters, vol. 77, no. 11, pp. 1137-1147.

13. R.D. Gupta, R.C. Gupta, Analyzing skewed data by power normal model, 2008, Test, vol. 17, no. 1, pp. 197-210.

14. M.M. Chen, J.H. Ma, N.N. Ji, The alpha normal distribution and its application to environment pollution, 2016, Statistics & Information Forum, vol. 31, no. 6, pp. 22-27.

15. B.C. Arnold, R.J. Beaver, A. Azzalini, Skewed multivariate models related to hidden truncation and/or selective reporting, 2002, Test, vol. 11, no. 1, pp. 7-54.

16. R.C. Gupta, R.D. Gupta, Generalized skew normal model, 2004, Test An Official Journal of the Spanish Society of Statistics & Operations Research, vol. 13, no. 2, pp. 501-524.

17. Y. Xue, L.P. Chen, R statistical modeling and R software, 2007, Tsinghua University Press,Peking, pp. 485-490.

18. S.S. Mao, Y.M. Cheng, X.L. Pu, Probability and Mathematical Statistics Tutorial, 2004, Higher Education Press, Peking, pp. 199-202.