Abstract
Let X be a random variable with finite second moment. We investigate the inequality: \(P\{|X-\textrm{E}[X]|\le \sqrt{\textrm{Var}(X)}\}\ge P\{|Z|\le 1\}\), where Z is a standard normal random variable. We prove that this inequality holds for many familiar infinitely divisible continuous distributions including the Laplace, Gumbel, Logistic, Pareto, infinitely divisible Weibull, Log-normal, Student’s t and Inverse Gaussian distributions. Numerical results are given to show that the inequality with continuity correction also holds for some infinitely divisible discrete distributions.
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References
Culham JR (2004) Error and complementary error functions. http://www.mhtlab.uwaterloo.ca/ courses/me755/web_chap2.pdf
Krishnamoorthy K (2016) Handbook of statistical distributions with applications, 2nd edn. CRC Press, Berlin
Rainville ED (1960) Special functions. The Macmillan Company, New York
Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge
Sun P, Hu Z-C, Sun W (2024) The infimum values of two probability functions for the Gamma distribution. J Inequal Appl 1:5. https://doi.org/10.1186/s13660-024-03081-w
Wackerley D, Mendenhall W, Scheaffer R (2008) Mathematical statistics with applications, 7th edn. Duxbury Press, New York
Wikipedia (2024) Error function. https://en.wikipedia.org/wiki/Error_function
Wikipedia (2024) Gumbel distribution, https://en.wikipedia.org/wiki/Gumbel_distribution; inverse Gaussian distribution, https://en.wikipedia.org/wiki/Inverse_Gaussian _distribution; Laplace distribution, https://en.wikipedia.org/wiki/Laplace_distribution; Logistic distribution, https://en.wikipedia.org/wiki/Logistic_distribution; Log-normal distribution, https://en.wikipedia.org/wiki/Log-normal_distribution; Pareto distribution, https://en.wikipedia.org/wiki/Pareto_distribution; Student’s \(t\)-distribution, https://en.wikipedia.org/wiki/Student%27s_t-distribution; Weibull distribution, https://en.wikipedia.org/wiki/Weibull_distribution
Zhang J, Hu Z-C, Sun W (2023) On the measure concentration of infinitely divisible distributions. Acta Math Sci, to appear
Acknowledgements
The authors would like to thank the anonymous reviewers and Professor Werner G. Müller, Editor-in-Chief, for suggestions, comments and criticisms leading to a significant improvement of the paper.
Funding
This work was supported by the National Natural Science Foundation of China (No. 12171335), the Science Development Project of Sichuan University (No. 2020SCUNL201) and the Natural Sciences and Engineering Research Council of Canada (No. 4394-2018).
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Sun, P., Hu, ZC. & Sun, W. Variation comparison between infinitely divisible distributions and the normal distribution. Stat Papers (2024). https://doi.org/10.1007/s00362-024-01561-1
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DOI: https://doi.org/10.1007/s00362-024-01561-1
Keywords
- Variation comparison inequality
- Infinitely divisible distribution
- Normal distribution
- Weibull distribution
- Log-normal distribution
- Student’s t-distribution
- Inverse Gaussian distribution