Variation comparison between infinitely divisible distributions and the normal distribution

Ping Sun (), Ze-Chun Hu () and Wei Sun ()
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Ping Sun: Chengdu University
Ze-Chun Hu: Sichuan University
Wei Sun: Concordia University

Statistical Papers, 2024, vol. 65, issue 7, No 14, 4405-4429

Abstract: Abstract Let X be a random variable with finite second moment. We investigate the inequality: $$P\{|X-\textrm{E}[X]|\le \sqrt{\textrm{Var}(X)}\}\ge P\{|Z|\le 1\}$$ P { | X - E [ X ] | ≤ Var ( X ) } ≥ P { | Z | ≤ 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.

Keywords: Variation comparison inequality; Infinitely divisible distribution; Normal distribution; Weibull distribution; Log-normal distribution; Student’s t-distribution; Inverse Gaussian distribution; 60E15; 62G32; 90C15 (search for similar items in EconPapers)
Date: 2024
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00362-024-01561-1

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