Abstract
The Cantelli inequality or the one-sided Chebyshev inequality is extended to the problem of the probability of multiple inequalities for events with more than one variable. The corresponding two-sided Cantelli inequality is extended in a similar manner. The results for the linear combination of several variables are also given as their special cases.
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References
Arnold BC (2015) Pareto distributions, 2nd edn. CRC Press, Boca Raton
Bollerslev T (1986) Generalized autoregressive conditional heteroscedasticity. J Econom 31:307–327
Cantelli FP (1910) Intorno ad un teorema foundamentale della teoria del rischio. Boll dell’ Assoc degli Attuari Ital 24:1–23
Chen X (2007) A new generalization of Chebyshev inequality for random vectors. arXiv: 0707.0805v1
Chen X (2011) A new generalization of Chebyshev inequality for random vectors. arXiv: 0707.0805v2
Cramér H (1946) Mathematical methods of statistics. Princeton University Press, Princeton
David FN, Barton DE (1962) Combinatorial chance. Charles Griffin, London
Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica 50:987–1007
Ghosh BK (2002) Probability inequalities related to Markov’s theorem. Am Stat 56:186–190
He S, Zhang J, Zhang S (2010) Bounding probability of small deviation: a fourth moment approach. Math Oper Res 35:208–232
Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley, New York
Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions. Models and applications, vol 1, 2nd edn. Wiley, New York
Laha RG, Rohatgi VK (1979) Probability theory. Wiley, New York
Loperfido N (2013) Skewness and the linear discriminant function. Stat Probab Lett 83:93–99
Loperfido N (2014) A probability inequality related to Mardia’s kurtosis. In: Perma C, Sibillo M (eds) Mathematical and statistical methods for actuarial sciences. Springer, Milan, pp 129–132
Lugosi G (2009) Concentration-of-measure inequalities: Lecture notes by Gábor Lugosi. http://84.89.132.1/~lugosi/anu.pdf
Mardia KV (1962) Multivariate Pareto distributions. Ann Math Stat 33:1008–1015
Mardia KV (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika 57:519–530
Ogasawara H (2019) Some improvements on Markov’s theorem with extensions. To appear in the American statistician. https://doi.org/10.1080/00031305.2018.1497539
Ogasawara H (2020) An expository supplement to the paper “The multiple Cantelli inequalities: Higher-order moments for Mardia’s bivariate Pareto distribution”. To appear in Economic Review (Otaru University of Commerce), 70 (4). http://www.res.otaru-uc.ac.jp/~emt-hogasa/, https://barrel.repo.nii.ac.jp/
Rohatgi VK, Saleh AKMdE (2015) An introduction to probability theory and mathematical statistics, 3rd edn. Wiley, New York
Tsay RS (2010) Analysis of financial time series, 3rd edn. Wiley, New York
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The author is indebted to the comments of the reviewers, which have led to some additional findings and improvements of an earlier version of this paper. This work was partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology (JSPS KAKENHI, Grant No.17K00042).
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Ogasawara, H. The multiple Cantelli inequalities. Stat Methods Appl 28, 495–506 (2019). https://doi.org/10.1007/s10260-019-00452-2
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DOI: https://doi.org/10.1007/s10260-019-00452-2