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The Mean-Median-Mode Inequality: Counterexamples

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  • Abadir, Karim M.

Abstract

Let x be a random variable whose first three moments exist. If the density of x is unimodal and positively skewed, then counterexamples are provided which show that the inequality mode ≤ median ≤ mean does not necessarily hold.I thank Andrey Vasnev for help with the graphs and Jan Magnus for various helpful discussions. I also thank Martin Bland, Paolo Paruolo, Peter Phillips, Michael Rockinger, and a referee for their comments. ESRC grant R000239538 is gratefully acknowledged.

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  • Abadir, Karim M., 2005. "The Mean-Median-Mode Inequality: Counterexamples," Econometric Theory, Cambridge University Press, vol. 21(2), pages 477-482, April.
    • Handle: RePEc:cup:etheor:v:21:y:2005:i:02:p:477-482_05
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    Cited by:

    1. Radu Tunaru, 2015. "Model Risk in Financial Markets:From Financial Engineering to Risk Management," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 9524, August.
    2. Robert Bordley & Marco Licalzi & Luisa Tibiletti, 2017. "A Target-Based Foundation for the “Hard-Easy Effect” Bias," Eurasian Studies in Business and Economics, in: Mehmet Huseyin Bilgin & Hakan Danis & Ender Demir & Ugur Can (ed.), Country Experiences in Economic Development, Management and Entrepreneurship, pages 659-671, Springer.
    3. ALTINAY, Galip, 2016. "A Simple Class of Measures of Skewness," MPRA Paper 72353, University Library of Munich, Germany.
    4. Bickel David R., 2013. "Simple estimators of false discovery rates given as few as one or two p-values without strong parametric assumptions," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 12(4), pages 529-543, August.
    5. Robert Bordley & Luisa Tibiletti, 2017. "Benchmark-Based Preferences Make Investors Loss Averse in Bull Markets and Gain Seeking in Bear Markets," International Journal of Business and Management, Canadian Center of Science and Education, vol. 13(1), pages 1-46, December.
    6. V. J. García & M. Martel & F.J. Vázquez-Polo, 2015. "Complementary information for skewness measures," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 69(4), pages 442-459, November.
    7. Yevgeniy Kovchegov, 2022. "A New Life of Pearson’s Skewness," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2896-2915, December.
    8. Ghitany, M.E. & Atieh, B. & Nadarajah, S., 2008. "Lindley distribution and its application," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 78(4), pages 493-506.
    9. Horan, Sean & Manzini, Paola & Mariotti, Marco, 2022. "When is coarseness not a curse? Comparative statics of the coarse random utility model," Journal of Economic Theory, Elsevier, vol. 202(C).

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