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Robust Properties of the Median of Absolute Differences and Family of Inter-α-Quantile Ranges

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Russian Physics Journal Aims and scope

When studying the properties of various physical objects, experimental data often contain gross errors (outliers) which can lead to significant distortions of the results of statistical processing of such data. For this reason, statistical procedures are being developed that are protected from the presence of outliers in observations. In this paper, two types of robust estimates of the scale parameter, which characterizes the spread (variability) of the random variable under study, are considered. The proposed estimates are asymptotically normally distributed, have bounded influence functions, and therefore, unlike the standard deviation estimates, are protected from the presence of outliers in the sample. The results of comparing the estimates of the scale parameter by the efficiency for different observation models, in particular, for the Gauss model with large-scale contamination, are presented.

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References

  1. P. J. Bickel and E. L. Lehmann, in: Proc. Prague Symp. Asymptotic Statist., Vol. 1, Charles University, Prague (1974), pp. 25–36.

  2. P. J. Bickel and E. L. Lehmann, Ann. Statist., 3, 1038–1044; 1045–1069 (1975).

  3. P. J. Bickel and E. L. Lehmann, Ann. Statist., 4, No. 6, 1139–1158 (1976).

    Article  MathSciNet  Google Scholar 

  4. P. J. Bickel and E. L. Lehmann, in: Contributions to Statistics. Hajek Memorial Volume, J. Jureckova, ed., Academia, Prague (1979), pp. 33–40.

  5. F. R. Hampel, J. Am. Statist. Assoc., 69, No. 346, 383–393 (1974).

    Article  Google Scholar 

  6. R. J. Serfling, Approximation Theorems of Mathematical Statistics, Wiley, New York (1980).

    Book  Google Scholar 

  7. R. J. Serfling, Ann. Statist., 12, 76–86 (1984).

    Article  MathSciNet  Google Scholar 

  8. V. P. Shulenin, Robust Methods of Mathematical Statistics, Publishing House of Scientific and Technology Literature, Tomsk (2020).

    Google Scholar 

  9. V. P. Shulenin, Introduction to Robust Statistics [in Russian], Publishing House of Tomsk State University, Tomsk (1993).

    Google Scholar 

  10. V. P. Shulenin, Mathematical Statistics. Part 2. Nonparametric Statistics: A Textbook [in Russian], Publishing House of Scientific and Technology Literature, Tomsk (2012).

  11. V. P. Shulenin, Mathematical Statistics. Part 3. Robust Statistics: A Textbook [in Russian], Publishing House of Scientific and Technology Literature, Tomsk (2012).

  12. V. P. Shulenin, Robust Methods of Mathematical Statistics [in Russian], Publishing House of Tomsk Scientific and Technology Literature, Tomsk (2016).

    Google Scholar 

  13. F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust Statistics: The Approach Based on Influence Functions [Russian Translation], Mir, Moscow (1989).

    MATH  Google Scholar 

  14. P. J. Huber, Robust Statistics [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  15. T. R. Hettsmansperger, Statistical Inference Based on Ranks [Russian translation], Finansy i Statistika, Moscow (1987).

    Google Scholar 

  16. A. P. Serykh and V. P. Shulenin, Russ. Phys. J., 36, No. 10, 1000–1007 (1993).

    Article  Google Scholar 

  17. V. P. Shulenin, Russ. Phys. J., 38, No. 9, 941–946 (1995).

    Article  MathSciNet  Google Scholar 

  18. V. P. Shulenin, Russian Physics J., 59, No. 6, 824–832 (2016).

    Article  ADS  Google Scholar 

  19. V. P. Shulenin, Russ. Phys. J., 63, No. 4, 574–590 (2020).

    Article  Google Scholar 

  20. F. P. Tarasenko and V. P. Shulenin, in: Trans. of the 12th Prague Conf. on Information Theory, Statistical Decision Functions and Random Processes, Prague (1994), pp. 220–223.

  21. V. P. Shulenin, in: Abstracts of Reports at the 6th Prague Symposium on Asymptotic Statistics Prague Stochastics’ 98, Prague (1998), p. 84.

  22. V. P. Shulenin, Reliability: Theory and Applications. Gnedenko Forum, Vol. 8, No. 2, 24–38 (2013).

    Google Scholar 

  23. V. P. Shulenin, Theory of Probability and Its Application, 37, No. 4, 816–818 (1992).

    Google Scholar 

  24. V. P. Shulenin, in: Materials of Int. Sci. Conf. Devoted to the 80 Anniversary of Gennadii Alekseevich Medvedev, Professor, Doctor of Physical and Mathematical Sciences [in Russian], Minsk (2015), pp. 372–377.

  25. F. P. Tarasenko and V. P. Shulenin, in: Proc. Int. Workshop on Applied Methods of Statistical Analysis. Nonparametric Approach AMSA’2015, Novosibirsk (2015), pp. 12–17.

  26. V. P. Shulenin, Additional Chapters of Mathematical Statistics (Course of Lectures) [in Russian], Publishing House of Scientific and Technology Literature, Tomsk (2018).

    Google Scholar 

  27. P. Janssen, R. Serfling, and M. Veraverbeke, Ann. Statist., 12, No. 4, 1369–1379 (1984).

    Article  MathSciNet  Google Scholar 

  28. P. Janssen, R. Serfling, and M. Veraverbeke, J. Stat. Plan. Infer., 16, 63–74 (1987).

    Article  Google Scholar 

  29. P. J. Rousseeuw and C. Croux, J. Am. Statist. Assoc., 88, No. 424, 1273–1283 (1993).

    Article  Google Scholar 

  30. V. P. Shulenin, Tomsk State Univ. J. of Control and Computer Science, Suppl. No. 9 (11), 184–190 (2004).

  31. V. P. Shulenin, Tomsk State Univ. J. of Control and Computer Science, No. 2 (11), 96–112 (2010).

  32. M. G. Kendall and A. Stuart, The advanced Theory of Statistics [Russian translation], Nauka, Moscow (1966).

    Google Scholar 

  33. V. P. Shulenin, Tomsk State Univ. J. of Control and Computer Science, No. 4 (37), 73–82 (2016).

  34. V. P. Shulenin, in: Stable Measures of the Scale Parameter, Their Estimation and Comparison. Mathematical Statistics and Its Applications [in Russian], Publishing House of Tomsk State University, Tomsk (1981), pp. 199–215.

  35. R. V. Hogg, J. Am. Statist. Assoc., 69, 909–923 (1974).

    Article  Google Scholar 

  36. V. P. Shulenin, in: Nonparametric and Robust Statistical Methods in Cybernetics and Computer Science [in Russian], Publishing House of Tomsk State University, Tomsk (1990), pp. 564–570.

    Google Scholar 

  37. F. P. Tarasenko and V. P. Shulenin, in: Abstracts of Reports Presented at the VI Int. Symp. on Information Theory of the USSR Academy of Sciences and the UzSSR Academy of Sciences. Vol. I, Moscow; Tashkent (1984), pp. 171–173.

  38. V. P. Shulenin, Tomsk State Univ. J. of Control and Computer Science, No. 2 (35), 62–69 (2016).

  39. V. P. Shulenin, in: Proc. of the V Meeting-Seminar on Nonparametric and Robust Statistical Methods in Cybernetics of the SB of the USSR Academy of Sciences, Vol. II [in Russian], Tomsk (1987), pp. 460–467.

  40. V. P. Shulenin, Mathematical statistics. Part 1. Parametrical Statistics: A Textbook [in Russian], Publishing House of Scientific and Technology Literature, Tomsk (2012).

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  1. National Research Tomsk State University, Tomsk, Russia

    V. P. Shulenin

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Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 124–137, December, 2020.

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Shulenin, V.P. Robust Properties of the Median of Absolute Differences and Family of Inter-α-Quantile Ranges. Russ Phys J 63, 2189–2204 (2021). https://doi.org/10.1007/s11182-021-02288-4

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  • DOI: https://doi.org/10.1007/s11182-021-02288-4

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