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Efficiency of a Certain Modification of the Studentized Range of Symmetric Stable Random Variables

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We study the properties of tests constructed by a simple modification of the studentized range of the sample from a symmetric stable population in the problem of testing the hypothesis \( \mathrm{\mathscr{H}} \) α (the stability index equals α, α ∈ (1, 2)) against the alternative \( \mathrm{\mathscr{H}} \) 2. We obtain approximate formulas for the calculation of critical values and estimation of the test power and develop a method for the estimation of the accuracy of these approximations. The major part of the paper deals with the construction of the approximations to the distribution function of the normalized sum of squares of symmetric stable random variables and estimation of the accuracy of approximations.

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References

  1. H. Bateman and A. Erdelyi, Higher Transcendental Functions, McGraw–Hill, New York (1953).

    MATH  Google Scholar 

  2. B. Gambrowski and S. Rachev, “Financial models using the stable laws,” Rev. Ind. Appl. Math., 2, No. 4, 556–604 (1995).

    Google Scholar 

  3. B. V. Gnedenko, “Sur la distribution limite du terme maximum d’une série aléatoire,” Ann. Math., No. 45, 23–53 (1944).

  4. V. M. Zolotarev, One-dimensional Stable Distributions, Nauka, Moscow (1983).

    MATH  Google Scholar 

  5. V. M. Zolotarev, “Analogue of the Cramer–Edgeworth expansion for the case of rapprochement with the stable laws,” in: Proc. VI All-Union Workshop on Prob. Theory and Math. Stat., Vilnius (1962), pp. 49–50.

  6. I. A. Ibragimov and Yu.A. Linnik, Independent and Stationary Associated Random Variables, Nauka, Moscow (1965).

    MATH  Google Scholar 

  7. V. V. Petrov, Sums of Independent Random Variables, Springer, Berlin (1975).

    Book  MATH  Google Scholar 

  8. C.-G. Esseen, “Fourier analysis of distribution functions,” Acta Math., No. 77, 1–125 (1944).

  9. E. F. Fama and R. Roll, “Parameter estimates for symmetric stable distributions,” J. Am. Stat. Assoc., No. 66, 331–338 (1971).

  10. G. Cristoph, “Űber notwendige und hinreichende Bedingungen fur Konvergentzgeschwindigkeitsaussagen im Falle einer stabilen Grenzverteilung,” Z. Wahrscheinlichkeitstheorie verw. Gebiete, No. 54, 29–40 (1980).

  11. B. Mandelbrot, “The variations of certain speculative prices,” J. Business, No. 36, 394–419 (1963).

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Authors and Affiliations

  1. Perm State National Research University, Perm, Russia

    P. N. Sapozhnikov

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  1. P. N. Sapozhnikov
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Correspondence to P. N. Sapozhnikov.

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Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, Vol. 21, pp. 47–61, 2008

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Sapozhnikov, P.N. Efficiency of a Certain Modification of the Studentized Range of Symmetric Stable Random Variables. J Math Sci 228, 522–532 (2018). https://doi.org/10.1007/s10958-017-3641-8

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  • DOI: https://doi.org/10.1007/s10958-017-3641-8

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