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Lower and upper bounds on the variances of spacings

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Abstract

For classic i.i.d. samples with an arbitrary nondegenerate and finite variance distribution, Papadatos (1995, Annals of the Institute of Statistical Mathematics, 47, 185–193) presented sharp lower and upper bounds on the variances of order statistics, expressed in population variance units. We provide here analogous results for spacings. Also, we describe the parent distributions which attain the bounds.

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References

  • Danielak, K. (2004). Sharp upper bounds for expectations of differences of order statistics in various scale units. Communications in Statistics—Theory and Methods, 33, 787–803.

  • Danielak, K., Rychlik, T. (2003). Sharp bounds for expectations of spacings from DDA and DFRA families. Statistics and Probability Letters, 65, 303–316.

  • Danielak, K., Rychlik, T. (2004). Sharp bounds for expectations of spacings from decreasing density and failure rate families. Applicationes Mathematicae (Warsaw), 31, 369–395.

  • David, H. A., Nagaraja, H. N. (2003). Order statistics (3rd ed.). Hoboken, NJ: Wiley.

  • Goroncy, A., Rychlik, T. (2015a). Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order. Metrika, 78, 175–204.

  • Goroncy, A., Rychlik, T. (2015b). Evaluations of expected order statistics and spacings based on the IFR distributions (submitted).

  • Hoeffding, W. (1940). Maßtabinvariante Korrelations-Theorie. Schriften des Mathematischen Instituts and Instituts für Angewandte Mathematik der Universität Berlin, 5, 181–233.

    Google Scholar 

  • Irwin, J. O. (1925). The further theory of Francis Galton’s individual difference problem. Biometrika, 17, 100–128.

    Article  MATH  Google Scholar 

  • Jasiński, K., Rychlik, T. (2013). Maximum variance of order statistics from symmetric populations revisited. Statistics, 47, 422–438.

  • Jones, M. C., Balakrishnan, N. (2002). How are moments and moments of spacings related to distribution function. Journal of Statistical Planning and Inference, 103, 377–390.

  • Kozyra, P. M., Rychlik, T. (2015). Sharp bounds on the expectations of \(L\)-statistics expressed in the Gini mean differences units. Communications in Statistics—Theory and Methods (to appear).

  • Lin, G. D., Huang, J. S. (1989). Variances of sample medians. Statistics and Probability Letters, 8, 143–146.

  • Moriguti, S. (1951). Extremal properties of extreme value distributions. Annals of Mathematical Statistics, 22, 523–536.

    Article  MathSciNet  MATH  Google Scholar 

  • Moriguti, S. (1953). A modification of Schwarz’s inequality, with applications to distributions. Annals of Mathematical Statistics, 24, 107–113.

    Article  MathSciNet  MATH  Google Scholar 

  • Papadatos, N. (1995). Maximum variance of order statistics. Annals of the Institute of Statistical Mathematics, 47, 185–193.

    Article  MathSciNet  MATH  Google Scholar 

  • Papadatos, N. (1997). A note on maximum variance of order statistics from symmetric populations. Annals of the Institute of Statistical Mathematics, 49, 117–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Pearson, K. (1902). Note on Francis Galton’s problem. Biometrika, 1, 390–399.

    Google Scholar 

  • Plackett, R. L. (1947). Limits of the ratio of mean range to standard deviation. Biometrika, 34, 120–122.

    Article  MathSciNet  MATH  Google Scholar 

  • Pyke, R. (1965). Spacings. (With discussion). Journal of the Royal Statistical Society: Series B, 27, 395–449.

    MathSciNet  MATH  Google Scholar 

  • Pyke, R. (1972). Spacings revisited. In L. M. Le Cam, J. Neyman, E. L. Scott (Eds.), Proceedings of the sixth Berkeley symposium on mathematical statistics and probability (Vol. 1, pp. 417–427) Theory of Statistics. Berkeley, CA: University of California Press.

  • Raqab, M. Z. (2003). Evaluating improvements for spacings of order statistics. Extremes, 6, 259–273.

    Article  MathSciNet  MATH  Google Scholar 

  • Rychlik, T. (2001). Projecting statistical functionals. Lecture Notes in Statistics (Vol. 160). New York: Springer.

  • Schoenberg, I. J. (1959). On variation diminishing approximation methods. In R. E. Langer (Ed.), Proceedings of a symposium on numerical approximation, Madison, April 21–23, 1958 (pp. 249–274). Madison, WI: University of Wisconsin Press.

  • Yang, H. (1982). On the variances of median and some other order statistics. Bulletin of the Institute of Mathematics Academia Sinica, 10, 197–204.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the associate editor and referee for helpful comments that allowed them to correct mistakes and improve the presentation.

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00656, Warsaw, Poland

    Paweł Marcin Kozyra & Tomasz Rychlik

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  1. Paweł Marcin Kozyra
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  2. Tomasz Rychlik
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Correspondence to Tomasz Rychlik.

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Kozyra, P.M., Rychlik, T. Lower and upper bounds on the variances of spacings. Ann Inst Stat Math 69, 417–428 (2017). https://doi.org/10.1007/s10463-015-0545-6

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  • DOI: https://doi.org/10.1007/s10463-015-0545-6

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