Brazilian Journal of Probability and Statistics

Order statistics and exceedances for some models of INID random variables

Ismihan Bayramoglu and Goknur Giner

Full-text: Open access


Order statistics and exceedances for some general models of independent but not necessarily identically distributed (INID) random variables are considered. The distributions of order statistics from INID sample are described in terms of symmetric functions. Some exceedance models based on order statistics from INID random variables are considered, the limit distributions of exceedance statistics are obtained. For the model of INID random variables referred as $F^{\alpha}$-scheme introduced by Nevzorov (Zapiski Nauchnykh Seminarov LOMI 142 (1985) 109–118) the limiting distribution of exceedance statistic has been derived. This distribution is expressed in terms of permutations with inversions, Gaussian Hypergeometric function and incomplete beta functions.

Article information

Braz. J. Probab. Stat., Volume 28, Number 4 (2014), 492-514.

First available in Project Euclid: 30 July 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

INID random variables exceedances


Bayramoglu, Ismihan; Giner, Goknur. Order statistics and exceedances for some models of INID random variables. Braz. J. Probab. Stat. 28 (2014), no. 4, 492--514. doi:10.1214/13-BJPS221.

Export citation


  • Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. New York: Wiley.
  • Bairamov, I. G. (1997). Some distribution free properties of statistics based on record values and characterizations of distributions through a record. Journal of Applied Statistical Science 5(1), 17–25.
  • Bairamov, I. G. (2007). Advances in Exceedance Statistics Based on Ordered Random Variables. Recent Developments in Ordered Random Variables. New York: Nova Science Publishers.
  • Bairamov, I. G. and Eryilmaz, S. N. (2000). Distributional properties of statistics based on minimal spacing and record exceedance statistics. Journal of Statistical Planning and Inference 90, 21–33.
  • Bairamov, I. G. and Eryilmaz, S. (2001). On properties of statistics connected with minimal spacing and record exceedances. Applied Statistical Science V (M. Ahsanullah, J. Kennyon and S. K. Sarkar, eds.), 245–254. Huntington, New York: Nova Science Publishers.
  • Bairamov, I. and Eryılmaz, S. (2004). Characterization of symmetry and exceedance models in multivariate FGM distributions. Journal of Applied Statistical Science 13(2), 87–99.
  • Bairamov, I. and Khan, M. K. (2007). On exceedances of record and order statistics. Proceedings of the American Mathematical Society 135(6), 1935–1945.
  • Bairamov, I. G. and Kotz, S. (2001). On distributions of exceedances associated with order statistics and record values for arbitrary distributions. Statistical Papers 42(2), 171–185.
  • Balakrishnan, N. (2007). Permanent, order statistics, outliers and robustness, review. Mathematics of Computation 1, 7–107.
  • Bateman, H. (1953). Higher Transcendental Functions. New York: MacGraw Hill.
  • Bourget, J. (1871). Des permutations. Nouvelles Anna les De Mathematiques 10, 254–268.
  • Comtet, L. (1974). Advanced Combinatorics. Dordrecht: Reidel.
  • David, H. A. (1981). Order Statistics, 2nd ed. New York: Wiley.
  • David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd ed. New York: Wiley.
  • Gnedenko, B. N. (1978) The Theory of Probability. Moscow: Mir Publishers.
  • Gurler, S. and Bairamov, I. (2009). Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions. Applied Mathematical Modelling 33, 1116–1125.
  • Knuth, D. E. (1973). The Art of Computer Programming, Vol. 3. Reading, MA: Addison-Wesley.
  • Margolius, B. H. (2001). Permutations with inversions. Journal of Integer Sequences 4(2), Article 01.2.4 (electronic).
  • Maurer, W. and Margolin, B. H. (1976). The multivariate inclusion–exclusion formula and order statistics from dependent variates. The Annals of Statistics 4(6), 1190–1199.
  • Muir, T. (1882). A Treatise on the Theory of Determinants with Graduate Sets of Exercises for Use in Colleges and Schools. London: MacMillian and Co.
  • Nevzorov, V. B. (1985). On record times and inter-record times for sequences of nonidentically distributed random variables. Zapiski Nauchnykh Seminarov LOMI 142, 109–118.
  • Nevzorov, V. B. (1987). Two characterizations using records. In Stability Problems for Stochastic Models. Theory Probab. Appl. Lecture Notes in Mathematics 1233, 79–85. Berlin: Springer.
  • Pfeifer, D. (1989). Extremal processes, secretary problems, and the l/e law. Journal of Applied Probability 26, 722–733.
  • Pfeifer, D. (1991). Some remarks on Nevzorov’s record model. Advances in Applied Probability 23, 823–833.
  • Vaughan, R. J. and Venables, W. N. (1972). Permanent expressions for order statistics densities. Journal of the Royal Statistical Society, Ser. B 34, 308–310.
  • Wesolowski, J. and Ahsanullah, M. (1998). Distributional properties of exceedance statistics. Annals of the Institute of Statistical Mathematics 50(3), 543–565.