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Metrika

, Volume 70, Issue 1, pp 19–33 | Cite as

Prediction of k-records from a general class of distributions under balanced type loss functions

  • Jafar Ahmadi
  • Mohammad Jafari Jozani
  • Éric Marchand
  • Ahmad Parsian
Article

Abstract

We study the problem of predicting future k-records based on k-record data for a large class of distributions, which includes several well-known distributions such as: Exponential, Weibull (one parameter), Pareto, Burr type XII, among others. With both Bayesian and non-Bayesian approaches being investigated here, we pay more attention to Bayesian predictors under balanced type loss functions as introduced by Jafari Jozani et al. (Stat Probab Lett 76:773–780, 2006a). The results are presented under the balanced versions of some well-known loss functions, namely squared error loss, Varian’s linear-exponential loss and absolute error loss or L 1 loss functions. Some of the previous results in the literatures such as Ahmadi et al. (Commun Stat Theory Methods 34:795–805, 2005), and Raqab et al. (Statistics 41:105–108, 2007) can be achieved as special cases of our results.

Keywords

Absolute value error loss Balanced loss function Bayes prediction Conditional median prediction Maximum likelihood prediction LINEX loss Record values 

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References

  1. Ahmadi J, Doostparast M (2006) Bayesian estimation and prediction for some life distributions based on record values. Stat Pap 47: 373–392MATHCrossRefMathSciNetGoogle Scholar
  2. Ahmadi J, Doostparast M, Parsian A (2005) Estimation and prediction in a two-parameter exponential distribution based on k-record values under LINEX loss function. Commun Stat Theory Methods 34: 795–805MATHCrossRefMathSciNetGoogle Scholar
  3. Ahsanullah M (1980) Linear prediction of record values for the two parameter exponential distribution. Ann Inst Stat Math 32: 363–368MATHCrossRefMathSciNetGoogle Scholar
  4. Ali Mousa MAM, Jaheen ZF, Ahmad AA (2002) Bayesian estimation, prediction and characterization for the Gumbel model based on records. Statistics 36: 65–74MATHMathSciNetGoogle Scholar
  5. Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, New YorkMATHGoogle Scholar
  6. Awad AM, Raqab MZ (2000) Prediction intervals for the future record values from exponential distribution: comparative study. J Stat Comput Simul 65: 325–340MATHCrossRefMathSciNetGoogle Scholar
  7. Basak P, Balakrishnan N (2003) Maximum likelihood prediction of future record statistic. Mathematical and statistical methods in reliability. In: Lindquist BH, Doksum KA(eds) Series on quality, reliability and engineering statistics. World Scientific Publishing, Singapore, pp 159–175Google Scholar
  8. Chandler KN (1952) The distribution and frequency of record values. J R Stat Soc Ser B 14: 220–228MATHMathSciNetGoogle Scholar
  9. Danielak K, Raqab MZ (2004a) Sharp bounds on expectations of kth record spacings from restricted families. Stat Probab Lett 69(2): 175–187MATHCrossRefMathSciNetGoogle Scholar
  10. Danielak K, Raqab MZ (2004b) Sharp bounds for expectations of kth record increments. Aus N Z J Stat 46(4): 665–673MATHCrossRefMathSciNetGoogle Scholar
  11. Doostparast M, Ahmadi J (2006) Statistical analysis for geometric distribution based on records. Comput Math Appl 52: 905–916MATHCrossRefMathSciNetGoogle Scholar
  12. Dziubdziela W, Kopocinski B (1976) Limiting properties of the kth record values. Zastosowania Matematyki 15: 187–190MATHMathSciNetGoogle Scholar
  13. Dunsmore IR (1983) The future occurrence of records. Ann Inst Stat Math 35: 267–277MATHCrossRefMathSciNetGoogle Scholar
  14. Jafari Jozani M, Marchand É, Parsian A (2006a) On estimation with weighted balanced-type loss function. Stat Probab Lett 76: 773–780MATHCrossRefMathSciNetGoogle Scholar
  15. Jafari Jozani M, Marchand É, Parsian A (2006b) Bayes estimation under a general class of balanced loss functions (It can also be found at Rapport de recherche 36. Département de mathématiques, Université de Sherbrooke, http://www.usherbrooke.ca/mathématiques/telechargement) (submitted)
  16. Jaheen ZF (2003) A Bayesian analysis of record statistics from the Gompertz model. Appl Math Comput 145: 307–320MATHCrossRefMathSciNetGoogle Scholar
  17. Jaheen ZF (2004) Empirical Bayes analysis of record statistics based on LINEX and quadratic loss functions. Comput Math Appl 47: 947–954MATHCrossRefMathSciNetGoogle Scholar
  18. Kamps U (1995) A concept of generalized order statistics. Teubner Scripten zur Mathematischen Stochastik. Teubner, StuttgartGoogle Scholar
  19. Klimczak M (2006) Prediction of kth records. Stat Probab Lett 76(2): 117–127MATHMathSciNetGoogle Scholar
  20. Klimczak M, Rychlik T (2004) Maximum variance of kth records. Stat Probab Lett 69(4): 421–430MATHCrossRefMathSciNetGoogle Scholar
  21. Lawless JL (2003) Statistical models and methods for lifetime data. 2nd edn. Wiley, New YorkMATHGoogle Scholar
  22. Madi MT, Raqab MZ (2004) Bayesian prediction of temperature records using the Pareto model. Environmetrics 15: 701–710CrossRefGoogle Scholar
  23. Nevzorov V (2001) Records: mathematical theory. Translation of Mathematical Monographs, vol 194, American Mathematical Society, ProvidenceGoogle Scholar
  24. Parsian A, Kirmani SNUA (2002) Estimation under LINEX loss function. In: Ullah A, Wan ATK, Chaturvedi A (eds) Handbook of Applied Econometrics and Statistical Inference, Marcel Dekker, Inc., vol. 165, pp 53–76Google Scholar
  25. Raqab MZ (2004) Projection P-norm bounds on the moments of kth record increments. J Stat Plan Inference 124(2): 301–315MATHCrossRefMathSciNetGoogle Scholar
  26. Raqab M, Nagaraja HN (1995) On some predictors of future order statistics. Metron 53(1–2): 185–204MATHMathSciNetGoogle Scholar
  27. Raqab M, Ahmadi J, Doostparast M (2007) Statistical inference based on record data from Pareto model. Statistics 41: 105–108MATHCrossRefMathSciNetGoogle Scholar
  28. Zellner A (1994) Bayesian and non-Bayesian estimation using balanced loss functions. In: Berger JO, Gupta SS(eds) Statistical decision theory and methods V. Springer, NewYork, pp 337–390Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Jafar Ahmadi
    • 1
  • Mohammad Jafari Jozani
    • 2
  • Éric Marchand
    • 3
  • Ahmad Parsian
    • 4
  1. 1.School of Mathematical SciencesFerdowsi University of MashhadMashhadIran
  2. 2.Department of Statistics, Faculty of Economics, Statistical Research and Training Center (SRTC)Allameh Tabatabaie UniversityTehranIran
  3. 3.Département de mathématiquesUniversité de SherbrookeSherbrookeCanada
  4. 4.School of Mathematics, Statistics and Computer ScienceUniversity of TehranTehranIran

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