Skip to main content
SpringerLink
Log in

Moments of Progressively Type-II Censored Order Statistics

  • Chapter
  • First Online:
The Art of Progressive Censoring

Part of the book series: Statistics for Industry and Technology ((SIT))

  • 1662 Accesses

Abstract

Results on moments of progressively Type-II censored order statistics are reviewed. After presenting general expressions and existence results, explicit expressions for particular population distributions are given. Further, results for symmetric population distributions are developed. The presentation is completed by a survey on reccurence relations, bounds, and first-order approximations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aggarwala R, Balakrishnan N (1996) Recurrence relations for single and product moments of progressive type-II right censored order statistics from exponential and truncated exponential distributions. Ann Inst Stat Math 48:757–771

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnold BC (1985) p-Norm bounds on the expectation of the maximum of possibly dependent sample. J Multivariate Anal 17:316–332

    Article  MATH  MathSciNet  Google Scholar 

  3. Arnold BC, Balakrishnan N (1989) Relations, bounds and approximations for order statistics. Lecture notes in statistics, vol 53. Springer, New York

    Google Scholar 

  4. Arnold BC, Balakrishnan N, Nagaraja HN (1992) A first course in order statistics. Wiley, New York

    MATH  Google Scholar 

  5. Balakrishnan N (1993) A simple application of binomial-negative binomial relationship in the derivation of sharp bounds for moments of order statistics based on greatest convex minorants. Stat Probab Lett 18:301–305

    Article  MATH  Google Scholar 

  6. Balakrishnan N, Aggarwala R (2000) Progressive censoring: theory, methods, and applications. Birkhäuser, Boston

    Book  Google Scholar 

  7. Balakrishnan N, Cohen AC (1991) Order statistics and inference: estimation methods. Academic, Boston

    MATH  Google Scholar 

  8. Balakrishnan N, Joshi PC (1984) Product moments of order statistics from the doubly truncated exponential distribution. Naval Res Logist Quart 31:27–31

    Article  MATH  MathSciNet  Google Scholar 

  9. Balakrishnan N, Rao CR (eds) (1998a) Order statistics: applications. Handbook of statistics, vol 17. Elsevier, Amsterdam

    Google Scholar 

  10. Balakrishnan N, Rao CR (eds) (1998b) Order statistics: theory and methods. Handbook of statistics, vol 16. Elsevier, Amsterdam

    Google Scholar 

  11. Balakrishnan N, Saleh HM (2011) Relations for moments of progressively Type-II censored order statistics from half-logistic distribution with applications to inference. Comput Stat Data Anal 55:2775–2792

    Article  MATH  MathSciNet  Google Scholar 

  12. Balakrishnan N, Saleh HM (2012) Relations for moments of progressively Type-II censored order statistics from log-logistic distribution with applications to inference. Comm Stat Theory Meth 41:880–906

    Article  MATH  Google Scholar 

  13. Balakrishnan N, Saleh HM (2013) Recurrence relations for single and product moments of progressively Type-II censored order statistics from a generalized half-logistic distribution with application to inference. J Stat Comput Simul 83:1704–1721

    Article  MathSciNet  Google Scholar 

  14. Balakrishnan N, Sultan KS (1998) Recurrence relations and identities for moments of order statistics. In: Balakrishnan N, Rao CR (eds) Order statistics: theory and Methods, Handbook of statistics, vol 16. Elsevier, Amsterdam, pp 149–228

    Google Scholar 

  15. Balakrishnan N, Cramer E, Kamps U (2001a) Bounds for means and variances of progressive type II censored order statistics. Stat Probab Lett 54:301–315

    Article  MATH  MathSciNet  Google Scholar 

  16. Balakrishnan N, Cramer E, Kamps U, Schenk N (2001b) Progressive type II censored order statistics from exponential distributions. Statistics 35:537–556

    Article  MATH  MathSciNet  Google Scholar 

  17. Balakrishnan N, AL-Hussaini EK, Saleh HM (2011a) Recurrence relations for moments of progressively censored order statistics from logistic distribution with applications to inference. J Stat Plan Infer 141:17–30

    Google Scholar 

  18. Balasooriya U, Saw SLC (1999) A note on approximate moments of progressively censored order statistics. Metron 57:117–130

    MATH  MathSciNet  Google Scholar 

  19. Burkschat M, Cramer E, Kamps U (2007a) Linear estimation of location and scale parameters based on generalized order statistics from generalized Pareto distributions. In: Ahsanullah M, Raqab M (eds) Recent developments in ordered random variables. Nova Science Publisher, New York, pp 253–261

    Google Scholar 

  20. Cramer E (2002) A note on moments of progressively Type II censored order statistics. Comm Stat Theory Meth 31:1301–1307

    Article  MATH  MathSciNet  Google Scholar 

  21. Cramer E, Kamps U (1998) Sequential k-out-of-n systems with Weibull components. Econ Qual Control 13:227–239

    MATH  Google Scholar 

  22. Cramer E, Kamps U (2001b) Sequential k-out-of-n systems, Chap. 12. In: Balakrishnan N, Rao CR (eds) Handbook of statistics: advances in reliability, vol 20. Elsevier, Amsterdam, pp 301–372

    Google Scholar 

  23. Cramer E, Kamps U, Rychlik T (2002a) Evaluations of expected generalized order statistics in various scale units. Appl Math 29:285–295

    MATH  MathSciNet  Google Scholar 

  24. Cramer E, Kamps U, Rychlik T (2002b) On the existence of moments of generalized order statistics. Stat Probab Lett 59:397–404

    Article  MATH  MathSciNet  Google Scholar 

  25. Cramer E, Kamps U, Rychlik T (2004b) Unimodality of uniform generalized order statistics, with applications to mean bounds. Ann Inst Stat Math 56:183–192

    Article  MATH  MathSciNet  Google Scholar 

  26. Dahmen K, Burkschat M, Cramer E (2012) A- and D-optimal progressive Type-II censoring designs based on Fisher information. J Stat Comput Simul 82:879–905

    Article  MATH  MathSciNet  Google Scholar 

  27. David FN, Johnson NL (1954) Statistical treatment of censored data: I. Fundamental formulae. Biometrika 41:228–240

    MATH  MathSciNet  Google Scholar 

  28. David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, Hoboken

    Book  MATH  Google Scholar 

  29. Govindarajulu Z (1963b) Relationships among moments of order statistics in samples from two related populations. Technometrics 5:514–518

    Article  MATH  MathSciNet  Google Scholar 

  30. Govindarajulu Z (1966) Best linear estimates under symmetric censoring of the parameters of double exponential population. J Am Stat Assoc 61:248–258

    MathSciNet  Google Scholar 

  31. Graybill FA (1983) Matrices with applications in statistics, 2nd edn. Wadsworth, Belmont

    MATH  Google Scholar 

  32. Gumbel EJ (1954) The maxima of the mean largest value of the range. Ann Math Stat 25:76–84

    Article  MATH  MathSciNet  Google Scholar 

  33. Hartley HO, David HA (1954) Universal bounds for mean range and extreme observation. Ann Math Stat 25:85–99

    Article  MATH  MathSciNet  Google Scholar 

  34. Huang JS (1997) Sharp bounds for the expected value of order statistics. Stat Probab Lett 33:105–107

    Article  MATH  Google Scholar 

  35. Joshi PC (1978) Recurrence relations between moments of order statistics from exponential and truncated exponential distributions. Sankhyā Ser B 39:362–371

    MATH  Google Scholar 

  36. Joshi PC (1979) A note on the moments of order statistics from doubly truncated exponential distribution. Ann Inst Stat Math 31:321–324

    Article  MATH  Google Scholar 

  37. Joshi PC (1982) A note on the mixed moments of order statistics from exponential and truncated exponential distributions. J Stat Plan Infer 6:13–16

    Article  MATH  Google Scholar 

  38. Kałuszka M, Okolewski A (2004) Tsallis entropy bounds for generalized order statistics. Probab Math Stat 24:253–262

    MATH  Google Scholar 

  39. Kamps U (1995a) A concept of generalized order statistics. Teubner, Stuttgart

    Book  MATH  Google Scholar 

  40. Kamps U, Cramer E (2001) On distributions of generalized order statistics. Statistics 35:269–280

    Article  MATH  MathSciNet  Google Scholar 

  41. Kotz S, Kozubowski TJ, Podgórski K (2001) The Laplace distribution and generalizations. A revisit with applications to communications, economics, engineering, and finance. Birkhäuser, Boston

    Google Scholar 

  42. Lieblein J (1955) On moments of order statistics from the Weibull distribution. Ann Math Stat 26:330–333

    Article  MATH  MathSciNet  Google Scholar 

  43. Ludwig O (1959) Ungleichungen für Extremwerte und andere Ranggrößen in Anwendung auf biometrische Probleme. Biom. Z 1:203–209

    Article  MATH  Google Scholar 

  44. Mahmoud MAW, Rostom NKA, Yhieha NM (2008) Recurrence relations for the single and product moments of progressively type-II right censored order statistics from the generalized Pareto distribution. J Egypt Math Soc 16:61–74

    MATH  Google Scholar 

  45. Mann NR (1971) Best linear invariant estimation for Weibull parameters under progressive censoring. Technometrics 13:521–533

    Article  MATH  MathSciNet  Google Scholar 

  46. Marohn F (2005) Exponential inequalities for tail probabilities of generalized order statistics. Metrika 61:251–260

    Article  MATH  MathSciNet  Google Scholar 

  47. Moriguti S (1953) A modification of Schwarz’s inequality with applications to distributions. Ann Math Stat 24:107–113

    Article  MATH  MathSciNet  Google Scholar 

  48. Papadatos N (1997) Exact bounds for the expectation of order statistics from non-negative populations. Ann Inst Stat Math 49:727–736

    Article  MATH  MathSciNet  Google Scholar 

  49. Raqab MZ (2003) P-norm bounds for moments of progressive type II censored order statistics. Stat Probab Lett 64:393–402

    Article  MATH  MathSciNet  Google Scholar 

  50. Rychlik T (1993) Sharp bounds for expectations of L-estimates from dependent samples. Comm Stat Theory Meth 22:1053–1068. Correction: Comm Stat Theory Meth 23:305–306

    MathSciNet  Google Scholar 

  51. Rychlik T (2001) Projecting statistical functionals. Lecture notes in statistics, vol 160. Springer, New York

    Google Scholar 

  52. Rychlik T (2010) Evaluations of generalized order statistics from bounded populations. Stat Papers 51:165–177

    Article  MATH  MathSciNet  Google Scholar 

  53. Saran J, Pande V (2012) Recurrence relations for moments of progressively Type-II right censored order statistics from half logistic distribution. J Stat Theory Appl 11:87–96

    MathSciNet  Google Scholar 

  54. Saran J, Pushkarna N (2001) Recurrence relations for moments of progressive type-II right censored order statistics from Burr distribution. Statistics 35:495–507

    Article  MATH  MathSciNet  Google Scholar 

  55. Sen PK (1959) On the moments of the sample quantiles. Calcutta Stat Assoc Bull 9:1–19

    MATH  Google Scholar 

  56. Shah BK (1966) On the bivariate moments of order statistics from a logistic distribution. Ann Math Stat 37:1002–1010

    Article  MATH  Google Scholar 

  57. Shah BK (1970) Note on moments of a logistic order statistics. Ann Math Stat 41: 2150–2152

    Article  MATH  Google Scholar 

  58. Tsallis C (1988) Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys 52: 479–487

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada

    N. Balakrishnan

  2. Institute of Statistics, RWTH Aachen University, Aachen, Germany

    Erhard Cramer

Authors
  1. N. Balakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erhard Cramer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Balakrishnan, N., Cramer, E. (2014). Moments of Progressively Type-II Censored Order Statistics. In: The Art of Progressive Censoring. Statistics for Industry and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-0-8176-4807-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation