Abstract
Further probabilistic results on progressively Type-II censored order statistics are summarized. First, characterization and stochastic ordering results are reviewed. After discussing aging properties, limiting and extreme value results are discussed. Finally, the notion of near minimum progressively Type-II censored order statistics is addressed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ahsanullah M (1981) On characterizations of the exponential distribution by spacings. Stat Papers 22:316–320
Ahsanullah M (1984) A characterization of the exponential distribution by higher order gap. Metrika 31:323–326
Ahsanullah M, Hamedani G (2010) Exponential distribution: theory and methods. Nova Science Publisher, New York
Arias-Nicolás JP, Belzunce F, Núñez Barrera O, Suárez-Llorens A (2009) A multivariate IFR notion based on the multivariate dispersive ordering. Appl Stoch Models Bus Ind 25:339–358
Arnold BC, Nagaraja HN (1991) Lorenz ordering of exponential order statistics. Stat Probab Lett 11:485–490
Arnold BC, Villaseñor JA (1991) Lorenz ordering of order statistics. Lecture notes–monograph series, vol 19. Institute of Mathematical Statistics, Hayward, pp 38–47
Arnold BC, Villaseñor JA (1998) Lorenz ordering of order statistics and record values. In: Balakrishnan N, Rao CR (eds) Order Statistics: theory and methods. Handbook of statistics, vol 16. Elsevier, Amsterdam, pp 75–87
Arnold BC, Balakrishnan N, Nagaraja HN (1992) A first course in order statistics. Wiley, New York
Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, New York
Bagai I, Kochar SC (1986) On tail-ordering and comparison of failure rates. Comm Stat Theory Meth 15:1377–1388
Balakrishnan N, Basu AP (eds) (1995) The exponential distribution: theory, methods, and applications. Taylor & Francis, Newark
Balakrishnan N, Dembińska A (2008) Progressively Type-II right censored order statistics from discrete distributions. J Stat Plan Infer 138:845–856
Balakrishnan N, Dembińska A (2009) Erratum to ‘progressively Type-II right censored order statistics from discrete distributions’ [J Stat Plan Infer 138:845–856 (2008)]. J Stat Plan Infer 139:1572–1574
Balakrishnan N, Malov SV (2005) Some characterizations of exponential distribution based on progressively censored order statistics. In: Balakrishnan N, Bairamov IG, Gebizlioglu OL (eds) Advances on models, characterizations and applications. Chapman & Hall/CRC, Boca Raton, pp 97–109
Balakrishnan N, Rao CR (eds) (1998a) Order statistics: applications. Handbook of statistics, vol 17. Elsevier, Amsterdam
Balakrishnan N, Rao CR (eds) (1998b) Order statistics: theory and methods. Handbook of statistics, vol 16. Elsevier, Amsterdam
Balakrishnan N, Stepanov A (2005) A note on the number of observations near an order statistic. J Stat Plan Infer 134:1–14
Balakrishnan N, Stepanov A (2008) Asymptotic properties of numbers of near minimum observations under progressive Type-II censoring. J Stat Plan Infer 138:1010–1020
Balakrishnan N, Belzunce F, Hami N, Khaledi BE (2010a) Univariate and multivariate likelihood ratio ordering of generalized order statistics and associated conditional variables. Probab Eng Inform Sci 24:441–455
Balakrishnan N, Cramer E, Dembińska A (2011b) Characterizations of geometric distribution through progressively Type-II right censored order statistics. Statistics 59:559–573
Balakrishnan N, Belzunce F, Sordo MA, Suárez-Llorens A (2012a) Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data. J Multivariate Anal 105:45–54
Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing, probability models. Holt-Rinehart and Winston, New York
Bartoszewicz J (1985) Moment inequalities for order statistics from ordered families of distributions. Metrika 32:383–389
Belzunce F, Lillo RE, Ruiz JM, Shaked M (2001) Stochastic comparisons of nonhomogeneous processes. Probab Eng Inform Sci 15:199–224
Belzunce F, Mercader JA, Ruiz JM (2003) Multivariate aging properties of epoch times of nonhomogeneous processes. J Multivariate Anal 84:335–350
Belzunce F, Mercader JA, Ruiz JM (2005) Stochastic comparisons of generalized order statistics. Probab Eng Inform Sci 19:99–120
Belzunce F, Ruiz JM, Suárez-Llorens A (2008) On multivariate dispersion orderings based on the standard construction. Stat Probab Lett 78:271–281
Bieniek M (2007a) On characterizations of distributions by regression of adjacent generalized order statistics. Metrika 66:233–242
Bieniek M, Szynal D (2003) Characterizations of distributions via linearity of regression of generalized order statistics. Metrika 58:259–271
Boland PJ, Shaked M, Shanthikumar JG (1998) Stochastic ordering of order statistics. In: Balakrishnan N, Rao CR (eds) Order statistics: theory and methods. Handbook of statistics, vol 16. Elsevier, Amsterdam, pp 89–103
Bon JL, Păltănea E (1999) Ordering properties of convolutions of exponential random variables. Lifetime Data Anal 5:185–192
Burkschat M (2006) Estimation with generalized order statistics. Ph.D. thesis, RWTH Aachen University, Aachen, Germany
Burkschat M (2008b) On optimality of extremal schemes in progressive Type-II censoring. J Stat Plan Infer 138:1647–1659
Burkschat M, Navarro J (2011) Aging properties of sequential order statistics. Probab Eng Inform Sci 25:449–467
Chan W, Proschan F, Sethuraman J (1991) Convex-ordering among functions, with applications to reliability and mathematical statistics. In: Block HW, Sampson AR, Savits TH (eds) Topics in statistical dependence. IMS lecture notes: monograph series. IMS, Hayward, pp 121–134
Chen J, Hu T (2007) Multivariate dispersive ordering of generalized order statistics. J Iran Stat Soc (JIRSS) 6:61–75
Čibisov DM (1964) Limit distributions for order statistics. Theory Probab Appl 9:142–148
Cramer E (2003) Contributions to generalized order statistics. Habilitationsschrift. University of Oldenburg, Oldenburg
Cramer E (2004) Logconcavity and unimodality of progressively censored order statistics. Stat Probab Lett 68:83–90
Cramer E (2014) Extreme value analysis for progressively Type-II censored order statistics. Comm Stat Theory Meth 43:2135–2155
Cramer E, Kamps U (2003) Marginal distributions of sequential and generalized order statistics. Metrika 58:293–310
Cramer E, Kamps U (2005) Characterization of the exponential distribution by conditional expectations of generalized spacings. In: Balakrishnan N, Bairamov IG, Gebizlioglu OL (eds) Advances on models, characterizations, and applications. Chapman & Hall/CRC, Boca Raton, pp 83–96
Cramer E, Kamps U, Raqab MZ (2003) Characterizations of exponential distributions by spacings of generalized order statistics. Appl Math 30:257–265
Cramer E, Kamps U, Keseling C (2004a) Characterizations via linear regression of ordered random variables: a unifying approach. Comm Stat Theory Meth 33:2885–2911
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
Dallas AC (1976) Characterizing the Pareto and power distributions. Ann Inst Stat Math 28:491–497
David HA, Groeneveld RA (1982) Measures of local variation in a distribution: expected length of spacings and variances of order statistics. Biometrika 69:227–232
David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, Hoboken
De Haan L, Ferreira A (2006) Extreme value theory. an introduction. Springer, New York
Dembińska A, Wesołowski J (1998) Linearity of regression from non-adjacent order statistics. Metrika 90:215–222
Dembińska A, Wesołowski J (2000) Linearity of regression from non-adjacent record values. J Stat Plan Infer 90:195–205
Dharmadhikari S, Joag-dev K (1988) Unimodality, convexity, and applications. Academic, Boston
El-Neweihi E, Govindarajulu Z (1979) Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics. J Stat Plan Infer 3:85–90
Falk M (1989) A note on uniform asymptotic normality of intermediate order statistics. Ann Inst Stat Math 41:19–29
Ferguson TS (1967) On characterizing distributions by properties of order statistics. Sankhyā A 29:265–278
Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math Proc Cambridge Phil Soc 24:180–190
Franco M, Ruiz JM (1995) On characterization of continuous distributions with adjacent order statistics. Statistics 26:375–385
Franco M, Ruiz JM (1996) On characterization of continuous distributions by conditional expectation of record values. Sankhyā 58:135–141
Franco M, Ruiz JM (1998) Characterization of discrete populations through conditional expectations of order statistics. Stat Papers 39:249–262
Franco M, Ruiz JM (1999) Characterization based on conditional expectations of adjacent order statistics: a unified approach. Proc Am Math Soc 127:861–874
Franco M, Ruiz JM, Ruiz MC (2002) Stochastic orderings between spacings of generalized order statistics. Probab Eng Inform Sci 16:471–484
Gajek L, Gather U (1989) Characterizations of the exponential distribution by failure rate and moment properties of order statistics. In: Hüsler J, Reiss RD (eds) Extreme value theory. Springer, Berlin, pp 114–124
Galambos J (1975) Characterizations of probability distributions by properties of order statistics II (discrete distributions). In: Patil GP, Kotz S, Ord JK (eds) Statistical distributions in scientific work, vol 3. D. Reidel, Boston, pp 89–102
Galambos J (1978) The asymptotic theory of extreme order statistics. Wiley, New York
Galambos J (1987) The asymptotic theory of extreme order statistics, 2nd edn. Krieger, Malabar
Gather U, Kamps U, Schweitzer N (1998) Characterizations of distributions via identically distributed functions of order statistics. In: Balakrishnan N, Rao CR (eds) Handbook of statistics, vol 16. Elsevier, Amsterdam, pp 257–290
Gnedenko BV (1943) Sur la distribution limite du terme maximum d’une série aléatoire. Ann Math 44:423–453
Govindarajulu Z (1980) Characterization of the geometric distribution using properties of order statistics. J Stat Plan Infer 4:237–247
Grudzień Z, Szynal D (1985) On the expected values of kth record values and associated characterizations of distributions. In: Konecny F, Mogyoródi J, Wertz W (eds) Probability and statistical decision theory, vol A. Reidel, Dordrecht, pp 119–127
Gupta PL, Gupta RC, Tripathi RC (1997) On the monotonic properties of discrete failure rates. J Stat Plan Infer 65:255–268
Hashemi M, Asadi M (2007) Some characterization results on generalized Pareto distribution based on progressive Type-II right censoring. J Iran Stat Soc (JIRSS) 6:99–110
Hashemi M, Tavangar M, Asadi M (2010) Some properties of the residual lifetime of progressively Type-II right censored order statistics. Stat Probab Lett 80:848–859
Hofmann G, Cramer E, Balakrishnan N, Kunert G (2005a) An asymptotic approach to progressive censoring. J Stat Plan Infer 130:207–227
Houben A (1998) Verallgemeinerte Ordnungsstatistiken bei zufälligem Stichprobenumfang. Ph.D. thesis, RWTH Aachen University, Aachen, Germany
Hu T, Wei Y (2001) Stochastic comparisons of spacings from restricted families of distributions. Stat Probab Lett 53:91–99
Hu T, Zhuang W (2003) Stochastic comparisons of m-spacings. Technical report. Department of Statistics and Finance, University of Science and Technology of China, Hefei
Hu T, Zhuang W (2005) A note on stochastic comparisons of generalized order statistics. Stat Probab Lett 72:163–170
Iwińska M (1986) On the characterizations of the exponential distribution by order statistics and record values. Fasc Math 16:101–107
Izadi M, Khaledi BE (2007) Progressive Type II censored order statistics and their concomitants: some stochastic comparisons results. J Iran Stat Soc (JIRSS) 6:111–124
Kamps U (1995a) A concept of generalized order statistics. Teubner, Stuttgart
Kamps U (1998a) Characterizations of distributions by 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 291–311
Kamps U (1998b) Subranges of generalized order statistics from exponential distributions. Fasc Math 28:63–70
Kamps U, Cramer E (2001) On distributions of generalized order statistics. Statistics 35:269–280
Kamps U, Keseling C (2003) A theorem of Rossberg for generalized order statistics. Sankhyā Ser A 65:259–270
Keilson J, Sumita U (1982) Uniform stochastic ordering and related inequalities. Can J Stat 10:181–198
Keseling C (1999a) Charakterisierungen von Wahrscheinlichkeitsverteilungen durch verallgemeinerte Ordnungstatistiken. Ph.D. thesis, RWTH Aachen University, Aachen, Germany
Keseling C (1999b) Conditional distributions of generalized order statistics and some characterizations. Metrika 49:27–40
Khaledi BE (2005) Some new results on stochastic orderings between generalized order statistics. J Iran Stat Soc (JIRSS) 4:35–49
Khaledi BE, Kochar SC (1999) Stochastic orderings between distributions and their sample spacings II. Stat Probab Lett 44:161–166
Khaledi BE, Kochar SC (2004) Ordering convolutions of gamma random variables. Sankhyā 66:466–473
Khaledi BE, Kochar SC (2005) Dependence orderings for generalized order statistics. Stat Probab Lett 73:357–367
Khan AH, Abu-Salih MS (1988) Characterization of the Weibull and the inverse Weibull distributions through conditional moments. J Inform Optim Sci 9:355–362
Khan AH, Abu-Salih MS (1989) Characterizations of probability distributions by conditional expectations of order statistics. Metron 47:171–181
Khan AH, Khan IA (1986) Characterization of the Pareto and the power function distributions. J Stat Res 20:71–79
Khan AH, Khan IA (1987) Moments of order statistics from Burr distribution and its characterization. Metron 45:21–29
Kochar SC (1996) Dispersive ordering of order statistics. Stat Probab Lett 27:271–274
Kochar SC (2006) Lorenz ordering of order statistics. Stat Probab Lett 76:1855–1860
Korwar R (2003) On the likelihood ratio order for progressive type II censored order statistics. Sankhyā Ser A 65:793–798
Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, New York
Li Y (1999) A note on the number of records near the maximum. Stat Probab Lett 43: 153–158
Lillo RE, Nanda AK, Shaked M (2001) Preservation of some likelihood ratio stochastic orders by order statistics. Stat Probab Lett 51:111–119
López-Blázquez F, Miño B (1998) A characterization of the geometric distribution. Stat Papers 39:231–236
López Blázquez F, Moreno Rebollo JL (1997) A characterization of distributions based on linear regression of order statistics and record values. Sankhyā 59:311–323
Marohn F (2002) A characterization of generalized Pareto distributions by progressive censoring schemes and goodness-of-fit tests. Comm Stat Theory Meth 31:1055–1065
Misra N, van der Meulen EC (2003) On stochastic properties of m-spacings. J Stat Plan Infer 115:683–697
Mohie El-Din MM, Mahmoud MAW, Abo Youssef SE (1991) Moments of order statistics from parabolic and skewed distributions and a characterization of Weibull distribution. Comm Stat Simul Comput 20:639–645
Mosteller F (1946) On some useful “inefficient” statistics. Ann Math Stat 17:377–408
Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, Chichester
Nagaraja HN (1977) On a characterization based on record values. Aust J Stat 19:70–73
Nagaraja HN (1988a) Some characterizations of continuous distributions based on regressions of adjacent order statistics and record values. Sankhyā Ser A 50:70–73
Nagaraja HN (1988b) Some characterizations of discrete distributions based on linear regressions of adjacent order statistics. J Stat Plan Infer 20:65–75
Nagaraja HN, Srivastava RC (1987) Some characterizations of geometric type distributions based on order statistics. J Stat Plan Infer 17:181–191
Nasri-Roudsari D (1996) Extreme value theory of generalized order statistics. J Stat Plan Infer 55:281–297
Nasri-Roudsari D, Cramer E (1999) On the convergence rates of extreme generalized order statistics. Extremes 2:421–447
Nevzorov VB (2001) Records: mathematical theory. Translations of mathematical monographs, vol 194. American Mathematical Society, Providence, Rhode Island
Oakes D, Dasu T (1990) A note on residual life. Biometrika 77:409–410
Ouyang LY (1993) Characterizations of the uniform distribution by conditional expectations. Inter J Inform Manag Sci 4:107–111
Ouyang LY (1995) Characterization through the conditional expectation of a function of one order statistic relative to an adjacent one. Sankhyā Ser A 57:500–503
Pakes AG, Li Y (1998) Limit laws for the number of near maxima via the Poisson approximation. Stat Probab Lett 40:395–401
Pakes AG, Steutel FW (1997) On the number of records near the maximum. Aust N Z J Stat 39:179–192
Pellerey F, Shaked M, Zinn J (2000) Nonhomogeneous Poisson processes and logconcavity. Probab Eng Inform Sci 14:353–373
Pfeifer D (1989) Einführung in die Extremwertstatistik. Teubner, Stuttgart (in german)
Pudeg A (1991) Charakterisierungen von Wahrscheinlichkeitsverteilungen durch Verteilungseigenschaften der Ordnungsstatistiken und Rekorde. Ph.D. thesis, RWTH Aachen University, Aachen, Germany
Puri PS, Rubin H (1970) A characterization based on the absolute difference of two i.i.d. random variables. Ann Math Stat 41:2113–2122
Rao CR, Shanbhag DN (1986) Recent results on characterization of probability distributions: a unified approach through extensions of Deny’s theorem. Adv Appl Probab 18:660–678
Rao CR, Shanbhag DN (1994) Choquet-Deny type functional equations with applications to stochastic models. Wiley, New York
Rao CR, Shanbhag DN (1998) Recent approaches to characterizations based on order statistics and record values. In: Balakrishnan N, Rao CR (eds) Order statistics: theory and methods. Handbook of statistics, vol 16. Elsevier, Amsterdam, pp 231–256
Raqab MZ, Amin WA (1996) Some ordering results on order statistics and record values. IAPQR Trans 21:1–8
Reiss RD (1989) Approximate distributions of order statistics. Springer, New York
Resnick SI (1987) Extreme values, regular variation, and point processes. Springer, New York
Rogers GS (1959) A note on the stochastic independence of functions of order statistics. Ann Math Stat 30:1263–1264
Rogers GS (1963) An alternative proof of the characterization of the density Ax B. Am Math Monthly 70:857–858
Rojo J, He GZ (1991) New properties and characterizations of the dispersive ordering. Stat Probab Lett 11:365–372
Ross SM (1996) Stochastic processes, 2nd edn. Wiley, New York
Rossberg HJ (1972) Characterization of distribution functions by the independence of certain functions of order statistics. Sankhyā Ser A 34:111–120
Saunders D (1984) Dispersive ordering of distributions. Adv Appl Probab 16:693–694
Sen PK, Singer JM (1993) Large sample methods in statistics. Chapman & Hall, New York
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, Berlin
Smirnov NV (1952) Limit distributions for the terms of a variational series. Am Math Soc Transl Ser 1:82–143
Smirnov NV (1967) Some remarks on limit laws for order statistics. Theory Probab Appl 12:337–339
Stacy EW (1962) A generalization of the gamma distribution. Ann Math Stat 33:1187–1192
Takahasi K (1988) A note on hazard rates of order statistics. Comm Stat Theory Meth 17:4133–4136
Tavangar M, Asadi M (2011) On stochastic and aging properties of generalized order statistics. Probab Eng Inform Sci 25:187–204
Tavangar M, Hashemi M (2013) On characterizations of the generalized Pareto distributions based on progressively censored order statistics. Stat Papers 54:381–390
Torrado N, Lillo R, Wiper M (2012) Sequential order statistics: ageing and stochastic orderings. Method Comput Appl Probab 14:579–596
Tran TTH (2006) Discrete generalized order statistics. Ph.D. thesis, University of Oldenburg, Oldenburg, Germany
Wu CY (1966) The types of limit distributions for some terms of variational series. Sci Sinica 15:749–762
Xie H, Hu T (2010) Some new results on multivariate dispersive ordering of generalized order statistics. J Multivariate Anal 101:964–970
Xie H, Zhuang W (2011) Some new results on ordering of simple spacings of generalized order statistics. Probab Eng Inform Sci 25:71–81
Zhuang W, Hu T (2009) Multivariate dispersive ordering of spacings of generalized order statistics. Appl Math Lett 22:968–974
Zijlstra M (1983) Characterizations of the geometric distribution by distributional properties. J Appl Probab 20:843–850
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Balakrishnan, N., Cramer, E. (2014). Further Distributional Results on Progressive Type-II Censoring. 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_3
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4807-7_3
Published:
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-0-8176-4806-0
Online ISBN: 978-0-8176-4807-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)