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Progressive censoring from heterogeneous distributions with applications to robustness

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Abstract

Progressively censored order statistics from heterogeneous distributions are introduced and their properties are investigated. After deriving the joint density function, some properties are established. In particular, the case of proportional hazards leads to an interesting connection to the model of generalized order statistics. Finally, the special case of exponential distribution is considered and some known results are generalized to this heterogeneous case, and their implications to robustness are highlighted.

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References

  • Arnold B.C., Balakrishnan N. (1989). Relations, bounds and approximations for order statistics. Lecture Notes in Statistics (Vol. 53). Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Arnold B.C., Balakrishnan N., Nagaraja H.N. (1992). A first course in order statistics. Wiley, New York

    MATH  Google Scholar 

  • Balakrishnan N. (1988). Recurrence relations for order statistics from n independent and non-identically distributed random variables. Annals of the Institute of Statistical Mathematics 40, 273–277

    Article  MATH  MathSciNet  Google Scholar 

  • Balakrishnan N. (1989). Recurrence relations among moments of order statistics from two related sets of independent and non-identically distributed random variables. Annals of the Institute of Statistical Mathematics 41, 323–329

    Article  MATH  MathSciNet  Google Scholar 

  • Balakrishnan N. (1994). Order statistics from non-identical exponential random variables and some applications (with discussion). Computational Statistics & Data Analysis 18, 203–253

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • Balakrishnan N., Cramer E., Kamps U. (2001). Bounds for means and variances of progressive Type II censored order statistics. Statistics & Probability Letters 54, 301–315

    Article  MATH  MathSciNet  Google Scholar 

  • Bapat R. (1990). Permanents in probability and statistics. Linear Algebra and its Applications 127, 3–25

    Article  MATH  MathSciNet  Google Scholar 

  • Bapat R., Beg M.I. (1989a). Identities and recurrence relations for order statistics corresponding to nonidentically distributed variables. Communications in Statistics – Theory and Methods 18, 1993–2004

    Article  MATH  MathSciNet  Google Scholar 

  • Bapat R., Beg M.I. (1989b). Order statistics for nonidentically distributed variables and permanents. Sankhyā, Series A 51, 79–93

    MATH  MathSciNet  Google Scholar 

  • Barlow R.E., Proschan F. (1975). Statistical theory of reliability and life testing: probability models. Holt-Rinehart and Winston, New York

    Google Scholar 

  • Barnett V., Lewis T. (1993) Outliers in statistical data (3rd ed.). Wiley, Chichester

    Google Scholar 

  • Burkschat M., Cramer E., Kamps U. (2006). On optimal schemes in progressive censoring. Statistics & Probability Letters 76, 1032–1036

    Article  MATH  MathSciNet  Google Scholar 

  • Cohen A.C. (1963). Progressively censored samples in life testing. Technometrics 5, 327–339

    Article  MATH  MathSciNet  Google Scholar 

  • Cramer E., Kamps U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58, 293-310

    Article  MATH  MathSciNet  Google Scholar 

  • Cramer E., Kamps U., Rychlik T. (2002). Evaluations of expected generalized order statistics in various scale units. Applied Mathematics 29, 285–295

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  • Gross A.J., Hunt H.H., Odeh R.E. (1986). The correlation coefficient between the smallest and largest observations when (n-1) of the n observations are i.i.d. exponentially distributed. Communications in Statistics – Theory and Methods 15, 1113–1123

    Article  MATH  MathSciNet  Google Scholar 

  • Gupta R.C., Kirmani S.N.U.A. (1988). Closure and monotonicity properties of nonhomogeneous Poisson processes and record values. Probability in Engineering and Information Sciences 2, 475–484

    Article  MATH  Google Scholar 

  • Harter H.L., Balakrishnan N. (1998). Order statistics: a historical perspective. In: Balakrishnan N., Rao C.R. (eds). Handbook of Statistics (Vol. 16) Elsevier, Amsterdam, pp. 25–64

    Google Scholar 

  • Joshi P.C. (1972). Efficient estimation of the mean of an exponential distribution when an outlier is present. Technometrics 14, 137–143

    Article  MATH  Google Scholar 

  • Joshi P.C. (1988). Estimation and testing under exchangeable exponential model with a single outlier. Communications in Statistics – Theory and Methods 7, 2315–2326

    Article  Google Scholar 

  • Kale B.K., Sinha S.K. (1971). Estimation of expected life in the presence of an outlier observation. Technometrics 13, 755–759

    Article  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kochar S.C., Kirmani S.N.U.A. (1995). Some results on normalized spacings from restricted families of distributions. Journal of Statistical Planning and Inference 46, 47–57

    Article  MATH  MathSciNet  Google Scholar 

  • Kochar S.C., Korwar R. (1996). Stochastic orders for spacings of heterogeneous exponential random variables. Journal of Multivariate Analysis 57, 69–83

    Article  MATH  MathSciNet  Google Scholar 

  • Kochar S.C., Rojo J. (1996). Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions. Journal of Multivariate Analysis 59, 272–281

    Article  MATH  MathSciNet  Google Scholar 

  • Marshall A.W., Olkin I. (1979). Inequalities: theory of majorization and its applications. Academic, New York

    MATH  Google Scholar 

  • Minc H. (1978). Permanents. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Pledger G., Proschan F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In: Rustagi J.S. (eds). Optimization methods in statistics. Academic, New York, pp. 89–113

    Google Scholar 

  • Sen P.K. (1970). A note on order statistics for heterogeneous distributions. Annals of Mathematical Statistics 41, 2137–2139

    Article  MathSciNet  MATH  Google Scholar 

  • Vaughan R.J., Venables W.N. (1972). Permanent expressions for order statistic densities. Journal of the Royal Statistical Society, Series B 34, 308–310

    MATH  MathSciNet  Google Scholar 

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Correspondence to Erhard Cramer.

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Balakrishnan, N., Cramer, E. Progressive censoring from heterogeneous distributions with applications to robustness. AISM 60, 151–171 (2008). https://doi.org/10.1007/s10463-006-0070-8

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  • DOI: https://doi.org/10.1007/s10463-006-0070-8

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