Schur-concave survival functions and survival analysis.

*(English)*Zbl 0773.62066Summary: We consider an \(N\)-tuple of exchangeable nonnegative random variables, which can, e.g., be interpreted as lifetimes of \(N\) similar units, and we assume that the joint survival function
\[
\overline F_ N(x_ 1,\dots,x_ N)=P\{X_ 1>x_ 1,\dots,X_ N>x_ N\}
\]
is, in particular, Schur-concave. This condition is relevant since, as it has been recently shown, it provides a probabilistic model for aging in the subjectivist set-up. We analyze general properties of Schur-concave survival functions and give representation theorems. In particular, we study properties of Schur-concave survival distributions which are a finite-population version of time-transformed exponential distributions. These distribution models are of interest in analyzing life data.

##### MSC:

62N05 | Reliability and life testing |

62E10 | Characterization and structure theory of statistical distributions |

60G09 | Exchangeability for stochastic processes |

##### Keywords:

life distributions; finite exchangeability; logarithmic concavity; exchangeable nonnegative random variables; Schur-concave survival functions; representation theorems; finite-population version of time- transformed exponential distributions; life data
PDF
BibTeX
XML
Cite

\textit{R. E. Barlow} and \textit{F. Spizzichino}, J. Comput. Appl. Math. 46, No. 3, 437--447 (1993; Zbl 0773.62066)

Full Text:
DOI

##### References:

[1] | R.E. Barlow and M. Mendel, De Finetti-type representations for life distributions, J. Amer. Statist. Assoc., to appear. · Zbl 0764.62077 |

[2] | Barlow, R.E.; Proschan, F., Statistical theory of reliability and life testing, (1981), To Begin With Silver Spring, MD |

[3] | de Finetti, B.; de Finetti, B., Foresight: its logical laws, its subjective sources, (), 7, 53-118, (1937), English translation |

[4] | Marshall, A.W.; Olkin, I., Majorization in multivariate distributions, Ann. statist., 2, 1189-1200, (1974) · Zbl 0292.62037 |

[5] | Marshall, A.W.; Olkin, I., Inequalities, theory of majorization and its applications, (1979), Academic Press New York · Zbl 0437.26007 |

[6] | Mendel, M.B., Bayesian parametric models for lifetimes, Proc. ivth internat. meeting on Bayesian statistics, Valencia, 1991, (1992), Oxford Univ. Press Oxford |

[7] | Nevius, S.E.; Proschan, F.; Sethuraman, J., Schur functions in statistics II. stochastic majorization, Ann. statist., 5, 263-273, (1977) · Zbl 0383.62039 |

[8] | Spizzichino, F., Reliability decision problems under conditions of aging, Proc. ivth internat. meeting on Bayesian statistics, Valencia, 1991, (1992), Oxford Univ. Press Oxford |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.