Asymptotic properties of maximum likelihood estimates for a bivariate exponential distribution and mixed censored data

Di Chen, Jye Chyi Lu, Jacqueline M. Hughes-Oliver, Chin-Shang Li

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Abstract

This article investigates asymptotic properties of the maximum likelihood estimators (MLE) of parameters in the bivariate exponential distribution (BVE) of Marshall and Olkin (1967) based on the following mixed censored data. In life-testing two-component parallel systems (A, B), a cost-saving procedure is to stop the testing experiment after observing the first r failure times of component A. The resulting data are (X(1), Y*[1]), (X(2), Y*[2]), . . . , (X(r), Y*[r]), (X*(r+1), Y*[r+1]), . . . , (X*(n) Y*[n]), where X(1) ≤ X(2) ≤ ⋯ ≤ X(r) are ordered lifetimes from component A, and Y[i] is the concomitant order statistic corresponding to X(i) from component B. The data X*(r+1), . . . , X*(n), and Y*[i], i = 1, . . . , n are all censored at time X(r) = x(r). Because of the complexity of the data type and the irregularity of the BVE distribution of (X, Y), there are no immediately applicable asymptotic results for the MLE. This article provides a rigorous treatment of the asymptotic behavior of the MLE, with a numerical example illustrating the mathematical derivations.

Original languageEnglish (US)
Pages (from-to)109-125
Number of pages17
JournalMetrika
Volume48
Issue number2
Publication statusPublished - 1998
Externally publishedYes

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Keywords

  • Asymptotic theory
  • Concomitants of order statistics
  • Life-testing
  • Marshall-Olkin distribution
  • Maximum likelihood estimation

ASJC Scopus subject areas

  • Statistics and Probability

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