### Abstract

This paper estimates component reliability from masked series-system life data, viz, data where the exact component causing system failure might be unknown. We extend the results of Usher & Hodgson (1988) by deriving exact maximum likelihood estimators (MLE) for the general case of a series system of 3 exponential components with independent masking. Their previous work shows that closed-form MLE are intractable, and they propose an iterative method for the solution of a system of 3 Nonlinear likelihood equations. They do not, however, prove convergence for their iterative method. As such, we show how this system of Nonlinear equations can be replaced by a single quartic equation, whose solution is straight-forward. Since it does not depend upon the convergence of numerical solution algorithms, the results are exact. Though the resulting estimators are somewhat lengthy & cumbersome to find manually, they can be written as a straightforward computer code. The calculations can then be easily performed on a personal computer. This method for reducing the likelihood equations to simpler-to-solve forms can be extended readily to a higher number of components. In many cases for more than 3 components it is easier while for others it is more complicated; even in the more complicated cases, this simplification makes the problems much more tractable.

Original language | English (US) |
---|---|

Pages (from-to) | 631-635 |

Number of pages | 5 |

Journal | IEEE Transactions on Reliability |

Volume | 42 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1993 |

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### All Science Journal Classification (ASJC) codes

- Safety, Risk, Reliability and Quality
- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Reliability*,

*42*(4), 631-635. https://doi.org/10.1109/24.273596

}

*IEEE Transactions on Reliability*, vol. 42, no. 4, pp. 631-635. https://doi.org/10.1109/24.273596

**Exact Maximum Likelihood Estimation Using Masked System Data.** / Lin, Dennis K.J.; Usher, John S.; Guess, Frank M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Exact Maximum Likelihood Estimation Using Masked System Data

AU - Lin, Dennis K.J.

AU - Usher, John S.

AU - Guess, Frank M.

PY - 1993/1/1

Y1 - 1993/1/1

N2 - This paper estimates component reliability from masked series-system life data, viz, data where the exact component causing system failure might be unknown. We extend the results of Usher & Hodgson (1988) by deriving exact maximum likelihood estimators (MLE) for the general case of a series system of 3 exponential components with independent masking. Their previous work shows that closed-form MLE are intractable, and they propose an iterative method for the solution of a system of 3 Nonlinear likelihood equations. They do not, however, prove convergence for their iterative method. As such, we show how this system of Nonlinear equations can be replaced by a single quartic equation, whose solution is straight-forward. Since it does not depend upon the convergence of numerical solution algorithms, the results are exact. Though the resulting estimators are somewhat lengthy & cumbersome to find manually, they can be written as a straightforward computer code. The calculations can then be easily performed on a personal computer. This method for reducing the likelihood equations to simpler-to-solve forms can be extended readily to a higher number of components. In many cases for more than 3 components it is easier while for others it is more complicated; even in the more complicated cases, this simplification makes the problems much more tractable.

AB - This paper estimates component reliability from masked series-system life data, viz, data where the exact component causing system failure might be unknown. We extend the results of Usher & Hodgson (1988) by deriving exact maximum likelihood estimators (MLE) for the general case of a series system of 3 exponential components with independent masking. Their previous work shows that closed-form MLE are intractable, and they propose an iterative method for the solution of a system of 3 Nonlinear likelihood equations. They do not, however, prove convergence for their iterative method. As such, we show how this system of Nonlinear equations can be replaced by a single quartic equation, whose solution is straight-forward. Since it does not depend upon the convergence of numerical solution algorithms, the results are exact. Though the resulting estimators are somewhat lengthy & cumbersome to find manually, they can be written as a straightforward computer code. The calculations can then be easily performed on a personal computer. This method for reducing the likelihood equations to simpler-to-solve forms can be extended readily to a higher number of components. In many cases for more than 3 components it is easier while for others it is more complicated; even in the more complicated cases, this simplification makes the problems much more tractable.

UR - http://www.scopus.com/inward/record.url?scp=0027842108&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027842108&partnerID=8YFLogxK

U2 - 10.1109/24.273596

DO - 10.1109/24.273596

M3 - Article

AN - SCOPUS:0027842108

VL - 42

SP - 631

EP - 635

JO - IEEE Transactions on Reliability

JF - IEEE Transactions on Reliability

SN - 0018-9529

IS - 4

ER -