For a minimally repaired system, whose failure process is described by a non-homogeneous Poisson process (NHPP), the classical maximum likelihood estimation procedures cannot be used when the system failures are hidden and detected only at inspection epochs. By assuming that the failure process follows a NHPP with power law intensity function, the Expectation-Maximization (EM) algorithm was recently proposed to estimate the model parameters and a procedure to test the presence of trend in the real failure data of some components of identical medical infusion pumps was discussed. However, the EM algorithm suffers in this application from some limitations due to its complexity and the large computational time required for convergence. This paper proposes a new estimation algorithm which is still iterative but, unlike the EM algorithm, is not based on the expectation of the log-likelihood function with respect to the conditional distribution of the unobserved data, but rather on the expectation of the conditioning variables, that is, of the unknown age of the system at the previous failure. This approach allows one to specify a simpler and much faster estimation procedure. A comparison between the proposed and the EM algorithms shows that the former performs as well as the latter, while requiring a drastically reduced computational burden. In addition, the proposed scheme can be applied to other intensity functions, such as the log-linear and the 2-parameter logarithmic functions. As a result, the test hypothesis of no trend in one of the analyzed datasets, which can not be rejected under the power law intensity function, is instead rejected under the alternative hypothesis of a log-linear intensity function.
A New Estimation Algorithm for Interval Censored Data from Repairable Systems
GUIDA, Maurizio;POSTIGLIONE, Fabio;
2012
Abstract
For a minimally repaired system, whose failure process is described by a non-homogeneous Poisson process (NHPP), the classical maximum likelihood estimation procedures cannot be used when the system failures are hidden and detected only at inspection epochs. By assuming that the failure process follows a NHPP with power law intensity function, the Expectation-Maximization (EM) algorithm was recently proposed to estimate the model parameters and a procedure to test the presence of trend in the real failure data of some components of identical medical infusion pumps was discussed. However, the EM algorithm suffers in this application from some limitations due to its complexity and the large computational time required for convergence. This paper proposes a new estimation algorithm which is still iterative but, unlike the EM algorithm, is not based on the expectation of the log-likelihood function with respect to the conditional distribution of the unobserved data, but rather on the expectation of the conditioning variables, that is, of the unknown age of the system at the previous failure. This approach allows one to specify a simpler and much faster estimation procedure. A comparison between the proposed and the EM algorithms shows that the former performs as well as the latter, while requiring a drastically reduced computational burden. In addition, the proposed scheme can be applied to other intensity functions, such as the log-linear and the 2-parameter logarithmic functions. As a result, the test hypothesis of no trend in one of the analyzed datasets, which can not be rejected under the power law intensity function, is instead rejected under the alternative hypothesis of a log-linear intensity function.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
