An Erlang-Coxian-Based Method for Modeling Accelerated Life Testing Data

  • Haitao Liao
  • Ye Zhang
  • Huairui Guo
Part of the Decision Engineering book series (DECENGIN)


Accelerated life testing (ALT) can be used to expedite failures of a product for predicting the product’s reliability under the normal operating conditions. The resulting ALT data are often modeled by a probability distribution along with a life-stress relationship. However, if the selected probability distribution cannot adequately describe the underlying failure process, the resulting reliability prediction would be misleading. It would be quite valuable if the distribution providing an adequate fit to the ALT data can be determined automatically. This chapter provides a new analytical method to assist reliability engineers in this regard. Essentially, this method uses Erlang-Coxian (EC) distributions, which belong to a particular subset of phase-type distributions, to characterize ALT data. Such distributions are quite efficient for approximating many non-negative distributions, such as Weibull, lognormal and gamma. The advantage of this method is that the best fit to the ALT data can be obtained by gradually changing the model structure, i.e., the number of phases of the associated continuous-time Markov chain (CTMC). To facilitate the implementation of this method, two statistical inference approaches are provided. First, a mathematical programming approach is formulated to simultaneously match the moments of the EC-based ALT model to the empirical moments at the corresponding test stress levels. This approach resolves the feasibility issue of the method of moments. In addition, the maximum likelihood estimation approach is presented, which can easily handle different types of censoring in ALT. Both approaches are accompanied with a stopping criterion for determining the number of phases of the resulting CTMC. Moreover, nonparametric bootstrap method is used to construct the pointwise confidence interval for the resulting reliability estimates. Numerical examples for constant-stress ALT with Type-I and multiple censoring schemes are provided to illustrate the capability of the method in modeling ALT data.


Accelerated life testing Phase-type distributions Erlang-Coxian distributions Method of moments Maximum likelihood estimation 



Accelerated life testing


Accelerated failure time


Akaike Information Criterion


Cumulative distribution function


Probability density function


Continuous-time Markov chain








Least square estimate


Maximum likelihood estimate


Cdf of failure time under stress Z

R(t; Z)

Reliability function under stress Z

f(t; Z)

Pdf of failure time under stress Z

h(t; Z)

Hazard rate function under stress Z

r(Z; θ)

Function of stress Z


Subgenerator matrix

\( \uplambda_{j} \)

The jth transition rate

\( p_{{^{j} }} \)

The jth transition probability

\( \pi \)

Initial probability

\( M_{l}^{i} \)

The lth empirical moment under stress level i

\( \tau_{\upiota} \)

Censoring time at stress level i

\( \delta_{ij}^{{[0,\tau_{i} ]}} \)

Indicator function for the jth failure time under stress level i

\( \varepsilon_{i,l} , s_{i,l} \)

Excess and slack variables for the lth moment for stress level i

\( w_{i,l} \)

Weights assigned to the deviation from the lth moment for stress level i



This work is supported in part by the U.S. National Science Foundation under grants CMMI-1238301 and CMMI-1635379.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Industrial EngineeringThe University of ArkansasFayettevilleUSA

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