Maximum Likelihood Estimation for Three-Parameter Weibull Distribution Using Evolutionary Strategy

Abstract

The maximum likelihood estimation is a widely used approach to the parameter estimation. However, the conventional algorithm makes the estimation procedure of three-parameter Weibull distribution difficult. Therefore, this paper proposes an evolutionary strategy to explore the good solutions based on the maximum likelihood method. The maximizing process of likelihood function is converted to an optimization problem. The evolutionary algorithm is employed to obtain the optimal parameters for the likelihood function. Examples are presented to demonstrate the proposed method. The results show that the proposed method is suitable for the parameter estimation of the three-parameter Weibull distribution.

Reliability analysis of gas turbine engine by means of bathtub‐shaped failure rate distribution

“…The three‐parameter Weibull distribution has been employed in this study since it is flexible for random data fitting, which makes it highly adaptable for various types of probability distribution . In the three‐parameter Weibull distribution, the parameter estimation is considered to be highly involved .…”

“…A Monte Carlo approach for the estimation has also been used in the past but was dependent on huge amount of data . Depending on the availability of data, the MLE is preferred, however, it is generally considered difficult to apply as it involves numerical iteration for parameter estimations . Conventional method to estimate Weibull parameters is to plot failure data on Weibull probability paper .…”

“…The three-parameter Weibull distribution has been employed in this study since it is flexible for random data fitting, which makes it highly adaptable for various types of probability distribution. 40 In the three-parameter Weibull distribution, the parameter estimation is considered to be highly involved. 40 Hence, different techniques are available for estimating the parameters such as graphic method, 41 maximum likelihood estimation, 42 least square method (LSM) with particle swarm optimization, 43 moment estimation, 44 gray model, 45 and so on.…”

“…40 In the three-parameter Weibull distribution, the parameter estimation is considered to be highly involved. 40 Hence, different techniques are available for estimating the parameters such as graphic method, 41 maximum likelihood estimation, 42 least square method (LSM) with particle swarm optimization, 43 moment estimation, 44 gray model, 45 and so on. For the present study, however, the characteristic life, γ (the third Weibull parameter), was predetermined that reduces the estimation to two parameters only, the detail of which is set out in Section 3.1.…”

“…25 Depending on the availability of data, the MLE is preferred, 25 however, it is generally considered difficult to apply as it involves numerical iteration for parameter estimations. 40,49 Conventional method to estimate…”