Parameters Estimation for Wear-out Failure Period of Three-Parameter Weibull Distribution

Takatoshi Sugiyama,Toru Ogura

doi:10.5539/ijsp.v11n1p40

Abstract

The shape parameter estimation using the minimum-variance linear estimator with hyperparameter (MVLE-H) method is believed to be effective for a wear-out failure period in a small sample. In the process of the estimation, our method uses the hyperparameter and estimate shape parameters of the MVLE-H method. To obtain the optimal hyperparameter c, it takes a long time, even in the case of the small sample. The main purpose of this paper is to remove the restriction of small samples. We observed that if we set the shape parameters, for sample size n and c, we can use the regression equation to infer the optimal c from n. So we searched in five increments and complemented the hyperparameter for the remaining sample sizes with a linear regression line. We used Monte Carlo simulations (MCSs) to determine the optimal hyperparameter for various sample sizes and shape parameters of the MVLE-H method. Intrinsically, we showed that the MVLE-H method performs well by determining the hyperparameter. Further, we showed that the location and scale parameter estimations are improved using the shape parameter estimated by the MVLE-H method. We verified the validity of the MVLE-H method using MCSs and a numerical example.

Highlights

The Weibull and exponential distributions are often used to analyse lifetime data (Huang, WT & Huang, HH, 2006; Ogura et al, 2020)
The shape parameter estimation using the minimum-variance linear estimator with hyperparameter (MVLE-H) method is believed to be effective for a wear-out failure period in a small sample
We extended the MVLE-H method removing the restriction of small samples

Summary

Introduction

The Weibull and exponential distributions are often used to analyse lifetime data (Huang, WT & Huang, HH, 2006; Ogura et al, 2020). An exponential distribution is applied when the probability of occurrence is high immediately after the start, and decreases monotonically. The Weibull distribution is applied when the probability of occurrence is high, and when it is low immediately after the start, and increases. This is called the wear-out failure period. We used the Weibull distribution because we were interested in a wear-out failure period. A three-parameter Weibull distribution that does not require location parameter constraint is widely used. The probability density and cumulative distribution functions of the three-parameter Weibull distribution are expressed as follows: g(x; m, η, γ)

Objectives

Methods

Conclusion