Reliability analysis for q‐Weibull distribution with multiply Type‐I censored data

Abstract

The widely used Weibull distribution could be generalized to be q-Weibull distribution. To fill out the gap in existing literature, the reliability is studied for q-Weibull distribution with multiply Type-I censored data, which is the general form of Type-I censored data. The point estimates and confidence intervals (CIs) for q-Weibull parameters and reliability parameters such as the reliability and remaining lifetime are all focused on. The maximum likelihood estimates (MLE) are obtained by maximizing the likelihood function and transforming it to an unconstrained optimization problem. The least-square estimates (LSEs) are proposed by minimizing the single-variable profile error function derived from reducing the previous multivariable error function. These improvements could make the computation of point estimates efficient. Concerning the CIs, the asymptotic normality of log-transformed MLE is used to guarantee they fall into the value ranges. Particularly, the closed form for the Fisher information matrix is derived using the missing information principal and is combined with the delta method to construct the CIs for reliability. Besides, the bias-corrected and accelerated (BCa) bootstrap method is also applied. Further, a Monte Carlo simulation study is conducted to compare different point estimates and CIs. Finally, an illustrative example is presented to show the application of the study in this paper.

Reliability analysis of computed tomography equipment using the q‐Weibull distribution

“…The q distributions are functions based on a non‐generalized formalism introduced in non‐generalized statistical mechanics. One of these functions is the q‐exponential function 34,35 : $$$$ {\exp}_q(x)=\left\{\begin{array}{c}{\left[1+\left(1-q\right)x\right]}^{\frac{1}{1-q}},\kern0.55em \mathrm{if}\ \left[1+\left(1-q\right)x\right]>0,\\ {}0,\kern6.78em \mathrm{otherwise}.\end{array}\ \right. $$$$ Since the Weibull model is most commonly used to describe lifetime data, the corresponding q‐Weibull distribution has received more attention than other q‐type distributions (q‐exponential, q‐gamma, etc.).…”

“…[31][32][33] The q distributions are functions based TA B L E 1 q-Weibull failure rate for different values of q and 𝛽 on a non-generalized formalism introduced in non-generalized statistical mechanics. One of these functions is the q-exponential function 34,35 :…”

“…In engineering applications, the most fundamental problem is the estimation of the q-Weibull parameters. The existing parameter estimation methods of the q-Weibull distribution include maximum likelihood estimates (MLEs), 34 least-squares estimates (LSEs), 35 probability plotting method, 39 adaptive hybrid artificial bee colony algorithm, 38,40 and the method of moments. 36 The disadvantage of the q-Weibull model is that it is difficult to estimate the parameters.…”

Framework for robust parameter design and optimization of reliability characteristics

“…Maximum likelihood estimation (MLE) has been found to have good consistency, asymptotic normality, and other characteristics that can effectively analyze non-normal and censored data. 43,44 Consequently, MLE has been widely applied in a range of reliability engineering applications. 3,45,46 The Weibull distribution has been commonly used in reliability engineering because it can flexibly model various failure mechanisms.…”