The Process Capability Index of Pareto Model under Progressive Type-II Censoring: Various Bayesian and Bootstrap Algorithms for Asymmetric Data

Abstract

It is agreed by industry experts that manufacturing processes are evaluated using quantitative indicators of units produced from this process. For example, the Cpy process capability index is usually unknown and therefore estimated based on a sample drawn from the requested process. In this paper, Cpy process capability index estimates were generated using two iterative methods and a Bayesian method of estimation based on stepwise controlled type II data from the Pareto model. In iterative methods, besides the traditional probability-based estimation, there are other competitive methods, known as bootstrap, which are alternative methods to the common probability method, especially in small samples. In the Bayesian method, we have applied the Gibbs sampling procedure with the help of the significant sampling technique. Moreover, the approximate and highest confidence intervals for the posterior intensity of Cpy were also obtained. Massive simulation studies have been performed to evaluate the behavior of Cpy. Ultimately, application to real-life data is seen to demonstrate the proposed methodology and its applicability.

Estimation of Marshall–Olkin Extended Generalized Extreme Value Distribution Parameters under Progressive Type-II Censoring by Using a Genetic Algorithm

“…In type-II progressively censored data, the maximum likelihood estimators (MLEs) for the parameters α, μ, σ and ζ are derived by setting the equations from (19) to (22) and from (23) to (25) to zero. The nonlinear characteristics of the likelihood equations Equation ( 19)- (22) and Equation ( 23)-( 25) are considered a challenge as they do not have a straightforward explicit solution.…”

“…In type-II progressively censored data, the maximum likelihood estimators (MLEs) for the parameters α, μ, σ and ζ are derived by setting the equations from (19) to (22) and from (23) to (25) to zero. The nonlinear characteristics of the likelihood equations Equation ( 19)- (22) and Equation ( 23)-( 25) are considered a challenge as they do not have a straightforward explicit solution. Given a specific set of a given value (N, n, M, m, x, y), obtaining the maximum likelihood estimators of parameters involves solving these complex equations, which may not be available by classical techniques.…”

“…Similarly, the observed Fisher information can be constructed by providing the MLE of parameters from Equation ( 19)- (22) for MO-GEVL and from Equation ( 23)- (25) for MO-Gumbel.…”

“…For more comprehensive information on this scheme, refer to the work by [18]. Also, many others have used type-II progressive censored scheme in their papers, such as [19][20][21][22][23]. This paper's structure is methodically arranged into sections, beginning with a discussion on the Marshall-Olkin extended generalized extreme value under linear normalization distribution, addressing PDF and CDF along with type-II progressive censoring schemes, in Section 2.…”