Markov chain Monte Carlo in Bayesian models for testing gamma and lognormal S-type process qualities

Abstract

The process capability index C pu is widely used to measure S-type process quality. Many researchers have presented adaptive techniques for assessing the true C pu assuming normality. However, the quality characteristic is often abnormal, and the derived techniques based on the normality assumption could mislead the manager into making uninformed decisions. Therefore, this study provides an alternative method for assessing C pu of non-normal processes. The Markov chain Monte Carlo, an emerging popular statistical tool, is integrated into Bayesian models to seek the empirical posterior distributions of specific gamma and lognormal parameters. Afterwards, the lower credible interval bound of C pu can be derived for testing the non-normal process quality. Simulations show that the proposed method is adaptive and has good performance in terms of coverage probability.

Efficient Technique for Assessing Actual Non‐normal Quality Loss: Markov Chain Monte Carlo

“…Because the prior information was insufficient in this case, we followed the study of Jiang et al . and Liao to adapt the non‐informative uniform distribution as the prior information, that is = π ( θ 1 , θ 2 , …, θ k ) = π ( θ 1 ) × π ( θ 2 ) × ⋯ × π ( θ k ) and π ( θ 1 ) = π ( θ 2 ) = ⋯ = π ( θ k ) = 1. Therefore, the joint posterior distribution of ( α , β ) is and the joint posterior distribution of ( γ , η ) is …”

“…In the Bayesian approach, the former distribution πθ must be given initially in order to obtain the posterior distribution pθjx , using the following function: The determination of the prior distribution is usually based on prior information about the parameters, including historical data, previous experience, expert suggestions, subjective supposition, or simply mathematical convenience. 37 Because the prior information was insufficient in this case, we followed the study of Jiang et al 28 and Liao 11 to adapt the noninformative uniform distribution as the prior information, that is πθ = π(θ 1 , θ 2 , …, θ k ) = π(θ 1 ) × π(θ 2 ) × ⋯ × π(θ k ) and π(θ 1 ) = π(θ 2 ) = ⋯ = π(θ k ) = 1. Therefore, the joint posterior distribution of (α, β) is…”

“…and Li and Meeker . However, assessing PCIs by employing the MCMC technique is very rare, one related study is Liao . When a statistical inference is made from a classical or Bayesian perspective, sometimes it is necessary to calculate the marginal density function or posterior moments; however, integrating over high‐dimensional PDFs for estimating the moments of interest is often difficult.…”

“…Many researchers have promoted the use of these indices and examined them with varying degrees of completeness. Important examples include Deleryd and Vännman, Spiring et al ., Castagliola and Castellanos, Vännman, Pearn and Cheng, Vännman and Albing, Pearn and Cheng, Pearn and Wu, Pearn et al ., Scagliarini, Liao, Karimi et al ., Lepore and Palumbo . By adequately analyzing these indices, engineers can trace and improve poor processes to enhance quality and satisfy customer requirements.…”