Abstract
In this paper, we discussed the estimation of the index Cpy for a 3-Burr-XII distribution based on Progressive Type-II censoring. The maximum likelihood and Bayes method have been used to obtain the estimating of the index Cpy. The Fisher information matrix has been used to construct approximate confidence intervals. Also, bootstrap confidence intervals (CIs) of the estimators have been obtained. The Bayesian estimates for the index Cpy have been obtained by the Markov Chain Monte Carlo method. Also, the credible intervals are constructed by using MCMC samples. Two real-datasets have been discussed using the proposed index.
Highlights
The 95% confidence intervals (CIs) based on maximum likelihood estimators (MLEs) and the 95% bootstrap (Boot-p and bootstrap-t method (Boot-t)) CIs of Cpy were determined, and the results are summarized in Tables 7 and 8
We considered the MLEs and bootstrap (Boot-p and Boot-t) for classical estimation methods in order to get the estimates of the unknown parameters and the Cpy index
Since theoretical comparison of these methods is not feasible, we have carried out comprehensive simulation study to compare these methods with different sample sizes and different combinations of the unknown parameters. erefore, we considered Bayesian inference of the unknown parameters of the TPBXIID and the index Cpy using MCMC approach
Summary
Statistician and quality control engineers in manufacturing industries often employ varied statistical process techniques to measure the capability of a manufacturing process and quantify the process behavior to identify contradictions between the actual process performance and the desired specifications. ese techniques include the process capability index (PCI), and the PCI compares the output of the process to customer’s specification. e objective of the PCI is to provide a numerical indicator of whether or not a production process is able to produce products within the specification limits. ese specifications are determined through the lower specification limit (L), the upper specification limit (U), and the target value (t) e most commonly used PCIs Cp, Cpk, Cpmk, and Cpm are based on the assumption that a given process may be described by a normal probability model with process mean and process standard deviation. Maiti et al [11] have established a generalized PCI Cpy which is directly or indirectly connected to most of the PCIs described in the literature It includes both normal and nonnormal and continuous as well as discrete random variables and is defined as follows: F(U) − F(L) p. Kargar et al [15] studied the Bayesian approach with normal prior depending on subsamples to check process capability via capability index Cpk. Maiti and Saha [16] obtained the Bayesian estimation of the index Cpy based on SE loss function for normal, exponential, and Poisson process distributions.
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