Process capability refers to a process's intrinsic ability to produce a good product as specified in the product design process. A crucial component of any continuous quality improvement endeavor is measuring the performance of a process and taking appropriate action based on the measurements Spiring [1]. Businesses evaluate the effectiveness of their processes using a variety of metrics. The process capability indices(PCIs) are the most prevalent of these metrics [2]. The key reason for its popularity is that businesses need quantitative measures of the process's performance in relation to its specification constraints [3]. PCIs are statistical quantifications that are unitless and are used to compare how well a process characteristic performs. Quality control literature includes many process capability indices applicable to various process scenarios. Some commonly used are: Cp by Juran et al. [4]; Cpk by Kane [5]; Cpm, Chan et al. [6]; Cpp by Greenwich and Jahr-Schaffrath [7]. There are some other PCIs introduced in literature, namely CNp, CNpk, CNpm, and CNpmk [see, Pearn and Chen [8]; Schneider et al., [9], and Tong and Chen [10]].

In statistics, finite mixture models have been used extensively. It was particularly valued to generalize distributional assumptions and to model heterogeneity in population. The finite mixture models have several uses in various fields including biomedical, engineering, social sciences, medicine, economics, marketing, reliability studies and life testing problems, because mixture distributions represent heterogeneous data set when there is evidence of multimodality or simply unimodality. Generally, a mixture distribution can be formulated by combining two or more distributions using mixing parameters and mixture distribution of two sub-populations could be a suitable model for characterizing the overall population. Moreover, multiple causes of failure can be studied simultaneously via mixture distributions. Usually, the failure time population comprises of weak and strong components corresponding to short and long lives, respectively. Recently, various authors discussed different types of mixtures of distributions. The readers may refer to Turkan and Calis [31]; Everitt [11]; Everitt and Hand [12]; Jiang and Murthy [13]; Jiang and Murthy [14]; Marin et al. [15]; McLachlan and Peel [16]; Titterington et al., [17]; Titterington et al., [18]; and Lindsy [19]. Ali and Riaz [20] studied the generalized capability indices using Bayesian approach for different loss functions for the simple and mixture of generalized lifetime models. Moreover, Ouyang et al., [21] and Lin et al., [22] developed credible intervals for the PCIs. Wu [32] employed non-informative priors to construct the PC estimator for subsamples gathered over time using the squared error loss function. Kargar et al., [23] employed a Bayesian technique with a normal prior based on subsamples to assess process capacity using the capability index Cpk. Saxena and Singh [24] studied Bayesian estimation of the Cp index for normal distributions. Kanwal and Abbas [25] developed confidence interval for PCIs using objective Bayesian and made comparison with classical approach. The PCIs Cpy and CNpmk were developed for non-normal process by Kanwal et al. [26].

The main objective of this study is to develop the Bayesian estimators for the vector of parameters (Θ = {ψ, α_j, β_j}, j = 1, 2) of mixture of Frechet distribution (MFD) and to estimate the PCIs S_pmk and C_PY for MFD. Furthermore, bootstrap confidence intervals(BCIs) are evaluated as a measure of performance for both maximum likelihood estimates(MLEs) and Bayesian estimates (BE). The novelty of present work can be assessed from the fact that no attempt has been made to study the PCIs for MFD. Our aim is to give recommendations for selecting the suitable approach for estimating Θ for MFD and further evaluation of effective PCIs. This study is hoped to be valuable to practicing engineers, quality experts, and applied statisticians. In this work, we propose a MFD that can be successfully used in evaluation of PCIs in real-life problems. Let a random variable T follows a finite mixture model with k components, then the probability density function (PDF) can be written as

where ψ is the mixing parameter which is a non-negative proportion such that ∑j=1kψj=1. The PDF of MFD is given as

The PDF of the jth component, j = 1, 2, is

where α_j and β_j are the shape and scale parameters of Frechet distribution(FD). The cumulative distribution function(CDF), reliability function, and hazard rate function of the MFD are, respectively, given by

In continuation of this introductory section, the rest of the article unfolds as: In Section 2, MLE for the parameters of MFD are obtained. Bayesian estimators are presented in Section 3, and PCIs are discussed in Section 4. Simulations study is given in Section 5, and BCIs are presented in Section 6. Finally, a real data set is analyzed in Section 7, and conclusions are presented in Section 8.

Let t₁, t₂, ..., t_n be a random sample of size n from MFD, then log-likelihood function is

Differentiating Equation 7 with respect to the mixing parameter ψ, α_j, β_j, where j = 1, 2 and equating to zero. The score equations are

where f₁(t) and f₂(t) are the PDF of the FD. The system of nonlinear equations cannot be written in closed form. Here, we use the BB package which is available in the R software to get the MLEs of model parameters.

For Bayesian estimation, we need prior distribution of ψ, α₁, α₂, β₁, and β₂. Assuming that ψ has the prior Beta(s₁, s₂) distribution, whereas α₁, α₂, β₁, and β₂ each have independent inverted gamma priors with PDFs, respectively, given by

where (d_j, c_j, n_j, l_j) are the hyperparameters. The joint prior density of the random vector Θ = {ψ, α_j, β_j}, j = 1, 2 is

The joint posterior density function for MFD is given by

The posterior distribution (Equation 10) assumes a ratio form that is not amenable to closure. Therefore, in absence of closed form results here, we are using Lindley [27] approximation to obtain the BE, which can be expressed as

where a_i = ρ₁σ_i1 + ρ₂σ_i2 + ρ₃σ_i3 + ρ₄σ_i4 + ρ₅σ_i5, i = 1, 2, 3, 4, 5

The Bayesian estimators for ψ, α₁, α₂, β₁, and β₂ under squared error loss function are

where ψ^,α1^,α2^,β1^ and β2^ are the maximum likelihood estimates of ψ, α₁, α₂, β₁ and β₂, respectively. Rest derivatives will be provided on request.

The most frequently used PCIs C_p, C_pk, C_pmk, and C_pm rely strictly on the normality assumption for a given process with mean and process dispersion. However, many production and service processes make the presumption of normality mostly invalid.

In this section, Monte Carlo simulation is conducted to examine the behavior of PCIs for both MLEs and BE in terms of BCIs, and mean squared errors (MSEs) against different sample sizes. The Monte Carlo simulation is performed as follows:

Franklin and Gray [29] developed the bootstrap method, and BCIs are typically employed to obtain the empirical distribution of different PCIs. The following steps are followed for bootstrapping, and the standard BCIs are considered for S_pmk and C_PY. :