Abstract This paper introduces a new capability index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>QT</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(C_{\rm QT})} which is based on an asymmetric loss function named Quadratic-Type Squared Error Loss Function (QTSELF). To estimate the proposed index <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mi>QT</m:mi> </m:msub> </m:math> {C_{\rm QT}} , we have used five different classical methods of estimation (namely maximum likelihood, least squares and weighted least squares, Cramér–von-Mises, Anderson–Darling and maximum product of spacing) and their performances are compared. Additionally, three bootstrap confidence intervals, namely standard bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒮</m:mi> </m:math> {\mathcal{S}} -boot), percentile bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒫</m:mi> </m:math> {\mathcal{P}} -boot) and Student’s t bootstrap ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒯</m:mi> </m:math> {\mathcal{T}} -boot) are constructed. Their performances are compared in terms of average widths and coverage probabilities through a Monte-Carlo simulation study. Finally, two datasets are taken from electronic industries to demonstrate the practical utilities of the proposed estimation techniques and confidence intervals.

Abstract This work considers the intuitionistic fuzzy approach, using the Rayleigh distribution to define failure parameters and trapezoidal intuitionistic fuzzy numbers (TrIFNs) to characterize fuzzy variables. We calculate the reliability function using the universal (probability) generating function. We further apply the method to a hydropower system subjected to a PID (proportional-integral-derivative) controller by evaluating intuitionistic fuzzy reliability and the sensitivity of fuzzy reliability. We also use the average operator of TrIFNs with equal weights to analyze intuitionistic fuzzy reliability and sensitivity more effectively. We present the results in graphical and tabular form for greater clarity.

Abstract In this article, we consider six process capability indices (PCIs) whose quality characteristics have a normal distribution, out of which first PCI 𝒞 p {\mathcal{C}_{p}} was developed in [J. M. Juran, Juran’s Quality Control Handbook, 3rd ed., McGraw-Hill, New York, 1974], next PCI 𝒞 p ⁢ m {\mathcal{C}_{pm}} was developed in [T. C. Hsiang and G. Taguchi, A tutorial on quality control and assurance, Annual Meeting on the American Statistical Association, Las Vegas 1985], third PCI 𝒞 p ⁢ m ⁢ c {\mathcal{C}_{pmc}} was developed in [M. Saha, S. Dey and L. Wang, Parametric inference of the loss based index C p ⁢ m C_{pm} for normal distribution, Qual. Reliab. Eng. Int. 38 2022, 10.1002/qre.2987], fourth PCI 𝒞 p ⁢ m ′ {\mathcal{C}^{\prime}_{pm}} was developed in [S. Dey, W. Wang and M. Saha, Modified estimation and confidence intervals of an asymmetric loss based process capability index 𝒞 p ⁢ m ′ \mathcal{C}^{\prime}_{pm} , Qual. Reliab. Eng. Int. 38 2022, 8, 4033–4048], fifth PCI 𝒞 p ⁢ c {\mathcal{C}_{pc}} and sixth one 𝒞 p ⁢ m ⁢ c ′ {\mathcal{C}^{\prime}_{pmc}} are newly proposed in the paper. Next, we estimate the cited process capability indices using the method of moment estimation (MOM) approach when the underlying process follows normal distribution. A simulation study is conducted to evaluate the performance of these indices with respect to their mean squared errors. Additionally, confidence intervals are constructed using the asymptotic confidence interval (ACI), the generalized confidence interval (GCI) and parametric bootstrap confidence interval (BCI). Using Monte Carlo simulation, the performance of the GCI and BCI is compared in terms of average width and associated coverage probabilities. Finally, four data sets, two related to electronic industries and two related to food industries are re-analyzed to show the applicability’s of the suggested indices, MOM estimation, GCI and BCI.

Abstract This paper investigates E-Bayesian estimation for the Weighted Power Function Distribution (WPFD), a modified version of the Power Function distribution which is widely applied in biosciences and engineering. Three distinct priors for hyper-parameters are considered, and E-posterior risks are derived using squared error, entropy (measuring uncertainty), and precautionary loss functions (penalizing extreme deviations). Properties of the E-Bayesian estimators are analyzed, supplemented by a simulation study and real data application. Results indicate that E-Bayesian estimators outperform Bayesian counterparts in terms of E-posterior risk, as demonstrated in simulation studies.

Abstract This paper presents a comparative study of reliability analysis of a gas production plant, before and after applying optimal age replacement policy for a specific machine. The gas production plant has six machines connected in the series configuration a sulfur (S) shaft, coal pit, reaction furnace, sulfur separation machine, adiabatic reactor, and cooling machine. Here, the optimal age replacement policy has been applied to the cooling machine as this machine is working under a wear-out period, i.e., it has completed its prescribed age (say T) and is still operational. Two different situations have been discussed to analyze the performance variation of the system (i) when the model is working with high risk because the cooling machine is operational even after completion of its prescribed age (ii) when the old cooling machine is replaced by a new one after completing the age, T. The failure of any machine can cause the model failure. Additionally, a situation of employee walkout is also considered a responsible factor for model failure. Supplementary variable technique and copula methodology have been applied to solve the Markov model.

Abstract In this work, we provide five different classical estimation methods for the gamma distribution and unknown parameters, which are used to estimate the newly constructed process capability index (PCI) C c C_{c} . Next, we use the aforementioned estimation techniques to consider five bootstrap confidence intervals: standard bootstrap (SB), percentile bootstrap (PB), student’s 𝑡 bootstrap (STB), bias-corrected percentile bootstrap (BCPB), and bias-corrected accelerated bootstrap (BCAB). These allow us to construct confidence intervals of the PCI C c C_{c} . Using the suggested techniques of estimate for small, moderate, and large sample sizes, Monte Carlo simulations are run to compare the performances of the bootstrap confidence intervals’ average widths and coverage probabilities. Simulation results indicate that MLE gives the least AW for all BCIs (SB, PB, BCPB, TB, and BCAB) when compared to all other estimation approaches evaluated in this paper. Further, simulation results demonstrate that, for almost all sample sizes and all of the examined estimation approaches, BCAB CI generates smaller AW and larger CP than their counterparts. Finally, for illustrative purposes, three real data sets are analysed.

Abstract The probability generating function (pgf) B ⁢ ( s ) {B(s)} of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) {(Z_{t}:t\geq 0)} (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 {B(0)=0} and B ′ ⁢ ( 0 ) &gt; 0 {B^{\prime}(0)&gt;0} . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 ( Z t = j ) {P_{1}(Z_{t}=j)} have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.

Abstract Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.

Abstract Recently, entropy and extropy-based tests for the uniform distribution have attracted the attention of some researchers. This paper proposes nonparametric entropy and extropy estimators based on progressive type-II censoring and investigates their properties and behavior. Performance of the proposed estimators is studied via simulations. Entropy and extropy-based goodness-of-fit tests for uniformity are developed by the well performed estimators. The powers of the proposed uniformity tests are compared also via simulations assuming various alternatives and censoring schemes.

Abstract In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable ( | z | ⁢ | ln ⁡ | z | | ) (\lvert z\rvert\sqrt{\lvert\ln\lvert z\rvert\rvert}) . We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.