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The purpose of this paper is to develop a cost model by the variable sampling interval and optimization of the average cost per unit of time. The paper considers an economic–statistical design of the X̅ control charts under the Burr shock model and multiple assignable causes were considered and compared with three types of prior distribution for the mean shift parameter.

The design of the modified X̅ chart is based on the two new concepts of adjusted average time to signal and average number of false alarms for X̅ control chart under Burr XII shock model with multiple assignable causes.

The cost model was examined through a numerical example, with the same cost and time parameters, so the optimal of design parameters were obtained under uniform and non-uniform sampling schemes. Furthermore, a sensitivity analysis was conducted in a way that the variability of loss cost and design parameters was evaluated supporting the changes of cost, time and Burr XII distribution parameters.

The economic–statistical model scheme of X̅ chart was developed for the Burr XII distributed with multiple assignable causes. The correlated data are among the assumptions to be examined. Moreover, the optimal schemes for the economic-statistic chart can be expanded for correlated observation and continuous process.

The economic–statistical design of control charts depends on the process shock model distribution and due to difficulties from both theoretical and practical aspects; one of the proper alternatives may be the Burr XII distribution which is quite flexible. Yet, in Burr distribution context, only one assignable cause model was considered where more realistic approach may be to consider multiple assignable causes.

This study presents an advanced theoretical model for cost model that improved the shock model that presented in the literature. The study obviously indicates important evidence to justify the implementation of cost models in a real-life industry.

Burr distribution has been proved to be a useful failure model. It can assume different shapes which allow it to be a good fit for various lifetimes data. Hybrid censoring is an important way of generating lifetimes data. The purpose of this paper is to estimate an unknown parameter of the Burr type XII distribution when data are hybrid censored.

The problem is dealt with through both the classical and Bayesian point of view. Specifically, the methods of estimation used to tackle the problem are maximum likelihood estimation method and Bayesian method. Empirical Bayesian approach is also considered. The performance of all estimates is compared through their mean square error values. The paper employs Monte Carlo simulation to evaluate the mean square error values of all estimates.

The key findings of the paper are that the Bayesian estimates are superior to the maximum likelihood estimates (MLE).

This work has practical importance. Indeed, the proposed methods are applied to real life data.

The paper is original and is quite applicable in lifetimes data analysis.

The purpose of this paper is to contribute to the empirical evidence about crop yield distributions that are often used in practical models evaluating crop yield risk and insurance. Additionally, a simulation approach is used to compare the performance of alternative specifications when the underlying form is not known, to identify implications for the choice of parameterization of yield distributions in modeling contexts.

Using a unique high-quality farm-level corn yield data set, commonly used parametric, semi-parametric, and non-parametric distributions are examined against widely used in-sample goodness-of-fit (GOF) measures. Then, a simulation framework is used to assess the out-of-sample characteristics by using known distributions to generate samples that are assessed in an insurance valuation context under alternative specifications of the yield distribution.

Bias and efficiency trade-offs are identified for both in- and out-of-sample contexts, including a simple insurance rating application. Use of GOF measures in small samples can lead to inappropriate selection of candidate distributions that perform poorly in straightforward economic applications. The β distribution consistently overstates rates even when fitted to data generated from a β distribution, while the Weibull consistently understates rates; though small sample features slightly favor Weibull. The TCMN and kernel density estimators are least biased in-sample, but can perform very badly out-of-sample due to overfitting issues. The TCMN performs reasonably well across sample sizes and initial conditions.

Economic applications should consider the consequence of bias vs efficiency in the selection of characterizations of yield risk. Parsimonious specifications often outperform more complex characterizations of yield distributions in small sample settings, and in cases where more demanding uses of extreme-event probabilities are required.

The study helps provide guidance on the selection of distributions used to characterize yield risk and provides an extensive empirical demonstration of yield risk measures across a high-quality set of actual farm experiences. The out-of-sample examination provides evidence of the impact of sample size, underlying variability, and region of the probability measure used on the performance of candidate distributions.

The purpose of this paper is to analyze mathematical aspects of the q‐Weibull model and explore the influence of the parameter q.

The paper uses analytical developments with graph illustrations and an application to a practical example.

The q‐Weibull distribution function is able to reproduce the bathtub shape curve for the failure rate function with a single set of parameters. Moments of the distribution are also presented.

The generalized q‐Weibull distribution unifies various possible descriptions for the failure rate function: monotonically decreasing, monotonically increasing, unimodal and U‐shaped (bathtub) curves. It recovers the usual Weibull distribution as a particular case. It represents a unification of models usually found in reliability analysis. Q‐Weibull model has its inspiration in nonextensive statistics, used to describe complex systems with long‐range interactions and/or long‐term memory. This theoretical background may help the understanding of the underlying mechanisms for failure events in engineering problems.

Q‐Weibull model has already been introduced in the literature, but it was not realized that it is able to reproduce a bathtub curve using a unique set of parameters. The paper brings a mapping of the parameters, showing the range of the parameters that should be used for each type of curve.

The purpose of this paper is to investigate a new sampling methodology previously proposed for systems with a known lifetime distribution: the Predetermined Sampling Intervals (PSI) method.

The methodology is defined on basis of system hazard cumulative rate, and is compared with other approaches, particularly those whose parameters may change in real time, taking into account current sample information.

For different lifetime distributions, the results obtained for adjusted average time to signal (AATS) using a control chart for the sample mean are presented and analysed. They demonstrate the high degree of statistical performance of this sampling procedure, particularly when used in systems with an increasing failure rate distribution.

This PSI method is important from a quality and reliability management point of view.

This methodology involves a process by which sampling instants are obtained at the beginning of the process to be controlled. Also this new approach allows for statistical comparison with other sampling schemes, which is a novel feature.

This chapter of the book seeks to use famous mathematical functions (statistical distribution functions) in evaluating and analyzing supply chain network data related to supply chain management (SCM) elements in organizations. In other words, the main purpose of this chapter is to find the best-fitted statistical distribution functions for SCM data. Explaining how to best fit the statistical distribution function along with the explanation of all possible aspects of a function for selected components of SCM from this chapter will make a significant attraction for production and services experts who will lead their organization to the path of competitive excellence. The main core of the chapter is the reliability values related to the reliability function calculated by the relevant chart and extracting other information based on other aspects of statistical distribution functions such as probability density, cumulative distribution, and failure function. This chapter of the book will turn readers into professional users of statistical distribution functions in mathematics for analyzing supply chain element data.

The aim of this research paper is to generalize the previous works on the design of accelerated life tests (ALTs) for periodic inspection and Type I censoring and to promote the use of an exponentiated Weibull (EW) distribution in accelerated life testing.

Statistically optimal ALT plans are suggested for items whose lifetime follows the EW distribution under periodic inspection and Type I censoring. It is assumed that the mean lifetime (scale parameter) is a log‐linear function of stress and that the shape parameters are independent of stress. Given shape parameters, design stress and high test stress, the test plan is optimized with respect to the low test stress and the proportion of test units are also allocated to this test stress. The asymptotic variance (AsVar) of the maximum likelihood estimator of log mean life at the design stress is used as an optimality criterion with equally spaced inspection times. A FORTRAN program was written to calculate the optimal plans. Procedures for planning of an ALT, including selection of sample size, have also been discussed. An illustration of the optimal ALT plans has been done through a numerical example.

Computational findings for various values of the shape parameters indicate that the AsVar of log mean life at the design stress is insensitive to the number of inspection times and to misspecifications of imputed failure probabilities at design and high test stresses. Computational findings also show that optimal designs of ALT previously obtained for exponential, Rayleigh, and Weibull distributions become special cases of the EW distribution. Thus, the EW distribution is a useful and widely applicable reliability model for optimal ALT plans.

The present investigation features the EW distribution of lifetimes of test items and it generalizes the previous works on accelerated life testing. Furthermore, the propose test plans can be applied to estimate the lifetime of highly reliable product or material, if a researcher designs a test under the assumption of this model.

The purpose of this paper is to consider the estimation of multicomponent stress-strength reliability. The system is regarded as alive only if at least s out of k (s<k) strengths exceed the stress. The reliability of such a system is obtained when strength, stress variates are from Erlang-truncated exponential (ETE) distribution with different shape parameters. The reliability is estimated using the maximum likelihood (ML) method of estimation when samples are drawn from strength and stress distributions. The reliability estimators are compared asymptotically. The small sample comparison of the reliability estimates is made through Monte Carlo simulation. Using real data sets the authors illustrate the procedure.

The authors have developed multicomponent stress-strength reliability based on ETE distribution. To estimate reliability, the parameters are estimated by using ML method.

The simulation results indicate that the average bias and average mean square error decreases as sample size increases for both methods of estimation in reliability. The length of the confidence interval also decreases as the sample size increases and simulated actual coverage probability is close to the nominal value in all sets of parameters considered here. Using real data, the authors illustrate the estimation process.

This research work has conducted independently and the results of the author’s research work are very useful for fresh researchers.

The study based on the estimation of the stress–strength reliability parameter plays a vital role in showing system efficiency. In this paper, considering independent strength and stress random variables distributed as inverted exponentiated Rayleigh model, the author have developed estimation procedures for the stress–strength reliability parameter R = P(X>Y) under Type II hybrid censored samples.

The maximum likelihood and Bayesian estimates of R based on Type II hybrid censored samples are evaluated. Because there is no closed form for the Bayes estimate, the author use the Metropolis–Hastings algorithm to obtain approximate Bayes estimate of the reliability parameter. Furthermore, the author construct the asymptotic confidence interval, bootstrap confidence interval and highest posterior density (HPD) credible interval for R. The Monte Carlo simulation study has been conducted to compare the performance of various proposed point and interval estimators. Finally, the validity of the stress–strength reliability model is demonstrated via a practical case.

The performance of various point and interval estimators is compared via the simulation study. Among all proposed estimators, Bayes estimators using MHG algorithm show minimum MSE for all considered censoring schemes. Furthermore, the real data analysis indicates that the splashing diameter decreases with the increase of MPa under different hybrid censored samples.

The frequentist and Bayesian methods are developed to estimate the associated parameters of the reliability model under the hybrid censored inverted exponentiated Rayleigh distribution. The application of the proposed stress–strength reliability model will help the reliability engineers and also other scientists to estimate the system reliability.

The purpose of this article is to derive inference for multicomponent reliability where stress-strength variables follow unit generalized Rayleigh (GR) distributions with common scale parameter.

The authors derive inference for the unknown parametric function using classical and Bayesian approaches. In sequel, (weighted) least square (LS) and maximum product of spacing methods are used to estimate the reliability. Bootstrapping is also considered for this purpose. Bayesian inference is derived under gamma prior distributions. In consequence credible intervals are constructed. For the known common scale, unbiased estimator is obtained and is compared with the corresponding exact Bayes estimate.

Different point and interval estimators of the reliability are examined using Monte Carlo simulations for different sample sizes. In summary, the authors observe that Bayes estimators obtained using gamma prior distributions perform well compared to the other studied estimators. The average length (AL) of highest posterior density (HPD) interval remains shorter than other proposed intervals. Further coverage probabilities of all the intervals are reasonably satisfactory. A data analysis is also presented in support of studied estimation methods. It is noted that proposed methods work good for the considered estimation problem.

In the literature various probability distributions which are often analyzed in life test studies are mostly unbounded in nature, that is, their support of positive probabilities lie in infinite interval. This class of distributions includes generalized exponential, Burr family, gamma, lognormal and Weibull models, among others. In many situations the authors need to analyze data which lie in bounded interval like average height of individual, survival time from a disease, income per-capita etc. Thus use of probability models with support on finite intervals becomes inevitable. The authors have investigated stress-strength reliability based on unit GR distribution. Useful comments are obtained based on the numerical study.