Statistical inference for a competing failure model based on the Wiener process and Weibull distribution.

Considering the competitive failure that exists in the operation of industrial systems, including degradation failure and sudden failure, this paper presents a reliability assessment method based on the three-parameter Weibull distribution and the Wiener process. Nonlinear functions are proposed to establish the relationship between the different failure processes, and the reliability model is derived. The Metropolis-Hastings (MH) sampling algorithm of the Monte Carlo Markov Chain (MCMC) method is employed to estimate the parameters in this study. The reliability assessment results are obtained by the real degradation samples. The results show that the reliability model incorporating the three-parameter Weibull distribution produces more comprehensive and dependable results. Furthermore, MH sampling can solve the issues of complex likelihood functions that cannot directly obtain parameter evaluation results. Additionally, this paper analyzes the sensitivity of the model parameters, thereby offering theoretical support for enhancing the safe operation of the system.

On the basis of the principle of degradation mechanism invariance, a Wiener degradation process with random drift parameter is used to model the data collected from the constant stress accelerated degradation test. Small‐sample statistical inference method for this model is proposed. On the basis of Fisher's method, a test statistic is proposed to test if there is unit‐to‐unit variability in the population. For reliability inference, the quantities of interest are the quantile function, the reliability function, and the mean time to failure at the designed stress level. Because it is challenging to obtain exact confidence intervals (CIs) for these quantities, a regression type of model is used to construct pivotal quantities, and we develop generalized confidence intervals (GCIs) procedure for those quantities of interest. Generalized prediction interval for future degradation value at designed stress level is also discussed. A Monte Carlo simulation study is used to demonstrate the benefits of our procedures. Through simulation comparison, it is found that the coverage proportions of the proposed GCIs are better than that of the Wald CIs and GCIs have good properties even when there are only a small number of test samples available. Finally, a real example is used to illustrate the developed procedures.

This paper investigates the performance of the confidence intervals for a single inverse mean and the difference of two inverse means of normal distributions with unknown coefficients of variation (CVs). The confidence intervals for the inverse mean with unknown coefficient of variation (CV) were constructed based on the generalized confidence interval (GCI) approach and the large sample approach. The generalized confidence interval and large sample confidence interval for the inverse mean with unknown CV were compared with the generalized confidence interval for the inverse mean of Niwitpong and Wongkhao [5]. Moreover, the confidence intervals for the difference of inverse means with unknown CVs were constructed using the GCI approach, large sample approach and method of variance estimates recovery (MOVER) approach and then compared with existing confidence interval for the difference of inverse means based on the GCI approach of Niwitpong and Wongkhao [6]. The coverage probability and average length of the confidence intervals were evaluated by a Monte Carlo simulation. Carrying out the simulation studies, the results showed that the generalized confidence interval provides the best confidence interval for the inverse mean with unknown CV. The generalized confidence interval and the MOVER confidence interval for the difference of inverse means with unknown CVs perform well in terms of the coverage probability and average length. Finally, two real datasets are analyzed to illustrate the proposed confidence intervals.

In this paper, we consider interval estimation for the inverse Gaussian (IG) distribution. Using generalized pivotal quantity method, we derive the generalized confidence intervals (GCIs) for the model parameters and some quantities such as the quantile, the reliability function of the lifetime, the failure rate function, and the mean residual lifetime. We verify that the GCI of the scale parameter is the same as its commonly used exact CI. We also obtain the generalized prediction intervals (GPIs) for future failure times based on the observed failure data set. In addition, we get the GCI for the reliability of the stress–strength model when the stress and strength variables follow the IG distributions with different parameters. We compare the proposed GCIs and GPIs with the Wald CIs and bootstrap‐ CIs by simulation. The simulation results show that the proposed GCIs and GPIs are superior to the Wald CIs and the bootstrap‐pCIs in terms of the coverage probability. Finally, two examples are used to illustrate the proposed procedures.

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.

PurposeConfidence intervals play a crucial role in economics and finance, providing a credible range of values for an unknown parameter along with a corresponding level of certainty. Their applications encompass economic forecasting, market research, financial forecasting, econometric analysis, policy analysis, financial reporting, investment decision-making, credit risk assessment and consumer confidence surveys. Signal-to-noise ratio (SNR) finds applications in economics and finance across various domains such as economic forecasting, financial modeling, market analysis and risk assessment. A high SNR indicates a robust and dependable signal, simplifying the process of making well-informed decisions. On the other hand, a low SNR indicates a weak signal that could be obscured by noise, so decision-making procedures need to take this into serious consideration. This research focuses on the development of confidence intervals for functions derived from the SNR and explores their application in the fields of economics and finance.Design/methodology/approachThe construction of the confidence intervals involved the application of various methodologies. For the SNR, confidence intervals were formed using the generalized confidence interval (GCI), large sample and Bayesian approaches. The difference between SNRs was estimated through the GCI, large sample, method of variance estimates recovery (MOVER), parametric bootstrap and Bayesian approaches. Additionally, confidence intervals for the common SNR were constructed using the GCI, adjusted MOVER, computational and Bayesian approaches. The performance of these confidence intervals was assessed using coverage probability and average length, evaluated through Monte Carlo simulation.FindingsThe GCI approach demonstrated superior performance over other approaches in terms of both coverage probability and average length for the SNR and the difference between SNRs. Hence, employing the GCI approach is advised for constructing confidence intervals for these parameters. As for the common SNR, the Bayesian approach exhibited the shortest average length. Consequently, the Bayesian approach is recommended for constructing confidence intervals for the common SNR.Originality/valueThis research presents confidence intervals for functions of the SNR to assess SNR estimation in the fields of economics and finance.

Signal-to-noise ratio (SNR) is a reciprocal of coefffficient of variation. The SNR is a measure of mean relative to the variability. Confidence procedures for common SNR of two-parameter exponential distributions were developed using generalized confidence interval (GCI) approach, large sample (LS) approach, adjusted method of variance estimates recovery (Adjusted MOVER) approach, and bootstrap approaches based on standard bootstrap (SB) and parametric bootstrap (PB). The performances of all approaches are measured by coverage probability and average length. Simulation studies show that all approaches have the coverage probabilities below the nominal confidence level of 0.95 when the common SNR is negative value. However, the coverage probabilities of all approaches are greater than the nominal confidence level of 0.95 when the common SNR is positive value. Moreover, the LS and AM approaches are the conservative confidence intervals. In addition, the GCI and PB approaches provide the confidence intervals with coverage probabilities close to the nominal confidence level of 0.95 when the sample sizes are large and the common SNR is positive value. The GCI and PB approaches are recommended to estimate the confidence intervals for the common SNR of two-parameter exponential distributions. Finally, all proposed approaches are employed in the data of the survival days of lung cancer patients for a demonstration.

In this chapter we describe a simple degradation model based on the Wiener process. A failure occurs when the degradation reaches a given level for the first time. In this case, the time to failure is inverse Gaussian distributed. The parameters of the lifetime distribution can be estimated from observation of degradation only, from observation of failures, or from observation of both degradation increments and failure times. In the chapter, statistical methods for estimating the parameters of degradation processes for different data structures are developed and compared.

As the products become more and more complicated, and the demands of users vary continuously, failure threshold of product may be probabilistic rather than determined. This paper develops an extension research based on probabilistic failure threshold of competing failure with degradation and random shocks. Firstly, general competing failure model is proposed based on the assumption that the degradation process and the cumulative shock-damage process are independent, and then reliability formula based on different distribution is elaborated. Secondly, another competing failure model based on probabilistic failure threshold is proposed with the consideration of dependent competing of degradation and random shocks based on the former model. Random shocks could affect degradation process with a damage factor, which adds to the probability of degradation failure to the product. At last, those two models are applied to two cases of accelerator with degradation data and vibrating-damage data, and the comparison reveals the differences between those two models as well as the traditional model with pre-determined threshold. It tests the reasonability and accuracy of the probabilistic threshold competing failure models. The result of this paper provides an extensional method for the product failure prediction and healthy management based on the competing failure modes of probabilistic threshold degradation and random shocks.

Log-normal distribution is used widely in application fields such as economics and finance. This paper considers confidence interval estimates for common signal-to-noise ratio of log-normal distributions based on generalized confidence interval (GCI), adjusted method of variance estimates recovery, and computational approaches. A simulation study is conducted to compare the performance of these confidence intervals. A Monte Carlo simulation is applied to report coverage probability and average length of the confidence intervals. Based on the simulation study, for $$k=$$ 3, the GCI can be used. For $$k=$$ 6, the results of GCI approach perform similarly to the results of computational approach. For $$k=$$ 10, the computational approach can be considered as an alternative to estimate the confidence interval. A numerical example based on real data is presented to illustrate the proposed approaches.

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.

The data used for the analysis were collected from multiple regions or years. Evaluating each region or year separately may be insufficient for drawing comprehensive inferences and may fail to reveal statistically significant differences. To ensure the reliability of the analysis and to enable overall conclusions, it is necessary to apply a statistical method known as simultaneous confidence intervals. This technique enables the simultaneous construction of confidence intervals for multiple parameters. Therefore, we proposed and evaluated methods for constructing simultaneous confidence intervals for all pairwise differences between the coefficients of variation in zero-inflated Birnbaum-Saunders distributions. The methods utilized for constructing simultaneous confidence intervals comprise the generalized confidence interval (GCI), the bootstrap confidence interval (BCI), the method of variance estimates recovery (MOVER), the MOVER based on GCI, the MOVER based on BCI, the Bayesian credible interval, and the highest posterior density interval (HPD). Monte Carlo simulations were employed to evaluate the performance of each method, which involved the assessment of coverage probabilities and average widths under a set of parameter configurations and sample sizes. The generalized confidence interval method was the most efficient overall, as indicated by the simulation results. Finally, all proposed methods were applied to real-world wind speed data to examine their practical applicability and to demonstrate the consistency of the results between the simulation study and real-world applications.

The aim of this study is to propose confidence intervals for the difference between the coefficients of variation of inverse Gaussian distributions based on the generalized confidence interval (GCI), the adjusted generalized confidence interval (AGCI), the bootstrap percentile confidence interval (BPCI), and the method of variance estimates recovery (MOVER). The performances of the proposed confidence intervals were evaluated using coverage probabilities and average lengths via Monte Carlo simulation. The results showed that the GCI and AGCI methods were higher than or close to the nominal level in all cases. For small sample sizes, MOVER was better than the other methods because it provided the narrowest average length. The performances of all the approaches were illustrated using two real data examples.KeywordsBootstrapCoefficients of variationGeneralized confidence intervalInverse Gaussian distributionMethod of variance estimates recovery

Processcapability indices (PCIs) are extensively used statistical measures to assess process performance in manufacturing industry. In this paper, confidence intervals for the difference between PCIs for two processes are derived by the generalized pivotal quantity method. The indices C pk , C pmk , and C pm are considered in this study. The performance of the proposed method is assessed using simulation study. The results are also illustrated using an example from industrial contexts.

This paper proposes confidence intervals for the coefficient of variation of the delta-lognormal distribution which are based on the generalized confidence interval (GCI) method, it is compared with the modified Fletcher method. The coverage probabilities and the expected lengths are used for assessing the performance of these confidence interval. Simulation results show that the GCI method perform best in term of the coverage probability and expected length.