Inverse Weibull Research Articles

A NEWCLASS OF TAN-G FAMILY OF DISTRIBUTIONS WITH PROPERTIES AND APPLICATIONS

This research article is dedicated to the exploration of a novel family of distributions that is based on a tangent transformation. The new class of distributions, which we have named the new class tan-G (NCT-G) family, has been developed using the ratio of the cumulative distribution function (CDF) G(x) and 1+G(x) of a baseline distribution. We provide an overview of the general properties of this family of distributions. To demonstrate the applicability of the NCT-G family, we have utilized the inverse Weibull distribution as a baseline and introduced a new member of the suggested family. This new distribution exhibits a reverse-J, increasing, or inverted bathtub-shaped hazard function. We have also explored some statistical properties of this distribution, as well as its associated parameters estimated through maximum likelihood estimation (MLE). To evaluate the accuracy of the estimation procedure, we have conducted a Monte Carlo simulation. We found that even for small sample sizes, biases and mean square errors decrease as the sample size increases. Additionally, we have applied the NCT-IW distribution to two real data sets. By using model selection criteria and goodness of fit test statistics, we empirically proved that our suggested model outperforms other existing models, most of which have a greater number of parameters.

Discriminating among inverse Weibull, lognormal, and inverse Gaussian distributions

AbstractInverse Weibull, lognormal, and inverse Gaussian are some commonly used statistical distributions for modeling positively skewed failure lifetime data. These distributions share some interesting properties among themselves like they all have uni‐modal hazard rates. In this paper, we address the problem of discriminating among these statistical distributions to consider a more appropriate lifetime model. We first consider the complete samples and make use of the maximized log‐likelihood approach for choosing the correct model. We also obtain the expressions for logarithmic of defined test statistics, and associated asymptotic distributions. We then extend our discussion based on the observed sample in the presence of some censoring. We perform a simulation study in both cases to compare the probabilities of correct selection. Furthermore, for a given probability of correct selection and user‐specified protection level, we present a discussion to determine the minimum sample size required to discriminate among the three lifetime models. Finally, two real data sets are analyzed to illustrate the proposed methodology. In our findings, we observed that model parameters and methods of estimating unknown parameters play very important roles in the discrimination process, and sample size and the proportion of censoring become key factors to ensure a high probability of selecting a correct model.

A versatile model for lifetime of a component under stress

In this study, a versatile model, called alpha-monotone inverse Weibull distribution (alphaIW), for lifetime of a component under stress is introduced by using the alpha-monotone concept. The alphaIW distribution is also expressed as a scale-mixture between the inverse Weibull distribution and uniform distribution on (0, 1). The alphaIW distribution includes alpha-monotone inverse exponential and alpha-monotone inverse Rayleigh distributions as submodels and converenges to the inverse Weibull, inverse exponential, and inverse Rayleigh distributions as limiting cases. Also, slash Weibull, slash Rayleigh, and slash exponential distribuitons can be obtained under certain variable transformation and parameter settings. The alphaIW distribution is characterized by its hazard rate function and characterizing conditions are provided as well. Maximum likelihood, maximum product of spacing, and least squares methods are used to estimate distribution parameters. A Monte-Carlo simulation study is conducted to compare the efficiencies of the considered estimation methods. In the application part, two practical data sets, Kevlar 49/epoxy and Kevlar 373/epoxy, are modeled via the alphaIW distribution. Modeling performance of the alphaIW distribution is compared with its rivals by means of some well-known goodness-of-fit statistics and results show that alphaIW distribution performs better modeling than them. Results of comparison also indicate that obtaining the alphaIW distribution by using the alpha-monotone concept is cost effective since the new shape parameter added to the distribution by using the alpha-monotone concept significantly increases the modeling capability of the IW distribution. As a result of this study, it is shown that the alphaIW distribution can be an alternative to the well-known and recently-introduced distributions for modeling purposes.

ON EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL CHART FOR TIME TRUNCATED LIFE TEST USING INVERSE WEIBULL DISTRIBUTION

Statistical control charts are widely used in industry for process and measurement control to monitor production processes in order to discover any problem or issues that may arise during the production process and to help in finding solutions for these issues. This research proposes the exponentially weighted moving average (EWMA) control chart, comparing it’s performance to NP and MA control chart for monitoring the number of nonconforming with the truncated lifetime of a product when the lifetimes of products follows inverse Weibull (IW) distribution. The number of failures during the life test is used as an indicator of the quality of the product. EWMA control chart was proposed for this specific situation, thus extending the applicability of control charts methodology to situations involving truncated life tests. The performance of these control charts was measured with the average run length (ARL). The simulation shows that the results obtained from the EWMA chart using IW distribution is more sensitive in detecting small shift and performs much better to monitoring shifts as when compared to other control charts considered in the study.

Adaptive Type-II Hybrid Progressive Censoring Samples for Statistical Inference of Comparative Inverse Weibull Distributions

Recently, there has been a lot of interest in comparative life testing for items under jointly censored schemes for products from multiple production lines. The inverse Weibull distribution (IWD) is commonly used in life testing and reliability theory. In this paper, we address the problem of statistical inference from comparative inverse Weibull distributions under joint samples. An adaptive type-II hybrid progressive censoring scheme (HPCS) is used to save the balance between the ideal test time and the number of observed failures. Under the adaptive type-II HPCS, unknown parameters of the inverse Weibull populations are estimated using both maximum likelihood and Bayesian approaches. Asymptotic confidence intervals are established using the observed Fisher information matrix and bootstrap confidence intervals. We suggest using Markov chain Monte Carlo (MCMC) techniques to compute credible intervals under independent gamma priors. Using Monte Carlo simulations, all theoretical conclusions are tested and contrasted. For illustration purposes, an actual sample from comparative populations is analysed.

Bayesian and non-Bayesian inference for inverse Weibull model based on jointly type-II hybrid censoring samples with modeling to physics data

In recent years, there has been a lot of interest in the research of cooperative censoring schemes. In this work, we compared the relative benefits of two competing length-of-life products using inverse Weibull lifetime products with a joint type-II hybrid censoring scheme (JHC-Type II). We initially examined the maximum likelihood estimators and their confidence intervals (CIs) for the unknown parameters based on JHC-Type II. Then, under the premise of independent gamma priors, we offer Bayes estimates of the parameters using squared error loss and LINEX loss functions. We used the Markov chain Monte Carlo method to create credible intervals and Bayesian estimates. Based on the parametric bootstrapping techniques known as Boot-p and Boot-t, we create two bootstrapping CIs. In addition, we do a Monte Carlo simulation experiment to track how well the aforementioned approaches work and to determine the corresponding confidence and credible intervals. Finally, to show how the approaches covered in this paper might be used, we consider a real physical dataset.

Odd Chen Exponential Distribution: Properties and Applications

In this study, a new statistical distribution with three parameters called the Odd-Chen Exponential has been proposed. The statistical properties of the proposed distribution, such as the quantile, moments, incomplete moments, moment-generating function, and mean residual life, were developed. The density shows different shapes, making it more flexible for analyzing different forms of data. The hazard function also exhibits different shapes, including the well-known bathtub shape, which means that the distribution is flexible with real-life data. To estimate the distribution parameters, ordinary least squares estimators, Cramér-von Mises estimators, and maximum likelihood estimators were derived. The results were compared using a Monte Carlo simulation. Two-time datasets; one from the mining field and the other from survival analysis, were used to check the applicability of the proposed distribution. The results revealed that the OCE distribution performed better than the Odd Chen Weibull, Odd Chen Rayleigh, Rayleigh, Cauchy, Generalised Inverse Weibull and the Modified Extended Chen distributions.

Forecasting COVID-19 new cases through the Mixed Generalized Inverse Weibull Distribution and time series model

In this paper, the MGIW-ARIMA model is proposed to predict the newly diagnosed cases of COVID-19. The MGIW model is used to predict the trend components, and the ARIMA model is used to predict the random components. Finally, the prediction results of trend components and random components are added together to get the estimation results. The data of COVID-19 in the United States is used to verify and evaluate the model. In the short-term prediction, the combined model of MGIW-ARIMA is better than the single MGIW model. In the long-term prediction, the prediction accuracy of the MGIW model is slightly higher than that of the MGIW-ARIMA model. The MGIW-ARIMA model proposed in this paper makes up for the shortcoming of the lack of randomness in the estimated values of the MGIW model. The combination model can effectively capture the short-term changes of the epidemic while grasping the long-term development trend.

Reliability Estimation of Inverse Weibull Distribution Based on Intuitionistic Fuzzy Lifetime Data

As a commonly used model in reliability analysis, the inverse Weibull distribution (IWD) is widely applied in various scientific fields. This paper considers the reliability estimation of the IWD based on intuitionistic fuzzy lifetime data. Firstly, the related concepts of the fuzzy set theory are reviewed, and the concepts of the intuitionistic fuzzy conditional density, intuitionistic fuzzy likelihood function, and intuitionistic fuzzy conditional expectation are obtained by extension. In classical estimations, the maximum likelihood estimators of parameters and reliability are derived. Due to the nonlinearity, the EM algorithm is used to obtain the maximum likelihood estimates. In the Bayesian estimation, the gamma prior is selected, and the Bayesian estimation of the parameters and reliability is conducted under the symmetric entropy and the scale square error loss function, respectively. Since the integrals are complicated, the Lindley approximation is used to approximate the Bayesian estimates. Then, the performance of these estimators is evaluated by the Monte Carlo simulation. The simulation results show that the Bayesian estimation is more suitable than the maximum likelihood estimation for the reliability estimation. Finally, a set of real data is used to prove the effectiveness of these proposed methods. Through these methods, the reliability of the intuitive fuzzy life data is accurately estimated, which provides an important reference for the reliability analysis in the scientific field.

Retracted: The Generalization Inverse Weibull Distribution Related to X-Gamma Generator Family: Simulation and Application for Breast Cancer

Retracted: The Generalization Inverse Weibull Distribution Related to X-Gamma Generator Family: Simulation and Application for Breast Cancer

On Entropy Estimation of Inverse Weibull Distribution under Improved Adaptive Progressively Type-II Censoring with Applications

This article utilizes improved adaptive progressively Type-II censored data to estimate the entropy of the inverse Weibull distribution. Rényi, q, and Shannon entropy measurements are used to define entropy to achieve this objective. Both point and interval estimations of the entropy quantities are investigated through the maximum likelihood and maximum product of spacing methods. Two parametric bootstrap confidence intervals based on the two estimation techniques are also considered for the various entropy measures. A Monte Carlo simulation study is conducted to investigate how estimates behave at various sample sizes and different censoring schemes based on some statistical measurements. The simulations demonstrate that, as anticipated, when the sample size grows, the estimation accuracy also grows. Furthermore, they show that the estimated entropy measures get closer to the actual entropy values when the censoring level decreases. For purposes of explanation, two applications to actual datasets are taken into consideration. The results verified that the adaptive or improved adaptive progressive censoring schemes give more information about data than the conventional progressive censoring scheme in terms of minimum entropy measures.

Optimal scheme and estimation for a bivariate step‐stress accelerated life test with the inverse Weibull distribution under type‐I progressive censored samples

AbstractIn this paper, we provide an optimization design for a step‐stress accelerated life test (ALT) with two stress variables when the lifespan of test units is assumed to follow the inverse Weibull (IW) distribution. We first utilize progressive type‐I censoring and accelerated life testing to shorten the time and reduce cost of testing and then adopt a cumulative exposure (CE) model to look at the impact of varying stress levels. A log‐linear relationship between the scale parameter of the IW distribution and stress has been postulated. We obtain the maximum likelihood estimators and Bayes estimators of the unknown model parameters. Under normal operating conditions, we design an optimal test plan via minimizing the asymptotic variance (AV) of the percentile life. We carry out simulation studies to illustrate the performance and optimality of the proposed model. Finally, a real‐life data is analyzed for illustrative purposes.

Statistical inference based on lower record values for the inverse Weibull distribution under the modified loss function

The main objective of this article is to develop a linear, exponential loss function (LXLF) to the dependability value and variables of the Inverse Weibull Distribution (IWD) according to Lowest Record Values. By weighting the LXLF to construct current function loss, the modified linear, exponential loss function (WLXLF) was named. Then use WLXLF to estimate the parameters and reliability functions of the (IWD). After that, we evaluated how well the prediction made in this article performed against the Bayesian estimator using the symmetric and asymmetric loss functions and maximum likelihood estimation (MLE). The credibility range for the values of the parameters and durability function has been created using the parametric bootstrap approach, and the outcomes have been contrasted via a Monte Carlo simulation. An actual data set from a realistic situation was utilized to explain the concept. The computer simulation results demonstrated that the suggested method, performed as expected, was the best for estimating the scale parameter and reliability function. The proposed technique performed well in estimating the shape parameter based on real data and had an acceptable performance for estimating scale parameters and reliability functions.

An extensive mathematical approach for wind speed evaluation using inverse Weibull distribution

Wind velocity and potential are frequently estimated using probability density functions (PDFs) such as the Inverse Weibull and Rayleigh. Determining whether they are suitable for modelling observed distributions is crucial. Many wind energy systems require a thorough understanding of wind speed probability distributions. This research presents an extensive mathematical approach for wind speed evaluation using the inverse Weibull distribution (IWD) and a modified maximum likelihood function. IWD is a valuable tool for modelling wind speed characteristics, providing insights into the PDF and cumulative distribution function (CDF) of wind speeds. The suggested modified maximum likelihood (MML) function includes tweaks and improvements to increase parameter estimate accuracy in the IWD. The IWD is computed using the MML function, which considers data features, outliers, and unique research demands. Model calibration techniques, including goodness-of-fit assessment, sensitivity analysis, and error analysis, are employed to refine and assess the model’s accuracy. The developed mathematical model based on the inverse Weibull distribution and modified maximum likelihood function offers a robust approach for wind speed evaluation and analysis, contributing to renewable energy resource assessment, wind farm design, and structural engineering applications.

Correlated compound geometric frailty models based on reversed hazard rate

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g. matched pairs experiments, twin or family data), the shared frailty models were suggested. Shared frailty models are used despite their limitations. To overcome their disadvantages correlated frailty models may be used. In this paper, we introduce the correlated compound geometric frailty models based on reversed hazard rate with three different baseline distributions namely, the generalized log-logistic type I, the generalized log-logistic type II and the modified inverse Weibull. We introduce the Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. We also apply the proposed models to the Australian twin data set and a better model is suggested.

Fuzzy vs. Traditional Reliability Model for Inverse Weibull Distribution

In this paper, fuzzy stress strengths RF=P(Y≺X) and traditional stress strengths R=P(Y<X) are considered and compared when X and Y are independently inverse Weibull random variables. When axiomatic fuzzy set theory is taken into account in the stress–strength inference, it enables the generation of more precise studies on the underlying systems. We discuss estimating both conventional and fuzzy models of stress strength utilizing a maximum product of spacing, maximum likelihood, and Bayesian approaches. Simulations based on the Markov Chain Monte Carlo method are used to produce various estimators of conventional and fuzzy dependability of stress strength for the inverse Weibull model. To generate both conventional and fuzzy models of dependability, we use the Metropolis–Hastings method while performing Bayesian estimation. In conclusion, we will examine a scenario taken from actual life and apply a real-world data application to validate the accuracy of the provided estimators.

Bayesian Estimations of Shannon Entropy and Rényi Entropy of Inverse Weibull Distribution

In this paper, under the symmetric entropy and the scale squared error loss functions, we consider the maximum likelihood (ML) estimation and Bayesian estimation of the Shannon entropy and Rényi entropy of the two-parameter inverse Weibull distribution. In the ML estimation, the dichotomy is used to solve the likelihood equation. In addition, the approximation confidence interval is given by the Delta method. Because the form of estimation results is more complex in the Bayesian estimation, the Lindley approximation method is used to achieve the numerical calculation. Finally, Monte Carlo simulations and a real dataset are used to illustrate the results derived. By comparing the mean square error between the estimated value and the real value, it can be found that the performance of ML estimation of Shannon entropy is better than that of Bayesian estimation, and there is no significant difference between the performance of ML estimation of Rényi entropy and that of Bayesian estimation.

Statistical Analysis of Inverse Lindley Data Using Adaptive Type-II Progressively Hybrid Censoring with Applications

This paper deals with the statistical inference of the unknown parameter and some life parameters of inverse Lindley distribution under the assumption that the data are adaptive Type-II progressively censored. The maximum likelihood method is considered to acquire the point and interval estimates of the distribution parameter, reliability, and hazard rate functions. The approximate confidence intervals are also addressed. The delta method is taken into consideration to approximate the variances of the estimators of the reliability and hazard rate functions to get the required intervals. Based on the assumption of gamma prior, we further consider Bayesian estimation of the different parameters. The Bayes estimates are obtained by considering squared error and general entropy loss functions. The Bayes estimates and highest posterior density credible intervals are obtained by employing the Markov chain Monte Carlo procedure. An exhaustive numerical study is conducted to compare the offered estimates with regard to their root means squared error, relative absolute biases, confidence lengths, and coverage probabilities. To explain the suggested methods, two applications are investigated. The numerical findings show that the Bayes estimates perform better than those obtained based on the maximum likelihood method. The Bayesian estimations using the asymmetric loss function give more efficient estimates than the symmetric loss. Finally, the inverse Lindley distribution is recommended to be used as a suitable model to fit airborne communication transceiver and wooden toys data sets when compared with some competitive models including inverse Weibull, inverse gamma and alpha power inverted exponential.

DUS Inverse Weibull Distribution and Parameter Estimation in Regression Model

This paper considers various estimation methods to estimate the unknown parameters of the DUS Inverse Weibull (DIW) distribution using the maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CVM) and the Anderson-Darling (AD) estimators. A Monte-Carlo simulation study is conducted to determine the most preferable estimators in terms of their efficiencies. Furthermore, the distribution of the error terms in the simple linear regression is assumed to be DIW to show the implementation of it to the linear models. We also carry out a simulation study for comparing the performances of the estimators of the unknown regression parameters.

Optimal test plan of discrete alpha power inverse Weibull distribution under censored data

When measuring the lifespan of a product or gadget becomes difficult or cumbersome, the discretization phenomena typically occur an unbroken scale. In this study, we have created a discrete alpha power inverse Weibull distribution, which is a discrete of the continuous alpha power inverse Weibull distribution. The probability mass and hazard rate functions of the proposed model have asymmetric shapes. Also, the statistical properties of our current model are calculated. These include quantile, median, moments, skewness, kurtosis, index of dispersion, entropy and order statistics. The numerical values of some statistical properties of discrete alpha power inverse Weibull are calculated, such as minimum, first quantile, median, mean, third quantile, maximum, standard deviation, skewness, kurtosis, index of dispersion. We have calculated the model parameters using the maximum likelihood method under type II censoring. The parameters' asymptotic confidence intervals are built. It is noted that Bootstrap algorithms of estimators are shorter than asymptotic confidence intervals. Four potential optimum test methods are also examined utilizing various optimal criteria. Sometimes, the optimal censoring scheme is the maximum value of the criterion while the optimal test plan of the others is corresponding to the lowest value of the criteria. Numerous simulations are run to evaluate the effectiveness of the methodology that is suggested in this article. Last but not least, the two real data sets are given to illustrate the flexibility of the suggested model in actual real life.