Analyzing k-Out-of-n Load Sharing Systems under Progressive Censoring

Study’s Excerpt:• A new extension of the Modified Inverted Kumaraswamy (MIK) distribution using the inverse power function is introduced.• Maximum likelihood estimation (MLE) and maximum product spacing (MPS) methods were applied, with MPS consistently outperforming MLE.• Simulation results demonstrated that as sample sizes increased, both estimation techniques showed improved accuracy.• The practical utility of the proposed model was validated using two real-life datasets.Full Abstract:This article considers the problem of estimating additional parameters of the modified inverted Kumaraswamy (MIK) distribution using the inverse power function based on the general Kumaraswamy distribution. The parameters' maximum likelihood (MLE) estimators are obtained, while the Bayesian estimates are obtained using the maximum product spacing (MPS). We obtained a new model for generalizing the existing ones to make them more flexible and to aid their application in various fields. An expression for reliability measures, order statistics, and some other important properties are derived. The maximum likelihood estimation method is used to estimate the proposed model's unknown parameters. Finally, a simulation study was reported concerning different sample sizes and method schemes. The practical utility of the proposed distribution is demonstrated using two real-life datasets: (i) survival times (in months) of 101 patients diagnosed with advanced acute myelogenous leukaemia, and (ii) strengths of 63 samples of 1.5 cm glass fibres, originally obtained by workers at the UK National Physical Laboratory. The results highlight the robustness and flexibility of the proposed model in reliability and survival analysis contexts.

In this study, we proposed some ridge estimators by considering the work of Mansson (Econ Model 29(2):178–184, 2012), Dorugade (J Assoc Arab Univ Basic Appl Sci 15:94–99, 2014) and some others for the gamma regression model (GRM). The GRM is a special form of the generalized linear model (GLM), where the response variable is positively skewed and well fitted to the gamma distribution. The commonly used method for estimation of the GRM coefficients is the maximum likelihood (ML) estimation method. The ML estimation method perform better, if the explanatory variables are uncorrelated. There are the situations, where the explanatory variables are correlated, then the ML estimation method is incapable to estimate the GRM coefficients. In this situation, some biased estimation methods are proposed and the most popular one is the ridge estimation method. The ridge estimators for the GRM are proposed and compared on the basis of mean squared error (MSE). Moreover, the outperforms of proposed ridge estimators are also calculated. The comparison has done using a Monte Carlo simulation study and two real data sets. Results show that Kibria’s and Mansson and Shukur’s methods are preferred over the ML method.

Load-sharing systems have several practical applications. In load-sharing systems, the event of a component failure will result in a higher load, therefore inducing a higher failure rate, in each of the surviving components. This introduces failure dependency among the load-sharing components, which in turn increases the complexity in analyzing these systems. In this chapter, we first discuss modeling approaches and existing solution methods for analyzing the reliability of load-sharing systems. We then describe tampered failure rate (TFR) load-sharing systems and their properties. Using these properties, we provide efficient solution methods for solving TFR load-sharing models. Because load-sharing k-out-of-n systems have several practical applications in reliability engineering, we provide a detailed analysis for various cases of these systems. The solution methods proposed in this chapter are applicable for both identical and non-identical component cases. The proposed methods are not restricted to the exponential failure distributions and are applicable for a wide range of failure time distributions of the components. In most cases, we provide closed-form analytical solutions for the reliability of TFR load-sharing k-out-ofn: G systems. As a special case, efficient solutions are provided for systems with identical components where all surviving components share the load equally. The efficiencies of the proposed methods are demonstrated through several numerical examples.

This paper discussed robust estimation for point estimation of the shape and scale parameters for generalized exponential (GE) distribution using a complete dataset in the presence of various percentages of outliers. In the case of outliers, it is known that classical methods such as maximum likelihood estimation (MLE), least square (LS) and maximum product spacing (MPS) in case of outliers cannot reach the best estimator. To confirm this fact, these classical methods were applied to the data of this study and compared with non-classical estimation methods. The non-classical (Robust) methods such as least absolute deviations (LAD), and M-estimation (using M. Huber (MH) weight and M. Bisquare (MB) weight) had been introduced to obtain the best estimation method for the parameters of the GE distribution. The comparison was done numerically by using the Monte Carlo simulation study. The two real datasets application confirmed that the M-estimation method is very much suitable for estimating the GE parameters. We concluded that the M-estimation method using Huber object function is a suitable estimation method in estimating the parameters of the GE distribution for a complete dataset in the presence of various percentages of outliers.

Maximum product spacing for stress–strength model based on progressive Type-II hybrid censored samples with different cases has been obtained. This paper deals with estimation of the stress strength reliability model R = P(Y < X) when the stress and strength are two independent exponentiated Gumbel distribution random variables with different shape parameters but having the same scale parameter. The stress–strength reliability model is estimated under progressive Type-II hybrid censoring samples. Two progressive Type-II hybrid censoring schemes were used, Case I: A sample size of stress is the equal sample size of strength, and same time of hybrid censoring, the product of spacing function under progressive Type-II hybrid censoring schemes. Case II: The sample size of stress is a different sample size of strength, in which the life-testing experiment with a progressive censoring scheme is terminated at a random time T ∈ (0,∞). The maximum likelihood estimation and maximum product spacing estimation methods under progressive Type-II hybrid censored samples for the stress strength model have been discussed. A comparison study with classical methods as the maximum likelihood estimation method is discussed. Furthermore, to compare the performance of various cases, Markov chain Monte Carlo simulation is conducted by using iterative procedures as Newton Raphson or conjugate-gradient procedures. Finally, two real datasets are analyzed for illustrative purposes, first data for the breaking strengths of jute fiber, and the second data for the waiting times before the service of the customers of two banks.

In this paper, the m-component stress-strength parameter is considered under progressive first failure (PFF) censoring scheme, assuming the stress and strength variables follow the Lomax distribution. To achieve this, various classical and Bayesian estimation methods are explored under two scenarios. In the first scenario, where the common parameters of the stress and strength variables are unknown, maximum likelihood estimation (MLE), Bayesian estimation using the Markov Chain Monte Carlo (MCMC) method, asymptotic confidence intervals, and highest posterior density (HPD) credible intervals are derived. In the second scenario, where the common parameters of the stress and strength variables are known, MLE, Bayesian estimation using the MCMC method and Lindley’s approximation, the uniformly minimum variance unbiased estimator (UMVUE), as well as the asymptotic confidence intervals and the HPD credible intervals, are derived. A Monte Carlo simulation study is conducted to evaluate and compare the performance of the proposed methods. Additionally, two real datasets are analyzed for illustrative purposes.

In this study, an alpha power inverted Kumaraswamy (APIK) distribution is introduced. The APIK distribution is de- rived by applying alpha power transformation to an inverted Kumaraswamy distribution. Some submodels and limiting cases of the APIK distribution are obtained as well. To the best of the authors’ knowledge, some of these distributions have not been introduced yet. Statistical inference of the APIK distribution, including survival and hazard rate func- tions, are obtained. Unknown parameters of the APIK distribution are estimated by using the maximum likelihood (ML), maximum product of spacings (MPS), and least squares (LS) methods. A Monte Carlo simulation study is con- ducted to compare the efficiencies of the ML estimators of the shape parameters α, β and λ of the APIK distribution with their MPS and LS counterparts. An application to a real data set is provided to show the implementation and modeling capability of the APIK distribution.

ABSTRACTIn this article, the generalized lifetime performance index (GLPI) is taken into consideration where the process distribution follows exponentiated exponential distribution. The different classical estimation procedures, namely, maximum likelihood estimation (MLE), least squares, weighted least squares, and maximum product spacing estimation (MPSE) methods have been discussed using progressively type‐II censored samples. Next, the Bayesian estimation of the considered index with gamma prior distribution has been derived using the symmetric as well as asymmetric loss functions. Further, the different parametric interval estimation techniques, namely, asymptotic confidence interval based on MLE and MPSE, bootstrap confidence intervals, and Bayes credible intervals have been obtained for the considered censoring schemes. Monte Carlo simulation study has been carried out to compare these estimators in terms of their mean squared errors and simulated posterior risks, respectively. Lastly, two real data sets have been re‐analyzed to illustrate applications of the proposed GLPI under progressively type‐II censored scheme.

This paper aims to propose a new right skewed, heavy tailed probability distribution with upside-down bathtub shape hazard rate. The related statistical characteristics and its measures are derived. The unknown parameter of the proposed distribution is estimated by using five estimation methods, namely maximum likelihood estimation (MLE) method, maximum product spacing estimation (MPSE) method, least square estimation (LSE) method, weighted least square estimation method and Cramer-Von-Mises estimation (CVME) method. Also, asymptotic confidence interval (ACI) and bootstrap confidence intervals (BCIs) namely, standard bootstrap (s-boot), percentile bootstrap (p-boot), and Student’s bootstrap (t-boot) of the parameter are also computed. The Monte Carlo simulation study has been performed to compare the performance of the proposed estimators and corresponding interval width along with coverage probability. At last, three real data sets have been used to demonstrate the suitability of the proposed study in real life scenario. The considered data sets also exhibit skewed, heavy tailed, upside-down bathtub shape hazard rate pattern.

In this paper, we propose the method of Maximum product of spacings for point estimation of parameter of generalized inverted exponential distribution (GIED). The aim of this paper is to analyse the small sample behaviour of proposed estimators. Further, we have also proposed asymptotic confidence interv als of the parameters and the estimates of reliability and ha zard function using Maximum Product Spacings (MPS) method and compared with corresponding asymptotic confidence intervals and the es timates of reliability and hazard function of Maximum Likelihood estimation (MLEs). A comparative study among the method of MLE, method of least square (LSE) and the method of maximum product of spacings (MPS) is performed on the basis of simulated sample of GIED. The MPS method outperforms the method of MLE and the method ofLSE. Furthermore, comparison of different estimation method have been proposed on the basis of K-S distance and AIC. For numerical illustration one real data set has been considered.

Shannon’s entropy is a fundamental concept in information theory that quantifies the uncertainty or information in a random variable or data set. This article addresses the estimation of Shannon’s entropy for the Maxwell lifetime model based on progressively first-failure-censored data from both classical and Bayesian points of view. In the classical perspective, the entropy is estimated using maximum likelihood estimation and bootstrap methods. For Bayesian estimation, two approximation techniques, including the Tierney-Kadane (T-K) approximation and the Markov Chain Monte Carlo (MCMC) method, are used to compute the Bayes estimate of Shannon’s entropy under the linear exponential (LINEX) loss function. We also obtained the highest posterior density (HPD) credible interval of Shannon’s entropy using the MCMC technique. A Monte Carlo simulation study is performed to investigate the performance of the estimation procedures and methodologies studied in this manuscript. A numerical example is used to illustrate the methodologies. This paper aims to provide practical values in applied statistics, especially in the areas of reliability and lifetime data analysis.

Researchers from various fields of science encounter phenomena of interest, and they seek to model the occurrences scientifically. An important approach of performing modeling is to use probability distributions. Probability distributions are probabilistic models that have many applications in different research areas, including, but not limited to, environmental and financial studies. In this paper, we study a quartic transmuted Weibull distribution from a general quartic transmutation family of distributions as a generalization and an alternative to the well-known Weibull distribution. We also investigate the practical application of this generalization by modeling climate-related data sets and check the goodness-of-fit of the proposed model. The statistical properties of the proposed model, which includes non-central moments, generating functions, survival function, and hazard function, are derived. Different estimation methods to estimate the parameters of the proposed quartic transmuted distribution: the maximum likelihood estimation method, the maximum product of spacings method, two least-squares-based methods, and three goodness-of-fit-based estimation methods. Numerical illustration and an extensive comparative Monte Carlo simulation study are conducted to investigate the performance of the estimators of the considered inferential methods. Regarding estimation methods, simulation outcomes indicated that the maximum likelihood estimation (MLE), Anderson-Darling estimation (ADE) and right Anderson-Darling (RADE) methods in general outperformed the other considered methods in terms of estimation efficiency for large sample size, while all considered estimation methods performed almost same in terms of goodness-of-fit regardless the values of shape and transmuted parameters. Two real-life data sets are used to demonstrate the suggested estimation methods, the applicability and flexibility of the proposed distribution compared to Weibull, transmuted Weibull, and cubic transmuted Weibull distributions. Weighted least squares estimation (WLSE) and least squares estimation (LSE) methods provided best model fitting estimates of the proposed distribution for Wheaton River and rainfall data respectively. The proposed quartic transmuted Weibull distribution provide significantly improved fit for the two datasets as compared with other distributions.

In this paper, we studied the problem of estimating the odd exponentiated half-logistic exponential (OEHLE) parameters using several frequentist estimation methods. Parameter estimation provides a guideline for choosing the best method of estimation for the model parameters, which would be very important for reliability engineers and applied statisticians. We considered eight estimation methods, called maximum likelihood, maximum product of spacing, least squares, Cramér–von Mises, weighted least squares, percentiles, Anderson–Darling, and right-tail Anderson–Darling for estimating its parameters. The finite sample properties of the parameter estimates are discussed using Monte Carlo simulations. In order to obtain the ordering performance of these estimators, we considered the partial and overall ranks of different estimation methods for all parameter combinations. The results illustrate that all classical estimators perform very well and their performance ordering, based on overall ranks, from best to worst, is the maximum product of spacing, maximum likelihood, Anderson–Darling, percentiles, weighted least squares, least squares, right-tail Anderson–Darling, and Cramér–von-Mises estimators for all the studied cases. Finally, the practical importance of the OEHLE model was illustrated by analysing a real data set, proving that the OEHLE distribution can perform better than some well known existing extensions of the exponential distribution.

It is known that various types of location privacy attacks can be carried out using a personalized transition matrix that is learned for each target user, or a population transition matrix that is common to all target users. However, since many users disclose only a small amount of location information in their daily lives, the training data can be extremely sparse. The aim of this paper is to clarify the risk of location privacy attacks in this realistic situation. To achieve this aim, we propose a learning method that uses tensor factorization (or matrix factorization) to accurately estimate personalized transition matrices (or a population transition matrix) from a small amount of training data. To avoid the difficulty in directly factorizing the personalized transition matrices (or population transition matrix), our learning method first factorizes a transition count tensor (or matrix), whose elements are the number of transition counts that the user has made, and then normalizes counts to probabilities. We focus on a localization attack, which derives an actual location of a user at a given time instant from an obfuscated trace, and compare our learning method with the maximum likelihood (ML) estimation method in both the personalized matrix mode and the population matrix mode. The experimental results using four real data sets show that the ML estimation method performs only as well as a random guess in many cases, while our learning method significantly outperforms the ML estimation method in all of the four data sets.

This chapter presents a competing risk model with two dependent failure causes whose latent failure times are Marshall–Olkin bivariate exponential family distribution that includes Burr XII, Gompertz, and Weibull distributions as special cases. Maximum likelihood estimation and Bayesian estimation methods for the model parameters are discussed. The existence and uniqueness of maximum likelihood estimates are established under some regular conditions for Weibull and Burr XII base distributions, respectively. A Markov-Chain Monte Carlo process is proposed for Bayesian estimation method. Due to the possible flaw of maximum likelihood estimation method, a Monte Carlo simulation study is only conducted to assess the performance of Bayesian estimations for model parameters, tenth percentile, median and ninety percentile lifetimes under square error, absolute error, and LINEX loss functions. A real data set is used for illustration.