Gompertz Distribution Research Articles

Multicomponent Stress-Strength Model Based on Generalized Progressive Hybrid Censoring Scheme: A Statistical Analysis.

The statistical inference of the reliability and parameters of the stress–strength model has received great attention in the field of reliability analysis. When following the generalized progressive hybrid censoring (GPHC) scheme, it is important to discuss the point estimate and interval estimate of the reliability of the multicomponent stress–strength (MSS) model, in which the stress and the strength variables are derived from different distributions by assuming that stress follows the Chen distribution and that strength follows the Gompertz distribution. In the present study, the Newton–Raphson method was adopted to derive the maximum likelihood estimation (MLE) of the model parameters, and the corresponding asymptotic distribution was adopted to construct the asymptotic confidence interval (ACI). Subsequently, the exact confidence interval (ECI) of the parameters was calculated. A hybrid Markov chain Monte Carlo (MCMC) method was adopted to determine the approximate Bayesian estimation (BE) of the unknown parameters and the high posterior density credible interval (HPDCI). A simulation study with the actual dataset was conducted for the BEs with squared error loss function (SELF) and the MLEs of the model parameters and reliability, comparing the bias and mean squares errors (MSE). In addition, the three interval estimates were compared in terms of the average interval length (AIL) and coverage probability (CP).

Modeling survival response using a parametric approach in the presence of multicollinearity

Survival times with valid distributional assumptions are hard to treat when the assumption of no multicollinearity between predictors is violated. The model that evaluates the performance of survival functions provides an efficient tool to determine accurate systems. However, the standard methods addressing survival functions have some limitations in long-term evaluations in the presence of high multicollinearity with valid distributional assumptions. Therefore, a method that can treat survival outcomes on complicated functions is introduced which flexibly estimates the parameters. This method is developed by integrating a flexible parametric algorithm with partial least squares (PLS-FPM) to estimate continuous functions for extrapolating the survival modeling in long-term evaluations. The integrated method based on simulated data generated from Gompertz distributions is compared with standard partial least squares-Cox regression (PLS-CoxR). The analysis reflects the flexibility and accuracy of PLS-FPM in addressing survival time response. The method is also executed for four real data sets of breast cancer having survival responses. The regression coefficients of significant genes are estimated. The applicability in addressing several probability distributions as a parametric tool validates the flexibility of the method to a wider range of survival responses and evidence an effective instrument for the data-oriented evaluation of survival systems.

Life expectancy for community-dwelling persons with dementia and severe disability.

Understanding life expectancy can help persons with dementia, their care partners, and policymakers plan for what lies ahead. We sought to determine life expectancy and predictors of mortality for community-dwelling persons with dementia and severe disability. Using the National Health and Aging Trends Study (NHATS) linked to Medicare claims, we identified community-dwelling respondents age 65+ who entered NHATS in 2011 with dementia and severe disability (defined as three impairments in activities of daily living), or who subsequently met criteria for dementia and then severe disability. We estimated time to death based on the timing of meeting severe disability criteria. We conducted parametric survival analyses using a Gompertz distribution to calculate risk of death and predicted median time to death. Predictors included demographic, functional, clinical characteristics, and behavioral symptoms (assessed among NHATS respondents with proxy interviews). Among 842 community-dwelling persons with dementia and severe disability, 80.5% died during the study period. After adjusting for age and gender, overall predicted median time to death was 1.7 years (25th percentile 0.6, 75th percentile 3.8 years). Six notable characteristics were associated with shorter life expectancy: 1) older age (90+), with a predicted median time to death of 1.0year (0.4, 2.1); 2) being bedbound, 1.1 years (0.4, 2.3); 3) being homebound, 1.2 years (0.5, 2.6); 4) having comorbid cancer, 1.2 years (0.5, 2.6); 5) unintended weight loss, 1.4 years (0.5, 3.1); and 6) comorbid depression, 1.5 years (0.6, 3.3). Community-dwelling persons with dementia and severe disability lived a median of 1.7 years. Clinicians can use the study findings to provide anticipatory guidance to patients and care partners, and policymakers to inform design of longitudinal supportive services.

Inference on adaptive progressive hybrid censored accelerated life test for Gompertz distribution and its evaluation for virus-containing micro droplets data

In this paper, we discuss the problem of estimating the unknown parameters of the Gompertz distribution and the acceleration factor under constant-stress partially accelerated life test model. Based on adaptive Type II progressive hybrid censored samples, the maximum likelihood estimates of the model parameters and acceleration factor are derived. From Bayesian aspect, the Bayes estimates of the unknown parameters are obtained using the Gibbs sampler together with Metropolis-Hastings (GMH) algorithm. Based on the interval estimation viewpoint, the asymptotic, Bayesian credible and bootstrap confidence intervals for model parameters and acceleration factor are also constructed. In order to investigate the impact of estimation procedures and methodologies adopted, a simulation study is performed. Finally, a real life example contains the persistence of the virus-containing micro droplets is analyzed to illustrate the application of the partially accelerated life test model. It is observed that the GMH algorithm is superior for estimating the parameters of partially accelerated life test model after comparison. The results of real example analysis show that the proposed model can be applied to the engineering problem and the considered methods are suitable for the modeling of the virus-containing micro droplets lifetime data under two co-current air flow velocities.

A reliability and survival model for one and two failure modes system with applications to complete and censored datasets

Lifetime distributions that have bathtub-shaped failure rates with low and long constant region at the mid-time period are more realistic for modelling failure time datasets with similar failure rates. In this paper, we introduce a new model named flexible additive Chen–Gompertz distribution. The model has increasing and bathtub-shaped failure rate with or without low and long constant region, and can be adequately used in reliability context. Some distributional properties of the model are obtained. We present a complete Bayesian analysis of the proposed model for censored and non-censored datasets and provide the Bayes estimators of the model parameters using the Metropolis–Hastings algorithm by utilizing the square error loss function. We have also given the maximum likelihood estimators of the model parameters for complete and right-censored datasets and performed simulations on the two methods for different settings of the parameters and sample sizes. The applicability of the proposed distribution is demonstrated by fitting three different failure time datasets each with bathtub failure rate from reliability and survival analysis. The results of the applications illustrate that the model provides a good fit to all the three datasets compared to the other well-known Chen and Weibull extended distributions as evidenced by some goodness-of-fit tests.

Numerical Methods on the Triple Informative Prior Distribution for the Failure Rate Basic Gompertz Model

In this paper, it has been dealt with basic Gompertz distribution. The maximum likelihood, Bayes methods of estimation were used to estimate the unknown shape parameter. The failure rate (hazard) function with the least loss was found using different priors (Gamma, exponential, chi-square and triple prior) under symmetric loss function (Degroot loss function). A comparison was made about the performance of these estimators with the numerical solution that was found using expansion methods (Bernstein polynomial and power function) which was applied to find the failure rate function numerically. The proficiency test of the proposed methods was conducted with a number of test examples. Finally, for computations the Matlab (R2015b) is used.

The Bivariate Defective Gompertz Distribution Based on Clayton Copula with Applications to Medical Data

In medical studies, it is common the presence of a fraction of patients who do not experience the event of interest. These patients are people who are not at risk of the event or are patients who were cured during the research. The proportion of immune or cured patients is known in the literature as cure rate. In general, the traditional existing lifetime statistical models are not appropriate to model data sets with cure rate, including bivariate lifetimes. In this paper, it is proposed a bivariate model based on a defective Gompertz distribution and also using a Clayton copula function to capture the possible dependence structure between the lifetimes. An extensive simulation study was carried out in order to evaluate the biases and the mean squared errors for the maximum likelihood estimators of the parameters associated to the proposed distribution. Some applications using medical data are presented to show the usefulness of the proposed model. Maximum likelihood and Bayesian methods were used to estimate the parameters of the model.

Cubic rank transmuted generalized Gompertz distribution: properties and applications

In this paper, we introduce a new lifetime distribution as an alternative to generalized Gompertz, Gompertz distribution and its modified ones. This new distribution is a special case of the family of distributions introduced by Granzotto et al. [D.C.T. Granzotto, F. Louzada and N. Balakrishnan, Cubic rank transmuted distributions: inferential issues and applications., J. Stat. Comput. Simul. 87 (2017), pp. 2760–2778]. We obtain some characteristic properties of suggested distribution such as hazard function, ordinary moments, coefficient of skewness, coefficient of kurtosis, moment generating function, quantile function and median. We discuss three different methods of estimation to estimate the parameters of proposed distribution. A comprehensive Monte Carlo simulation study is performed in order to compare the performances of estimators according to mean square errors and biases. Finally, three real data applications are performed to illustrate usefulness of suggested distribution.

Statistical inference of Gompertz distribution under general progressive type II censored competing risks sample

Gompertz distribution is a significant and practical continuous lifetime distribution, which plays an important role in reliability engineering. A Gompertz distribution competing risks model is discussed and studied under general progressive censoring in this paper. When the lifetime model fails for different latent reasons, the maximum likelihood estimates are given for the unknown parameters. The approximate confidence intervals through Fisher information matrix and bootstrap confidence intervals are established, containing bootstrap-p and bootstrap-t techniques. In addition, the Bayes estimation of unknown parameters is investigated under the condition of squared error loss. Moreover, the Bayes credible interval is derived. Finally, a numerical simulation is conducted to evaluate the performances of the proposed methods and real-life data is analyzed by applying the proposed inference methods.

A Family of Bayesian Estimators for the Two-Parametric Burr Type II Distribution

This study discusses the posterior estimation for the parameters of the Burr type II distribution (BIID). The informative and noninformative priors along with different loss functions have also been assumed for the posterior estimation. The applicability of the proposed distribution has also been discussed. The modeling capability of the proposed model has been compared with seven classes of the lifetime distributions using real data. The generalizations of Weibull, exponential, Rayleigh, gamma, log normal, Pareto, Maxwell, Levy, Laplace, inverse gamma, Gompertz, chi-square, inverse chi-square, half normal, and log-logistic distributions have been considered for the comparison. The comparison has been made based on different goodness-of-fit criteria, such as Akaike information criteria (AIC), Bayesian information criteria (BIC), and Kolmogorov-Smirnov (KS) test. Based on the results from the study, it can be suggested that the BIID can efficiently replace commonly used lifetime distributions and their modifications. The results under this model were comparable with different conventional/modified distributions having up to six parameters.

Improving CubeSat reliability: Subsystem redundancy or improved testing?

The objective of this paper is to investigate which approach would lead to more reliable CubeSats: full subsystem redundancy or improved testing. Based on data from surveys, the reliability of satellites and subsystems is estimated using a Kaplan–Meier estimator. Subsequently, a variety of reliability models is defined and their maximum likelihood estimates are compared. A product of a Lognormal distribution addressing immaturity failure and a Gompertz distribution addressing wear-out is found to best represent CubeSat reliability. Bayesian inference is used to find realistic wear-out parameters by using failure data of small satellites. Subsystem reliability estimates are subsequently found using a similar approach. A reliability model for CubeSats with redundant subsystems is established, verified and applied in a Monte Carlo simulation. The results are compared with a model for reduced immaturity failure. Allocating resources to reduction of immaturity failures through improved testing is considered to be superior to allocating these resources to the implementation of subsystem redundancy.

Generalized Kumaraswamy Generalized Power Gompertz Distribution: Statistical Properties, Application, and Validation Using a Modified Chi-Squared Goodness of Fit Test

A new six-parameter continuous distribution called the Generalized Kumaraswamy Generalized Power Gompertz (GKGPG) distribution is proposed in this study, a graphical illustration of the probability density function and cumulative distribution function is presented. The statistical features of the Generalized Kumaraswamy Generalized Power Gompertz distribution are systematically derived and adequately studied. The estimation of the model parameters in the absence of censoring and under-right censoring is performed using the method of maximum likelihood. The test statistic for right-censored data, criteria test for GKGPG distribution, estimated matrix Ŵ, Ĉ, and Ĝ, criteria test Y2n, alongside the quadratic form of the test statistic is derived. Mean simulated values of maximum likelihood estimates and their corresponding square mean errors are presented and confirmed to agree closely with the true parameter values. Simulated levels of significance for Y2n (γ) test for the GKGPG model against their theoretical values were recorded. We conclude that the null hypothesis for which simulated samples are fitted by GKGPG distribution is widely validated for the different levels of significance considered. From the summary of the results of the strength of a specific type of braided cord dataset on the GKGPG model, it is observed that the proposed GKGPG model fits the data set for a significance level ε = 0.05.

Competing Risks Step-Stress Model with Lagged Effect under Gompertz Distribution

In many survival analysis studies, the failure of a product may be attributed to one of several competing risks. In addition, if survival time is long, researchers can adopt accelerated life tests, causing devices to fail more quickly. One popular type of accelerated life tests is the step-stress test, and in this test, the stress level changes at a predetermined point time. The manner that stress levels change abruptly and increase discontinuously has been studied extensively. This paper considers a more realistic situation where the effect of stress increases cannot be achieved all at once, but with a lag time, and we propose a step-stress model consisting of two independent competing risks with a lag period in which the failure time caused by different risks at different stress levels obey Gompertz distribution, and the range of lag period is predetermined. The unknown parameters are estimated by maximum likelihood estimation and least squares estimation. For comparison, asymptotic confidence intervals and percentile bootstrap confidence intervals are constructed. By using Monte-Carlo simulations, we obtain the means and mean square errors of the maximum likelihood estimates and the least squares estimates, as well as the mean lengths and coverage rates of the two confidence intervals, which show the performance of various methods. Finally, in order to illustrate the model and proposed methods, we analyze a dataset from a solar energy experiment.

On estimating of the stress-strength model with more than two states under Gompertz distribution

The Gompertz distribution, one of statistical distributions, is considered in survival analysis because it has an increasing hazard rate function. It has many applications in various fields such as medical and reliability. The problem of estimation of stress-strength model with more than two states is considered by other authors under exponential and Weibull distribution. The main aim of this paper is to consider the same problem under Gompertz distribution based on progressive type-II censoring data. Three estimation methods are considered to estimate the reliability parameter, maximum likelihood (MLE), uniformly minimum variance unbiased estimator (UMVE) and Bayes estimator. To compare and study the performance of three estimators, Monte Carlo simulation study is performed.

A study on exponentiated Gompertz distribution under Bayesian discipline using informative priors

Abstract The exponentiated Gompertz (EGZ) distribution has been recently used in almost all areas of human endeavours, starting from modelling lifetime data to cancer treatment. Various applications and properties of the EGZ distribution are provided by Anis and De (2020). This paper explores the important properties of the EGZ distribution under Bayesian discipline using two informative priors: the Gamma Prior (GP) and the Inverse Levy Prior (ILP). This is done in the framework of five selected loss functions. The findings show that the two best loss functions are the Weighted Balance Loss Function (WBLF) and the Quadratic Loss Function (QLF). The usefulness of the model is illustrated by the use of real-life data in relation to simulated data. The empirical results of the comparison are presented through a graphical illustration of the posterior distributions.

Bayesian and Classical Inference under Type-II Censored Samples of the Extended Inverse Gompertz Distribution with Engineering Applications.

In this article, we introduce a new three-parameter distribution called the extended inverse-Gompertz (EIGo) distribution. The implementation of three parameters provides a good reconstruction for some applications. The EIGo distribution can be seen as an extension of the inverted exponential, inverse Gompertz, and generalized inverted exponential distributions. Its failure rate function has an upside-down bathtub shape. Various statistical and reliability properties of the EIGo distribution are discussed. The model parameters are estimated by the maximum-likelihood and Bayesian methods under Type-II censored samples, where the parameters are explained using gamma priors. The performance of the proposed approaches is examined using simulation results. Finally, two real-life engineering data sets are analyzed to illustrate the applicability of the EIGo distribution, showing that it provides better fits than competing inverted models such as inverse-Gompertz, inverse-Weibull, inverse-gamma, generalized inverse-Weibull, exponentiated inverted-Weibull, generalized inverted half-logistic, inverted-Kumaraswamy, inverted Nadarajah–Haghighi, and alpha-power inverse-Weibull distributions.

The Gamma–Gompertz distribution: Theory and applications

In this article, we explore a new model with three parameters called Gamma–Gompertz (GGo) distribution which is generated by the gamma-X family. The GGo distribution provides better fits than some competing distributions such as three-parameter Gompertz, Gompertz–Lindley, Gompertz–geometric, Gompertz–Poisson, and inverse Gompertz distributions. Some well-known distributions are included in the GGo distribution as special sub-models. The density function of the distribution can be decreasing, unimodal and decreasing–increasing–decreasing shaped while the failure rate function can be increasing and unimodal shaped. Various properties of the GGo distribution are obtained. An extensive simulation study is carried out to assess the effectiveness of some classical estimation approaches which are discussed to estimate the model parameters. To demonstrate the potentiality of the GGo distribution, two real data sets are used and the bootstrap percentile confidence intervals are also obtained by bootstrap resampling. In addition, a new regression model is proposed via re-parametrization of the GGo distribution, called the log-GGo distribution. The maximum likelihood method is considered to estimate the unknown parameters of re-parametrized log-GGo distribution. The potentiality of log-GGo regression model is analyzed for HIV+ censored data set.

Bayesian and Classical Estimation of Strength-Stress Reliability for Gompertz Distribution Based on Upper Record Values

In this paper, we consider the problem of estimating stress-strength reliability R=P(X>Y) for Gompertz lifetime models having the same shape parameters but different scale parameters under a set of upper record values. We obtain the maximum likelihood estimator (MLE), the approximate Bayes estimator and the exact confidence intervals of stress-strength reliability when the shape parameter is known. Also, when the shape parameter is unknown, the MLE, the asymptotic confidence interval and some bootstrap confidence intervals of stress-strength reliability are studied. Furthermore, a Bayesian approach is proposed for estimating the parameters and then the corresponding credible interval are achieved using Gibbs sampling technique via OpenBUGS software. Monte Carlo simulations are performed to compare the performance of different proposed estimation methods.

Estimation of Cumulative Incidence Function in the Presence of Middle Censoring Using Improper Gompertz Distribution

In this paper we deal with the modelling of cumulative incidence function using improper Gompertz distribution based on middle censored competing risks survival data. Together with the unknown parameters, cumulative incidence function also estimated. In classical set up, we derive the point estimates using maximum likelihood estimator and midpoint approximation methods. The asymptotic confidence interval are obtained based on asymptotic normality properties of maximum likelihood estimator. We also derive the Bayes estimates with associated credible intervals based on informative and non-informative types of priors under two loss functions such as squared error and LINEX loss functions. A simulation study is conducted for comprehensive comparison between various estimators proposed in this paper. A real life data set is also used for illustration.

The Marshall-Olkin Gompertz Distribution: Properties and Applications

This article introduces three parameters class for lifetime Poisson processes in the Marshall-Olkin transformation family that are increasing, bathtub and skewed. Some structural mathematical properties of the Marshall-Olkin Gompertz (MO-G) model were derived. The MO-G model parameters were established by maximum likelihood approach. The flexibility, efficiency, and behavior of the MO-G model estimators were examined through simulation. The empirical applicability, flexibility and proficiency of the MO-G model was scrutinized by a real-life dataset. The proposed MO-G model provides a better fit when compared to existing models in statistical literature and can serve as an alternative model to those appearing in modeling Poisson processes.