Gompertz Distribution Research Articles

Health inequalities in cause-specific mortality in Costa Rica: a population-based cohort study.

To analyze health inequalities in cause-specific mortality in Costa Rica from 2010 to 2018, observing the main causes for inequality in the country. The National Electoral Rolls were used to follow-up all Costa Rican adults aged 20 years or older from 2010 to 2018 (n = 2,739,733) in an ecological study. A parametric survival model based on the Gompertz distribution was performed and the event death was classified according to the ICD-10. After adjustment for urbanicity, the poorest districts had a higher mortality than the wealthier districts for most causes of death except neoplasms, mental and behavioral disorders, and diseases of the nervous system. Urban districts showed significantly higher mortality than mixed and rural districts after adjustment for wealth for most causes except mental and behavioral disorders, diseases of the nervous system, and diseases of the respiratory system. Differences according to wealth were more frequent in women than men, whereas differences according to urbanicity were more frequent in men than in women. The study's findings were consistent, but not fully similar, to the international literature.

A New Bivariate Odd Generalized Exponential Gompertz Distribution

The objective of this study was to present a novel bivariate distribution, which we denoted as the bivariate odd generalized exponential gompertz(BOGE-G) distribution. Other well-known models included in this one include the gompertz, generalized exponential, odd generalized exponential, and odd generalized exponential gompertz distribution. The model introduced here is of Marshall-Olkin type [16]. The marginals of the new bivariate distribution have odd generalized exponential gompertz distribution which proposed by[7]. Closed forms exist for both the joint probability density function and the joint cumulative distribution function. The bivariate moment generating function, marginal moment generating function, conditional distribution, joint reliability function, marginal hazard rate function, joint mean waiting time, and joint reversed hazard rate function are some of the properties of this distribution that have been discussed. The maximum likelihood approach is used to estimate the model parameters. To demonstrate empirically the significance and adaptability of the new model in fitting and evaluating real lifespan data, two sets of real data are studied using the new bivariate distribution. Using the software Mathcad, a simulation research was conducted to evaluate the bias and mean square error (MSE) characteristics of MLE. We found that the bias and MSE decrease as the sample size increases.

Cure rate and statistical inference for the Marshall-Olkin generalized defective Gompertz distribution under censored survival data

Cure fraction modeling usually requires the addition of an extra parameter. Defective modeling, alternatively, allows cure fraction modeling with less complexity. A new extension of the defective version of Gompertz distribution is studied in this paper. This distribution is compared to other well-founded distributions and its distinguished performance to fit censored survival data with and without a cure fraction is presented. A computational study is carried out to highlight the correlation between the censorship level and the proportion of long-term survivors. Interesting mathematical properties are established. The practical relevance of the model is illustrated by applying various real and simulated datasets.

Survival Function Estimation for Fuzzy Gompertz Distribution with neutrosophic data

A relatively recent area of research known as neutrosophic statistics deals with data that are ambiguous, indeterminate, and inconsistent. By embracing the idea of neutrosophy, which denotes the existence of three components in a statement: truth, falsity, and indeterminacy, it broadens the application of classical statistics. One of the significant offshoots of statistics is life-time data analysis. Traditional statistical methods only account for variation within the data and calculate lifetime observations as accurate numbers. Actually, there are two different kinds of uncertainty in data: fluctuation between observations and fuzziness. As a result, analysis techniques that solely employ precise lifetime data and ignore fuzziness use incomplete information and produce false results. This paper sought to generalize hazard rates, survival functions, and parameter estimates for fuzzy Gompertz Distribution. Simulation studies are implemented to examine the performance of the fuzzy Gompertz Distribution. The results show that the fuzzy Gompertz Distribution has better flexibility in handling over the standard Gompertz Distribution.

Exploring the Properties, Simulation, and Applications of the Odd Burr XII Gompertz Distribution

Exploring the Properties, Simulation, and Applications of the Odd Burr XII Gompertz Distribution

The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference

Abstract In this article, we propose a Gompertz-two-parameter-Lindley distribution by mixing the frailty parameter of the Gompertz distribution with a two-parameter Lindley distribution. The structural properties of the model, such as shape properties, cumulative distribution, quantile functions, moment, moment generating function, failure rate function, mean residual function, and stochastic orders, were derived. Moreover, the unknown parameters are estimated by the profile log likelihood algorithm, and their performance is examined by simulation studies. Finally, a real data example is used to demonstrate the application of the proposed model.

Statistical Inference of Stress-Strength Reliability of Gompertz Distribution under Type II Censoring

This paper develops the problem of estimating stress-strength reliability for Gompertz lifetime distribution. First, the maximum likelihood estimation (MLE) and exact and asymptotic confidence intervals for stress-strength reliability are obtained. Then, Bayes estimators under informative and noninformative prior distributions are obtained by using Lindley approximation, Monte Carlo integration, and MCMC. Bayesian credible intervals are constructed under these prior distributions. Also, simulation studies are used to illustrate these inference methods. Finally, a real dataset is analyzed to show the implementation of the proposed methodologies.

Estimating the Parametric Functions and Reliability Measures for Exponentiated Lifetime Distributions Family

The paper aims are to extend the theory to estimate the parameters of the exponentiated lifetime distribution. For it, in this paper, we derived the probability density function, cumulative density, reliability function and the stress-strength parameter of the distribution. To estimate the parameters of such distribution, we considered the maximum likelihood and uniformly minimum variance unbiased methods. The validity of the proposed work has been conducted over the simulation study of both estimation methods under the special sub model as exponentiated inverse Gompertz distribution. Finally, some real data has been taken to conduct an analysis and to discuss the effectiveness and advantages of the established work by comparing with other methods.

PP47 Modelling Non-small Cell Lung Cancer Treatment: Predicted and Observed Impact Of Immunotherapy In The Netherlands

IntroductionPatients treated with immunotherapy are divided into two subgroups: (i) long-term survivors (LTS) and (ii) moderate survivors. Nevertheless, clinical trials (RCTs) report only average treatment effects such as hazard rate (HRs). Health economic-models often only input average treatment effects, even though it has been shown that accounting for the LTS subgroup is crucial for accurate projection of long-term survival under immunotherapy. We investigated the incorporation of a statistical mixture cure model (MCM) in a health-economic model for lung cancer as a way to account for LTS while incorporating reported average RCT-based treatment effects.MethodsWe developed a microsimulation model describing disease progression under three treatment lines in advanced lung cancer using Dutch real-world data of chemotherapies treated patients. Here we focus on first-line treatment, for which we used gompertz distribution to simulate time-to-progression. To simulate the impact of immunotherapy, we adjusted base-model assuming MCM for first-line treatment, where the LTS subgroup was not at risk to progress, but instead die from background mortality. The subgroup of moderate survivors on the other hand are at risk to progress with adjusted progression-free HR (PF-HR). We simulated the model with size of LTS (prop_LTS) ranging from 14-34 percent (keynote-001 five-year overall survival [OS], 95% confidence interval) while fixing average RCT PF-HR at 0.5. Model predictions under the different prop_LTS were compared to real-world Dutch OS as well as the long-term RCT five-year OS.ResultsWith respect to observed short-term survival outcomes, model predictions were insensitive to assumptions regarding the size of the LTS subgroup. However, to match the five-year RCT OS rate reported (32%), the prop_LTS had to be equal to 34 percent. Under this latter setting for the prop_LTS, the progression HR in the subgroup of moderate survivors was calibrated to be 1.1.ConclusionsThe use of a mixture cure model improves long-term model-based projections with the implicit assumption that moderate survivors have little or no treatment benefit.

Gompertz zero‐inflated cure rate regression models applied to credit risk data

AbstractIn the financial market, it is important to consider that there is a proportion of customers that have settled their debt in time zero, immediately recovering their ability to pay. In this context, in this paper, we propose a survival analysis methodology that allows the insertion of times equal to zero in scenarios where credit risk is observed. The proposed model addresses the survival analysis model of the zero‐inflated cure rate which incorporates the heterogeneity of three subgroups (individuals having events in the initial time, and individuals not susceptible and susceptible to the event). In our proposal, all available survival data of customers are modeled considering that the number of competitive causes follows a Poisson distribution and the baseline risk function follows a Gompertz distribution. The model parameter estimation is obtained by the maximum likelihood estimation procedure and simulation studies are conducted to evaluate the estimators' performance. The studied methodology will be applied to a credit database provided by a financial institution in Brazil.

Statistical Inference for Competing Risks Model with Adaptive Progressively Type-II Censored Gompertz Life Data Using Industrial and Medical Applications

This study uses the adaptive Type-II progressively censored competing risks model to estimate the unknown parameters and the survival function of the Gompertz distribution. Where the lifetime for each failure is considered independent, and each follows a unique Gompertz distribution with different shape parameters. First, the Newton-Raphson method is used to derive the maximum likelihood estimators (MLEs), and the existence and uniqueness of the estimators are also demonstrated. We used the stochastic expectation maximization (SEM) method to construct MLEs for unknown parameters, which simplified and facilitated computation. Based on the asymptotic normality of the MLEs and SEM methods, we create the corresponding confidence intervals for unknown parameters, and the delta approach is utilized to obtain the interval estimation of the reliability function. Additionally, using two bootstrap techniques, the approximative interval estimators for all unknowns are created. Furthermore, we computed the Bayes estimates of unknown parameters as well as the survival function using the Markov chain Monte Carlo (MCMC) method in the presence of square error and LINEX loss functions. Finally, we look into two real data sets and create a simulation study to evaluate the efficacy of the established approaches.

The effect of exponentiating generalized models

A common practice in statistical distribution theory involves exponentiating existing distribution functions to include some extra parameters that increase the flexibility of the distribution. This paper examines the effect of exponentiating some generalized models by adding three extra parameters to their probability distribution. Particularly, a new generalized distribution that is a member of the inverted Kumaraswamy family of distributions is considered. Afterward, three additional parameters are applied to enhance this generalized distribution, which results in a novel distribution referred to as the new generalized exponentiated generalized inverted Kumaraswamy Gompertz distribution (NGEGIKGD). Some of the statistical and mathematical characteristics of this distribution were derived. Additionally, parametric estimation of the new distribution parameters was considered using the maximum likelihood method. Several Monte Carlo simulation studies were conducted in order to explore the usefulness of the estimation method. The proposed distribution is then compared with its corresponding sub-models in order to assess the effects of the exponentiation. Further evaluation of the distribution is accomplished by comparing it to some relative distributions. Specifically, three real-world datasets were analyzed to demonstrate the potentiality of the suggested new modeling approach in enhancing the goodness of fit of the generalized models. Results indicate that exponentiating a generalized model significantly improves its fit compared to the non-exponentiating distributions.

Asymptotic analysis of the SIR model and the Gompertz distribution

The SIR (Susceptible–Infected–Removed) is one of the simplest models for epidemic outbreaks. The present paper derives a novel, simple, analytical asymptotic solution for the I-variable, which is valid on the entire real line. Connections with the Gompertz and Gumbel distributions are also demonstrated. The approach is applied to the ongoing coronavirus disease 2019 (COVID-19) pandemic in four European countries — Belgium, Italy, Sweden, and Bulgaria. The reported raw incidence data from the outbreaks in 2020–2021 have been fitted using constrained least squares. It is demonstrated that the asymptotic solution can be used successfully for parametric estimation either in stand-alone mode or as a preliminary step in the parametric estimation using numerical inversion of the exact parametric solution.

Statistical inference for Gompertz distribution under adaptive type-II progressive hybrid censoring

Gompertz distribution is a significant and commonly used lifetime distribution, which plays an important role in reliability engineering. In this paper, we study the statistical inference of Gompertz distribution based on adaptive Type-II hybrid progressive censored schemes. From the perspective of frequentist, we derive the point estimations through the method of maximum likelihood estimation (MLE) and the existence of MLE is proved. Besides MLE, we propose the stochastic EM algorithm to reduce complexity and simplify computing. We also apply the method of Bootstraps (Bootstrap-p and Bootstrap-t) to construct confidence intervals. From Bayesian aspect, the Bayes estimates of the unknown parameters are evaluated by applying the MCMC method, the average length and coverage rate of credible intervals are also carried out. The Bayes inference is based on the squared error loss function and LINEX loss function. Furthermore, a numerical simulation is conducted to assess the performance of the proposed methods. Finally, a real-life example is considered to illustrate the application and development of the inference methods. In summary, the Bayesian method seems to perform the best among all approaches, while other approaches also present different advantages.

Choosing the best method for estimating the survival function of inverse Gompertz distribution by using Integral mean squares error (IMSE)

In this research, we study the inverse Gompertz distribution (IG) and estimate the survival function of the distribution, and the survival function was evaluated using three methods (the Maximum likelihood, least squares, and percentiles estimators) and choosing the best method estimation, as it was found that the best method for estimating the survival function is the squares-least method because it has the lowest IMSE and for all sample sizes. 

Choosing the best method for estimating the survival function of inverse Gompertz distribution by using Integral mean squares error (IMSE)

In this research , we study the inverse Gompertz distribution (IG) and estimate the survival function of the distribution , and the survival function was evaluated using three methods (the Maximum likelihood, least squares, and percentiles estimators) and choosing the best method estimation ,as it was found that the best method for estimating the survival function is the squares-least method because it has the lowest IMSE and for all sample sizes

Mixture model for dependent competing risks data and application for diabetic retinopathy treatment

This paper proposes a flexible mixture modeling approach to analyze competing risks data with multiple causes of failure under a progressive first-failure censoring scheme where the causes may not be independent. The failure time distributions corresponding to different causes of failure are assumed to belong to the flexible proportional hazard family of distributions, some important members of which are Weibull, Burr XII and Gompertz distributions. We address the problem here from a classical viewpoint. The point estimates of the parameters can be easily obtained by employing the maximum likelihood principle. For interval estimation purposes, we resort to using the asymptotic confidence intervals. An extensive simulation study is carried out to validate our proposed model. Finally, a diabetic retinopathy treatment dataset is used as an application to study the effectiveness of our proposed model assuming some important members of the proportional hazard family.

Comparison of Entropy Measures for Gompertz and Truncated Gompertz Distribution

Comparison of Entropy Measures for Gompertz and Truncated Gompertz Distribution

Statistical Analysis of Joint Progressive Censoring Data from Gompertz Distribution

Statistical Analysis of Joint Progressive Censoring Data from Gompertz Distribution

Extended Gompertz Distribution: Properties and Estimation under Complete and Censored Data

Extended Gompertz Distribution: Properties and Estimation under Complete and Censored Data