Environmental stochasticity of gompertz model with time delay

We consider the environmental stochasticity of the Gompertz model with time delay when the parameters are assumed to be described by correlated Gaussian white noise processes. The delay kernel is assumed to be exponentially decaying and the Fokker-Planck equation is obtained in both the Stratonovich and Ito calculi. The exact expressions for the first and second order moments of the logarithm of population size are evaluated and the stability of the system in mean as well as In mean-square is discussed.     

中国服务贸易发展的实证研究——基于产品生命周期理论视角

介绍了改革开放以来中国服务贸易的发展情况,从产品生命周期视角,对中国服务贸易的各项竞争性指标的分析与已有研究相比照,结论较为相似,即发现中国服务贸易的竞争力、结构以及创新性等方面都存在一定问题。运用非线性最小二乘法,龚珀兹曲线对中国服务贸易周期趋势拟合的结果表明：目前服务贸易存在的种种竞争力较低的表象,均是其在我国成长生命周期中“发展阶段前期”的特征;这种服务贸易发展境况背后的阶段性本质,表明中国服务贸易尚存在着很大的发展与调整空间;同时也提出了中国服务贸易的发展方向,即在整体结构以及创新密度上要有所作为。     

改进的Gompertz模型参数估计方法比较研究

讨论改进的Gompertz模型两种参数估计方法:三和法和非线性最小二乘估计法,并通过蒙特卡洛实验比较两种估计方法的精度和收敛率,得出非线性最小二乘估计法在估计精度和估计的成功率两方面都优于三和法的结论;利用Gompertz曲线拟合中国电影票房数据并对其未来发展作出预测:中国电影票房最终可以在2025年左右到达饱和状态,饱和状态总规模大约为1 676.5亿元。     

Maximum likelihood estimation for mixtures of two gompertz distributions when censoring occurs

In estimating the proportion ‘cured’ after adjuvant treatment, a population of cancer patients can be assumed to be a mixture of two Gompertz subpopulations, those who will die of other causes with no evidence of disease relapse and those who will die of their primary cancer. Estimates of the parameters of the component dying of other causes can be obtained from census data, whereas maximum likelihood estimates for the proportion cured and for the parameters of the component of patients dying of cancer can be obtained from follow-up data.

This paper examines, through simulation of follow-up data, the feasibility of maximum likelihood estimation of a mixture of two Gompertz distributions when censoring occurs. Means, variances and mean square error of the maximum likelihood estimates and the estimated asymptotic variance-covariance matrix is obtained from the simulated samples. The relationship of these variances with sample size, proportion censored, mixing proportion and population parameters are considered.

Moderate sample size typical of cooperative trials yield clinically acceptable estimates. Both increasing sample size and decreasing proportion of censored data decreases variance and covariance of the unknown parameters. Useful results can be obtained with data which are as much as 50% censored. Moreover, if the sample size is sufficiently large, survival data which are as much as 70% censored can yield satisfactory results.     

Latent cure rate model under repair system and threshold effect

In this paper, we formulate a simple latent cure rate model with repair mechanism for a cell exposed to radiation. This latent approach is a flexible alternative to the models proposed by Klebanov et al. [A stochastic model of radiation carcinogenesis: latent time distributions and their properties. Math Biosci. 1993;18:51–75], Kim et al. [A new threshold regression model for survival data with a cure fraction. Lifetime Data Anal. 2011;17:101–122], and is along the lines of the destructive cure rate model formulated recently by Rodrigues et al. [Destructive weighted Poisson cure rate model. Lifetime Data Anal. 2011b;17:333–346]. A new version of the modified Gompertz model and the promotion cure rate model that takes into account the first passage time of reaching the critical point are discussed, and the estimation of tumor size at detection is then addressed from the Bayesian viewpoint. In addition, a simulation study and an application to real data set illustrate the usefulness of the proposed cure rate model.     

An insight into technology diffusion of tractor through Weibull growth model

Most of the technological innovation diffusion follows an S-shaped curve. But, in many practical situations this may not hold true. To this end, Weibull model was proposed to capture the diffusion of new technological innovation, which does not follow any specific pattern. Nonlinear growth models play a very important role in getting an insight into the underlying mechanism. These models are generally ‘mechanistic’ as the parameters have meaningful interpretation. The nonlinear method of estimation of parameters of Weibull model fails to converge. Taking this problem into consideration, we propose the use of a powerful technique of genetic algorithm for parameter estimation. The methodology is also validated by simulation study to check whether parameter estimates are closer to the real value. For illustration purpose, we model the tractor density time-series data of India as a whole and some major states of India. It is seen that fitted Weibull model is able to capture the technology diffusion process in a reasonable manner. Further, comparison is also made with Logistic and Gompertz model; and is found to perform better for the data sets under consideration.     

A maximum likelihood estimator for left-truncated lifetimes based on probabilistic prior information about time of occurrence

In forestry, many processes of interest are binary and they can be modeled using lifetime analysis. However, available data are often incomplete, being interval- and right-censored as well as left-truncated, which may lead to biased parameter estimates. While censoring can be easily considered in lifetime analysis, left truncation is more complicated when individual age at selection is unknown. In this study, we designed and tested a maximum likelihood estimator that deals with left truncation by taking advantage of prior knowledge about the time when the individuals enter the experiment. Whenever a model is available for predicting the time of selection, the distribution of the delayed entries can be obtained using Bayes' theorem. It is then possible to marginalize the likelihood function over the distribution of the delayed entries in the experiment to assess the joint distribution of time of selection and time to event. This estimator was tested with continuous and discrete Gompertz-distributed lifetimes. It was then compared with two other estimators: a standard one in which left truncation was not considered and a second estimator that implemented an analytical correction. Our new estimator yielded unbiased parameter estimates with empirical coverage of confidence intervals close to their nominal value. The standard estimator leaded to an overestimation of the long-term probability of survival.     

Test for comparing complete expectations of life of two groups

Abstract

Complete expectation of life of an individual gives an intuitive and interesting perspective on the ageing process and is an important concept in the insurance sector for determination of premium. We propose a new test for testing equality of complete expectations of life of two groups/populations. Power of the new test is calculated through simulations and compared with the power of the tests given by Berger, Boos, and Guess (1988 Berger, R. L., D. D. Boos, and F. M. Guess. 1988. Tests and confidence sets for comparing two mean residual life functions. Biometrics 44 (1):103–15.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Aly (1997 Aly, E. E. A. A. 1997. Nonparametric tests for comparing two mean residual life functions. Lifetime Data Analysis 3 (4):353–66.[Crossref], [PubMed] , [Google Scholar]). It is observed that the proposed test statistic is more powerful than the competing tests for the cases considered in this paper. A real life illustration is included.     

Covariate effect on the life expectancy and percentile residual life functions under the proportional hazards and the accelerated life models

The present paper is concerned with statistical models for the dependence of survival time or time to occurrence of an event, such as time to tumor, on a vector X of covariates or prognostic variables such as age, sex, blood pressure, length of exposure to a toxic material, etc., measured on a group of individuals in biomedical investigations. It is assumed that the covariates influence the distribution of time to tumor only through a linear predictor μ =β^′X.

The object of our paper is to investigate the effect due to the covariates on the Life Expectancy and the Percentile Residual Life (PRL) function of a family of organisms under the proportional hazards and the accelerated life models. The key result is that the families of survival distributions under these models have the 'setting the clock back to zero' property if the family of baseline survival distributions does. This property is a generalization of the lack of memory property of the exponential distribution. Simple examples of the members of this family are the linear hazard exponential, Pareto and Gompertz life distributions.

As a simple application of the main results obtained in the present paper, we have considered a stochastic survival model recently proposed by Chiang and Conforti (1989) for the time-to-tumor distribution in the context of a large-scale serial sacrifice experiment by the National Center of Toxicological Research (NCTR). This involves some mice that were fed 2-AAF from infancy and those that developed bladder and/or liver neoplasms, see Farmer et al (1980). It is shown that their stochastic model for tumor incidence intensity at time t leads to a family of survival models that has the setting the clock back to zero property. The survival functions and the effect of the vector X of covariates on the PRL and the tumor-free life expectancies are evaluated for the proportional hazards and accelerated life models.     

Age patterns of mortality for older women: An analysis using the age‐specific rate of mortality change with age

The age‐specific rate of mortality change with age, defined by k(x) = d Inμ(x)/dx, where μ(x) is the age‐specific death rate at exact age x, is estimated for middle and old ages in ten selected populations that are considered to have relatively accurate age data. For females in each of the study populations, k(x) follows a bell‐shaped curve that usually peaks around age 75. In some of the populations, the age pattern of k(x) for males is confounded with substantial cohort variations, which seem to reflect long‐term impacts of their World War I experiences.

Among the mathematical models proposed by Gompertz, Makeham, Perks and Beard, only the Perks model is consistent with the bell‐shaped pattern of k(x). It is shown that, if the risk of death for every individual follows the Makeham equation and if the individual frailty is gamma‐distributed, then the age‐specific death rate follows the Perks equation.