Gompertz-Makeham Distribution Research Articles

Usual stochastic ordering of the sample maxima from dependent distribution‐free random variables

In this paper, we discuss stochastic comparison of the largest order statistics arising from two sets of dependent distribution‐free random variables with respect to multivariate chain majorization, where the dependency structure can be defined by Archimedean copulas. When a distribution‐free model with possibly two parameter vectors has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we obtain the first sample maxima is larger than the second sample maxima with respect to the usual stochastic order, based on certain conditions. Applications of our results for scale proportional reverse hazards model, exponentiated gamma distribution, Gompertz–Makeham distribution, and location‐scale model, are also given. Meanwhile, we provide two numerical examples to illustrate the results established here.

Optimal dividends for regulated insurers with a nonlinear penalty

This study investigates an optimal dividend problem for an insurer with random time regulation and a nonlinear penalty from the payment of dividends. The surplus process of the insurer is a jump–diffusion model and the goal of the insurer is to maximise the expected accumulated discount dividend payment until ruin occurs or regulatory time arrives. The introduction of the external random regulatory time leads that the HJB equation associated with this control problem is a time in-homogeneous parabolic variational inequality. There is no common explicit solution to this variational inequality. We derived explicit solutions for optimal policies when both the regulatory time and the claims follow an exponential distribution. For general cases, this paper provides a Markov chain approximation method for the value function and optimal policies and presents an example to illustrate the algorithm when the random regulatory time follows the Gompertz–Makeham distribution.

Optimal active lifetime investment

Optimal decision-making regarding investments is important. In this study, we examine how an individual aims to find optimal investment policies to overcome a benchmarked opponent during their lifetime. The individual evaluates the trade-off between the achievement of a performance goal and the risk of a shortfall. The dynamic programming method is applied in this study. When the individual's lifetime follows an exponential distribution, the associated Hamilton–Jacobi–Bellman (HJB) equation is an ordinary differential equation, and the explicit optimal policies for several important performance functions are derived. When the lifetime follows a general distribution, the HJB equation is a parabolic partial differential equation, and a Markov chain approximating numerical scheme is presented to estimate the value function. We further illustrate the numerical method when the random lifetime follows the Gompertz–Makeham distribution.

ON GENERALISATION OF GOMPERTZ-MAKEHAM DISTRIBUTION

In this paper an exponentiated generalised Gompertz-Makeham distribution. An exponentiated generalised family was introduced by Codeiro, et. al., which allows greater flexibility in the analysis of data. Some Mathematical and Statistical properties including cumulative distribution function, hazard function and survival function of the distribution are derived. The estimation of model parameters are derived via the maximum likelihood estimate method.

A Study on Properties and Applications of a Lomax Gompertz-Makeham Distribution

The article presents an extension of the Gompertz-Makeham distribution using the Lomax generator of probability distributions. This generalization of the Gompertz-Makeham distribution provides a more skewed and flexible compound model called Lomax Gompertz-Makeham distribution. The paper derives and discusses some Mathematical and Statistical properties of the new distribution. The unknown parameters of the new model are estimated via the method of maximum likelihood estimation. In conclusion, the new distribution is applied to two real life datasets together with two other related models to check its flexibility or performance and the results indicate that the proposed extension is more flexible compared to the other two distributions considered in the paper based on the two datasets used.

An extended Gompertz-Makeham distribution with application to lifetime data

ABSTRACTIn this paper, we propose an extension of the Gompertz-Makeham distribution. This distribution is called the transmuted Gompertz-Makeham (TGM). The new model which can handle bathtub-shaped, increasing, increasing-constant and constant hazard rate functions. This property makes TGM is useful in survival analysis. Various statistical and reliability measures of the model are obtained, including hazard rate function, moments, moment generating function (mgf), quantile function, random number generating, skewness, kurtosis, conditional moments, mean deviations, Bonferroni curve, Lorenz curve, Gini index, mean inactivity time, mean residual lifetime and stochastic ordering; we also obtain the density of the ith order statistic. Estimation of the model parameters is justified by the method of maximum likelihood. An application to real data demonstrates that the TGM distribution can provides a better fit than some other very well known distributions.

Performance Rating of the Exponentiated Generalized Gompertz Makeham Distribution: An Analytical Approach

We developed a five parameter distribution known as the Generalized Exponentiated Gompertz Makeham distribution which is quite flexible and can have a decreasing, increasing and bathtub-shaped failure rate function depending on its parameters making it more effective in modeling survival data and reliability problems. Some comprehensive properties of the new distribution, such as closed-form expressions for the density function, cumulative distribution function, hazard rate function, moment generating function and order Statistics were provided as well as maximum likelihood estimation of the Generalized Exponentiated Gompertz Makeham distribution parameters and at the end, in order to show the distribution flexibility, an application using a real data set was presented.

ON ORDER STATISTICS FROM THE GOMPERTZ–MAKEHAM DISTRIBUTION AND THE LAMBERT W FUNCTION

The aim of this paper is twofold. First, we show that the expected value of the minimum order statistic from the Gompertz-Makeham distribution can be expressed in closed form in terms of the incomplete gamma function. We also give a general formula for the moments of the minimum order statistic in terms of the generalized integro-exponential function. As a consequence, the moments of all order statistics from this probability distribution can be more easily evaluated from the moments of the minimum order statistic. Second, we show that the maximum and minimum order statistics from the Gompertz-Makeham distribution are in the domains of attraction of the Gumbel and Weibull distributions, respectively. Lambert W function plays an important role in solving these problems.

Estimation Methods for the Gompertz–Makeham Distribution Under Progressively Type-I Interval Censoring Scheme

Gompertz–Makeham distribution has been widely used in describing human mortality, establishing actuarial tables and growth models. In some real applications researchers are faced with incomplete data. Employing the efficient estimation method in this situations is very important. In this paper, we propose an EM algorithm-based estimator (EME), moment-based estimator (MME) and mid-point estimator (MPE) for the Gompertz-Makeham distribution under the well-known progressively type-I interval censoring scheme. Then, some comparisons are made of the following estimators: the maximum likelihood estimator (MLE), EME, MME and MPE to choose the best one.

A closed-form expression for the quantile function of the Gompertz–Makeham distribution

In this note, we give a closed-form expression in terms of the Lambert W function for the quantile function of the Gompertz–Makeham distribution. This probability distribution has frequently been used to describe human mortality and to establish actuarial tables. The analytical expression provided for the quantile function is helpful to generate random samples drawn from the Gompertz–Makeham distribution by means of the inverse transform method.

仙台市における木質系解体材発生量予測と地域の特性に関する研究

This study reports on the prediction for the wood construction wastes of Sendai city based on the house ledgers of fixed asset tax rolls. The survival rate of wooden houses has been supposed to follow a Gompertz-Makeham distribution. The estimated data indicates that the survival rate in 1997 was lower than in 2002-2004. The life expectancy for wooden houses in exclusive residential district was estimated to be approximately 50 years, which is roughly two years longer than the average of whole Sendai city. The prediction for wooden construction have been classified according to destruction rate and new-building rate.

Modeling human mortality using mixtures of bathtub shaped failure distributions

Aging and mortality is usually modeled by the Gompertz–Makeham distribution, where the mortality rate accelerates with age in adult humans. The resulting parameters are interpreted as the frailty and decrease in vitality with age. This fits well to life data from ‘westernized’ societies, where the data are accurate, of high resolution, and show the effects of high quality post-natal care. We show, however, that when the data are of lower resolution, and contain considerable structure in the infant mortality, the fit can be poor. Moreover, the Gompertz–Makeham distribution is consistent with neither the force of natural selection, nor the recently identified ‘late life mortality deceleration’. Although actuarial models such as the Heligman–Pollard distribution can, in theory, achieve an improved fit, the lack of a closed form for the survival function makes fitting extremely arduous, and the biological interpretation can be lacking. We show, that a mixture, assigning mortality to exogenous or endogenous causes, using the reduced additive and flexible Weibull distributions, models well human mortality over the entire life span. The components of the mixture are asymptotically consistent with the reliability and biological theories of aging. The relative simplicity of the mixture distribution makes feasible a technique where the curvature functions of the corresponding survival and hazard rate functions are used to identify the beginning and the end of various life phases, such as infant mortality, the end of the force of natural selection, and late life mortality deceleration. We illustrate our results with a comparative analysis of Canadian and Indonesian mortality data.

An improved failure-free time estimation method

In this paper a simple order-statistics-based iterative procedure for estimating the failure-free time is put forward. The method is applicable when the population lifetime follows the Weibull, C-Weibull or Gompertz–Makeham distribution. Copyright © 1999 John Wiley & Sons, Ltd.

Bahadur efficiency and local optimality of a test for the exponential distribution based on the gini statistic

The sample scale-free Gini index is known to be a powerful test of exponentiality against a broad class of alternatives. To understand better the efficiency properties of this test we calculate its Bahadur efficiency for most commonly used parametric alternatives to the exponential distribution. Using variational arguments and the Bahadur-Raghavachari inequality for exact slopes we find the conditions of local Bahadur optimality of the Gini test. It turns out that this property surprisingly holds for a family of alternative distributions including the well-known Gompertz-Makeham distribution.