Gompertz Law Research Articles

A calendar year mortality model in continuous time

AbstractThis article proposes a continuous time mortality model based on calendar years. Mortality rates belong to a mean-reverting random field indexed by time and age. In order to explain the improvement of life expectancies, the reversion level of mortality rates is the product of a deterministic function of age and of a decreasing jump-diffusion process driving the evolution of longevity. We provide a general closed-form expression for survival probabilities and develop it when the mean reversion level of mortality rates is proportional to a Gompertz–Makeham law. We develop an econometric estimation method and validate the model on the Belgian population.

Annual Premium Determination for Joint Life Insurance with De Moivre and Gompertz’s Mortality Laws

This research discusses how to determine the annual premium for joint life status. The death rate (mortality rate) is one of the factors that can be considered when calculating the premium. The quantitative method used in this study was to estimate the parameters of the De Moivre and Gompertz’s Mortality laws using secondary data from the 2011 Indonesian Mortality Table (TMI III) for men and women. This calculation generates a formula for calculating the annual premium for joint life insurance based on De Moivre and Gompertz mortality law. Starting with estimating the parameters of Gompertz's mortality law for the Indonesian Mortality Table in 2011 using the maximum likelihood estimation method, then calculating the combined life probability, death benefit APV, continuous life annuity APV, and annual premium for joint life insurance. The value of the annual premium on joint life insurance with the mortality law of De Moivre and Gompertz for a simulated term life insurance n = 10 years with age x (husband) 28 years and y (wife) 25 years, the death benefit (R) is Rp. 50,000,000; and the interest rate is 3.50 percent with the Indonesian Mortality Table in 2011. According to the calculations, the annual premium value of joint life insurance based on Gompertz's mortality law is greater than De Moivre's mortality law.Keywords: Join Life Insurance, Mortality Laws, Annual Premium.

A Hermite spline approach for modelling population mortality

AbstractOne complication in mortality modelling is capturing the impact of risk factors that contribute to mortality differentials between different populations. Evidence has suggested that mortality differentials tend to diminish over age. Classical methods such as the Gompertz law attempt to capture mortality patterns over age using intercept and slope parameters, possibly causing an unjustified mortality crossover at advanced ages when applied independently to different populations. In recent research, Richards (Scandinavian Actuarial Journal2020(2), 110–127) proposed a Hermite spline (HS) model that describes the age pattern of mortality differentials using one parameter and circumvents an unreasonable crossover by default. The original HS model was applied to pension data at individual level in the age dimension only. This paper extends the method to model population mortality in both age and period dimensions. Our results indicate that in addition to possessing desirable fitting properties, the HS approach can produce accurate mortality forecasts, compared with the Gompertz and P-splines models.

Using grey Gompertz model to explore the carbon emission and its peak in 16 provinces of China

As the largest carbon emitter in the world, China's carbon emissions and the time of carbon peak have always been a hot topic at home and abroad. However, in previous studies, the predicted period of carbon emissions is relatively short. In order to obtain the predicted data of China’s carbon emission and the specific time of carbon peak, the carbon emissions in 30 provinces of China are predicted and analyzed. For 16 provinces with increasing carbon emissions, a novel grey model is proposed to predict the limit of growth data. This model improves the structure of the traditional grey prediction model based on the Gompertz's law and the differential information principle. Its predicted period is longer than that of the traditional grey model. The predicted trend is more in line with the law of natural development. Moreover, this model can predict the limits of a variety of growth data, such as the emissions of various air pollutants. After the prediction, the carbon emissions in these 16 provinces from 2020 to 2032 are obtained. According to the forecast results, eight of the 16 provinces will reach the carbon peak by 2030, namely Jiangsu, Guangdong, Shanxi, Liaoning, Hubei, Guizhou, Guangxi and Jiangxi. The five provinces namely Shandong, Hebei, Anhui, Shaanxi and Fujian will reach the carbon peak in 2031 or 2032. The three provinces namely Inner Mongolia, Xinjiang and Ningxia will not reach their carbon peak by 2032. Based on the above forecasts and analysis, it can be inferred that China will reach the carbon peak in 2032, with emissions of 11.8 billion tons. Scientific predicted data will play a guiding role in carbon emission control. At the end of the paper, the current situations of carbon emissions in 30 provinces are analyzed and suggestions on carbon emission control are put forward.

On the Gompertz–Makeham law: A useful mortality model to deal with human mortality

The Gompertz–Makeham model was introduced as an extension of the Gompertz model in the second half of the 19th century by the British actuary William M. Makeham. Since then, this model has been successfully used in biology, actuarial science, and demography to describe mortality patterns in numerous species (including humans), determine policies in insurance, establish actuarial tables and growth models. In this paper, we derive some structural properties of the Gompertz–Makeham model in statistics, demography, and actuarial sciences, and present some other ones already introduced in the literature. All structural properties we provide are expressed in closed-form, which eliminates the need to evaluate them with numerical integration directly. In addition, we study the estimation of the Gompertz–Makeham model parameters through the discrete Poisson and Bell distributions. In particular, we verify that the recently introduced discrete Bell distribution can be an interesting alternative to the Poisson distribution, mainly because it is suitable to deal with over dispersion, unlike the Poisson distribution. On the basis of real mortality datasets, we compute the remaining life expectancy for several countries and verify that the Gompertz–Makeham model, especially under the Bell distribution, provides proper results to deal with human mortality in practice.

Mortality modeling and regression with matrix distributions

In this paper we investigate the flexibility of matrix distributions for the modeling of mortality. Starting from a simple Gompertz law, we show how the introduction of matrix-valued parameters via inhomogeneous phase-type distributions can lead to reasonably accurate and relatively parsimonious models for mortality curves across the entire lifespan. A particular feature of the proposed model framework is that it allows for a more direct interpretation of the implied underlying aging process than some previous approaches. Subsequently, towards applications of the approach for multi-population mortality modeling, we introduce regression via the concept of proportional intensities, which are more flexible than proportional hazard models, and we show that the two classes are asymptotically equivalent. We illustrate how the model parameters can be estimated from data by providing an adapted EM algorithm for which the likelihood increases at each iteration. The practical feasibility and competitiveness of the proposed approach, including the right-censored case, are illustrated by several sets of mortality and survival data.

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.

Does Social Distancing Matter for Infectious Disease Propagation? An SEIR Model and Gompertz Law Based Cellular Automaton.

In this paper, we present stochastic synchronous cellular automaton defined on a square lattice. The automaton rules are based on the SEIR (susceptible → exposed → infected → recovered) model with probabilistic parameters gathered from real-world data on human mortality and the characteristics of the SARS-CoV-2 disease. With computer simulations, we show the influence of the radius of the neighborhood on the number of infected and deceased agents in the artificial population. The increase in the radius of the neighborhood favors the spread of the pandemic. However, for a large range of interactions of exposed agents (who neither have symptoms of the disease nor have been diagnosed by appropriate tests), even isolation of infected agents cannot prevent successful disease propagation. This supports aggressive testing against disease as one of the useful strategies to prevent large peaks of infection in the spread of SARS-CoV-2-like diseases.

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.

Retracted] Classical and Bayesian Inference of Marshall‐Olkin Extended Gompertz Makeham Model with Modeling of Physics Data

The purpose of this study is to present the Marshall‐ Olkin extended Gompertz Makeham (MOEGM) lifetime distribution, which has four parameters. As a result, we will describe some of the structural elements that are introduced for this model. The maximum likelihood approach is used to estimate the model parameters, and it is well known that likelihood estimators for unknown parameters are not always available. As a result, we examine the prior distributions, which allow for prior dependence among the components of the parameter vector, as well as the Bayesian estimators derived with respect to the squared error loss function. A Monte Carlo simulation research is carried out to examine the performance of the likelihood estimators and the Bayesian technique. Finally, we demonstrate the significance of the new model. And to conclude, we illustrate the importance of the new model by exploring some of the empirical applications of physics to show it’s flexibility and potentiality of a new model.

A confirmation of the expected interval: Ab initio estimation technique of parsimonious Gompertz mortality parameters

The Gompertz law states a functional relationship on exponential scale between instantaneous intensity and age. The objective is to first estimate the model parameters by using mortality data and then confirm the interval of validity for the estimated parameters. The parsimonious model is implicitly expressed in terms of age and level of mortality while the force of mortality is the dependent variable. Current contributions in actuarial literature have made it tractable to obtain life span from the actuarial point of view, making the life table invaluable analytical tool for insurers. Mortality functions which have been developed recently possess sophisticated actuarial techniques with many parameters hence they are very complex to estimate numerically making it difficult to fit to mortality data. In order to overcome this problem, we need to employ numerical algebraic method to estimate the appropriate values of model parameters and which may enable us fit the function to mortality data. In this paper, the direct algebraic method offers simpler perspective of approximating mortality parameter and was decomposed into systems of algebraic equations. We observed that mortality over all ages for males is lower than that of females while the initial mortality for male is higher than that of female. The R-language software was employed in the computation. In view of actuarial benchmarks, our results confirm that the values of and for both males and females lie within the expected interval. (see PDF for further information).

Frequent, quantitative bone planar scintigraphy for determination of bone anabolism in growing mice.

BackgroundTo provide insight into bone turnover, quantitative measurements of bone remodeling are required. Radionuclide studies are widely used in clinical care, but have been rarely used in the exploration of the bone in preclinical studies. We describe a bone planar scintigraphy method for frequent assessment of bone activity in mice across the growing period. Since repeated venous radiotracer injections are hardly feasible in mice, we investigated the subcutaneous route.MethodsRepeated 99mTc-hydroxymethylene diphosphonate (HMDP) tracer bone planar scintigraphy studies of the knee region and µCT to measure femur growth rate were performed in eight mice between week 6 and week 27 of life, i.e., during their growth period. Three independent investigators assessed the regions of interest (ROI). An index was calculated based on the counts in knees ROI (normalized by pixels and seconds), corrected for the activity administered, the decay between administration and imaging, and individual weights.ResultsA total of 93 scintigraphy studies and 85 µCT were performed. Repeated subcutaneous tracer injections were well tolerated and allowed for adequate radionuclide studies. Mean scintigraphic indexes in the knees ROI decreased from 87.4 ± 2.6 × 10−6 counts s−1 pixel−1 MBq−1 g−1 at week 6 to 15.0 ± 3.3 × 10−6 counts s−1 pixel−1 MBq−1 g−1 at week 27. The time constant of the fitted exponential decay was equal to 23.5 days. As control mean femur length assessed by µCT increased from 12.2 ± 0.8 mm at week 6 to 15.8 ± 0.2 mm at week 22. The time constant of the fitted Gompertz law was equal to 26.7 days. A correlation index of −0.97 was found between femur growth and decrease of bone tracer activity count between week 6 and 24.ConclusionThis methodological study demonstrates the potential of repeated bone planar scintigraphy in growing mice, with subcutaneous route for tracer administration, for quantitative assessment of bone remodeling.

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.

A novel method for carbon emission forecasting based on Gompertz's law and fractional grey model: Evidence from American industrial sector

With the manufacturing reshoring to the US, increasing attention are focus on its energy consumption and environmental effects and accurate prediction of carbon emissions is vital to controlling growth from the source. Considering the slowing growth in carbon emissions with the Gompertz's law, this paper establishes a Gompertz differential equation. According to the differential information principle and fractional accumulation operator, this differential equation is transformed into a fractional accumulation grey Gompertz model. Furthermore, the chaotic whale optimization algorithm is used to optimize the order of accumulation generation and the grey background value in the proposed model. Then the Gompertz's datasets and six validation cases about carbon emissions are used to show that the proposed model demonstrates better accuracy in all cases and efficiency in the carbon emissions forecasting with several existing models. Three case studies indicate that the proposed model can fit the trend of American industrial carbon emissions better. The model results also reveal the recent policy changes have promoted the uptrend of the industrial and the total carbon emissions in the U.S. The future forecasting suggests that U.S. carbon emission is estimated to be 17.01% (in total emissions) or 17.89% (in industrial emission) percent below 2005 levels by 2025 under current policies, falling short of its commitment submitted to the United Nations Framework Convention on Climate Change.

Non-Homogeneous Tumor Growth and Its Implications for Radiotherapy: A Phenomenological Approach.

Tumor regrowth and heterogeneity are important clinical parameters during radiotherapy, and the probability of treatment benefit critically depends on the tumor progression pattern in the interval between the fractional irradiation treatments. We propose an analytic, easy-to-use method to take into account clonal subpopulations with different specific growth rates and radiation resistances. The different strain regrowth effects, as described by Gompertz law, require a dose-boost to reproduce the survival probability of the corresponding homogeneous system and for uniform irradiation. However, the estimate of the survival fraction for a tumor with a hypoxic subpopulation is more reliable when there is a slow specific regrowth rate and when the dependence on the oxygen enhancement ratio of radiotherapy is consistently taken into account. The approach is discussed for non-linear two-population dynamics for breast cancer and can be easily generalized to a larger number of components and different tumor phenotypes.

A joint explanation of infant and old age mortality.

Infant deaths and old age deaths are very different. The former are mostly due to severe congenital malformations of one or a small number of specific organs. On the contrary, old age deaths are largely the outcome of a long process of deterioration which starts in the 20s and affects almost all organs. In terms of age-specific death rates, there is also a clear distinction: the infant death rate falls off with age, whereas the adult and old age death rate increases exponentially with age in conformity with Gompertz's law. An additional difference is that whereas aging and old age death have been extensively studied, infant death received much less attention. To our knowledge, the two effects have never been inter-connected. Clearly, it would be satisfactory to explain the two phenomena as being two variants within the same explanatory framework. In other words, a mechanism providing a combined explanation for the two forms of mortality would be welcome. This is the purpose of the present paper. We show here that the same biological effects can account for the two cases provided there is a difference in their severity: death triggered by isolated lethal anomalies in one case and widespread wear-out anomalies in the second. We show that quite generally this mechanism leads indeed, respectively, to a declining and an upgoing death rate. Moreover, this theoretical framework leads to the conjecture that the severity of the death effects, whether in infancy or old age, is higher for organisms which comprised a larger number of organs. Finally, let us observe that the main focus of the paper is the drastic difference of the age-specific death rates (i.e., decreasing versus increasing) because this difference is found in many species, whereas the question of the best fit (e.g., Gompertz versus Weibull) is rather specific to human mortality.

Modified leaky competing accumulator model of decision making with multiple alternatives: the Lie-algebraic approach

In this communication, based upon the stochastic Gompertz law of population growth, we have reformulated the Leaky Competing Accumulator (LCA) model with multiple alternatives such that the positive-definiteness of evidence accumulation is automatically satisfied. By exploiting the Lie symmetry of the backward Kolmogorov equation (or Fokker–Planck equation) assoicated with the modified model and applying the Wei–Norman theorem, we have succeeded in deriving the N-dimensional joint probability density function (p.d.f.) and marginal p.d.f. for each alternative in closed form. With this joint p.d.f., a likelihood function can be constructed and thus model-fitting procedures become feasible. We have also demonstrated that the calibration of model parameters based upon the Monte Carlo simulated time series is indeed both efficient and accurate. Moreover, it should be noted that the proposed Lie-algebraic approach can also be applied to tackle the modified LCA model with time-varying parameters.

Nutrient supply, cell spatial correlation and Gompertzian tumor growth.

Gompertzian tumor growth can be reproduced by mitosis, related to nutrient supply, with local spatial cell correlations. The global energy constraint alone does not reproduce in vivo data by the observed values of the nutrient expenditure for the cell activities. The depletion of the exponential growth, described by the Gompertz law, is obtained by mean field spatial correlations or by a small word network among cells. The well-known interdependence between the two parameters of the Gompertz growth naturally emerges and depends on the cell volume and on the tumor density.

Gompertz Law in Clean Foam Coalescence

Clean foams tend to age with time through sequential coalescence events. This study evaluates aging dynamics in clean foams by measuring bubble populations from coalescence simulation experiments and adopting biological population dynamics analysis. The population dynamics of bubbles in clean foams during coalescence show that the mortality rates of individual bubbles change exponentially with time, regardless of initial simulation conditions, consistent with the Gompertz mortality law commonly observed in biological aging. This result would be beneficial in understanding the aging dynamics of clean foams.

Ordering properties of the smallest and largest lifetimes in Gompertz–Makeham model

The Gompertz–Makeham (GM) distribution, which is used commonly to represent lifetimes based on laws of mortality, is one of the most popular choices for mortality modeling in the field of actuarial science. This paper investigates ordering properties of the smallest and largest lifetimes arising from two sets of heterogeneous groups of insurees following respective GM distributions. Some sufficient conditions are provided for comparing the smallest and largest lifetimes from two sets of dependent variables in the sense of usual stochastic ordering. Comparison results on the smallest lifetimes, in the sense of hazard rate ordering and ageing faster ordering, are established for two groups of heterogeneous independent lifetimes. Under a similar set-up, no reversed hazard rate ordering is shown to exist between the largest lifetimes with the use of a counter-example. Finally, sufficient conditions are presented for comparing two sets of independent heterogeneous lifetimes under random shocks by means of usual stochastic ordering. Such comparisons for the smallest lifetimes are also carried out in terms of hazard rate ordering.