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The Population Dynamics of Endangered Blue Whales: Past, Present, and Future
(2024-05) Pflaumer, Peter
Blue whales (Balaenoptera musculus) are the largest animals that have ever lived on earth, but their populations were nearly driven to extinction due to industrial hunting at the beginning of the 20th century. Their numbers were estimated to be between 250,000 and 300,000 before the hunting, but this drastically declined over the years. Continuous models of population dynamics are used to estimate the intrinsic growth rate and other demographic characteristics of their populations. The Euler-Lotka equation is used to determine the mean annual growth rate, and with the help of the piecewise exponential distribution as a life table model, simple formulas can be derived for the calculation of important demographic parameters such as the age structure, life expectancy, and maximum age. The pre-exploitation abundance of Antarctic blue whales is found using the logistic function, assuming a minimum abundance of 1,000 in 1970, an intrinsic growth rate of 4.1%, and documented annual catches between 1904 and 1972. The estimated pre-exploitation abundance is forecast as 280,471 in 1904. Using the logistic model to forecast the population, it is calculated that it will take nearly 140 years for the population to recover to even half of its pre-exploitation abundance at current assumed rates. To preserve the endangered blue whales, it is essential to monitor their populations continuously, develop effective conservation strategies, and reduce the anthropogenic pressures on the species. Through collaborative efforts and conservation measures, it may be possible to help these magnificent creatures recover and thrive once again.
Probabilistic Perspectives on Sex Ratio at Birth Dynamics
(2024-08) Pflaumer, Peter
Predicting trends in Sex Ratio at Birth (SRB) is crucial in demographic research, shedding light on evolving population dynamics. This study conducts a thorough investigation into the selection and evaluation of optimal forecasting models for SRB data. Utilizing historical SRB records from selected countries, we meticulously assess various models, including Autoregressive Integrated Moving Average (ARIMA), Autoregressive (AR), and White Noise models. Our empirical analysis reveals the prominence of the AR(2) model in capturing intricate SRB dynamics. Additionally, we explore the White Noise model's role in understanding and predicting SRB fluctuations. Our findings emphasize the AR(2) model's efficacy, attributed to its parsimonious complexity, empirical validation, theoretical alignment, and superior statistical performance. Extending projections to 2070 for Germany, our study not only offers foresight into future SRB trends but also contributes a robust methodology to the broader field of time series analysis.
Tackling the Challenge of Aging Populations: The Impact of Increasing Life Expectancy and Low Fertility on the Old-Age Dependency Ratio
(2023-09) Pflaumer, Peter
The old-age dependency ratios are indicators of the number of elderly people who are generally economically inactive compared to the number of people of working age. They significantly affect the financial burden of social public pension schemes, making it essential to analyze the influence of mortality on this ratio. In this paper, the Gompertz model is used to investigate the effect of mortality and fertility on the old-age dependency ratio, with a focus on the impact of changes in life expectancy. Elasticity formulas are derived to analyze this effect, and the results indicate that an increase in life expectancy leads to a considerable rise in the old-age dependency ratio.
Refining Mortality Projections at Advanced Ages: Evaluating the Significance of Wittstein's Mortality Law
(2023-11) Pflaumer, Peter
Age-specific mortality rates for semi-supercentenarians and supercentenarians play a pivotal role in comprehending longevity and population dynamics at advanced ages. In this study, we introduce a modified Wittstein Model, offering an alternative to the conventional S-shaped curve models used in mortality forecasting. The Wittstein Model, originally formulated by Theodor Wittstein, has been adapted to suit contemporary contexts. Utilizing life table data for German women from 2019/2021, we project age-specific mortality rates, construct life tables commencing from age 100, and conduct a sensitivity analysis to assess the impact of model parameters on mortality patterns. The sensitivity analysis unveils the influence of parameter values on the shape of age-specific mortality rates. This study contributes to research in mortality forecasting, with a specific focus on semi-supercentenarians and supercentenarians, shedding light on an understudied population segment. Accurate projections carry profound implications for public health, healthcare planning, and social policy. Further research should explore the model's applicability in different contexts, providing a deeper understanding of mortality patterns at advanced ages. As the empirical database of centenarians expands, the model is expected to enhance its precision and reliability in forecasting age-specific mortality rates at advanced ages.
Analyzing the Historical Life Table of Thomas Young
(2023-08) Pflaumer, Peter
Thomas Young (1773-1829) is one of the greatest thinkers and polymaths. His scientific work includes significant contributions in the fields of medicine, physics, anthropology and ancient history. Less well known, however, is Young's demographic contribution. In 1826, Thomas Young examined graphical curves of mortality of his epoch (decrement tables of the deceased) to see if they matched a formula he had developed. Looking for a law of mortality, he created a high order polynomial for the function of mortality. We use modern demographic methods to analyze and criticize his life table. Young's discrete life table is fitted by a continuous life table function (Lazarus distribution) in order to calculate important parameters. It is shown that Young's formula is an early and successful method of determining a model life table. It corresponds to a particular life table of Coale and Demeny. The article concludes with an exploration of Young's mortality formula of 1816, a concise yet foundational model, showcasing its ability to facilitate calculations of vital functions like life expectancy and the force of mortality, despite its lesser-known status.
Leonhard Euler’s Research on the Multiplication of the Human Race with Models of Population Growth
(2023-06) Pflaumer, Peter
The renowned Swiss mathematician Leonhard Euler created three variations of a simple population projection model, including one stable model and two non-stable models, that consider a couple with different fertility behaviors and life-spans. While one of the models was published by a German demographer, Johann Peter Süßmilch, in his book “The Divine Order”, the other two are not widely known in contemporary literature. This paper compares and reanalyzes the three variants of Euler's population projections using matrix algebra, providing diagrams and tables of the population time series and their growth rates, as well as age structures of selected years. It is demonstrated that the non-stable projection models can be explained in the long run by their geometric trend component, which is a special case of strong ergodicity in demography as described by Euler. Additionally, a continuous variant of Euler's stable model is introduced, allowing for the calculation of the age structure, intrinsic growth rate, and population momentum in a straightforward manner. The effect of im¬mortality on population size and age structure at high growth rates is also examined.
Risk and Return of the Tontine: A Brief Discussion
(2021-07-16) Pflaumer, Peter
This article analyzes the stochastic aspects of a tontine using a Gompertz distribution. In particular, the probabilistic and demographic risks of a tontine investment are examined. The expected value and variance of tontine payouts are calculated. Both parameters increase with age. The stochastic present value of a tontine payout is compared with the present value of a fixed annuity. It is shown that only at very high ages the tontine is more profitable than an annuity. Finally, the demographic risks associated with a tontine are discussed. Elasticities are used to calculate the impact of changes in modal age on the tontine payout. It is shown that the tontine payout is very sensitive to changes in modal age.
Euler and Süßmilch’s Population Growth Model
(2019-12-02) Pflaumer, Peter
In 1761, the German demographer Johann Peter Süßmilch published a simple population growth model that starts with a couple, in the eighth chapter of his book "Die göttliche Ordnung". With the help of the Swiss mathematician Leonhard Euler, he projected the population for 300 years. He demonstrated that after that time the population will be growing approximately geometrically. In this paper, the population projection of Euler and Süßmilch is reanalyzed using matrix algebra. Graphs and tables show the time series of the population and its growth rates. Age structures of selected years are presented. The solution of the projection equation is derived. It is shown that the projection model can be described by a geometric trend model which is superimposed by six cyclical components. In the long run, the population time series can be explained quite well by the sum of only two components, the trend component and one component with explosive cycles of a period of about 24 years. In the very long run, the influence of the cyclical component diminishes, and the series can be solely explained by its geometric trend component, as has been also recognized by Euler and Süßmilch.
Risk Analysis in Capital Investment Appraisal with Correlated Cash Flows: Simple Analytical Methods
(2017-07) Pflaumer, Peter
Since uncertainty is the crucial point of a capital investment decision, risk analysis in capital budgeting is often applied. Usually risk analysis is carried out by a Monte Carlo simulation. The aim of this article is to present simple analytical methods which allow us to calculate the standard deviation of a project with correlated cash flows as a risk measure. These methods are compared with simulation procedures carried out with R, and it is shown that the proposed simple analytical methods are indeed a quick and efficient procedure for assessing the risk of an investment project where the cash flows are correlated.
A Statistical Analysis of the Roulette Martingale System: Examples, Formulas and Simulations with R
(2019-06) Pflaumer, Peter
Some gamblers use a martingale or doubling strategy as a way of improving their chances of winning. This paper derives important formulas for the martingale strategy, such as the distribution, the expected value, the standard deviation of the profit, the risk of a loss or the expected bet of one or multiple martingale rounds. A computer simulation study with R of the doubling strategy is presented. The results of doubling to gambling with a constant sized bet on simple chances (red or black numbers, even or odd numbers, and low (1–18) or high (19–36) numbers) and on single numbers (straight bets) are compared. In the long run, a loss is inevitable because of the negative expected value. The martingale strategy and the constant bet strategy on a single number are riskier than the constant bet strategy on a simple chance. This higher risk leads, however, to a higher chance of a positive profit in the short term. But on the other hand, higher risk means that the losses suffered by doublers and by single number bettors are much greater than that suffered by constant bettors.
Projecting Age-Specific Death Probabilities at Advanced Ages Using the Mortality Laws of Gompertz and Wittstein
(2018-12-18) Pflaumer, Peter
In this paper, death probabilities derived from the Gompertz and Wittstein models are used to project mortality at advanced ages beginning at the age of 101 years. Life table data of Germany from 1871 to 2012 serve as a basis for the empirical analysis. Projections of the death probabilities and life table survivors will be shown. The increase of the death probabilities slows down at very old ages. Finally, Wittstein´s formula will be regarded as a distribution function. Its reversed hazard rate function, which will be derived together with the median and the modal value, will clarify the significance of the parameters of the Wittstein distribution.
Distributions of Age at Death from Roman Epitaph Inscriptions in North Africa
(2017-11-01) Pflaumer, Peter
Thousands of inscriptions of age at death from Roman epitaphs in North Africa are statistically analyzed. The Gompertz distribution is used to estimate survivor functions. The smoothed distributions are classified according to the estimation results. Similarities and differences can be detected more easily. Parameters such as mean, mode, skewness, and kurtosis are calculated. Cluster analysis provides three typical distributions. The analysis of the force of mortality function of the three clusters shows that the epigraphic sample is not representative of the mortality in North Africa. The results are compared with data from epitaphs from the European provinces. Africa is quite different. The general mortality level is much lower. The African cluster is much more homogeneous than the European cluster. The distributions are determined by three factors: mortality levels, commemorative processes, and population growth rates.
Distributions of Age at Death from Roman Epitaph Inscriptions: An Application of Data Mining
(American Statistical Association, 2016-12-02) Pflaumer, Peter
Thousands of age at death inscriptions from Roman epitaphs are statistically analyzed. The Gompertz distribution is used to estimate survivor functions. The smoothed distributions are classified according to the estimation results. Similarities and differences can be detected more easily. Parameters such as mean, mode, skewness, and kurtosis are calculated. Cluster analysis provides three typical distributions. The analysis of the force of mortality function of the three clusters yields that the epigraphic sample is not representative of the mortality in the Roman Empire. However, the data is not worthless. It can be used to show and to explain the differences in the burial and commemorative processes. Finally, the bias due to a growing population is discussed. A simple formula is proposed for estimating the growth rate. The paper also discusses some special parameter constellations of the Gompertz distribution, since in this special application it cannot be approximated by the Gumbel distribution (as is often done in life table analysis).
Estimations of the Roman Life Expectancy Using Ulpian´s Table
(American Statistical Association, 2015-12) Pflaumer, Peter
In this paper a life table for the Roman population is constructed using Ulpian’s table. This table can be regarded as a tool to compute the value of an annuity taking into account the age of the beneficiary. The Gompertz distribution and some of its extensions are applied for the life table construction. It is shown that the Roman life table can be represented by a five-parameter formula, which consists of three terms. Since the life expectancy at birth depends on the unknown infant mortality, different assumptions are made. Simulations show that a range of the life expectancy between 20 and 30 years is quite possible. Finally, it is discussed whether Ulpian´s table represents annuities or life expectancies. It cannot be excluded that the values in Ulpian’s table represent annuities premiums based on an interest rate of about 1.5%.
Evaluating the Accuracy of Population Forecasts
(1992-06) Pflaumer, Peter
In this paper the accuracy of population forecasts is discussed. Various papers on errors of population forecasting are reviewed and summarized. The results are stated in six theses. The main findings show that no clear dominance of any one forecasting method can be determined, that the logarithmic forecast errors are more or less independent of the length of the forecast horizon, and that saturation models underestimate the population development in the long term, whereas geometric and polynomial trend models overestimate it. Finally, the accuracy of population forecasts is compared with the accuracy of short-term and long-term economic forecasts. It is found that the error of population forecasts is smaller than that of economic forecasts. However, the logarithmic error decreases with the length of the forecast period for most economic variables.
How Migration can Contribute to Achieving a Stationary Population
(1994-08) Pflaumer, Peter
Methods from mathematics of finance and demography are presented in order to investigate the influence of migration on the long-term population development. Methods from mathematics of finance do not take the age structure of a population into consideration and can therefore only be used as an approximation. The less the age structures in question deviate from those of stable populations, the more exact the approximation will be. In the empirical section quantitative measures for population policy are described and analyzed using the population of Germany and of the world as examples. The long-term goal of quantitative population policy is zero growth. Whereas in less developed countries, this goal can be achieved for the most part only by a reduction of fertility, it is possible in more developed countries with below-replacement fertility to achieve stationarity by means of immigration. Under the assumptions made here, Germany would have to take in between 350.000 and 500.000 immigrants each year for the population to remain at the present level. Immigration has demographic consequences for the age structure and the composition of the population which will be described at the end.
A Demometric Analysis of Ulpian’s Table
(American Statistical Association, 2014-12) Pflaumer, Peter
Ulpian’s table is a famous ancient text that is preserved in edited form in Justinian’s Digest, a compendium of Roman law compiled by order of the emperor Justinian I in the sixth century AD. This passage probably provides a rough estimation of Roman life expectancy in the early third century AD. The paper begins with a discussion of the demographic properties and peculiarities of Ulpian´s table. Then the Gompertz distribution and some of its extensions are used to fit life expectation functions to Ulpian´s data. The model can be used to estimate important demographic functions and parameters of the Roman life table. Inter alia, the average and median remaining life expectancies are calculated, and compared with the results of other investigations, e.g., Frier’s life table for the Roman Empire. It turns out that Ulpian´s life table is characterized by a steep decline of the life expectancy function in the advanced age classes, which is much steeper than in life expectancy functions of other life tables based on data. The modal or normal age at death, which is between 55 and 60 years, is comparatively high.
Life Table Forecasting with the Gompertz Distribution
(American Statistical Association, 2007-11) Pflaumer, Peter
First, this paper investigates the properties of the Gompertz distribution and the relationships of their constants. Then the use of Gompertz´s law to describe mortality is discussed with male and female period life table data of the United States between 1900 and 2000. For this purpose a model incorporating time trends has been formulated with age, time and the product of age and time as independent variables and the force of mortality as the dependent variable. The parameters of the model are estimated using the least squares method. Since the mortality of modern developed population is largely the mortality of old age this generalized Gompertz model provides a good approximation of life tables in these populations, and can be used to estimate and forecast many parameters of the life table and the stationary population like expectation of life, modal age, Keyfitz´entropy or old age dependency ratios. These and other parameters are forecast up to the year 2100 and compared with recent mortality forecasts of the Social Security Administration. While similar results for the male population can be observed, a greater difference between male and female mortality are forecast. Although the time dependent Gompertz model reveals systematic underestimation of mortality at young ages and overestimation at the oldest ages it is a very useful, an easy, and a quick tool for obtaining forecasts of important parameters of life tables with low mortality.
Forecasting the U.S. Population with the Gompertz Growth Curve
(American Statistical Association, 2012-11) Pflaumer, Peter
Population forecasts have received a great deal of attention during the past few years. They are widely used for planning and policy purposes. In this paper, the Gompertz growth curve is proposed to forecast the U.S. population. In order to evaluate its forecast error, population estimates from 1890 to 2010 are compared with the corresponding predictions for a variety of launch years, estimation periods, and forecast horizons. Various descriptive measures of these forecast errors are presented and compared with the accuracy of forecasts made with the cohort component method (e.g., the U.S. Census Bureau) and other traditional time series models. These models include quadratic and cubic trends, which were used by statisticians at the end of the 19th century (Pritchett and Stevens). The measures of errors considered are based on the differences between the projected and the actual annual growth rate. It turns out that the forecast accuracies of the models differ greatly. The accuracy of some simple time series models is better than the accuracy of more complex models.
Measuring the Rectangularization of Life Tables Using the Gompertz Distribution
(American Statistical Association, 2010-11) Pflaumer, Peter
The rectangularization of life tables is defined as a trend towards a more rectangular shape of the survival curve due to increased survival and concentration of deaths around the mean age at death. Since the mortality of modern developed population is largely the mortality of old age, the Gompertz model provides a good approximation of life tables in these populations and can be used to estimate and forecast many parameters of the life table and the stationary population, such as expectation of life, modal age, age dependency ratios, and indices of the rectangularization of life tables. Formulas of known rectangularization indices are developed assuming the Gompertz distribution, whereas some new indices are proposed, too. The mathematical relationships between the single indices are shown. It is demonstrated that some mentioned indices are a function of the coefficient of variation.

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