Witold Rzymowski , Agnieszka Surowiec
ARTICLE

(English) PDF

ABSTRACT

In this paper, we present a new method of the econometric model construction: the difference equation method. We illustrate the proposed approach using an application example from human population dynamic study. We find out that proposed method is very useful to find one of the three forms of proposed models of human population satisfying the small maximal relative errors. The maximal relative error is a measure to verify the model of human population. The proposed method is tested for all available data referring to the human population in the OECD countries as well as in selected non-OECD countries.

KEYWORDS

difference equations, nonlinear models, parameter estimation, relative error, demography

REFERENCES

Austin A. L., Brewer J. W., (1971–1972), World Population Growth and Related Technical Problems, Technological Forecasting and Social Change, 3, 23–49.

Bard Y., (1974), Nonlinear Parameter Estimation, Academic Press, New York.

Chen J., Wu H., (2008), Efficient Local Estimation for Time-varying Coefficients in Deterministic Dynamic Models with Applications to HIV-1 Dynamics, Journal of the American Statistical Association, 103, 369–384.

Dorn H. F., (1962), World Population Growth: An International Dilemma, Science, 135, 283–290.

Hemker P. W., (1972), Numerical Methods for Differential Equations in System Simulation and in Parameter Estimation, in: Hemker H. C., Hess B., (eds.), Analysis and Simulation of Biochemical Systems, North Holland Publ. Comp, 59–80.

Huang Y., (2010), A Bayesian Approach in Differential Equation Dynamic Models Incorporating Clinical Factors and Covariates. Journal of Applied Statistics, 37 (2), 181–199.

Huang Y., Liu D., Wu H., (2006), Hierarchical Bayesian Methods for Estimation of Parameters in a Longitudinal HIV Dynamic System, Biometrics, 62 (2), 413–23.

Koźniewska I., (1972), Równania rekurencyjne, PWN, Warszawa.

Lanczos C., (1964), Applied Analysis, Prentice Hall. Inc., USA.

Li L., Brown M. B., Lee K. H., Gupta S., (2002), Estimation and Inference for a Spline-Enhanced Population Pharmacokinetic Model, Biometrics, 58 (3), 601–611.

Li Z., Osborne M. R., Prvan T., (2005), Parameter Estimation of Ordinary Differential Equations, IMA Journal of Numerical Analysis, 25 (2), 264–285.

Liang H., Wu H., (2008), Parameter Estimation for Differential Equation Models Using a Framework of Measurement Error in Regression Models, Journal of the American Statistical Association, 103 (484), 1570–1583.

Malthus T. R., (1798), An Essay on the Principal of Population, J. Johnson, in St. Paul’s Church-Yard: London.

Miao H., Dykes C., Demeter L. M., Wu H., (2009), Differential Equation Modeling of HIV Viral Fitness Experiments: Model Identification, Model Selection, and Multimodel Inference, Biometrics, 65 (1), 292–300.

Murray J. D., (1989), Mathematical Biology, 19 of Biomathematics Texts, Springer: Berlin.

Nowak E., (2006), Zarys metod ekonometrii. Zbiór zadań, PWN, Warszawa.

Ogunnaike B. A., Ray W. H., (1994), Process Dynamics, Modeling, And Control, Series: Topics in Chemical Engineering, Oxford University Press, New York.

Pearl R., Reed L. J., (1924), The Growth of Human Population, in: Pearl R., (ed.), Studies in Human Biology, Wiliam and Wilkins: Baltimore, 584–637.

Poyton A. A., Varziri M. S., McAuley K. B., McLellan P. J., Ramsay J. O., (2006), Parameter Estimation in Continuous-Time Dynamic Models Using Principal Differential Analysis, Computer and Chemical Engineering, 30 (4), 698–708.

Ramsay J. O., Hooker G., Campbell D., Cao J., (2007), Parameter Estimation for Differential Equations: A Generalized Smoothing Approach (with Discussions), Journal of the Royal Statistical Society: Series B, 69 (5), 741–796.

Rao C. R., (1982), Modele liniowe statystyki matematycznej, PWN, Warszawa.

Robertson J. S., Bond V. P., Cronkite E. P., Hutton W. E., Howland W. E., Shinbrot M., von Foerster H., Mora P. M., Amiot L. W., (1961), Doomsday, Science, 133, 936–946.

Rzymowski W., Surowiec A., (2012), Method of Parameters Estimation of Pseudologistic Model, in: Zieliński Z. E., (ed.), Rola informatyki w naukach ekonomicznych i społecznych. Innowacje i implikacje interdyscyplinarne, 2, WSH, Kielce, 256–265.

Serrin J., (1975), Is ‘Doomsday’ on target? (Letter), Science, 189, 86–88.

Sierpiński W. F., (1946), Zasady algebry wyższej, PWN, Warszawa–Wrocław.

Smith D. A., (1977), Human Population Growth: Stability or Explosion? Mathematics Magazine, 50 (4), 186–197.

Varah J. M., (1982), A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations, SIAM Journal on Scientific and Statistical Computing, 3 (1), 28–46.

Verhulst P. F., (1838), Notice sur la loi que la population suit dans son d’accroissement. Correspondance Mathématique et Physique Publiée par A. Quételet, 10, 113–121.

von Foerster H., Mora P. M., Amiot L. W., (1960), Doomsday: Friday, 13 November, A. D. 2026, Science, 132 (3436), 1291–1295.

Back to top
© 2019–2022 Copyright by Statistics Poland, some rights reserved. Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0) Creative Commons — Attribution-ShareAlike 4.0 International — CC BY-SA 4.0