Nonparametric evaluation of continuous r-year deferred m-year term life annuity using information on probabilistic characteristics of lifetime
The theory of pension annuities is closely related to the ideology of net premiums of the life insurance theory. The mathematical theory of insurance is widely used to solve many problems that are determined by the requirements of the market economy. The requirements of practice stimulate the development of insurance theory and the closely related theory of annuities and force researchers to turn to more complex mathematical models in this area. New methods of calculating annuities appear that reduce the time for making optimal decisions in the absence of sufficient information about the markets of new types of pension services. The article considers the problem of estimating continuous r-year deferred m-year term life annuity with making use of information on probabilistic characteristics of lifetime. Insurance companies often offer their clients to conclude contracts of r-year deferred m-year annuities. Nonparametric estimators of life annuities are constructed from data on the lifetimes of individuals. Found the principal terms of the asymptotic mean squared errors (MSEs) of the proposed estimators; their asymptotic normality is proved. It is shown that the use of auxiliary information often leads to a lower MSE of the modified estimator compared to the MSE of the traditional estimator. An adaptive estimator is proposed that can be applied in practice. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.
Keywords
r-year deferred m-year term life annuity,
nonparametric evaluation,
auxiliary information,
mean squared error,
asymptotic normality,
adaptive estimatorAuthors
| Dmitriev Yury G. | National Research Tomsk State University | dmit@mail.tsu.ru |
| Koshkin Gennady M. | National Research Tomsk State University | kgm@mail.tsu.ru |
Всего: 2
References
Falin, G.I. (2002) Mathematical foundations of the theory of life insurance and pension schemes. Moscow: Ankil.
Al-Nator, M.S., Al-Nator, S.V. & Soloviev, A.K. (2018) Actuarial and statistical substantiation of the retirement age in the context of socio-economic uncertainty. Modern Mathematics and Concepts of Innovative Mathematical Education. 5(1). pp. 144-148.
Ovanesyan, N.M. & Midler, E.A. (2016) Long-term life insurance as a factor in the sustainable development of the insurance services market in an inflationary environment. Financial Research. 1(50). pp. 128-134.
Roik, V.D. (2013) Ways to improve insurance of occupational disease risks. Occupational Safety and Economics. 2(23). pp. 4-14.
Soloviev, A.K. (2018) Risks of forecasting pension reform in the context of digitalization. Bulletin of the Department of Statistics of the Plekhanov Russian University of Economics: Statistical studies of the socio-economic development of Russia and prospects for sustainable growth: materials and reports. Moscow. pp. 257-260.
Winter, P. & Planchet, F. (2022) Modern tontines as a pension solution: a practical overview premium. European Actuarial Journal. 12. pp. 3-32.
Gubina, O.V. & Koshkin, G.M. (2015) Estimation of actuarial present value of the whole continuous life annuity Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaya tekhnika i informatika - Tomsk State University Journal of Control and Computer Science. 30(1). pp. 38-43.
Bowers, N., Gerber, H., Hickman, J., Jones, D. & Nesbitt C. (1986) Actuarial Mathematics. Itasca: Society of Actuaries.
Gerber, H. (1997) Life Insurance Mathematics. 3rd ed. New York: Springer-Verlag.
Kokina, E.P. & Tregubova A.A. (2015) Formation of insurance tariff classes: application of statistical methods. Financial Research. 2(47). pp. 115-122.
Awondo, S., Ramirez, O., Datta, G.S., Colson, G. & Fonsah, E.G. (2018) Estimation of grop yields and insurance premiums using a shrinkage estimator. North American Actuarial Journal. 22(2). pp. 289-308.
Gubina, O.V. & Koshkin, G.M. (2020) Estimation of the present value of n-year life annuity for endowment insurance. Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaya tekhnika i informatika - Tomsk State University Journal of Control and Computer Science. 50. pp. 39-46.
Hu, J. & Hong, L (2022) A nonparametric sequential learning procedure for estimating the pure premium. European Actuarial Journal. 2. pp. 485-502.
Koshkin, G.M. (1999) Deviation moments of the substitution estimator and its piecewise smooth approximations. Siberian Mathematical Journal. 40(3). pp. 515-527.
Ibragimov, I.A. & Khasminskii, R.Z. (1981) Statistical Estimation: Asymptotic Theory. Berlin - New York: Springer.
Borovkov, A.A. (1998) Mathematical Statistics. New York: Gordon & Breach.
Dmitriev, Yu.G. & Koshkin, G.M. (1987) Using additional information in nonparametric estimation of density functionals. Automation and Remote Control. 48(10). pp. 1307-1316.
Dmitriev, Y.G. & Koshkin, G.M. (2018) Nonparametric estimators of probability characteristics using unbiased prior conditions. Statistical Papers. 59(4), pp. 1559-1575. doi: 10.1007/s00362-018-1044-7.
Rao, C.R. (1965) Linear Statistical Methods and Its Applications. New York: Willey.
Koshkin, G.M. (2014) Smooth estimators of the reliability functions for non-restorable elements.Russian Physics Journal. 57(5). pp. 672-681.
