Estimation of present value of whole life annuity using information about expectation of life | Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaja tehnika i informatika – Tomsk State University Journal of Control and Computer Science. 2018. № 45. DOI: 10.17223/19988605/45/3

Estimation of present value of whole life annuity using information about expectation of life

The paper deals with the estimation problem of the actuarial present value of the continuous whole life annuity using auxiliary information about the expectation of life. Nonparametric estimators of life annuity are constructed by individuals' death moments. It is shown that the usage of such auxiliary information can often provide the mean squared error (MSE) smaller than that of standard estimators. An adaptive estimator is also proposed. The asymptotic normality of all these estimators is proved.

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Keywords

непараметрическая оценка, пожизненная рента, дополнительная информация, среднеквадратическая ошибка, асимптотическая нормальность, nonparametric estimation, whole life annuity, auxiliary information, mean squared error, asymptotic normality

Authors

Name	Organization	E-mail
Dmitriev Yury Glebovich	Tomsk State University	dmit70@mail.ru
Koshkin Gennady Mikhailovich	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 Publ.

Koshkin, G.M. & Gubina, O.V. (2016) Estimation of the present values of life annuities for the different actuarial models. Proc. The Second International Symposium on Stochastic Models, in Reliability Engineering, Life Science, and Operations Management. SMRLO 2016. Beer Sheva, Israel, February 15-18, 2016. Conference Publishing Services The Institute of Electrical and Electronics Engineers, Inc. 2016. pp. 506-510.

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.

Koshkin, G.M. (1999) Deviation moments of the substitution estimator and its piecewise smooth approximations. Siberian Mathematical Journal. 40(3). pp. 515-527. DOI: 10.1007/BF02679759

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) On the use of a priori information in nonparametric regression estimation. IFAC Proc. Series. 2. pp. 223-228.

Dmitriev, Yu.G. & Tarasenko, P.F. (2014) The use of a priori information in the statistical processing of experimental data. Russian Physics Journal. 35(9). pp. 888-893. DOI: 10.1007/BF00560063

Dmitriev, Yu.G., Tarassenko, P.F. & Ustinov, Y.K. (2014) On estimation of linear functional by utilizing a prior guess. In: Dudin, A., Nazarov, A., Yakupov, R., Gortsev, A. (eds) Information Technologies and Mathematical Modelling. ITMM 2014. Communications in Computer and Information Science. Vol. 487. Springer, Cham. pp. 82-90.

Dmitriev, Yu.G. & Koshkin, G.M. (2017) On distribution functionals estimation with auxiliary information. Applied Methods of Statistical Analysis. pp. 9-18.

Dmitriev, Yu.G. & Koshkin, G.M. (2018) Nonparametric estimators of probability characteristics using unbiased prior conditions. Statistical Papers. pp. 1559-1575. DOI 10.1007/s00362-018-1044-7.

Levit, B.Ya. (1975) Conditional estimation of linear functionals. Problems of Information Transmission. 1(4). pp. 291-302.

Owen, A.B. (1991) Empirical likelihood for linear models. The Annals of Statistics. 19. pp. 1725-1747.

Qin, J. & Lawless, J. (1994) Empirical likelihood and general estimating equations. The Annals of Statistics. 22. pp. 300-325.

Tarima, S. & Pavlov, D. (2006) Using auxiliary information in statistical function estimation. ESAIM: Probability and Statistics. 10. pp. 11-23. DOI: 10.1051/ps:2005019

Zhang, B. (1995) M-estimation and quantile estimation in the presence of auxiliary information. Journal of Statistical Planning and Inference. 44. pp. 77-94. DOI: 10.1016/0378-3758(94)00040-3

Rao, C.R. (1965) Linear statistical methods and its applications. New York: Willey.

Nadaraya, E.A. (1964) Some new estimates of distribution function. Theory of Probability and its Applications. 9(3). pp. 497-500. DOI: 10.1137/1109069

Azzalini, A. (1981) A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika. 68(1). pp. 326-328. DOI: 10.1093/biomet/68.1.326

Reiss, R.-D. (1981) Nonparametric estimation of smooth distribution functions. Scand. J. Statist. 8. pp. 116-119.

Falk, M. (1983) Relative efficiency and deficiency of kernel type estimators of smooth distribution functions. Statistica Neerlandica. 37. pp. 73-83. DOI: 10.1111/j.1467-9574.1983.tb00802.x

Swanepoel, J.W.H. (1988) Mean integrated squared error properties and optimal kernels when estimating a distribution function. Communication in Statistics - Theory and Methods. 17(11). pp. 3785-3799. DOI: 10.1080/03610928808829835

Jones, M.C. (1990) The performance of kernel density functions in kernel distribution function estimation. Statistics and Probability Letters. 9(2). pp. 129-132. DOI: 10.1016/0167-7152(92)90006-Q

Shirahata, S. & Chu, I.S. (1992) Integrated squared error of kernel-type estimator of distribution function. Annals of the Institute of Statistical Mathematics. 44(3). pp. 579-591. DOI: 10.1007/BF00050707

Sarda, P. (1993) Smoothing parameter selection for smooth distribution functions. Journal of Statistical Planning and Inference. 35. pp. 65-75. DOI: 10.1016/0378-3758(93)90068-H

Altman, N. & Leger, C. (1995) Bandwidth selection for kernel distribution function estimation. Journal of Statistical Planning and Inference. 46. pp. 195-214. DOI: 10.1016/0378-3758(94)00102-2

Bowman, A., Hall, P. & Prvan, T. (1998) Trust bandwidth selection for the smoothing of distribution functions. Biometrika. 85(4). pp. 799-808.

Chu, I.S. (1995) Bootstrap smoothing parameter selection for distribution function estimation. Math. Japon. 41(1). pp 189-197.

Shao, Y. & Xiang, X. (1997) Some extensions of the asymptotics of a kernel estimator of a distribution function. Statistics & Probability Letters. vol. 34. pp. 301-308.

Una-Alvarez, J., Gonzalez-Manteiga, W. & Cadarso-Suarez, C. (2000) Kernel distribution function estimation under the Koziol-Green model. Journal of Statistical Planning and Inference. 87. pp. 199-219. DOI: 10.1016/S0378-3758(99)00192-5

Koshkin, G.M. (2014) Smooth estimators of the reliability functions for non-restorable elements. Russian Physics Journal. 57(5). pp. 672-681. DOI: 10.1007/s11182-014-0290-y