Nonparametric estimation of actuarial present value of deferred life annuity Текст научной статьи по специальности «Математика»

ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА 2019 Управление, вычислительная техника и информатика № 46

УДК 369:519.2

DOI: 10.17223/19988605/46/6

O.V. Gubina, G.M. Koshkin

NONPARAMETRIC ESTIMATION OF ACTUARIAL PRESENT VALUE OF DEFERRED LIFE ANNUITY

The paper deals with the estimation problem of the actuarial present value of the deferred life annuity. The nonpara-metric estimator of the deferred life annuity was constructed. We found the principal term of the asymptotic mean squared error (MSE) of the proposed estimator and proved its asymptotic normality. The simulations show that the empirical MSE of the annuity estimator decreases when the sample size increases. Keywords: nonparametric estimation; deferred life annuity; mean squared error; asymptotic normality.

Let x be the age of an individual and at the moment t = 0 payments start. The idea of the r-year deferred life annuity in accordance with [1. P. 174] is this: from the moment t + r = r, an individual starts receiving money once a year, which we take as a monetary unit, and payments are made only during the lifetime of an individual. It is known that the deferred life annuity is associated with the appropriate type of insurance. Thus, the average total cost of the present continuous r-year deferred life annuity is given by the following formula (see [1. P. 184]):

1 - A

a (5) = —±-i,

r' 5

да

where A = j e-5 fx (t)dt is the net premium (the expectation of the present value of an insured unit sum for

r

the deferred life insurance at age x), 5 is a force of interest, f ( t) = f (x + 5) is a probability density of future

x S (x)

lifetime of an individual T = X -x [1. P. 62], f (x) is a probability density of lifetime of an individual X, S(x) = P(X > x) is a survival function. Introduce the random variable

, -5T 1 — e '

z(x) = ——-, Tx > r. (1)

5

Then, by averaging z(x) (1), we get the formula of the deferred life annuity (see [2-4]):

a (s)=e( z)=1 f i'

r| x( ) ( ) Sl S(x)

(2)

where E is the symbol of the mathematical expectation,

to

0(x, S, r) = eSx J e~5tdF(t),

x+r

F(x) = P(X < x) = 1 - S(x) is a distribution function.

Note that the whole li fe annuity ax (S) [2] is the special case of the deferred life annuity (2) at r = 0.

1. Construction of the Deferred Annuity Estimator

Assume that we have a random sample X1,...,XN of N individuals' lifetimes. Using the empirical survival function

1 N

Sn (X) = -11(X, > x), N 1=1

where I(A) is the indicator of an event A, obtain the following estimator of (2):

aN (5)=-

r x w 5

1 --

S„(x) • N~1

exp(-5Xi. )I(X > x + r)

1

1-

o n (x, 5, r ) sn (x)

A

05x N

On(x, 5, r) = — X exp(-5X..)I(X > x + r).

(3)

2. Bias and Mean Squared Error of the Estimator r^aN(5)

Here we will obtain the principal term of the asymptotic MSE and the bias convergence rate of the estimator (3). Introduce the notation according to [5]: t^ = (t1JV, t2N ,---,tsN )T is an s-dimensional vector with

the components tjN = tjN (x) = tjN(x; X1,...,XN), j = 1, s, x e Ra, Ra is the a-dimensional Euclidean space;

H(t):Rs ^R1 is a function, where t = t(x) = (t(x),...,ts(x))T is an s-dimensional bounded vector function; N a) is the s-dimensional normally distributed random variable with a mean vector and covariance matrix

dH (z)

a = a(x); VH(t) = (Hi(t),...,Hs(t))T, Hj(t) = -

, j = 1, s; ^ is the symbol of convergence in distri-

bution; || x|| is the Euclidean norm of a vector x; ^ is the set of natural numbers.

Definition 1. The function H(t): Rs ^ R1 and the sequence {H(tN)j are said to belong to the class N s (t; Y), provided that:

1) there exists an s-neighborhood

o = {z :| z -1 < £,, = 1,sj, in which the function H (z) and all its partial derivatives up to order v are continuous and bounded;

2) for any values of variables Xx,..., X^ the sequence {H(tN )j is dominated by a numerical sequence

C0dY, such that dN t®, as N ^ ®, and 0 < y <<x>. Theorem 1 [5]. Let the conditions

1) H(z) , {H(tN)j eN2,s(t;y),

2) E\\tN - t|f = O(dN~i/2) hold for all i e Then, for every k e^,

E[H(tN) -H(t)]k -E[VH(t) • (tN -t)f = O(dN (k+1)/2).

(4)

If in formula (4) k = 1, we obtain the principal term of the bias for H(tn) , and at k = 2, we have the principal term of the MSE.

Theorem 2. If S(x) > 0 and S(t) is continuous at a point x, then 1) for the bias of (3), the following relation holds:

b( raN (s))

e( a (s) - a (s))

2) the MSE of (3) is given by the formula

O(N-1);

u2( a(5)) = E( a (5)- a (5))2 =

N,

O( x,25, r ) -O2( x, 5, r)/ S (x) N52 S 2( x)

+

O (N "3/2 ).

V

i=1

Proof. For the estimator r|ax (8) (3) in the notation of Theorem 1, we have: s = 2;

tN = (tiN,t2N)T = (^(x,5,r),S^(x))T; ^ =N; t = (ti,t2>T = (O(x,S,r),S(x))T;

H (t ) = -

5

t

1 - AV t2 y

1-

O( x, 5, r ) 5 ( x)

= a (5); H (tN ) = -

1 -

O N ( x, 5, r ) Sn (x)

= r\aXN (5);

VH (t ) = ( Hi(t), H2(t ) )T =

1 O(x, 5, r )

T

5S ( x) 5S 2 ( x)

* 0.

The sequence {H(tN )} satisfies the condition 1) of Theorem 1 with C0 = ^-(1 + e r ), y = 0. Indeed,

|H (tN )| =

1 -

O N ( x, 5, r )

SN ( x)

O N ( x, 5, r) Sn ( x)

< -

1

N

e S exp(-5Xt )I(Xt > x + r)

1 + —-

N

SI(Xt >x)

i=1

<

1

<8

5x -5(x+r) N

e e ( )

1 + -

SI(Xt >x + r)

t=1

V.

N

SI ( Xt > x)

t=1

< ^(1 + e~5 r ). 8

Further, the function H(t) satisfies the condition 1) in view of t2 = S(x) > 0. Also, this function satisfies the condition 2) due to Lemma 3.1 [6], as for all i e^ such inequalities hold: E{I(X >x)} = S(x) < 1, E{ei8xe~iSX Ii (X > x+r)} < ei5xe~i5xS(x + r) = S (x + r) < 1

It is well known that SN (x) is the unbiased and consistent estimator of S(x). Show that ON (x, 8, r) is the unbiased estimator of O(x, 8, r) and calculate the variance of ON (x, 8, r):

e8x fN )

EON (x, 8, r) = — E12 exp(-8X, )I(X, > x + r) U O(x, 8,r), (x,8,r) = ^ 2D{l(X; > x + r)e_5X } = 1 (o(x,2 8,r) -O2(x,8,r)).

DO

N

Considering that E(tN -1) = 0 and all the conditions of Theorem 1 are fulfilled, in accordance with (4) we get the order of the bias of i axN (5) :

E( a (S) - a (S)) - E[VH(t)(tN -1)]

E( AaN(S) - A(S))

■■O ( N-1 ).

Find the components of the covariance matrix o( ax (5)) =

for the statistics ON (x, 5, r) and

Sn ( x):

J11

= ND {ON(x,5,r)} = O(x,2 5,r) -O2(x, 5,r); o22 = ND {Sn(x)} = S(X)(1 - S(x));

°12 = 021 =

= N cov(Sn (x), O N (x, 8, r)) = N ( E {Sn (x)O n (x, 8, r)}- E {Sn (x)} E {O n (x, 8, r)}) = (1 - S (x))O( x, 8, r).

Using the previous results on the bias and covariance matrix, we obtain u2(a(8)) = E[VH(t)(tN -1)]2 + O(N-3/2)= Hi2(t)on + H22(t)022 + 2Hi(t)H2(t)^ + O(n-3/2) =

O( x,2 8, r) -O2 (x,8, r)/ S (x)

N52 S2 ( x)

■ +

O ( N-3/2 ).

(5)

The proof of Theorem 2 is completed.

<

3. Asymptotic Normality of the Estimator ^«^(S)

We need the following two Theorems.

Theorem 3 [7, Appendix 5]. If ^£2,...£N,... is a sequence of independent and identically distributed

T 1 N I—

s-dimensional vectors, E{£k} = 0, ct(x) = k}, tN = — £ k, then, as N ^ ®, y/NtN ^Ns(0,o(x)).

N k=1

Theorem 4 [8]. Let 4N • tN ^Ns {|,o(x)}, H(z) be differentiable at the point VH(|) * 0. Then

4N (H (tN) - H (|)) ^N1

( ^

£H., £ £ H} (l)ojpHp(|) j=1 p=1 j=1

Theorem 5. Under the conditions of Theorem 2

VN( «N (5) - (5)) ^ N [00(x,25,r)-2^2((xxj5,r)/S (x)

Proof. In the notation of Theorem 3, we have s = 2, ct(x) = o(r| ax (5)). Thus,

4N {(O n (x, 5, r), Sn (x)) - (O( x, 5, r), S ( x))}^N ((0,0), o(«(5))).

The function H(z) is differentiable at the point t = (O(x, S), S(x)) and VH(t) ^ 0. Consequently, all the

conditions of Theorem 4 hold, and using (5), we obtain the desired result. The proof of Theorem 5 is completed.

4. Simulations

[1, if x e (a, b]

Introduce the denotation I (a,b \ = [ . Consider de Moivre's model, for which the in-

J [0, if x € (a, b]

dividual's lifetime X is uniformly distributed in the interval (0, ro), where © is a limiting age. For this model the probability density and survival function are defined by the following formulas:

Ix (0, ro) xlx (0, ro)

f (x) = —-, S (x) = Ix (-®, ro)--x-. (6)

ro CO

Now, using (6), we obtain

f (x +1) _ It (0, ro - x)

fx (t) ^

S (x) ro - x

1 / ® N 1

1 I - - -5t - - * - 1 1

5 V r /5

f

1 ro-x \ 1

1 -Rt , 1

ax(5) =11 1 -Je-tfx(t)dt 1 =1 1--— J e-tdt 1 =

ro-x r /5

f -5r -5(ro-x)

e - e 1 --

(7)

y ro - a r yu^ 5( ro - x) /

The present value of the 5-years deferred annuity for a person at the age x = 45 years when ro = 100 years, S = 0,09531 (9,531%), and monthly payments in the size of 1000 rubles, is equal to

12000 • 5«j(0,09531) = 12000 • 9,581854 = 114982 rubles.

Note that for such S the effective annual interest rate i = e5 -1 = 0,1 (10%).

The simulations were carried out for de Moivre's model under the above presented conditions. The annuities and their estimators are presented in Fig. 1 for random samples Xt,...,XN of the sizes N = 50, 100, 500, uniformly distributed in the interval (0, 100).

We will characterize the quality of estimators presented in Fig. 1 using the empirical MSE

95 M 9

£( r|«x (S) - r\«N (S))2 G( N, r, S) = ^^

Fig. 1. Dependence on the age x of the 5-years deferred annuity (S = 0,09531) and its estimators for the sample sizes N: a - 50; b - 100, c - 500

a

c

The calculation results are given in Table I.

Table 1

Simulation results for the different sample sizes

N 25 50 100 250 500

G(N; 5; 0,09531) 1,632 0,815 0,413 0,117 0,052

So, according to Table 1, the quality of the deferred annuity estimators by the criterion G(N; 5; 0,09531) is improving when the sample size N is increasing.

Conclusion

In the paper, we found the principal term of the asymptotic MSE of the estimator ^aN (5). Also, the

following asymptotic properties of the estimator are proved: unbiasedness, consistency, and normality. Statistical modeling within the framework of de Moivre's model shows that the quality of estimation according to the empirical criterion G(N,r,5) improves with the growth of the sample size. Note that the improved estimators of life annuities (3) can be obtained by substituting of empirical survival functions by the smooth empirical survival functions (cf. [9-24]) and using auxiliary information of the different type [25-33], for example, connected with random variables X, Tx = X - x, Tx - r.

REFERENCES

1. Falin, G.I. (2002) Mathematical foundations of the theory of life insurance and pension schemes. Moscow: Ankil.

2. Koshkin, G.M. & Gubina, O.V. (2016) Estimation of the present values of life annuities for the different actuarial models. SMRLO

2016. Proc. The Second International Symposium on Stochastic Models, in Reliability Engineering, Life Science, and Operations Management. Beer Sheva, Israel. February 15-18, 2016. Conference Publishing Services The Institute of Electrical and Electronics Engineers. pp. 506-510.

3. Bowers, N., Gerber, H., Hickman, J., Jones, D. & Nesbitt, C. (1986) Actuarial Mathematics. Itasca: Society of Actuaries.

4. Gerber, H. (1997) Life Insurance Mathematics. 3rd ed. New York: Springer-Verlag.

5. Koshkin, G.M. (1999) Deviation moments of the substitution estimator and its piecewise smooth approximations. Sibirskiy

Matematicheskiy Zhurnal - Siberian Mathematical Journal. 40(3). pp. 515-527.

6. Ibragimov, I.A. & Khasminskii, R.Z. (1981) Statistical Estimation: Asymptotic Theory. Berlin; New York: Springer.

7. Borovkov, A.A. (1998) Mathematical Statistics. New York: Gordon & Breach.

8. Koshkin, G.M. (1990) Asymptotic properties of functions of statistics and their application to nonparametric estimation. Automation

and Remote Control. 51(3). pp. 345-357.

9. 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

10. Azzalini, A. (1981) A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika. 68(1). pp. 326328. DOi: 10.1093/biomet/68.1.326

11. Reiss, R.-D. (1981) Nonparametric estimation of smooth distribution functions. Scandinavian Journal of Statistics. 8. pp. 116-119.

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

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

14. Jones, M.C. (1990) The performance of kernel density functions in kernel distribution function estimation. Statist. Probab. Lett. 9. pp. 129-132. DOI: 10.1016/0167-7152(92)90006-Q

15. Shirahata, S. & Chu, I.S. (1992) Integrated squared error of kernel-type estimator of distribution function. Ann. Inst. Statist. Math. 44(3). pp. 579-591. DOI: 10.1007/BF00050707

16. Sarda, P. (1993) Smoothing parameter selection for smooth distribution functions. J. Statist. Plann. Inference Inf. 35. pp. 65-75. DOI: 10.1016/0378-3758(93)90068-H

17. Altman, N. & Leger, C. (1995) Bandwidth selection for kernel distribution function estimation. J. Statist. Plann. Inference. 46. pp. 195-214. DOI: 10.1016/0378-3758(94)00102-2

18. Bowman, A., Hall, P. & Prvan, T. (1998) Trust bandwidth selection for the smoothing of distribution functions. Biometrika. 85(4). pp. 799-808. DOI: 10.1093/biomet/85.4.799

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

20. Shao, Y. & Xiang, X. (1997) Some extensions of the asymptotics of a kernel estimator of a distribution function. Statist. Probab. Lett. 34. pp. 301-308. DOI: 10.1016/S0167-7152(96)00194-0

21. Una-Alvarez, J., Gonzalez-Manteiga, W. & Cadarso-Suarez, C. (2000) Kernel distribution function estimation under the Koziol-Green model. J. Statist. Plann. Inference. 87. pp. 199-219.

22. 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

23. Koshkin, G.M. (2015) Smooth recurrent estimators of the reliability functions. Russian Physics Journal. 58(7). pp. 1018-1025. DOI: 10.1007/s11182-015-0603-9

24. Fuks, I. & Koshkin, G. (2015) Smooth recurrent estimation of multivariate reliability function. Proceedings of the International Conference on Information and Digital Technologies 2015. IDT 2015. Zilina, Slovakia. July 7-9, 2015. pp. 84-89.

25. Dmitriev, Yu.G. & Koshkin, G.M. (1987) On the use of a priori information in nonparametric regression estimation. IFAC Proceedings Series. 2. pp. 223-228. DOI: 10.1016/S1474-6670(17)59798-0

26. 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.

27. 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.

28. Dmitriev, Yu.G., Tarassenko, P.F. & Ustinov, Y.K. (2014) On estimation of linear functional by utilizing a prior guess. In: Dudin, A. et al. (eds.) Communications in Computer and Information Science. ITMM2014. Vol. 487. pp. 82-90.

29. Levit, B.Ya. (1975) Conditional estimation of linear functionals. Problems of Information Transmission. 19(4). pp. 291-302.

30. Owen, A.B. (1991) Empirical likelihood for linear models. Ann. Statist. 19. pp. 1725-1747.

31. Qin, J. & Lawless, J. (1994) Empirical likelihood and general estimating equations. Ann. Statist. 22. pp. 300-325. DOI: 10.1214/aos/1176325370

32. 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

33. 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

Received: August 14, 2018

Gubina O.V., Koshkin G.M. (2019) NONPARAMETRIC ESTIMATION OF ACTUARIAL PRESENT VALUE OF DEFERRED LIFE ANNUITY. Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie vychislitelnaja tehnika i informatika [Tomsk State University Journal of Control and Computer Science]. 46. pp. 49-55

DOI: 10.17223/19988605/46/6

Губина О.В., Кошкин Г.М. НЕПАРАМЕТРИЧЕСКОЕ ОЦЕНИВАНИЕ АКТУАРНОЙ СОВРЕМЕННОЙ СТОИМОСТИ ОТСРОЧЕННОЙ РЕНТЫ. Вестник Томского государственного университета. Управление, вычислительная техника и информатика. 2019. № 46. С. 49-55

Рассматривается проблема оценивания актуарной современной стоимости отсроченной ренты. Синтезируется непараметрическая оценка отсроченной ренты. Находится главная часть асимптотической среднеквадратической ошибки оценки и ее предельное распределение. Моделирование показывает, что эмпирическая среднеквадратическая ошибка оценки ренты уменьшается с ростом объема выборки.

Ключевые слова: непараметрическое оценивание; отсроченная пожизненная рента; среднеквадратическая ошибка; асимптотическая нормальность.

GUBINA Oxana Viktorovna (Post-graduate student of National Research Tomsk State University, Russian Federation). E-mail: [email protected]

KOSHKIN Gennady Mikhailovich (Doctor of Physics and Mathematics, Professor of the Department of Theoretical Cybernetics of National Research Tomsk State University, Russian Federation). E-mail: [email protected]