Оценивание пожизненной ренты с использованием информации о средней продолжительности жизни Текст научной статьи по специальности «Математика»

ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА 2018 Управление, вычислительная техника и информатика № 45

УДК 369:519.2

DOI: 10.17223/19988605/45/3

Yu.G. Dmitriev, G.M. Koshkin

ESTIMATION OF PRESENT VALUE OF WHOLE LIFE ANNUITY USING INFORMATION ABOUT EXPECTATION OF LIFE

This research was supported by "The Tomsk State University competitiveness improvement programme"

under grant No. 8.1.37.2018.

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.

Keywords: nonparametric estimation; whole life annuity; auxiliary information; mean squared error; asymptotic normality.

The idea of life annuity in accordance with [1. P. 170] is this: from the moment t = 0 an individual once a year begins to get a certain money, which we take as the unit of money, and payments are made only for the lifetime of an individual. It is known that the calculation of the characteristics of life annuity is based on the characteristics of the respective type of insurance. Thus, the average total cost of the present continuous annuity is defined by the following formula (see [1. P. 184]):

« (5) = ЦА'

о

where Ax is a net premium (the average of the present value of a single sum of money in the insurance lifetime

at the age x), 5 is a force of interest. Let x be an individual's age on the moment of payments start,

Xbe his lifetime, Tx = X —x be his future lifetime. Let us introduce the random variable

, —от 1 — e '

z(x) = ——-, Tx > 0. (1)

о

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

«x (5) = E( z( x)) =1 5

( rrVv 1 —

0(x, 5)

S(x) j

where E is the symbol of the mathematical expectation, S(x) = P(X > x) is a survival function,

(2)

O(x, 5) = e5x J e'5'dF(t),

x

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

1. Estimation of Annuity

Suppose we have a random sample X1,...,XN of N individuals' lifetimes. Now, separately estimate the numerator and denominator in (2). The substitution of unknown function <S(x) for its nonparametric estimator

1 N

SN (x) = — ZI(> x),

N i=i

where I(A) is the indicator of an event A, gives us the following estimators of the whole life annuity:

—N/vs 1 ax (8) =-0

Sx

1 —

N

S^ (X) • N ;=1

Z exp(-5X, )I(X, > x)

^ 1'

1-

O N (X, 8) SN( x)

(3)

Sx N

On(x,8) = — Zexp(-5X¿)I(X, > x).

2. Bias and MSE of aN (8)

In this section, we will obtain the principal term of the asymptotic MSE and the bias convergence rate of the estimator (3). Now introduce the notation according to [5]: tw = (tm ,t2N ,~,tsN )T is an s-dimensional

vector with 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) = (tx(x),...,ts(x))T is an s-dimensional bounded vector function; Ns g) is the s-dimensional normally distributed random variable with a mean vector and covariance

dH (z)

matrix a = a(x); VH(t) = (H1(t),...,Hs(t))T, H (t) =

fe,

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

distribution (weak convergence); || 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)} are said to belong to class N s (t; Y), provided that:

1) there exists an s-neighborhood

a = {z-t, <s, i =1,s}, 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)} is dominated by a numerical sequence

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

1) H(z) , {H(tN )}e N2,s(t; y),

2) E^ - t|f = O(dN-/2)

hold for all i e Then, for every k e^,

E[H(tN)-H(t)]k -E[VH(t)• (tN -1)]\ = O(dN (k+1)/2). (4)

Note, 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 x, then

1) for the bias of (3), the following relation holds:

b(aXN (S))| = \E(aXN (8) - ^ (8)) = O (N"1);

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

u2(aN (5)) = E(aN (5) - ax (5))2 =

■N,

,2 O(x, 25) -02( x, 5)/ S (x)

N52 S 2( x)

+ O

( N "3/2 )

Proof. For the estimator ax (5) (3) in the notation of Theorem 1, we have: s = 2;

tN = (tiN,t2N)T = (On(x,5),Sn(x))T; dN = N; t = (ti,t2)T = (O(x,5),S(x))T;

H(t) = 1 8

1 - ^ v h j

i -

O( x, 8) S (x) ,

= ax (8); H (tN ) =1 8

1 -

O N ( x, 8)

SN (x)

=aN (8);

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

1 O( x, 8)

T

8S (x) 8S 2 (x)

* 0.

The sequence {H(^ )} satisfies the condition 1) of Theorem 1 with C0 = ^ and y = 0. Indeed,

|H (tN )| =

N ( x, 8)

SN ( x)

<-

A

< 1 8

O N ( x ,8)

SN (x) j

N

<

e8 Z exp(-8X )I(X > x)

1 + —-i=1-

N

ZI(x >x)

i=1

N

\

<

1+-

v

e5xe-sx zi(x. > x) i=i_

Z I(x, > x)

i=1

2 8 '

Further, the function H(t) satisfies the condition 1) in view of t2 = S(x) > 0, the condition 2) due to

Lemma 3.1 [6], provided that E{I (X > x)} = S(x) < 1, E{ei8xe~i8XI (X > x)} < ei8xe~i8xS(x) = S(x) < 1 for all i eM.

We know that SN (x) is an unbiased and consistent estimator of S(x). Show that ON (x, 8) is an unbiased estimator of the functional O(x, 8) :

e5x fN

EON (x, 8) = — E\ Z exp(-8Xt )I(Xt > x) \ = O(x, 8).

N b=1

Now, calculate the variance of ON (x, 8) :

f e 8x N

DO N ( x, 8) = D f — Z I( xt > x)e-I N /=1

028x

N2 ¿=1

ZD{l(X. >x)e~SX } = ^(o(x,28)-O2(x,8)).

The ratio of two unbiased estimators can have a bias. Considering that all the conditions of Theorem 1 are fulfilled and E(tN -1) = 0, in accordance with (4) we get the order of the bias of a^ (8) :

|E(aN (8) - ax(8)) - E[VH(t)(tN -1)]| = |e(ÔN(8) - a(8)) = O(N3 ).

Find the components of the covariance matrix o(ax (8)) =

o,,or

for the statistics ON (x, 8), SN ( x):

°12 = ° 21 =

on = ND {O N (x,5)} = O(x,25)-O2 (x,5); o 22 = ND{SN (x)} = S(X)(1 - S(x)); = N cov(Sn (x), O n (x, 5)) = N (E {Sn (x)O n (x, 5)} - E {Sn (x)} E {O n (x, 5)}) = (1 - S (x))O( x, 5). Using the previous results on the bias and the covariance matrix, we obtain

u2a (8)) = e [Vh(t)(tN -1)]2 + O (n-/2 )

3/2n C{âxm

+

O ( N3/2 ),

C(âM)=H{(t)ou +Hl{t)Q22+lHl{t)H2{t)Ql2 = The proof is completed.

N

O( x,28) -O2 ( x,8) / S ( x) 82 S2(x) .

3. Asymptotic Normality of a^ (S)

To find the limit distribution of (3), we need the following two Theorems.

Theorem 3 (The usual central limit theorem) [7, Appendix 5]. If ,...,<N,... is a sequence of inde-

T 1 N

pendent and identically distributed s-dimensional vectors, E(<^k} = 0, a(x) = E((tlkt^k}, tN =—X , then,

N k=1

as N , yfNtN ^N, (0, a(x)).

Theorem 4 (asymptotic normality of H(tN)) [2]. Let

1) 4N • tN ^N,, (^,o(x)|;

2) H(z) be differentiable at the point VH^ 0. Then

VN (H (tN ) - H M)

z h. , SE H. ]pHp fo) j=1 p=ij=1

Theorem 5. Under the conditions of Theorem 2

VN(aN (5) - ax (5)) ^ni [

Proof. In the notation of Theorem 3, we have s = 2, c(x) = a(ax (5)). Thus,

4N{(ON (x, 5), Sn (x)) - (O(x, 5), S(x))} ^ N2 ((0,0), a(ax(5))). The function H(z) is differentiable at the point t = (®(x,5),S(x)) and VH(t) ^ 0. Consequently, all the conditions of Theorem 4 hold, and using (5), we have s/n (flv'(ô) -flv(ô)) => N, (o,C(<7.(S))). Theorem 5 is proved.

4. Construction of Estimators Using Expected Lifetime

Suppose we know the expected lifetime

EX = a. (6)

The estimator by making use of such information according to [8-17] can be taken in the following form:

aN (SA)=1

S

1 -On (x,S) - x - a) SN ( x)

A

(7)

1 N

where x = — X Xi is an estimator of a, parameter 1 we will find minimizing the principal term of the

asymptotic MSE of ax (s,x) (7). The estimator (7) combines the available empirical information containing in (3) and prior information (6).

For the estimator a^ (s,x) in the notation of Theorem 1, we have: s = 3,

tN = (tiN, t2N ,t3N )T = (On (x, S), Sn (x), x)T ; dN = N ; t = (ti, t2,t3)T = (O(x, S), S(x), a)T ;

H (t ) = H (ti, t2, t3) = 1 S

1 - -1 - X(t3 - a)

V t2

O( x, S) ^ _ 1--:--X(a - a) = ax (S);

H (tN ) = -

1 -On (x,S) - ^x - a)

Sn ( x )

S ( x)

=ax (SA);

VH (t ) = ( H1(t ), H 2 (t ), H3(t ) )T =

1 O( x, S) X

SS ( x)'SS 2 ( x) ' S

* 0.

T

5. Bias and MSE of aN (5,1)

Arguing as in Section 1, it is easy to show that the sequence [H(tN)} satisfies the condition 1) of Theorem 1 with c0 = 2 + + a) , to < w is the limiting age, and y = 0; also, the statistic tN satisfies the con-5

dition 2) due to Lemma 3.1 [6], provided that EX1 < to < w for all i e From here, for the bias of (7) we obtain the following result:

E(aN (5,1) - ax(5)) - E [VH(t)(tN -1)]| = |E(af (5,1) - ax(5)) = |b(af (5,1))| = O(N-1).

Now, find the covariance matrix o(ax (5,1)) =

О 21O22O 23 O31O32O33

for statistics Фn (x, 5), SN (x), x :

ст33 = ND{x} = D(X); ст13 = ст31 = Ncov(x,Фу(x,5)) = C1(x,ô)-aФ(x,ô);

да да

023 =аз2 = Ncov(SN(x),x) = C2(x)-аВД, where Q(x,ô) = eôxje~5uudF(u), C2(x) = judF(u), and on,

x x

o12 , o21 , o22 are defined in Section 1. Using (5), the above results for the bias and the covariance matrix o(a (5,1)), we obtain:

л2 . „ÎT.r-3/2\ C(âT(ô,l))

u2 (ax (ô, x)) = E [VH(t)(tN - г)]2 + о (n-/2 ) =

N

- + о

( N-3/2 ),

3 3

C(âT(ô,l)) = Z Z HAt)BipHJt) =Я2(0ап + H2 (t)<522 +#2(f)a33 + 2H1(t)H2(t)au +

p=1 j=1

+2Я1(0Я3(0о1з +2 H2(t)H3(t)a23 =C(âx(5)) +

X C33 21HjCj3 2X^2^23

ô2

O33 ^ n n - H1O13 + H2O23

where Q = > 0, Q2 =

52

5

= C(âx(8)) + X2Ol-2W2, (8)

. Then the minimum of C(äx(5,1)) with respect to '/. is achived at

Q.

10 = —. Such minimizes the principal term of MSE u' l(aN (5, x)), and this minimum is as follows:

Q1

С(аД5,10))_ 1

N

N

2 Л

У1

<

с(Ш ô))

N

(9)

6. Bias, MSE, and Asymptotic Normality of a^ (5,10)

In accordance with (9), the estimator

aN (5Л )=1

5

1 ( -Ф^ - ( x - a;

v SN

Sn ( x)

(10)

Q2

will be called the optimal (in the mean square sense) estimator. The non-negative quantity in (9) determines the decrease of the principal term of MSE for the optimal estimator by using auxiliary information (6). Theorem 6. If S(x)>0 and S(t) is continuous at x, then 1) for the bias of (10), the following relation holds:

b ( aN (5Л ) )= О ( N1 );

2) the MSE of (10) is given by the formula

where C(ax(5, A,0)) is defined by the formula (9).

Theorem 7. Under the conditions of Theorem 2

(S,X0)" «,(8)) => Nx (0,C(fl,(5A0)))-

Proof. The statements of Theorems 6 and 7 follow from Theorems 1 and 4 with the usage of the arguments of Sections 3-5.

7. Adaptive Estimator

The statistic a^ (8,10) can be used as an estimator for ax (8) if we know 10; otherwise, it is required to construct an adaptive estimator. We need a more detailed formula for 10:

1

^ =

S (x)DX

(C2 (x) - aS (x)) - Q (x,8) + aO(x, 8)

o (x)

Using (11), we consider the following adaptive estimator:

( ^ ^^ . ^

aN (8,1 o)=1

8

with

O N (x, 8)

■n,b 1 i-ON(x,8) -^(x-a)

Sn (x)

1n =

0 s2sn (x)

Sn ( x)

(C2 (x) - aSN (x)) - CC1 (x, 8) + aO N (x, 8)

(11)

(12)

(13)

2 1 N _ , where s =-X (X - x) is an unbiased estimator of the variance DX,

N -1 Xi i

.N „ )N

C2 (x) = N~1X XI(X > x), C (x, 8) = N-1 X e_8Xi Xtl(Xt > x). i=1 t=1

Theorem 8. Under the conditions of Theorem 2,

^(af(8A0)-aT(8))^>N1(0,C(aT(8A0))).

Proof. The following equality holds:

4N(ax (8,10) - ax(8)) = VN(aN (8, 10 - a(8)) + Rn ,

where RN =8-1 (l0 -10 )VN (x - a). All the estimators, used in (13), converge almost surely to their true values according to the strong law of large numbers (the Second Theorem of Kolmogorov [18]). Thus, from the First Continuity Theorem of Borovkov [7], estimator 10 converges almost surely to 10. Based on the central limit theorem yfN(x - a) ^Nx(0, DX), we retrieve RN ^ 0. Now, the statement of Theorem 8 is proved by making use of Theorem 7.

Conclusion

The paper deals with the problem of estimating the present values of the continuous whole life annuity using auxiliary information about the expectation of life. It is shown that the usage of such auxiliary information can often provide the MSE smaller than that of standard estimators. We proved the results on asymptotic properties of the proposed estimators: unbiasedness, consistency and normality. Also, the main parts of the asymptotic MSEs of the estimators were found. An adaptive estimator is constructed; such estimator is equivalent (in the sense of asymptotic distribution) to the estimator with the optimal weight coefficient 10.

Note that the improved estimators of life annuities (3), (10) and (12) can be obtained by substituting of empirical survival functions by the smooth empirical survival functions (cf. [19-32]).

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Received: March 29, 2018

Dmitriev Yu.G., Koshkin G.M. (2018) 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]. 45. pp. 22-29

DOI: 10.17223/19988605/45/3

Дмитриев Ю.Г., Кошкин Г.М. ОЦЕНИВАНИЕ ПОЖИЗНЕННОЙ РЕНТЫ С ИСПОЛЬЗОВАНИЕМ ИНФОРМАЦИИ О СРЕДНЕЙ ПРОДОЛЖИТЕЛЬНОСТИ ЖИЗНИ. Вестник Томского государственного университета. Управление, вычислительная техника и информатика. 2018. N° 45. С. 22-29.

Рассматривается проблема оценивания актуарной непрерывной пожизненной ренты с использованием дополнительной информации о средней продолжительности жизни. По данным продолжительностей жизни индивидуумов строятся непараметрические оценки пожизненной ренты. Показано, что использование дополнительной информации приводит к среднеквад-ратической ошибке, меньшей, чем у стандартной оценки. Также предлагается адаптивная оценка. Показано, что адаптивная оценка эквивалентна в смысле асимптотического распределения оптимальной оценке. Доказана асимптотическая нормальность всех оценок.

Ключевые слова: непараметрическая оценка; пожизненная рента; дополнительная информация; среднеквадратическая ошибка, асимптотическая нормальность.

DMITRIEV Yury Glebovich (Doctor of Physics and Mathematics, Head of the Department of Theoretical Cybernetics of National Research Tomsk State University, Russian Federation). E-mail: dmit70@mail.ru

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: kgm@mail.tsu.ru