Estimation of present value of whole life annuity using information about expectation of life.pdf 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]): ^ (8) = ^, 8 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, X be his lifetime, Tx = X - x be his future lifetime. Let us introduce the random variable , -8T 1 - e x z(x) = ---, Tx > 0. (1) 8 Then, by averaging the random variable z(x) (1), we get the formula of the whole life annuity (see [2-4]): ax (5) = E( z( x)) =1 5 ад у where E is the symbol of the mathematical expectation, s(x) = p(x > x) is a survival function, ( (Wv 1 - 0(x, 5) (2) Ф(x, 8) = e& J e'5fdF(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: 5x ФN (x, 5) SN( x) -N 1 ax (5) =T О N 1 Z exp(-5X; )I(Xt > x) 1 - 1- (3) Sw (x) • N i=i Sx N ®n(x,5) = - Zexp(-5Xi)I(Xi > x). 2. Bias and MSE of aN (5) 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 = (tlN ,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 (ц, g) is the s-dimensional normally distributed random variable with a mean vector and covariance dH (z) , j = 1, s, ^ is the symbol of convergence in matrix a = a(x); VH(t) = (H1(t),...,Hs(t))T, H (t) = BZ, 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:|Z -t | x) i=1 < 1 8 2 8' 1 + - V e&xe-&x zi(x. > x) i=1_ Z I(X. > x) i=1 Л л Х_Ф N (x, 8) ФN (x,8) SN (x) у N |H (tw )| = 0, the condition 2) due to Lemma 3.1 [6], provided that E{I (X > x)} = S(x) < 1, E{ei8xe-8XI (X > x)} < ei8xe-8xS(x) = S(x) < 1 for all i еЯ. We know that SN (x) is an unbiased and consistent estimator of S(x). Show that ФN (x, 8) is an unbiased estimator of the functional Ф^, 8): e5x ГN ЕФN (x, 8) = - E\ Z exp(-8X )I(X, > x) !> = Ф(x, 8). N h=1 Now, calculate the variance of ФN (x, 8): f e Sx N БФN(x, 8) = Df - N I(X > x)e" I N /=1 028x ZD{l(Xt >x)e"SX } = ^(Ф(x,28) - Ф2(x,8)). N2 г=1 The ratio of two unbiased estimators can have a bias. Considering that all the conditions of Theorem 1 are fulfilled and E(tN - t) = 0, in accordance with (4) we get the order of the bias of aN (5): |E( aW( 8) - ax (8)) - E [VH (t )(tN -1)]| = |E(axN(8) - ax (8)) = О (N-). for the statistics ФN (x, 8), SN (x): Find the components of the covariance matrix a(ax (8)) = 011 = ND^w, (x, 8)} = Ф( x, 28) - Ф2 (x,8); o 22 = ND{S^ (x)} = S(X)(1 - S(x)); = N cov(Sw (x), Ф N (x, 8)) = N (E {Sw (x^ w (x, 8)} - E {Sn (x)} E {Ф w (x, 8)}) = (1 - S (x)^( x, 8). Using the previous results on the bias and the covariance matrix, we obtain °12 = 0 21 = u2(axw (8))=E [VH (t)(tN - t)]2 + О (n-/2 ) 3/2n C(aT(S)) О (w-/2), + W Ф( x,28) - Ф2 (x,8) / S (x) 82 S2(x) . C(aM)=H{(t)ou + Hi (t )o22 + 2HX (t )H2 (t)al2 = The proof is completed. 3. Asymptotic Normality of a^ (8) 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, (o/~(«v(S))j. 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: Л 1 -Фn (x,8) - A(x - a) Sn (x) aN (8,A) = 1 8 (7) 1 N where x = - X Xt is an estimator of a, parameter 1 we will find minimizing the principal term of the asymptotic MSE of ax (8, A) (7). The estimator (7) combines the available empirical information containing in (3) and prior information (6). For the estimator a^ (8, A) in the notation of Theorem 1, we have: s = 3, tN = (t1N,t2N,t3N)T = Ф(x,8), Sn(x), x)T; ^ = N; t = ft,t2,t3)T = (Ф(^8),S(x),a)T; Ф( x, 8) Л _ 1--:--A(a - a) = ax (8); H (t) = H (t1, t2, t3) = 1 8 1 - -1 - A(t3 - a) V t2 S ( x) VH (t) = (H1(t), H 2 (t), H3(t) )Т = Т 1 Ф( x, 8) a * 0. 8S (x)8S 3 (x)' 8 5. Bias and MSE of aW(8,X) Arguing as in Section 1, it is easy to show that the sequence {H(^)} satisfies the condition 1) of Theorem 1 with C0 = 2 + + a) , ю < да is the limiting age, and у = 0; also, the statistic tN satisfies the con-8 dition 2) due to Lemma 3.1 [6], provided that EX4 < ю! 0, Q2 = 82 8 X0 = -. Such X0 minimizes the principal term of MSE u' L(aW (5, X)), and this minimum is as follows: Q. Q1 2 Л С(аД5,Х0))_ 1 < N N N (9) С{(ах(Ь))-Щ-У1 6. Bias, MSE, and Asymptotic Normality of (8,X 0) In accordance with (9), the estimator 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 (aW (8Л) )= О ( N1); 2) the MSE of (10) is given by the formula where С(ад.(5,А0)) is defined by the formula (9). Theorem 7. Under the conditions of Theorem 2 (5Ло) - ад) Nx (о,с(ад A,0))). 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, A 0) can be used as an estimator for ax (5) if we know A0; otherwise, it is required to construct an adaptive estimator. We need a more detailed formula for A0: 1 A0 = (11) S (x)DX (C (x) - aS (x))- C1(x,8) + оФ(^8) о(x) Using (11), we consider the following adaptive estimator: ( л /Ч-ХЛ . aN (8, a 0)=1 (12) Sn (x) 8 with - 1 1 -Ф N (x,8) - A0 (x - a) ФN (x, 8) An = (13) 0 s2 Sn (x) C (x) - aSN (x)) - CC1 (x, 8) + aФ N (x, 8) Sn ( x) 2 1 N - 2 where s =-X (X - x) is an unbiased estimator of the variance DX, N -1 Xi г .N „ ,N C2 (x) = N~1X XI(Xt > x), C (x, 5) = N-1 X e_5X Xtl(Xt > x). t=1 t=1 Theorem 8. Under the conditions of Theorem 2, ^(af(5,a0)-at(5))^>n1(0,c(at(5,a0))). Proof. The following equality holds: VN(af (5, A0) - ax(5)) = (5, A0 - «x(5)) + RN, where RN =5-1 (a0 -AA0 ^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 A0 converges almost surely to A0. Based on the central limit theorem VN(x - a) (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 A0. 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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