相依正隨機變數之合其分佈函數之最佳化超立方體估計

國 立 交 通 大 學

統 計 學 研 究 所

博 士 論 文

相依正隨機變數之合其分佈函數之最佳化超立方體估計

An optimal hypercube approximation of the distribution

function of sum of positive dependent random variables

研 究 生：盧信銘

指導教授：彭南夫 博士

相依正隨機變數之合其分佈函數之最佳化超立方體估計

An optimal hypercube approximation of the distribution

function of sum of positive dependent random variables

研 究 生：盧信銘 Student：Hsin Ming Lu

指導教授：彭南夫 博士 Advisor：Nan Fu Peng

國 立 交 通 大 學 理 學 院

統 計 學 研 究 所

博 士 論 文

Submitted to Institute of Statistics College of Science

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in Statistics December 2007

Hsinchu, Taiwan, Republic of China

相依正隨機變數之合其分佈函數之最佳化超立方體估計

學生：盧信銘 指導教授：彭南夫 博士

國立交通大學統計學研究所 博士班

摘 要

在給定聯合分佈函數之下，對於計算多個相依的正隨機變數合之

分佈函數以及機率函數，我們提供一個最佳化的幾何數值方法。這個

方法包含對超立方體的積分並估計”半”超立方體的體積。同時我們也

提供了數值分析的結果。

An optimal hypercube approximation of the distribution

function of sum of positive dependent random variables

Student: Hsin Ming Lu Advisor: Dr. Nan Fu Peng

Institute of Statistics

Natinal Chiao Tung Unveristy

Abstract

We present optimal geometric numerical methods for computing the

distribution function and the density function of sum of several positive

dependent random variables with known joint distribution function. This

method involves integration on high dimensional hypercubesm, and

estimating the volume of “half” hypercubes. Numerical results are also

presented.

致 謝

待在交大的日子從碩士班到博士班轉眼之間，已經度過六年多的

光景。這六年多來感謝交大統研所師長的指導與照顧，讓我學到如何

由零到有的研究方法。在此尤其感謝我的指導教授彭南夫老師，感謝

他在這六年內對我的幫助，未來不論我在教書或是工作，絕對不會忘

記交大師長與彭老師對我的教誨。也感謝口試委員陳鄰安教授、洪慧

念教授、林宗儀教授以及許英麟教授給予我寶貴的意見。在此要感謝

我的家人，尤其是我的母親對我的支持，讓我能拿到博士學位。最後

要感謝我的女朋友，感謝淑珍能夠陪伴在我身邊，在失落時能安慰

我，給我研究的動力。在此衷心地希望大家都能平安順利的享受人生。

盧信銘 謹誌于

國立交通大學統計研究所

中華民國九十六年十二月

Contents

1. INTRODUCTION ……… 2

2. DECOMPOSITION OF

AND THE VOLUME OF

……… 5

3. AN OPTIMAL APPROXIMATION OF

……… 8

4. AN OPTIMAL APPROXIMATION OF

……… 14

5. NUMERICAL RESULTS ………19

6. DISCUSSION ……… 24

List of Tables

1. The optimal

α

and

β

……… 10

2. The optimal

γ

and

δ

……… 18

3. Sum of independent Gamma random variables, each with scale

and shape parameter 1, 1, 2, 3 and 4,respectively.……19

4. Sum of independent Gamma random variables, each with scale

and shape parameter 1, 1, 2, 3 and 4,respectively.(compared

approximations)……… 20

5. Sum of i.i.d. Weibull random variables, each with the same

shape parameter 2 and scale parameter 1 ……… 21

6. Sum of dependent random variables with joint survival

function

∏

∑

……… 22

7. Sum of dependent random variables with joint survival

function

1 i i

i=ti − = t

−

∑

……… 23

8. Sum of dependent random variables with joint survival

function

∑

……… 24

9. Sum of dependent random variables with joint survival

function

∑

…… 25

10. Sum of independent Gamma random variables, each with scale

and shape parameter 1, 1, 2, 3 and 4,respectively.

List of Figures

1. The left is type-1 set and the right is type-2 set …… 3

2. The unions of type-1 set and type-2 set in 3 dimension… 6

3. The left is optimal case, the middle is case 1 and the right

is case 2 ……… 8

4. The distance between the point

X

and the hyperplane in the

optimal case……… 18

5. Comparison of the exact cdf and pdf with the hypercube

approximations………20

6. The alternative volume-invariant set……… 26

An optimal hypercube approximation of the

distribution function of sum of positive dependent

random variables

Student: Hsin-Ming Lu Advisor: Dr. Nan-Fu Peng

Institute of Statistics National Chiao Tung University

Hsinchu, Taiwan

Abstract

We present optimal geometric numerical methods for computing the distri-bution function and the density function of sum of several positive dependent random variables with known joint distribution function. This method in-volves integration on high dimensional hypercubes, and estimating the volume of ”half” hypercubes. Numerical results are also presented.

1.

INTRODUCTION

The aim of this paper is to present some geometric ideas on numerically computing the distribution function and the density function of sum of k positive dependent continuous random variables. This problem is originated from acquiring the distri-bution function and the density function of a renewal in a renewal process in which a renewal contains several dependent steps, each is Weibull distributed. Unlike inde-pendent r.v.’s, there have been few papers concentrated on sum of deinde-pendent r.v.’s. The result of Serfozo (1986) need strong restrictions on dependent r.v.’s. We offer an innovative idea to express the approximations of distribution function and den-sity function of sum of positive dependent random variables in a closed form. Our numerical method deal with sum of arbitrary k positive dependent r.v.’s as long as their joint distribution function or joint survival function are known and their joint density function is uniformly continuous.

Assume X1, X2, ..., Xk are r.v.’s with joint distribution function

G(x1, x2, ..., xk) or joint survival function S(x1, x2, ..., xk). Let

Rk₊ =©x ∈ Rk : xi ≥ 0, for i = 1, 2, ..., k

ª

.

For a ∈ Rk

+ and b > 0, we define a hypercube in Rk+ that

Bb, a =

©

x ∈ R₊k : xi ∈ [ai, ai+ b], for i = 1, 2, ..., k

ª

, (1)

and define a ”half” hypercube for s ∈ (0, k)

Bs b, a= © x ∈ Rk +: x ∈ Bb, a and 10a ≤ 10x ≤ 10a + bs ª (2) where 1 ∈ Rk

+ is a column vector with each element being 1. The hyperplane that

is tangent to Bs b, a is

©

x ∈ Rk _{: 1}0_{x = 1}0_{a + bs}ª_{. We call the ”half” hypercube B}s

b, a a type-s set with size b and tail a. Without loss of generality, let k = 3 and we present type-1 set and type-2 set with size 1 and tail 0 in figure 1.

To approximate the distribution function F (t) =R_B1

t, 0g(x)dx where g is the joint

density function, we in section 2 decompose B1

t, 0 into union of subsets of the form (1) and (2) and then estimate B1

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 1: The left is type-1 set and the right is type-2 set.

Integration of g on a hypercube can be obtained easily in terms of G, see Durrett (1994) p120 and p121 as the following formula.

P (ai ≤ Xi ≤ bi, for i = 1, 2, ..., k) = k X i=0 (−1)i X ci∈Di G(c1i, c2i, ..., cki), where ci = (c1i, c2i, ..., cki) ∈ Di and

Di = {(d1, d2, ..., dk) : exact i d0s are a0s and (k − i) d0s are b0s} .

The key of our method is to find a suitable hypercube, or a few hypercubes, to replace a Bs

b, a contained in Bt, 01 . To do so, the volume of that Bb, as has to be known. Unlike Albert (2002), Barrow and Smith (1979) and Mitra (1971) in which are presented the distribution function of sum of uniformly distributed r.v.’s, we propose a recursive formula which is easier to obtain. In section 3, we present an optimal approximation and some compared approximations of F (t). In section 4, we present an optimal approximation of f (t) = dF (t)_dt . Numerical results are attached in section 5. The first example is of sum of independent Gamma r.v.’s, and it’s shown that the excellent performance to the hypercube approximation. The second example involves sum of i.i.d. Weibull r.v.’s, and it’s well known that the Weibull model plays an important role in many fields such as reliability applications and so on, see Pham and Lai (2007). Santos Filho and Yacoub(2006) deal with the

approximation of probability density function and distribution function of sum of i.i.d. Weibull r.v.’s in a simple and closed form. In our method, not only gives a simple and closed form also offers a good performance in precision of Weibull sums. The third example is about the dependent r.v.’s and some cases can be found in Nelson (1999). Some remaining discussions are listed in section 6.

2.

DECOMPOSITION OF B

t, 0

AND THE

VOL-UME OF B

In this section, we present a decomposition of B1

t, 0 and furthermore this decompo-sition can be used to obtain the volume of Bs

b, a contained in Bt, 01 . For a positive integer j, defining

ω(j) =©x ∈ Rk₊ : 10x = j and xi ∈ Z+∪ {0} , for i = 1, 2, ..., k

ª

,

we have the following decomposition of B1

t, 0. Theorem 1. Let c ∈ (0, 1). If £1 c ¤ < k, then B1 t, 0 = [1 c] [ j=0 [ y∈ω(j) B1c−j ct,cty , (3) and if £1 c ¤ ≥ k, then B_{t, 0}1 =    [1 c_[]−k j=0 [ y∈ω(j) Bct,cty   [    [1 c] [ j=[1_c]−k+1 [ y∈ω(j) B1c−j ct,cty    . (4)

Furthermore, the intersection of any two sets in (3) and (4) has Lebesque measure zero.

PROOF. We prove (4) only, for (3) can be obtained from (4). Suppose £1_c¤ ≥ k

and let ¯ B1_{t, 0} =    [1 c_[]−k j=0 [ y∈ω(j) Bct,cty   [    [1 c] [ j=[1_c]−k+1 [ y∈ω(j) Bct,cty    . By the structure of ¯B1

t, 0, it is obvious that the intersection of any two sets in ¯Bt, 01 has Lebesque measure zero. For x ∈ B1

t, 0, we observe that there exists k nonnegative integers d1, d2, ..., dk such that xi ∈ [ctdi, ct(di+ 1)]. Since t ≥

P_k i=1xi ≥ P_k i=1ctdi, we have Pk_i=1di ≤ £₁ c ¤ . Hence x ∈ Bct,ctd ⊂ ¯Bt, 01 . Therefore, Bt, 01 ⊂ ¯Bt, 01 . For 0 ≤ j ≤£1 c ¤ − k, we have 1

c ≥ j + k. Therefore, if y ∈ ω(j) and x ∈ Bct,cty,

then xi ∈ [ctyi, ct(yi+ 1)] ⊂ [0, t] and

0 ≤ ctj = ct10_{y ≤ 1}0_{x ≤ ct1}0_{(y + 1) = ct(j + k) ≤ ct ·} 1

Figure 2: The unions of type-1 set and type-2 set in three-dimension.

Hence x ∈ B1

t, 0. We have Bct,cty ⊂ B1t, 0 or Bct,cty T B1 t, 0 = Bct,cty. For £1 c ¤ − k + 1 ≤ j ≤ £1 c ¤ and y ∈ ω(j)_{, if x ∈ B} ct,cty T B1 t, 0, then xi ∈

[ctyi, ct(yi+ 1)] and

ct10_{y ≤ 1}0_{x ≤ t = ctj + ct} µ 1 c− j ¶ = ct10_{y + ct} µ 1 c − j ¶ . Hence x ∈ B1c−j

ct,cty and we have Bct,cty

T B1 t, 0 ⊂ B 1 c−j ct,cty. Conversely, if x ∈ B 1 c−j ct,cty then xi ∈ [ctyi, ct(yi+ 1)] ⊂ [0, t] and

0 ≤ ctj = ct10_{y ≤ 1}0_{x ≤ ct1}0_{y + ct} µ 1 c − j ¶ = t. Hence x ∈ B1 t, 0 and so B 1 c−j ct,cty ⊂ Bct,cty T B1 t, 0. We have B 1 c−j ct,cty = Bct,cty T B1 t, 0, then (4) holds. 2

Corollary 1. For a positive integer n ≥ k,

B1 t, 0=  n−k[ j=0 [ y∈ω(j) Bt n, t ny  [   [n j=n−k+1 [ y∈ω(j) Bn−jt n, t ny   =   n−k_[ j=0 [ y∈ω(j) Bt n,nty  [   k−1_[ j=0 [ y∈ω(n−j) Bjt n,nty   . (5)

According to corollary 1, for any ”section size” n ∈ N and n ≥ k, we can decom-pose B1

t, 0 into unions of hypercubes and unions of type-j set, for j = 0, 1, 2, ..., k −1. In figure 2, we show the unions of type-1 set and type-2 set in three dimension. Note that type-0 set has Lebesque measure zero. Denote mk the Lebesque measure in

Rk_{. It is obvious that}

mk(Bb, as ) = bkmk(B1, as ) = bkmk(B1, 0s ).

The following theorem is an easy consequence of theorem 1.

Theorem 2. For s ∈ (0, 1] mk(Bs1, 0) = sk k!, (6) and for s ∈ (1, k) mk(B1, 0s ) = sk k! − [s] X x=1 Ck+x−1 x mk(B1, 0s−x). (7)

PROOF. (6) is trivial and we prove (7) only. For s ∈ (1, k), we let c = 1

s and apply (3) to obtain mk(Bs, 01 ) = [s] X x=0 X y∈ω(x) mk(B1, ys−x) = [s] X x=0 X y∈ω(x) mk(B1, 0s−x) = [s] X x=0 Ck+x−1 x mk(B1, 0s−x).

The last equality holds since the number of y ∈ ω(x) _{is C}k+x−1

x , and mk(Bs, 01 ) = s

k

k!, we obtain (7). 2

Note that the volume of Bs

1, 0 also represents the distribution function of sum

of uniformly distributed r.v.’s that have been shown in Albert (2002), Barrow and Smith (1979) and Mitra (1971). In a geometric viewpoint, we present the volume of Bs

1, 0 through a recursive formula in theorem 2 above. Therefore, we have the

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 3: The left is optimal case, the middle is case 1 and the right is case 2

3.

AN OPTIMAL APPROXIMATION OF F (t)

We in this section provide an optimal volume-invariant approximation to F (t). That is, we present an optimal set ˆB1

t, 0 such that mk ³ ˆ B1 t, 0 ´ = mk ¡ B1 t, 0 ¢ and approx-imate F (t) = R_B1 t, 0g(x )mk(dx ) by R ˆ B1

t, 0g(x )mk(dx ). We first find an optimal

estimate of Bjt n,

t ny

the type-j set with tail t

ny and size

t

n and, without loss of gen-erality, manipulate this method in B_{1, 0}j the type-1 set with tail 0 and size 1, for 1 ≤ j ≤ k − 1 and j ∈ Z+_{. Let} Aj = © Bl, 0\ Bd, (l−d)1 : 0 < d < l ≤ 1 and mk ¡ Bl, 0\ Bd, (l−d)1 ¢ = mk ¡ B_{1, 0}j ¢ª (8)

be a class of sets, each of which is the difference of two hypercubes and volume-invariant from Bj_{1, 0}. Given l, the d in (8) is d = ¡lk_{− m}

k ¡

Bj_{1, 0}¢¢1k_{. First, We} present two different volume-invariant sets as the following.

Case 1: d1j = 0 and l1j =

¡

mk

¡

B_{1, 0}j ¢¢1k_;

Case 2: d2j and l2j satisfy l2j − d2j = j_k and lk2j − dk2j = mk ¡

B_{1, 0}j ¢, where Blij, 0\

Bdij, (lij−dij)1 ∈ Aj, i = 1, 2.

Next, finding the optimal αj and βj as the following. Let 0 < αj < βj ≤ 1 satisfy

mk ¡ B_{1, 0}j \¡Bβj, 0\ Bαj, (βj−αj)1 ¢¢ =mk ¡¡ Bβj, 0\ Bαj, (βj−αj)1 ¢ \ B_{1, 0}j ¢ = inf Bl, 0\Bd, (l−d)1∈Aj mk ¡¡ Bl, 0\ Bd, (l−d)1 ¢ \ B_{1, 0}j ¢.

The first equality holds since mk ¡ Bβj, 0\ Bαj, (βj−αj)1 ¢ = mk ¡ B_{1, 0}j ¢. It is easy to see that the αj and βj can be minimized the nonoverlapping part of B1, 0j and

Bl, 0\ Bd, (l−d)1 ∈ Aj. In figure 3, we present the optimal volume-invariant set and

two different cases as above in two-dimension. The dark parts in figure 3 represents the estimate of B_{1, 0}j . To obtain the optimal αj and βj, we need the following theorem.

Theorem 3. For Bl, 0\ Bd, (l−d)1 ∈ Aj, we have

mk ¡¡ Bl, 0\ Bd, (l−d)1 ¢ \ B_{1, 0}j ¢ =mk ¡ B_{1, 0}j ¢− lk_m k ³ Bjl 1, 0 ´ + I µ l − d < j k ¶ dk_m k µ Bj−k(l−d)d 1, 0 ¶ , where I is an indicator function.

PROOF. If l − d ≥ _kj, then mk ¡ Bd, (l−d)1 T B_{1, 0}j ¢= 0. We have mk ¡¡ Bl, 0\ Bd, (l−d)1 ¢ \ B_{1, 0}j ¢ =mk ¡¡ Bl, 0\ B1, 0j ¢ \ Bd, (l−d)1 ¢ =mk ¡ Bl, 0\ B1, 0j ¢ − mk ³¡ Bl, 0\ B1, 0j ¢ \ Bd, (l−d)1 ´ =mk(Bl, 0) − mk ³ Bl, 0 \ B_{1, 0}j ´ − mk ¡ Bd, (l−d)1 ¢ =lk_{− l}k_m k ³ B1, 0 \ Bj1 l, 0 ´ − dk =lk_{− d}k_{− l}k_m k ³ Bjl 1, 0 ´ =mk ¡ B_{1, 0}j ¢− lk_m k ³ Bjl 1, 0 ´ .

The second equality from the bottom holds since 1

l > 1. If l − d < j k, then mk ³ Bd, (l−d)1 \ B_{1, 0}j ´= mk µ Bj−k(l−d)d d, (l−d)1 ¶ = dk_m k µ Bj−k(l−d)d 1, 0 ¶ .

Hence we have theorem 3. 2

Some numerical results of αj and βj are listed in table 1, for j = 0, 1, ..., k − 1. Note that for j = 0, we let α0 = β0 = 0.

Let ˆ B1 t, 0 =  n−k[ j=0 [ y∈ω(j) Bt n,nty  [  k−1[ j=0 [ y∈ω(n−j) ³ Bt nβj,nty\ Bntαj,nt[y+(βj−αj)1] ´ 

Table 1: The optimal αj and βj k j βj αj k j βj αj 2 1 0.816497 0.408249 6 1 0.349801 0.276119 2 0.687728 0.541633 3 1 0.602942 0.374505 3 0.927982 0.719393 2 0.999999 0.550319 4 0.999999 0.657172 5 1.000000 0.334008 4 1 0.483559 0.337726 2 0.895692 0.615614 7 1 0.307856 0.251513 3 1.000000 0.451797 2 0.610611 0.498411 3 0.856245 0.693457 5 1 0.405585 0.305058 4 0.988259 0.783328 2 0.782639 0.585206 5 0.999999 0.586961 3 0.992610 0.716314 6 1.000000 0.295597 4 1.000000 0.383834 be our estimate of B1

t, 0. The second part of the right hand side of ˆBt, 01 represents the estimate of the unions of Bjt

n, t ny

, for y ∈ ω(n−j) _{and j = 0, 1, ..., k − 1. In each}

j, we have mk ³ Bt nβj,nty \ Bntαj,nt[y +(βj−αj)1] ´ = µ t n ¶_k mk ¡ Bβj, y \ Bαj, y+(βj−αj)1 ¢ = µ t n ¶_k mk ¡ Bβj, 0\ Bαj, (βj−αj)1 ¢ = µ t n ¶_k mk ¡ Bj_{1, 0}¢ = mk ³ Bjt n,nty ´ . Therefore, we obtain mk ¡ B1 t, 0 ¢ = mk ³ ˆ B1 t, 0 ´

. In the other hand, the nonoverlap-ping parts of B1

t, 0 and ˆBt, 01 are minimized, thus ˆBt, 01 would be a reasonable and optimal estimate of B1

Furthermore, our approximation ˆF (t) to F (t) is ˆ F (t) = Z ˆ B1 t, 0 g(x )mk(dx ) = n−k X j=0 X y∈ω(j) Z Bt n ,n yt g(x )mk(dx ) + k−1 X j=0 X y∈ω(n−j)   Z Bt n βj ,n yt g(x )mk(dx ) − Z Bt n αj ,n [y+(βj −αj )1]t g(x )mk(dx )   = n−k X j=0 X y∈ω(j) P µ t nyi ≤ Xi ≤ t n(yi+ 1), i = 1, 2, ..., k ¶ + k−1 X j=0 X y∈ω(n−j) · P µ t nyi ≤ Xi ≤ t n(yi+ βj), i = 1, 2, ..., k ¶ −P µ t n[yi+ (βj − αj)] ≤ Xi ≤ t n(yi+ βj), i = 1, 2, ..., k ¶¸ . (9)

The following theorem shows that ˆF (t) converges to F (t) when section size n tends to

infinity under the condition that the joint density function g is uniformly continuous. Before the theorem, we need the following lemma which is elementary.

Lemma 1. Given n and t, suppose that g is uniformly continuous in Rk

+, and for all y ∈ Rk + and z, w ∈ B_nt,t ny, then we have | g(z) − g(w) |≤ M n , for some M > 0.

Theorem 4. Suppose that g is uniformly continuous in [0, t]k_{, then}

| ˆF (t) − F (t) |−→ 0 as n −→ +∞.

Furthermore, the convergence rate of ˆF (t) is n−2_.

PROOF. Note that

Bjt n, t ny \³Bt nβj, t ny \ B t nαj, t n[y+(βj−αj)1] ´ ⊂ Bt n, t ny and ³ Bt nβj,nty \ Bntαj,nt[y +(βj−αj)1] ´ \ Bjt n,nty ⊂ B t n,nty.

For each y ∈ ω(n−j)_{, and fixing a point z ∈ Bt}

n, t

ny, we have by lemma 1 that

| ˆF (t) − F (t) | = | Z ˆ B1 t, 0 g(x )mk(dx ) − Z B1 t, 0 g(x )mk(dx ) | = | Z ˆ B1 t, 0\B1t, 0 g(x )mk(dx ) − Z B1 t, 0\ ˆBt, 01 g(x )mk(dx ) | = | k−1 X j=0 X y∈ω(n−j)   Z  Bt n βj ,n yt \Bn αj ,t n [y +(βj −αj )1]t  \Bjt n ,n yt g(x )mk(dx ) − Z Bjt n ,n yt \  Bt n βj ,n yt \Bn αj ,t n [y+(βj −αj )1]t g(x )mk(dx )   | ≤ k−1 X j=0 X y∈ω(n−j) | Z  Bt n βj ,n yt \Bn αj ,t n [y +(βj −αj )1]t  \Bjt n ,n yt g(x )mk(dx ) − Z Bjt n ,n yt \  Bt n βj ,n yt \Bn αj ,t n [y+(βj −αj )1]t g(x )mk(dx ) | ≤ k−1 X j=0 X y∈ω(n−j) | Z  Bt n βj ,n yt \Bn αj ,t n [y +(βj −αj )1]t  \Bjt n ,n yt µ g(z ) + M n ¶ mk(dx ) − Z Bjt n ,n yt \  Bt n βj ,n yt \Bn αj ,t n [y+(βj −αj )1]t  µ g(z ) − M n ¶ mk(dx ) | = k−1 X j=0 X y∈ω(n−j) 2M n mk ³ Bjt n, t ny \ ³ Bt nβj, t ny\ B t nαj, t n(y+βj−αj)1 ´´ =2M n µ t n ¶_{k k−1X} j=0 X y∈ω(n−j) mk ¡ Bj_{1, 0}\¡Bβj, 0\ Bαj, (βj−αj)1 ¢¢ =2M n µ t n ¶_{k k−1X} j=0 X y∈ω(n−j) · mk ¡ B_{1, 0}j ¢− βk jmk µ B j βj 1, 0 ¶ +I µ βj − αj < j k ¶ αk jmk  B j−k(_{βj −αj}) αj 1, 0     ≤2M n µ t n ¶_k kc1nk−1 −→ 0 as n −→ +∞,

elements in ω(n−j) _{is C}k+n−j−1

n−j when n is suitably larger than k. 2

In order to show the priority of the optimal approximation of F (t), we also present two compared volume-invariant estimates of B1

t, 0. For j = 0, 1, ..., k − 1, the first estimate ˆB1

t, 0(1), is to let d1j = 0 and l1j = ¡

mk

¡

B_{1, 0}j ¢¢1k_{, and the second} estimate ˆB1

t, 0(2), is to let d2j and l2j satisfy lk2j− dk2j = mk ¡ B_{1, 0}j ¢ and l2j− d2j = j_k, where Blij, 0\ Bdij, (lij−dij)1 ∈ Aj, i = 1, 2. That is ˆ B_{t, 0(1)}1 =   n−k_[ j=0 [ y∈ω(j) Bt n, t ny  [   k−1_[ j=0 [ y ∈ω(n−j) Bt nl1j, t ny   and ˆ B1 t, 0(2)=   n−k_[ j=0 [ y∈ω(j) Bt n, t ny  [   k−1_[ j=0 [ y∈ω(n−j) ³ Bt nl2j, t ny \ B t nd2j, t n[y+(l2j−d2j)1] ´  . Therefore, we have ˆ F(1)(t) = Z ˆ B1 t, 0(1) g(x )mk(dx ) = n−k X j=0 X y ∈ω(j) P µ t nyi ≤ Xi ≤ t n(yi+ 1), i = 1, 2, ..., k ¶ + k−1 X j=0 X y∈ω(n−j) P µ t nyi ≤ Xi ≤ t n(yi+ l1j), i = 1, 2, ..., k ¶ and ˆ F(2)(t) = Z ˆ B1 t, 0(2) g(x )mk(dx ) = n−k X j=0 X y∈ω(j) P µ t nyi ≤ Xi ≤ t n(yi+ 1), i = 1, 2, ..., k ¶ + k−1 X j=0 X y∈ω(n−j) · P µ t nyi ≤ Xi ≤ t n(yi+ 12j), i = 1, 2, ..., k ¶ −P µ t n[yi+ (l2j − d2j)] ≤ Xi ≤ t n(yi+ l2j), i = 1, 2, ..., k ¶¸

to be the compared approximations with respect to ˆF (t). Some numerical results of

ˆ

4.

AN OPTIMAL APPROXIMATION OF f (t)

Our optimal estimate ˜f (t) of f (t) is the derivative of ˜F (t) where ˜F (t) is a slight

modification of ˆF (t). Therefore, points of Rk−1

+ show up throughout this section.

In order to be consistent with the notations in the previous sections, we denote without alteration a vector in Rk _{by a or x , etc, and denote a vector in R}k−1 _by

a(k−1) _{or x}(k−1)_{, etc. Furthermore, we let 1}

i ∈ Rk and 1i = (0, ..., 0, 1, 0, ..., 0), the

ith component of which is one and the others are zero, for i = 1, 2, ..., k. For any

fixed component index i ∈ {1, 2, ..., k}, let

Di b, a+(s−ai)1i = © x ∈ Rk + : aj ≤ xj ≤ aj + b, j 6= i, xi = s ª (10) be a ”hyperdisc” with ith component equals to s and the others fall in the interval [aj, aj + b], for j 6= i. The hyperdisc Db, a+(s−ai i)1iin (10) is in fact (k−1)-dimensional and we denote mk−1 the Lebesque measure in Rk−1. The next lemma is elementary. Lemma 2. d dt Z Btb,ta g(x)mk(dx) = k X i=1 " (ai+ b) Z Di tb,t(a+b1i) g(z)mk−1(dz) − ai Z Di tb,ta g(z)mk−1(dz) # . (11) Let B1

t, 0(k−1) ⊂ Rk−1+ be similarly defined as (2), we have

f (t) = Z B1 t, 0(k−1) g(x(k−1)_{, t − 1}(k−1)0 x(k−1)_)m k−1 ¡ dx(k−1)¢_{. (12)}

Note that the following notation y_(j)=¡y(j),1, y(j),2, ..., y(j),k

¢

∈ ω(j) _{that coincides}

ˆ f (t) = d ˆF (t) dt = n−k X j=0 X y(j)∈ω(j) k X i=1  y(j),i+ 1 n Z Di t n ,n (y(j)+1i)t g(z )mk−1(dz ) −y(j),i n Z Di t n ,n y (j)t g(z )mk−1(dz )   + k−1 X j=0 X y(n−j)∈ω(n−j) k X i=1     y(n−j),i+ βj n Z Di t n βj ,n (y (n−j)+βj 1i)t g(z )mk−1(dz ) −y(n−j),i n Z Di t n βj ,n y(n−j)t g(z )mk−1(dz )   −  y(n−j),i+ βj n Z Di t n αj ,n (y(n−j)+(βj −αj )1+αj 1i)t g(z )mk−1(dz ) − y(n−j),i+ (βj− αj) n Z Di t n αj ,n (y (n−j)+(βj −αj )1)t g(z )mk−1(dz )     . Since y(j),i+ 1 = y(j+1),iand y(j)+ 1i ∈ ω(j+1), we can reduce the first mathematical expression of the right hand side of the above formula to obtain

ˆ f (t) = k X i=1 X y(n−k+1)∈ω(n−k+1) y(n−k+1),i n Z Di t n ,n y(n−k+1)t g(z )mk−1(dz ) + k X i=1 k−1 X j=0 X y(n−j)∈ω(n−j)     y(n−j),i+ βj n Z Di t n βj ,n (y (n−j)+βj 1i)t g(z )mk−1(dz ) − y(n−j),i n Z Di t n βj ,n y (n−j)t g(z )mk−1(dz )   −  y(n−j),i+ βj n Z Di t n αj ,n (y(n−j)+(βj −αj )1+αj 1i)t g(z )mk−1(dz ) − y(n−j),i+ (βj − αj) n Z Di t n αj ,n (y(n−j)+(βj −αj )1)t g(z )mk−1(dz )     . (13)

Recall that we called ˆF (t) in (9) volume-invariant, and ˆf (t) has similar property as

we derive as the following. Set g = 1 and mk ¡ B1 t, 0 ¢ = mk ³ ˆ B1 t, 0 ´ , we have mk−1 ³ B_{t, 0}1 (k−1) ´ = dmk ¡ B1 t, 0 ¢ dt = dmk ³ ˆ B1 t, 0 ´ dt = k X i=1 X y(n−k+1)∈ω(n−k+1) y(n−k+1),i n mk−1 ³ Di t n,nty(n−k+1) ´ + k X i=1 n X j=n−k+1 X y(j)∈ω(j) ½· y(j),i+ βn−j n mk−1 ³ Di t nβn−j,nt(y(j)+βn−j1i) ´ − y(j),i n mk−1 ³ Dit nβn−j, t ny(j) ´i − · y(j),i+ βn−j n mk−1 ³ Di t nαn−j,nt(y(j)+(βn−j−αn−j)1+αn−j1i) ´ − y(j),i+ (βn−j− αn−j) n mk−1 ³ Di t nαn−j,nt(y(j)+(βn−j−αn−j)1) ´¸¾ . (14)

Noting that in (14) the D’s with negative sign are not necessarily subsets of the

D’s with positive sign, we thus from (14) call ˆf (t) = d ˆF (t)_dt weighted-signed-measure

invariant. The derivative of any other volume-invariant approximation of F (t) of the form (9) should also be weighted-signed-measure invariant. The next theorem shows that ˆf (t) is a candidate of approximation of f (t). The proof of theorem 5

below is similar to that of theorem 4, and is hence omitted it here.

Theorem 5. Suppose that g is uniformly continuous, then

| ˆf (t) − f (t) |−→ 0 as n −→ +∞.

Furthermore, the convergence rate of ˆf (t) is n−1_.

Now we proceed acquiring an optimal approximation of f (t). Let D_k(j)=©x ∈ Rk

+

: 10_{x = j} and observe that for y ∈ ω}(n−j)_{, D}(t)

k mounting on B j

t

n, nty is equivalent to D(j)_k mounting on B_{1, 0}j . Denote dis

³

x , D_k(j)

´

= |j−1√0x |

k the distance between the point x and the hyperplane D(j)_k . Recall that a set Bl, 0 \ Bd, (l−d)1 in Aj of (8)

plays the major role in estimating Bj_{1, 0}, where d =¡lk_{− m} k

¡

B_{1, 0}j ¢¢k1_{. Let the set} of vertices of Bl, 0\ Bd, (l−d)1 is Z_l(j)= ( 0, l m X b=1 1ib, (l − d)1, d m X b=1 1ib+ (l − d)1 with {i1, i2, ..., im} ⊂ {1, 2, ..., k} , m = 1, 2, ..., k − 1} . Next theorem makes our criterion for choosing optimal ˜f (t) possible.

Theorem 6. For each j, 1 ≤ j ≤ k − 1 and j ∈ Z+_{, we have}

X x∈Z_l(j) dis³x, D_k(j)´= √1 k Ã_k−1 X m=0 Ck m|j − ml| + k−1 X m=0 Ck m|j − kl + (k − m)d| ! . PROOF. Since dis à l m X b=1 1ib, D (j) k ! = | j − ml |√ k and dis à d m X b=1 1ib + (l − d)1, D (j) k ! = | j − [kl − (k − m)d] |√ k

for all (i1, i2, ..., im), we have the theorem. 2

Our optimal approximation ˜f (t) of f (t) must satisfy ˜f (t) = d ˜F (t)_dt and ˜F (t) =

R ˜ B1 t, 0g(x )mk(dx ) where ˜ B1 t, 0 =  n−k[ j=0 [ y ∈ω(j) Bt n, t ny  [  k−1[ j=0 [ y∈ω(n−j) ³ Bt nγj, t ny \ B t nδj, t n[y+(γj−δj)1] ´ 

with γj and δj satisfying δj = ¡ γk j − mk ¡ B_{1, 0}j ¢¢1k _and X x ∈Z_γj(j) dis ³ x , D(j)_k ´ = inf l X x ∈Z_l(j) dis ³ x , D(j)_k ´ .

In figure 4, given the optimal γ1 and δ1, we show the distance between the point x

and the hyperplane in two-dimension. The optimal γj and δj are listed in table 2, for j = 0, 1, ..., k − 1. Note that γ0 = δ0 = 0. Numerical results of ˜f (t) are shown in

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 4: The distance between the point x and the hyperplane in the optimal case

Table 2: The optimal γj and δj

k j γj δj k j γj δj 2 1 0.750004 0.250011 6 1 0.348916 0.273116 2 0.682740 0.524109 3 1 0.590962 0.341192 3 0.905298 0.607947 2 0.977867 0.466793 4 1.000000 0.657177 5 0.999996 0.333149 4 1 0.470973 0.294631 2 0.863099 0.484134 7 1 0.304235 0.237667 3 0.999999 0.451796 2 0.603599 0.471527 3 0.848451 0.662802 5 1 0.399080 0.282191 4 0.972706 0.702235 2 0.771495 0.545529 5 0.997605 0.495816 3 0.971687 0.619478 6 0.999981 0.251129 4 0.999138 0.331933

Table 3: Sum of independent Gamma random variables, each with scale parameter 1 and shape parameter 1, 1, 2, 3 and 4, respectively.

t n F (t) F (t)ˆ RE( ˆF (t)) f (t) f (t)˜ RE( ˜f (t)) 3 25 0.000292 0.000293 0.003556 0.000810 0.000813 0.002956 50 0.000293 0.000894 0.000811 0.000637 6 25 0.042621 0.042621 0.000014 0.041303 0.041232 0.001732 50 0.042621 0.000002 0.041275 0.000675 9 25 0.294012 0.293557 0.002190 0.118580 0.118156 0.003575 50 0.293897 0.000552 0.118431 0.001255 12 25 0.652771 0.651690 0.004767 0.104837 0.104576 0.002487 50 0.652499 0.001196 0.104723 0.001093 15 25 0.881536 0.880589 0.009064 0.048611 0.048691 0.001652 50 0.881299 0.002264 0.048602 0.000172 18 25 0.969634 0.969163 0.015991 0.014985 0.015120 0.009002 50 0.969517 0.003974 0.015008 0.001523 21 25 0.993749 0.993588 0.025957 0.003485 0.003554 0.019769 50 0.993709 0.006411 0.003499 0.004011

5.

NUMERICAL RESULTS

We present numerical computation on three types of distributions, each involves sum of five random variables. In each case, we choose the section size n = 25 and

n = 50.

Example 1: (Independent but non-identically distributed r.v.’s) The first in-cludes five independent Gamma r.v.’s, each with scale parameter 1 and shape pa-rameter 1, 1, 2, 3 and 4, respectively. It’s well known that the sum of five Gamma r.v.’s is still Gamma distributed with scale parameter 1 and shape parameter 11. The main results and the compared approximations are listed in table 3 and 4, re-spectively. For i = 1, 2, let RE( ˆF (t)) = _{F (t)(1−F (t))}|F (t)− ˆF (t)| , RE( ˆF(i)(t)) = |F (t)− ˆ_{F (t)(1−F (t))}F(i)(t)| and

RE( ˜f (t)) = |f (t)− ˜_{f (t)}f (t)| stand for corresponding relative error of ˆF (t), ˆF(i)(t) and ˜f (t)

when F (t) and f (t) are available. Figure 5 compares the exact cumulative distribu-tion funcdistribu-tion and probability density funcdistribu-tion with the approximadistribu-tion results (box symbol) in table 3 for section size n = 50. It’s shown that the hypercube

approxi-0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 t 0 0.02 0.04 0.06 0.08 0.1 0.12 0 5 10 15 20 25 t

Figure 5: Comparison of the exact cdf and pdf with the hypercube approximations

Table 4: Sum of independent Gamma random variables, each with scale parameter 1 and shape parameter 1, 1, 2, 3 and 4, respectively. (compared approximations)

t n F (t) Fˆ(1)(t) RE( ˆF(1)(t)) Fˆ(2)(t) RE( ˆF(2)(t)) 3 25 0.000292 0.000294 0.005904 0.000294 0.005557 50 0.000293 0.001527 0.000293 0.001438 6 25 0.042621 0.042618 0.000083 0.042618 0.000070 50 0.042620 0.000011 0.042621 0.000009 9 25 0.294012 0.293211 0.003857 0.293257 0.003636 50 0.292810 0.000972 0.293822 0.000916 12 25 0.652771 0.650877 0.008353 0.650980 0.007901 50 0.652295 0.002100 0.652321 0.001982 15 25 0.881536 0.879870 0.015948 0.879956 0.015125 50 0.881120 0.003981 0.881143 0.003763 18 25 0.969634 0.968799 0.028344 0.968840 0.026941 50 0.969427 0.007006 0.969438 0.006632 21 25 0.993749 0.993461 0.046448 0.993474 0.044239 50 0.993679 0.011345 0.993682 0.010752

Table 5: Sum of i.i.d. Weibull random variables, each with the same shape param-eter 2 and scale paramparam-eter 1.

t n F (t)ˆ Fˆ(1)(t) Fˆ(2)(t) f (t)˜ 2.814097 25 0.050203 0.050410 0.050384 0.601481 50 0.050026 0.050075 0.050068 0.601295 3.357615 25 0.150178 0.150540 0.150498 1.249670 50 0.149882 0.149965 0.149955 1.251369 3.702083 25 0.250187 0.250554 0.250516 1.630926 50 0.249910 0.249991 0.249982 1.634485 3.987351 25 0.350015 0.350294 0.350271 1.841351 50 0.349838 0.349896 0.349890 1.846241 4.251523 25 0.449991 0.450116 0.450116 1.916361 50 0.449970 0.449987 0.449987 1.921880 4.513044 25 0.549681 0.549611 0.549647 1.871391 50 0.549849 0.549818 0.549824 1.876739 4.789864 25 0.649437 0.649152 0.649204 1.712146 50 0.649806 0.649722 0.649735 1.716465 5.106096 25 0.749394 0.748904 0.748978 1.435130 50 0.749945 0.749813 0.749832 1.437538 5.511143 25 0.849149 0.848519 0.848605 1.027061 50 0.849807 0.849644 0.849666 1.026800 6.217922 25 0.949411 0.948872 0.948938 0.438777 50 0.949942 0.949808 0.949825 0.435943

mation results match the exact cdf and pdf well. An interesting numerical result is that RE( ˜f (t)) is much smaller than n−1_{, the convergence rate of ˜f (t), as shown in} theorem 5.

Example 2: (Independent and identically distributed r.v.’s) The second includes five i.i.d. Weibull r.v.’s with shape parameter 2 and scale parameter 1. The results are listed in table 5. Although relative error is not available in table 5. We see that

n = 25 is good enough when compared to double the section size from n = 25 to n = 50. The t’s chosen in table 5 are estimated quantiles. It’s also shown that the

Table 6: Sum of dependent random variables with joint survival function exp(−P5_i=1ti− 0.5Π5i=1ti)

t n F (t)ˆ Fˆ(1)(t) Fˆ(2)(t) f (t)˜ 1 25 0.004369 0.004367 0.004367 0.017522 50 0.004371 0.004370 0.004371 0.017534 3 25 0.183282 0.183133 0.183153 0.161367 50 0.183434 0.183396 0.183401 0.161537 5 25 0.555022 0.554577 0.554635 0.183577 50 0.555484 0.555369 0.555384 0.183728 7 25 0.833752 0.833208 0.833275 0.085304 50 0.834315 0.834177 0.834194 0.085122 9 25 0.941615 0.941326 0.941356 0.034339 50 0.941860 0.941794 0.941802 0.034266 11 25 0.984243 0.984160 0.984174 0.011102 50 0.984368 0.984334 0.984338 0.011050 13 25 0.996348 0.996298 0.996305 0.002749 50 0.996409 0.996403 0.996405 0.002726

In Santus Filho and Yacoub (2006), some moments of sum of Weibull r.v.’s are needed to solve some parameters, and moreover a simple and closed approximation form can be obtained. In stead of Santus Filho and Yacoub (2006), our method is based on decomposing B1

t, 0 and making some estimates on type-j set, for j = 0, 1, ..., k − 1. We also offer a simple and closed approximated distribution function and density function of sum of not only Weibull but also arbitrary positive random variables while their joint distribution function or joint survival function are given. Example 3: (Dependent r.v.’s) In the third example, we present four kinds of dependence r.v.’s. These examples can be found in Nelson (1999) p20, p29, p46 and p51, and involve five dependent r.v.’s with joint survival functions, S(t1, t2, t3, t4, t5) =

Table 7: Sum of dependent random variables with joint survival function exp(−P5_i=1ti− 0.1 maxi=1∼5ti).

t n F (t)ˆ f (t)˜ 1 25 0.026341 0.041180 50 0.026410 0.041218 5 25 0.629634 0.163108 50 0.630152 0.163180 9 25 0.960979 0.025592 50 0.961216 0.025492 13 25 0.997747 0.001714 50 0.997780 0.001692 17 25 0.999905 0.000078 50 0.999908 0.000076 21 25 0.999997 0.000003 50 0.999997 0.000003 25 25 1.000000 0.000000 50 1.000000 0.000000

of two dimensional copula. (1) S(t1, t2, t3, t4, t5) = exp(− 5 X i=1 ti− 0.5Π5i=1ti), (2) S(t1, t2, t3, t4, t5) = exp(− 5 X i=1 ti− 0.1 max i=1∼5ti), (3) S(t1, t2, t3, t4, t5) = Ã 1 + 5 X i=1 ti1{ti≥0} !_−1.5 , (4) S(t1, t2, t3, t4, t5) = exp(− max i=1∼5ti) + 1 3exp(−2 5 X i=1 ti) h 1 − exp(3 min i=1∼5ti) i .

The results are listed in table 6 to table 9, respectively. Note that we only present the optimal approximations of F (t) and f (t) in table 7 to table 9. Some dependent examples about the Weibull model are provided in Lai and Xie (2006) p164 and p165, and thus our hypercube method can be also used to obtain the approximation of distribution function and density function of sum of dependent Weibull r.v.’s.

Table 8: Sum of dependent random variables with joint survival function ¡ 1 +P5_i=1ti1{ti≥0} ¢−1.5 t n F (t)ˆ f (t)˜ 1 25 0.064330 0.018105 50 0.064482 0.018121 5 25 0.591137 0.080680 50 0.591353 0.080733 9 25 0.775959 0.085038 50 0.776119 0.085083 13 25 0.854939 0.081448 50 0.855056 0.081485 17 25 0.896567 0.076873 50 0.896656 0.076905 21 25 0.921547 0.072559 50 0.921617 0.072588 25 25 0.937890 0.068724 50 0.937947 0.068750

6.

DISCUSSION

Some numerical results in section 5 show that the more overlapping with these hypercubes , our estimators are, the more accurate our approximation reach. In fact, ¯B1

t, 0 in the proof of theorem 1 can be used to be our estimator of Bt, 01 , yet this kind of estimator provides a less precision unless the section size n is huge. Each type-j set, for j = 1, 2, ..., k − 1, hides behind k type-(j − 1) sets, hence the roofs belong to the out most hypercubes of ¯B1

t, 0 leaving others unattended. We alternatively estimate all type-j sets individually and thus there are some ”holes” in our estimate ˆB1

t, 0, for j = 1, 2, ..., k − 1.

The volume-invariant set Aj in (8) can be extended to

A∗ j = ( ¡ Bl, 0\ Bd, (l−d)1 ¢ [Ã[k m=1 B k− 1kd, llm ! : 0 < d < l ≤ 1 and mk à ¡ Bl, 0\ Bd, (l−d)1 ¢ [Ã[k m=1 B k− 1k_{d, llm} !! = mk ¡ B_{1, 0}j ¢ ) ,

Table 9: Sum of dependent random variables with joint survival function exp(− maxi=1∼5ti) + 1₃exp(−2

P₅

i=1ti) [1 − exp(3 mini=1∼5ti)].

t n F (t)ˆ f (t)˜ 1 25 0.176449 0.181858 50 0.175551 0.182028 5 25 0.631992 0.074024 50 0.631940 0.074011 9 25 0.834713 0.033051 50 0.834712 0.033050 13 25 0.925727 0.014855 50 0.925727 0.014855 17 25 0.966627 0.006675 50 0.966627 0.006675 21 25 0.985004 0.002999 50 0.985004 0.002999 25 25 0.993262 0.001348 50 0.993262 0.001348

where lm = (0, ..., 0, 1, 0, ..., 0) the mth component is 1 and the others are zero. For instance, we take l3j = ¡ mk ¡ B_{1, 0}j ¢¢k1 _{and d} 3j = l3j − j_k, and ˆ B1 t, 0(3)=  n−k[ j=0 [ y∈ω(j) Bt n, t ny   [  k−1_[ j=0 [ y∈ω(n−j) Ã ³ Bt nl3j,nty\ Bntd3j,nt[y+(l3j−d3j)1] ´ [Ã_[k m=1 Bt nk − 1_k d3j, _nt(y+l3j1m) !! 

be our estimate of B_{1, 0}j . Figure 6 shows the alternative volume-invariant set in two-dimension. The dark part represents the estimate of B_{1, 0}j .

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 6: The alternative volume-invariant set

Table 10: Sum of independent Gamma random variables, each with scale parameter 1 and shape parameter 1, 1, 2, 3 and 4, respectively. (alternative volume-invariant set) t n F (t) Fˆ(3)(t) RE( ˆF(3)(t)) 3 25 0.000292 0.000294 0.005141 50 0.000293 0.001331 6 25 0.042621 0.042618 0.000082 50 0.042620 0.000010 9 25 0.294012 0.293313 0.003366 50 0.292836 0.000848 12 25 0.652771 0.651124 0.007265 50 0.652356 0.001829 15 25 0.881536 0.880092 0.013825 50 0.881174 0.003463 18 25 0.969634 0.968913 0.024473 50 0.969455 0.006087 21 25 0.993749 0.993501 0.039914 50 0.993688 0.009841

Let ˆF(3)(t), based on this kind of estimate, be the approximation of F (t), and ˆ F(3)(t) = Z ˆ B1 t, 0(3) g(x )mk(dx ) = n−k X j=0 X y∈ω(j) P µ t nyi ≤ Xi ≤ t n(yi+ 1), i = 1, 2, ..., k ¶ + k−1 X j=0 X y∈ω(n−j) · P µ t nyi ≤ Xi ≤ t n(yi+ 13j), i = 1, 2, ..., k ¶ − P µ t n[yi+ (l3j − d2j)] ≤ Xi ≤ t n(yi+ l3j), i = 1, 2, ..., k ¶ + k X m=1 P µ t nyi ≤ Xi ≤ t n ³ yi + k− 1 kd_3j ´ , i 6= m, t n(ym+ l3j) ≤ Xm ≤ t n ³ ym+ l3j + k− 1 kd_3j ´¸ .

Some numerical results are presented in table 10. The results, which compare with ˆ

F(1)(t) or ˆF(2)(t), show a good performance in precision. Therefore, many kinds of

the volume-invariant sets can be considered, and in each case the common goal is to increase the overlapping part of the estimate and B1

t, 0.

Note that our method is suitable for small sample size k. For large k, there are many well-known methods that can be used to obtain the approximated distribution function of sum of r.v.’s, such as the central limit theorem and Edgeworth expansion. This paper present an innovative idea which combines probability and geometry to approximate the distribution function and density function of sum of positive r.v.’s while the their joint distribution function G is given. Further analysis will be concentrated on applying our method to some relative fields such as biology, reliability and so on.

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