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, ..., X_k are r.v.’s with joint distribution function G(x₁, x₂, ..., x_k) or joint survival function S(x₁, x₂, ..., x_k). Let

R^k₊ =©

x ∈ R^k : xi ≥ 0, for i = 1, 2, ..., kª . For a ∈ R^k₊ and b > 0, we define a hypercube in R^k₊ that

Bb, a =©

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

, (1)

and define a ”half” hypercube for s ∈ (0, k) B_{b, a}^s =©

x ∈ R^k₊: x ∈ B_{b, a} and 1⁰a ≤ 1⁰x ≤ 1⁰a + bsª

(2) where 1 ∈ R₊^k is a column vector with each element being 1. The hyperplane that is tangent to B_{b, a}^s is©

x ∈ R^k : 1⁰x = 1⁰a + bsª

. We call the ”half” hypercube B_{b, a}^s 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

B_{t, 0}¹ g(x)dx where g is the joint density function, we in section 2 decompose B_{t, 0}¹ into union of subsets of the form (1) and (2) and then estimate B¹_{t, 0} by some unions and exclusions of hypercubes.

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 (a_i ≤ X_i ≤ b_i, for i = 1, 2, ..., k) = Xk

i=0

(−1)ⁱ X

ci∈Di

G(c_1i, c_2i, ..., c_ki),

where c_i = (c_1i, c_2i, ..., c_ki) ∈ D_i 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 B_{b, a}^s contained in B_{t, 0}¹ . To do so, the volume of that B_{b, a}^s 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}¹

AND THE VOL-UME OF B

_{b, a}^s

In this section, we present a decomposition of B_{t, 0}¹ and furthermore this decompo-sition can be used to obtain the volume of B_{b, a}^s contained in B_{t, 0}¹ . For a positive integer j, defining

ω^(j) =©

x ∈ R^k₊ : 1⁰x = j and xi ∈ Z⁺∪ {0} , for i = 1, 2, ..., kª , we have the following decomposition of B_{t, 0}¹ .

Theorem 1. Let c ∈ (0, 1). If £₁

Furthermore, the intersection of any two sets in (3) and (4) has Lebesque measure zero. has Lebesque measure zero. For x ∈ B_{t, 0}¹ , we observe that there exists k nonnegative integers d₁, d₂, ..., d_k such that x_i ∈ [ctd_i, ct(d_i+ 1)]. Since t ≥P_k

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

Corollary 1. For a positive integer n ≥ k,

B_{t, 0}¹ =

According to corollary 1, for any ”section size” n ∈ N and n ≥ k, we can decom-pose B_{t, 0}¹ 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 R^k. It is obvious that

mk(B_{b, a}^s ) = b^kmk(B_{1, a}^s ) = b^kmk(B_{1, 0}^s ).

The following theorem is an easy consequence of theorem 1.

Theorem 2. For s ∈ (0, 1]

m_k(B^s_{1, 0}) = s^k

k!, (6)

and for s ∈ (1, k)

m_k(B_{1, 0}^s ) = s^k k! −

X[s]

x=1

C_x^k+x−1m_k(B_{1, 0}^s−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(B_{s, 0}¹ ) = X[s]

x=0

X

y∈ω^(x)

mk(B_{1, y}^s−x)

= X[s]

x=0

X

y∈ω^(x)

m_k(B_{1, 0}^s−x)

= X[s]

x=0

C_x^k+x−1m_k(B_{1, 0}^s−x).

The last equality holds since the number of y ∈ ω^(x) is C_x^k+x−1, and mk(B_{s, 0}¹ ) = ^s_k!^k, we obtain (7). 2

Note that the volume of B_{1, 0}^s 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 B_{1, 0}^s through a recursive formula in theorem 2 above. Therefore, we have the volume of type-s set with size b and tail a, that is m_k(B_{b, a}^s ).

0

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 ˆB_{t, 0}¹ such that mk 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

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

l^k− m_k¡

B^j_{1, 0}¢¢¹_k

. First, We present two different volume-invariant sets as the following.

Case 1: d_1j = 0 and l_1j =¡

The first equality holds since m_k¡ 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 where I is an indicator function.

PROOF. If l − d ≥ _k^j, then m_k¡ Hence we have theorem 3. 2

Some numerical results of α_j and β_j are listed in table 1, for j = 0, 1, ..., k − 1. Note

Table 1: The optimal αj and βj

k j β_j α_j k j β_j α_j

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 B_{t, 0}¹ . The second part of the right hand side of ˆB_{t, 0}¹ represents the estimate of the unions of B^jt

n,_n^ty, for y ∈ ω^(n−j) and j = 0, 1, ..., k − 1. In each j, we have

m_k

³ B^t

nβj,_n^ty \ B^t

nαj,_n^t[y +(βj−αj)1]

´

= µt

n

¶_k m_k¡

B_{βj, y} \ B_{αj, y+(βj−αj)1}¢

= µt

n

¶_k m_k¡

B_{βj, 0}\ B_{αj, (βj−αj)1}¢

= µt

n

¶_k m_k¡

B^j_{1, 0}¢

= m_k

³ B^jt

n,_n^ty

´ .

Therefore, we obtain m_k¡ B_{t, 0}¹ ¢

= m_k

³Bˆ_{t, 0}¹

´

. In the other hand, the nonoverlap-ping parts of B_{t, 0}¹ and ˆB_{t, 0}¹ are minimized, thus ˆB_{t, 0}¹ would be a reasonable and optimal estimate of B_{t, 0}¹ of (5).

Furthermore, our approximation ˆF (t) to F (t) is

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 R^k₊, and for all y ∈ R^k₊ and z, w ∈ B^t

n,_n^ty, 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

B^jt

For each y ∈ ω^(n−j), and fixing a point z ∈ B^t

for some c1 > 0. The last inequality holds due to the fact that the number of

elements in ω^(n−j) is C_n−j^k+n−j−1 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 ¢¢¹_k

, and the second estimate ˆB_{t, 0(2)}¹ , is to let d2j and l2j satisfy l^k_2j− d^k_2j = mk

Therefore, we have Fˆ₍₁₎(t) =

to be the compared approximations with respect to ˆF (t). Some numerical results of F (t), ˆˆ F₍₁₎(t) and ˆF₍₂₎(t) are shown in section 5.

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 R₊^k−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 R^k 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 1i ∈ R^k 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 ∈ R₊^k : a_j ≤ x_j ≤ a_j + b, j 6= i, x_i = sª

(10) be a ”hyperdisc” with ith component equals to s and the others fall in the interval [a_j, a_j + b], for j 6= i. The hyperdisc D_{b, a+(s−a}ⁱ _i)1iin (10) is in fact (k−1)-dimensional and we denote m_k−1 the Lebesque measure in R^k−1. The next lemma is elementary.

Lemma 2.

d dt

Z

Btb,ta

g(x)mk(dx)

= Xk

i=1

"

(a_i+ b) Z

Dⁱ

tb,t(a+b1i)

g(z)m_k−1(dz) − a_i Z

Dⁱ_tb,ta

g(z)m_k−1(dz)

# .

(11)

Let B_{t, 0}¹ (k−1) ⊂ R^k−1₊ be similarly defined as (2), we have

f (t) = Z

B¹

t, 0(k−1)

g(x^(k−1), t − 1^(k−1)0x^(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 with y = (y₁, y₂, ..., y_k) ∈ ω^(j) and it is not difficult to see from (9) and (11) that

f (t) =ˆ d ˆF (t) expression of the right hand side of the above formula to obtain

f (t) =ˆ

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

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 ∈ R^k₊

plays the major role in estimating B^j_{1, 0}, where d =¡ 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

PROOF. Since

dis

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 γ_j and δ_j are listed in table 2, for j = 0, 1, ..., k − 1. Note that γ₀ = δ₀ = 0. Numerical results of ˜f (t) are shown in section 5.

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 γ_j and δ_j

k j γ_j δ_j k j γ_j δ_j

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 three different hypercube approximations display a good performance in precision.

Table 6: Sum of dependent random variables with joint survival function exp(−P₅

i=1ti− 0.5Π⁵_i=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 B_{t, 0}¹ 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) = P (X₁ > t₁, X₂ > t₂, X₃ > t₃, X₄ > t₄, X₅ > t₅), each of which is a simple imitation

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(−

X5 i=1

t_i− 0.5Π⁵_i=1t_i),

(2) S(t1, t2, t3, t4, t5) = exp(−

X5 i=1

ti− 0.1 max

i=1∼5ti), (3) S(t₁, t₂, t₃, t₄, t₅) =

à 1 +

X5 i=1

t_i1_{ti≥0}

!_−1.5 ,

(4) S(t1, t2, t3, t4, t5) = exp(− max

i=1∼5ti) + 1

3exp(−2 X5

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, ¯B_{t, 0}¹ in the proof of theorem 1 can be used to be our estimator of B_{t, 0}¹ , 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 ¯B_{t, 0}¹ leaving others unattended. We alternatively estimate all type-j sets individually and thus there are some ”holes”

in our estimate ˆB_{t, 0}¹ , for j = 1, 2, ..., k − 1.

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

A^∗_j = (¡

B_{l, 0}\ B_{d, (l−d)1}¢ [Ã _k [

m=1

Bk^{− 1k}d, llm

!

: 0 < d < l ≤ 1

and m_k á

B_{l, 0}\ B_{d, (l−d)1}¢ [Ã [k m=1

Bk^{− 1k}d, llm

!!

= m_k¡

B_{1, 0}^j ¢) ,

Table 9: Sum of dependent random variables with joint survival function exp(− maxi=1∼5ti) + ¹₃exp(−2P₅

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 l_m = (0, ..., 0, 1, 0, ..., 0) the mth component is 1 and the others are zero. For instance, we take l_3j =¡

m_k¡

B_{1, 0}^j ¢¢_k¹

and d_3j = l_3j − ^j_k, and

Bˆ_{t, 0(3)}¹ =



^n−k[

j=0

[

y∈ω^(j)

B^t

n,_n^ty





[



k−1[

j=0

[

y∈ω^(n−j)

ó B^t

nl3j,_n^ty\ B^t

nd3j,_n^t[y+(l3j−d3j)1]

´ [Ã _k [

m=1

Bt

nk^{− 1k}d3j, _n^t(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

Some numerical results are presented in table 10. The results, which compare with Fˆ₍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 B_{t, 0}¹ .

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.

References

[1] Albert, J. (2002). Sums of uniformly distributed variables: A combinatorial approach. College Math J. 33 201–206.

[2] Barrow, D.L. and Smith, P.W. (1979). Spline notation applied to a volume problem. Amer. Math. Monthly 86 50–51.

[3] Durrett, R. (1994). The essentials of probability. Duxbury Press, Belmont, CA.

[4] Lai, C.D. and Xie, M. (2006). Stochastic aging and dependence for reliability.

Springer-Verlag, New York.

[5] Mitra, S.K. (1971). On the probability distribution of the sum of uniformly distributed random variables. SIAM J. Appl. Math. 20 195–198.

[6] Nelson, R.B. (1999). An introduction to copulas. Springer-Verlag, New York.

[7] Pham, H. and Lai, C.D. (2007). On recent generalizations of the Weibull distribution. IEEE Trans. Reliability. 56 no.3, 454–458.

[8] Santus Filho J.C.S. and Yacoub, M.D. (2006). Simple precsion approxi-mations to Weibull sums IEEE Commun. Lett. 10 no.8, 614–616.

[9] Serfozo, R.F. (1986). Compound poisson approximations for sums of random variables. Ann. Probab. 14 1391–1398.

在文檔中 相依正隨機變數之合其分佈函數之最佳化超立方體估計 (頁 11-0)