有限個數的i.i.d. Weibull分布隨機變數之和的機率分布函數的近似值

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行政院國家科學委員會專題研究計畫成果報告

有限個數的 i.i.d. Weibull 分布隨機變數之和的機率分布

函數的近似值

計畫類別：個別型計畫計畫編號： NSC93-2118-M-009-018- 執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日執行單位：國立交通大學統計學研究所計畫主持人：彭南夫報告類型：精簡報告處理方式：本計畫可公開查詢

中華民國 94 年 10 月 31 日

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有限個數的 i.i.d. 韋伯隨機變數之和的累積機率函數近似解

關鍵字 : 韋伯隨機變數 , 累積機率函數

An Approximation of the Distribution Function of the Sum of Finite

i.i.d. Weibull Random Variables

Keywords : Weibull random variables, Cumulative Distribution

function

1.INTRODUTION

The main purpose in this discussion is to find the distribution function of the sum

of independent and identical distributed Weibull random variables. The Weibull

distribution is a two-parameter distribution which by adjusting a scale parameter

(denoted by c) and a shape parameter (denoted by β ) we can obtain a variety of shapes to fit experimental data. Thus, this distribution is highly adaptable and widely

used in practice. Without loss of generality, we can assume the scale parameter to be

one. In addition, the Weibull distribution is widely used in reliability modeling since

other distributions such as exponential (c=1), Rayleigh (c=2), and normal (if a

suitable value for the shape parameter is chosen) are special cases of the Weibull

distribution. In the version of hazard-rate function, when c>1, the hazard rate is a

monotonically increasing function with no upper bound that describes the wear-out

region, and when c=1, the hazard rate becomes constant (constant failure-rate region),

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failure-rate region). This enables the Weibull distribution to describe the failure rate of

many failure data in many kinds of application.

Unlike the renewal processes with the exponential inter arrivals, those with the

Weibull interarrivals do not possess elegant properties. It is well known that by the

nature of the renewal processes, the distribution of the process is related to the

distribution of a finite sum of the independent and identically distributed inter arrivals,

i.e. suppose that inter arrivals X₁,X₂,...,X_n are a random sample of Weibull(c,β ) distributed with probability density function

β β c x c _e x c x f = −1 − ) ( . (1)

The probability of interest is

P(S_n ≤ where t) S_n = X₁+...+X_n, (2) which has no simple formula to evaluate. The major result of this report is to present a

simple, elegant and relatively accurate form of an approximation.

2. METHODOLOGY

In this section, we discuss the approximation of F_n(t)=P(S_n ≤t), which could be seen as the convolution of n functions. When n is large, the evaluation of n-fold

convolution includes n-dimensional integration, thus the evaluation becomes difficult.

Even using the numerical analysis, the results of evaluation are obtained pointwisely.

That is, when fixed one value of t, we can numerically evaluate F_n(t), but just only one value in every evaluation. The numerical computation of the inverse Laplace

transform [8] still directs to the answer pointwisely. Another approach is to use

Edgeworth Expansion [3 and 7], but the shortage is that we have to includ numerical

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would not behave well relatively. Here we use three quantiles to obtain a simple

approximation method, and the evaluation of approximation is obtained

functionwisely, not pointwisely. Our approach describes as the following:

Suppose X is a random variable from _i Weibull( c, 1)distribution. Via the transform of variable Y_i = X_ic, we obtain that Y is a random variable from _i Exponential(1) distribution with i=1,...,n. Thus, we expect to use the distribution function of

n Y

Y₁+...+ to approximate the distribution function of X₁+ ...+X_n. Since the two distribution functions are continuously differentiable functions, for every t≥0, we expect to find some continuously differentiable function w(t), such that

P(X₁+...+X_n ≤t)=P(Y₁+...+Y_n ≤w(t)), (3) with 0w(0)= . Thus, we consider the form of third degree polynomial

) (t

w =αt3c +γt2c +τtc. (Note that we can consider the form αt2c +γtc, but the approximation isn’t well relatively.) Three unknown parameters above can be

determined by three quantiles of X₁+ ...+X_n and Y₁ + ...+Y_n. First, let t satisfy _p

P(X₁ +...+X_n ≤t_p)= p, (4) and θ satisfy _p

P(Y₁+...+Y_n ≤θ_p)= p, (5) Thus, t and _p θ are the p-quantile of _p X₁ + ...+X_n and Y₁+ ...+Y_n, respectively. When p 0.1, 0.5, and 0.9, we can apply = αt_p3c+γ t_p2c +τt_pc =θ_p to construct three linear functions. Thus, three unknown parameters can be evaluated through the three

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Lemma 1:

Suppose Y₁,Y₂,...,Y_n are a random sample from Exponential(1) distributed. Then

∑

− = − − = ≤ + + 1 0 1 ... ) 1 / ! ( n i i n e i Y Y P θ θθ . (6) Proof: First let

∑

= = n i i n Y S 1 *

, there is a counting process {N(θ)=sup(n:S_n* ≤θ),θ ≥0} which is a renewal process, conclude that

S_n* ≤θ ⇔ N(θ)≥n. (7) As inter arrivals are a random sample from Exponential(1) distributed, we know that )N(θ is a random variable from Poisson(θ) distributed. Thus

P(Y₁+...+Y_n ≤θ)=P(S_n* ≤θ)=P(N(θ)≥n)

∑

− = − − = < − = 1 0 ! / 1 ) ) ( ( 1 n i i _i e n N P θ θθ (8) ■

Via lemma 1, the value of θ can be obtained by numerical method (We use the _p Newton-Raphson method). Different to θ which follows lemma1, we must _p simulate the value of t . We interpret the procedure of the approach as the follows: _p

(1) Through Newton-Raphson method we can obtain the solutions of the equation

p / i! θ e n i i p θP = −

∑

− = − 1 0

1 , where p 0.1, 0.5, 0.9, i.e. = θ₀_.₁ ,θ₀_.₅,and θ₀_.₉.

(2) Determine the numbers of simulate data N =107, let T₁,T₂,...,T_N be a random sample from F_n(x)=P(X₁ +...+X_n ≤x) , and order them become to

) ( ) 2 ( ) 1 ( ,T ,...,TN

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where p 0.1, 0.5, 0.9. = (3) Via the three linear equations

1 0 1 0 2 1 0 3 1 0 ˆ ˆ ˆ c _. . c . c . γ t τ t θ t α + + = , 2 ₀₅ ₀₅ 5 0 3 5 0 ˆ ˆ ˆ c _. . c . c . γ t τ t θ t α + + = , (9) 9 0 9 0 2 9 0 3 9 0 ˆ ˆ ˆ c _. . c . c . γ t τ t θ t α + + = .

Three of unknown parameters α γ,, τ can be determined.

(4) Under fixed n and shape parameter c, the estimated form of w(t) is wˆ(t)=αt3c +γ t2c +τtc

. (10)

(5) Thus the approximation of the distribution function of interest is

∑

− = − − = ≤ + + ≅ ≤ + + 1 0 ) ( ˆ 1 1 ... ) ( ... ˆ( )) 1 ˆ( ) / ! ( n i i t w n n t P Y Y wt e w t i X X P . (11)

We choose the number of simulate data is N =107, and T_{( Np}₎ is the p-th sample quantile. From [6] we can know that T_{( Np}₎ is asymptotically following the

) ) ( ) 1 ( , ( ₂ N t f p p t N p n p −

distributed, the error is proportional to 2 7

10

1 −

=

N , thus T( Np)

would close to the true p-th quantile. Equation (11) is the most important result of this

dissertation and offers a simple approximation via the correlation between the

Exponential and the Weibull distributed. Moreover, the approximation approach will

reduce a lot of time relative to simulation. In the next section we will discuss the

accuracy of the approach.

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In order to appraise the accuracy of our approximation, the criterion that be used is the

relative error. We denote the relative error is P

P P₋ ˆ

, where p is the true probability and ))Pˆ =P(Y₁+...+Y_n ≤wˆ(T₍_Np₎ is the approximation of p. Total cases that we have done are as n=2, 3,..., 10, c=0.1, 0.2,..., 0.9 and c=2, 3,..., 9. In table 3.1, we list the relative error table includes only cases of n=2, 5and9, c = 0.5 and 5, the others have the analogous result and tendency. Fixed n and c, in every block of

table 3.1 includes the relative errors of probabilities 0.05, 0.1, …0.95. Further, we list

the same cases of three of parameters in table A.1 of Appendix A.

We can discover that when c=0.5, the maximum relative error is 0.14%; and c=5, the

maximum relative error is 1.4%, noted that happened as p=0.05, concludes that the

absolute error isn’t very large. Hence, we believe that Equation (11) brings a nice

accuracy of approximation. [Table 3.1] Fixed as n=2 p c 0.5 5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.000726445242646 -0.000000002616051 0.000119820278476 -0.000194908815901 0.000138893117596 -0.000011260430451 0.000043543016760 0.000186524455173 0.000024819173472 -0.000000000567722 -0.000148473532176 -0.000059519398334 -0.010161527424567 -0.000000000000001 0.003828824693817 0.005143137693705 0.005324085016400 0.005005107033746 0.004334074875617 0.002847272084006 0.001416814541063 0.000000000000000 -0.001696971491006 -0.003003892633153

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0.65 0.7 0.75 0.8 0.85 0.9 0.95 -0.000040240442348 -0.000105114490566 -0.000172923165788 -0.000064298286200 0.000007023458372 -0.000000001380996 0.000165231376151 -0.004332428288305 -0.005246866270862 -0.005520852102506 -0.005102997297659 -0.003342411510635 0.000000000000000 0.004920445151018 Fixed as n=5 p c 0.5 5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 -0.000028135251537 -0.000000007634631 0.000008069914397 0.000509988574720 0.000340433284307 0.000628625741176 0.000642236022542 0.000125130856831 0.000196852924741 -0.000000001023538 0.000325840009388 0.000290632220714 0.000252098046966 0.000277480011670 0.000254161667295 0.000199371124861 0.000174674060825 -0.000000001927334 -0.000021837121247 -0.014064661989290 0.000000000000002 0.003784904432964 0.005825262499093 0.006454284943475 0.005655125010712 0.004201874492865 0.002810794471418 0.001418848371266 0.000000000000002 -0.001174124992608 -0.002266358442895 -0.002949416858121 -0.003413516974510 -0.003491304489550 -0.003091183240431 -0.001950997481086 0.000000000000001 0.002886342734995 Fixed as n=9 p c 0.5 5 0.05 0.001436234621018 -0.010512189365780

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0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 -0.000000012003200 0.000557973789439 0.000524827975903 0.000174871687116 0.000275117740350 0.000340236827250 0.000406740368423 0.000277355403994 -0.000000001422286 0.000113761481000 -0.000128186125497 -0.000099682351945 0.000002286246061 0.000047694801632 -0.000065649121020 0.000010740192212 -0.000000002419210 0.000015724643715 0.000000000000005 0.004208106793534 0.004523974335599 0.004347404044753 0.003550649249438 0.002914461327970 0.001891905365702 0.000952014957686 -0.000000000000003 -0.000865972087613 -0.001469193796314 -0.002062921597401 -0.002299894298618 -0.002261558612072 -0.001909441671796 -0.001183948276717 -0.000000000000003 0.001799676396769

APPENDIX A

[Table A.1]

Fixed n and c, there are three parameters in every block, where α,γ and τ are in the upper, middle and down side, respectively.

n c 0.5 5 2 0.000280751644006 -0.010114574919690 1.255212150831596 0.000004835157895 -0.000513436509664 0.087023676809991 3 -0.000395335224320 -0.012406564768126 1.464308967719061 0.000000025577837 -0.000019005934388 0.018978874641112 4 -0.000224864735582 -0.016947170359548 1.650435209948320 0.000000000555125 -0.000001658936240 0.006292385415443

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5 -0.000493569188358 -0.018315452950023 1.816576635488150 0.000000000026227 -0.000000236200742 0.002643130317068 6 -0.000726355289725 -0.018704570924749 1.967622246201590 0.000000000002166 -0.000000047912910 0.001296831369222 7 -0.000497097072477 -0.022139525917010 2.112546353444382 0.000000000000270 -0.000000012598155 0.000709991929183 8 -0.000863384052326 -0.020185034372005 2.239924284448030 0.000000000000042 -0.000000003807886 0.000419152653162 9 -0.000835110482110 -0.021605902558135 2.367244177097372 0.000000000000008 -0.000000001360679 0.000263824260089 10 -0.000897769786661 -0.021741736990287 2.484964323969154 0.000000000000002 -0.000000000536868 0.000174175974127

REFERENCES

1. Bickel, P. J. & Doksum, K. A. (2001). Mathematical Statistics, Basic Ideas and Selected Topics. 2nd edition. Volume1. New Jersey: Prentice-Hall.

2. Elsayed, A. (1996). Reliability Engineering. MA: Addison-Wesley.

3. Hall, P. (1992). The Bootstrap and Edgeworth Expansion. New York: Springer-Verlag.

4. Ross, S. M. (1996). Stochastic Processes. 2nd edition. New York: Wiley. 5. Ross, S. M. (1997). Simulation. 2nd edition. New York: Academic Press.

6. Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. New York: J. Wiley & sons.

7. Stuart, A. & Ord, J. K. (1987). Kendall’s Advanced Theory of Statistics. 5th ed., Volume 1, Distribution Theory. London: Charles Griffin & Co.

8. Trivedi, K. S. (2002). Probability and Statistics with Reliability, Queuing and Computer Science Applications. 2nd edition. New York: Wiley.

有限個數的i.i.d. Weibull分布隨機變數之和的機率分布函數的近似值

行政院國家科學委員會專題研究計畫 成果報告