A Multivariate Weibull Disitribution

Cheng K. Lee [email protected] Charlotte, North Carolina, USA

Miin-Jye Wen National Cheng Kung University, City Tainan, Taiwan, R.O.C.

Summary.

A multivariate survival function of Weibull Distribution is developed by expanding the theorem by Lu and Bhattacharyya (1990). From the survival function, the probability density function, the cumulative probability function, the determinant of the Jacobian Matrix, and the general moment are derived. The proposed model is also applied to the tumor appearance data of female rats.

Key words: general moment, multivariate survival function, set partition

1. Introduction

Lu and Bhattacharyya (1990) developed a joint survival function by letting hx1 () and

hy2 () be two arbitrary failure rate functions on [0,∞) , and Hx1 ( ) and Hy2 () their corresponding cumulative failure rate. Given the stress S=s > 0, the joint survival function conditioned on s, as they defined, is γ Fxys(, |)=− exp ⎡ H x + H y⎤ s, { ⎣ 12() ()⎦ } where g measures the conditional association of X and Y. Further, based on the joint survival function, they proved a theorem that a bivariate survival function Fxys(, |) can be derived with the marginals F x and F y given the assumption that the Laplace transform of the stress S exists on [0,∞) and is strictly decreasing. From the theorem, they derived a bivariate Weibull Distribution α ⎧ ⎡ γγ12⎤ ⎫ ⎪ ⎛⎞xyαα ⎛⎞ ⎪ Fxy(, )=− exp ⎢ + ⎥ , ⎨ ⎢⎜⎟ ⎜⎟⎥ ⎬ ⎪ ⎝⎠λλ12 ⎝⎠ ⎪ ⎩⎭⎣⎢ ⎦⎥

where 0 < α ≤ 1, 0 < λ1, λ2 < ∞, and 0 < γ1, γ 2 < ∞. This bivariate Weibull Distribution is exactly the same as developed by Hougaard (1986).

By the same steps, the theorem can be expanded to more than two random variables, and, therefore, a multivariate survival function of Weibull distribution is constructed as α ⎧⎫⎡⎤γγ12 γ n ⎪⎪⎛⎞xxαα ⎛⎞ ⎛⎞x α Sx( , x ,..., x )=− exp⎢⎥12 + ++ ... n , (1) 12 n ⎨⎬⎢⎥⎜⎟ ⎜⎟ ⎜⎟ ⎪⎪⎝⎠λλ12 ⎝⎠ ⎝⎠ λn ⎩⎭⎣⎦⎢⎥ whereα measures the association among the variables, 0 < α ≤ 1, 0 < λ1, λ2,…,λn < ∞, and 0 < γ1, γ 2,…, γ n < ∞. This model can also be derived by using a copula construction in which the generator is α ()−log() . (Frees and Valdez, 1998). Equation 1 is similar to the “genuine multivariate Weibull distribution” developed by Crowder (1989) who studied another version extended from the genuine multivariate Weibull distribution. In this paper, we mathematically intensively studied the proposed multivariate Weibull model of Equation 1 by giving the probability density function in section 2, the Jacobian matrix in section 3, the general moment in section 4 and an application in section 5.

2. Probability Density Function of The Multivariate Weibull Distribution

The multivariate probability density function f (xx12 , ,..., xn ) of a multivariate distribution function can be obtained by differentiating the multivariate survival function with respect to each variable. Li (1997) has shown that n n ∂ Sx(12 , x ,..., xn ) f (xx12 , ,..., xn ) =()−1 . ∂∂xx12... ∂ xn Using Li’s derivation and one of the special cases of the multivariate Faa di Bruno formula by Constantine and Savits (1996), the probability density function is

α ⎧ γγ12 γ n ⎫ n ⎡ ⎤ ⎛⎞−1 ⎪ ⎛⎞xxαα ⎛⎞ ⎛⎞x α ⎪ fxx( , ,..., x )=−+++ exp⎢ 12 ... n ⎥ 12 n ⎜⎟ ⎨ ⎢⎜⎟ ⎜⎟ ⎜⎟⎥ ⎬ ⎝⎠αλλλ⎪ ⎝⎠12 ⎝⎠ ⎝⎠n ⎪ ⎩⎭⎣⎢ ⎦⎥ γγ γ ⎡⎤12−−11 n −1 ⎡⎤⎛⎞⎛⎞γγ ⎛⎞γ ⎛⎞xxαα ⎛⎞ ⎛⎞x α ⋅ 12...nn⎢⎥ 1 2 ... ⎢⎥⎜⎟⎜⎟⎜⎟⎢⎥ ⎜⎟ ⎜⎟ ⎜⎟ ⎣⎦⎝⎠⎝⎠λλ12⎝⎠ λnn ⎝⎠ λ 1 ⎝⎠ λ 2 ⎝⎠ λ ⎣⎦⎢⎥

kniα − ⎧⎫⎡⎤γγ12 γ n Pn() ki αα α ⎪⎪k ⎛⎞n ⎛⎞xx ⎛⎞ ⎛⎞x ⋅−1i Pni ,α j ⎢⎥12 + ++ ... n , (2) ∑ ⎨⎬()s ( )⎜⎟∏ ⎢⎥⎜⎟ ⎜⎟ ⎜⎟ i=1 ⎪⎪⎝⎠j=1 ⎝⎠λλ12 ⎝⎠ ⎝⎠ λn ⎩⎭⎣⎦⎢⎥ where ki is the number of summands of the ith partition of n such n that nn+++" n = n, nn≥≥≥" n >0 ,1≤ kn≤ ;α j is equal 12 ki 12 ki i toαα()−−+1 ...() α n j 1 , the falling factorial of α (Kunth, 1992); Pn( ) is the total number of partitions of n; Pnis (), is the total number of set partitions of the set Sn ={1,…,n} corresponding to the ith partition of n. The specific way of partitioning n and Sn is given by McCullagh and Wilks (1988). In their paper, partitions of n are in increasing number of summands and ordering all the summands in inverse lexicographic order when a partition has the same number of summands, and Sn ={1,…, n}is partitioned by “listing the blocks from the largest to the smallest and by breaking the ties of equal sized blocks by ordering them lexicographically” and the number of blocks in a set partition is equal to the number of summands of the corresponding partition of n. For example, the total number of blocks of the partition of Sn corresponding to the ith partition is k and the numbers of elements in each block are equal to nn,,," n. i 12 ki

3. The Jacobian Matrix

Similar to the derivation of the Bivariate Weibull Distribution by Lu and Bhattacharyya

(1990), let()yy12, ,..., yn

⎛⎞zz12 zn−1 α =⎜⎟, ,..., ,()zz12+++ ... zn (3) ⎝⎠zz12+++... zzznn 12 +++ ... z zz 12 +++ ... z n

γ1 γ 2 γ n α α α ⎛⎞x1 ⎛⎞x2 ⎛⎞xn where z1 = ⎜⎟, z2 = ⎜⎟,…, zn = ⎜⎟. ⎝⎠λ1 ⎝⎠λ2 ⎝⎠λn α 1 α 1 α 1

γγ11 γγ22 γγnn−−11 Then, xyy11= n λ 1, xyy22= n λ 2,…, xyynnnn−11= −−λ 1, 1 α γ γ n xyyyynnnn=−−()1 12 −−" − 1n λ .

Note that z1 , z2 ,..., zn > 0, and

1−−−−y12yy ... n− 1 zz+++... z =1− 12n− 1 zz12+++... zn z = n > 0. zz12+++... zn

The Jacobian matrix is

∂∂x xx ∂ 11" 1 ∂∂y12yy ∂n ∂∂x xx ∂ 22" 2 J = ∂∂y12yy ∂n . ## # ∂∂x xx ∂ nn" n ∂∂y12yy ∂n

Let C(i,j) be the ith row and jth column in the Jacobian matrix, then α 1 −1 αλ y γ iiy γ C(i,i)= ii n , i=1,2,…,n-1, γ i α 1 −1 λ yyγγii C(i,n)= ii n , i=1,2,…,n-1, γ i 1 α −1 γ n αλ ()1−−−−y yyy" γ n C(n,i)= − nnn12− 1 , i=1,2,…,n-1, γ n 1 α −1 γ n λ ()1−−−−yy" yγ n y C(n,n)= nnn12− 1 , γ n C(i,j)=0 when i ≠ j, i=2,…,n-1, j=2,…,n-1.

The determinant of the Jacobian matrix can be obtained using Gaussian elimination to construct an upper triangle matrix. Then, the determinant is equal to the product of the diagonal elements. ⎛⎞n−1 ⎛⎞n−1 CnjC( ,,) ( jn) |J| =⎜⎟∏Cii(),,⎜⎟ Cnn ( )− ∑ ⎝⎠i=1 ⎝⎠j=1 Cjj(), α αα α −1 11 1 −−11 −11γ n +++−" αλλλn−1 ""yyγγ12 yγγγγn−112()1−−−− yy " y y n = 12nn 1 2−− 1 1 2 nn 1 . (4) γγ12" γn

After the derivation of the Jacobian, the PDF in terms of y12,,,yy" n ,

gyy()12,,," yn = αα11 α 11 ⎛⎞α γγ11 γγ 2 2 γγnn−−11 γ n f ⎜⎟yyλλ,,,,1 y y"" y y λ−−−− y y yγ n y λ J ⎜⎟1122nnnnn−− 1 112() nnn − 1 ⎝⎠

⎧⎫Pn()⎛⎞ki nk⎪⎪⎛⎞n j = −−1αα−1 y ki −1 1i Pni , exp( − y ) ()⎨⎬∑ ⎜⎟ns () ( )⎜⎟∏ n ⎪⎪⎩⎭i=1 ⎝⎠⎝⎠j=1 = Γ−−−−ny11−− y 11... y 11 − 1 y y ... y11− fy ( ) , (5) ()()12nnn−− 1 ( 1 2 1 ) where y1 , y2 ,…, yn−1 has a Dirichlet distribution with the probability density equal 11−− 11 11 − 11− to Γ−−−−()ny12 y... ynn−− 1 ( 1 y 1 y 2 ... y 1 ) , and n ⎧⎫Pn()⎛⎞ki ()−1 ⎪⎪ki ⎛⎞n j f ()y = αα−1 y ki −1 −−1Pni , exp( y ) (6) n ⎨⎬∑ ⎜⎟ns() ( )⎜⎟∏ n Γ()n ⎩⎭⎪⎪i=1 ⎝⎠⎝⎠j=1 has a mixture distribution of the exponential distribution and Gamma distribution.

Equation (6) can be rewritten as n Pn() ki −1 ⎧⎫⎛⎞k ⎛⎞n ()⎪⎪−1 ki −1 i j exp(−yn ) f ()yn = ⎨⎬⎜⎟ααykPninis()−Γ1, ()( )⎜⎟ . ΓΓnk∑ ⎜⎟∏ ()⎩⎭⎪⎪i=1 ⎝⎠⎝⎠j=1 ()i When it is integrated over the range of yn, it becomes n ⎧⎫Pn()⎛⎞ki ()−1 ⎪⎪ki ⎛⎞n j αα−1 −Γ1,kPni =1. ⎨⎬∑ ⎜⎟() ()(is )⎜⎟∏ Γ()n ⎪⎪⎩⎭i=1 ⎝⎠⎝⎠j=1 That is the weights of yn are summed to 1. The probability density function of yn is the mixed Gamma distribution by Downton (1969). Following his derivation, the cumulative density function of yn is n ⎛⎞n i−1 Pn()⎛⎞k ()−1 y k ⎛⎞n j 1−−Γ−⎜⎟n ⎜⎟αα−1 1kPni , exp( y ) . (7) ⎜⎟∑∑⎜⎟() ()(s )⎜⎟∏ n ΓΓ()ni⎝⎠iki=≥1 () ⎝⎠⎝⎠j=1

4. The General Moment

The general moment of x12,,,xx" n is ⎡⎤ii12" in Exx⎣⎦12 xn =

αα11ii12α 11iin−1 n ⎡⎛⎞⎛⎞⎛⎞⎛α ⎞⎤ γγ11 γ 2 γ 2 γγnn−−11 γn Eyy⎢⎜⎟⎜⎟λλ yy""⎜⎟⎜ y y λ()1−−−− yy yγ n y λ ⎟⎥ ⎢⎜⎟⎜⎟1122nnnnn⎜⎟⎜−− 1 1 12 nnn − 1 ⎟⎥ ⎣⎝⎠⎝⎠⎝⎠⎝ ⎠⎦

ii12αα in−1α ii12 in ⎡⎤inα ⎡ ++" ⎤ ii i γγ γ γγ γ 12nn 1 2 −1 γ 12 n = λλ12"" λnnEy⎢⎥ 1 y 2 y−− 1()1−−−− y 1 y 2 " y n 1n Ey⎢ n ⎥ ⎣⎦⎢⎥⎣⎢ ⎦⎥ Then,

ii12αα in−1α ⎡⎤inα γγ γ 12 n−1 γ Ey⎢⎥12 y"" ynn−− 1()1−−−− y 1 y 2 y 1n ⎣⎦⎢⎥

ii12αα in−1α inα γγ12 γ n−1 = Γ−−−−nyyy...""" 1 yyydydyγ n ()∫∫12nnn−−− 1() 1 2 1 1 1 ⎛⎞⎛⎞iiαα⎛⎞i α ΓΓ()n ⎜⎟⎜⎟12 +Γ11 +" Γ⎜⎟n + 1 rr r = ⎝⎠⎝⎠12⎝⎠n ⎛⎞⎛⎞ii i Γ++++⎜⎟α ⎜⎟12" n n ⎝⎠⎝⎠γγ12 γn which is the Dirichlet integral (Rao, 1954.)

ii i ⎡⎤12++" n ii i For Ey⎢⎥γγ12 γn , let 12+++" n =c, then n γ γγ ⎣⎦⎢⎥12 n ⎡⎤c Ey⎣⎦n n ∞ Pn() ki −1 ⎧⎫k ⎛⎞n () c ⎪⎪−1 ki −1 i j = ynn⎨⎬ααyPniydy()−−1 s ( , )⎜⎟ exp( nn ) Γ n ∫ ∑ ∏ ()0 i=1 ⎩⎭⎪⎪⎝⎠j=1 n Pn() ki ∞ −1 ⎧⎫⎡⎤k ⎛⎞n () ⎪⎪−1 i j ck+−i 1 = ⎨⎬⎢⎥αα()−−1Pnis ( , )⎜⎟ ynnn exp( ydy ) Γ n ∑ ∏ ∫ ()i=1 ⎩⎭⎪⎪⎢⎥⎣⎦⎝⎠j=1 0 n Pn()⎡⎤ki ()−1 −1 ki ⎛⎞n j = ∑ ⎢⎥αα()−Γ+1,Pnisi ( )⎜⎟∏ () c k Γ()n i=1 ⎣⎦⎢⎥⎝⎠j=1 n Pn() ki −1 ⎡⎤k ⎛⎞n () −1 i j ki = ∑ ⎢⎥αα()−Γ1,Pnis ( )⎜⎟∏ () cc (8) Γ()n i=1 ⎣⎦⎢⎥⎝⎠j=1

ki where c is the rising factorial defined as cc( +11)"( c+− ki ) by Knuth (1992).

Considering Pnis (), in the above equation, it is the total number of set partitions corresponding to the ith partition of n such that nn+ ++" n = n, n , n ,…, n > 0. 12 ki 1 2 ki It has been shown by McCullaph and Wilks (1988) that n! Pni(), = s nn!!"" n !! mm ! m ! 12kdi 1 2 where m1 , m2 ,…, md are the number of each distinct summand.

ki ⎛⎞n j Then, the product Pnis (), ⎜⎟∏α ⎝⎠j=1

n! nn nk = α 12αα" i nn!!"" n !! mm ! m ! 12kdi 1 2 n! α !!αα ! = " mm12!!" md !nnnn1122!!!!()αα−−()nn!!α − kk()i n! ⎛⎞⎛⎞αα ⎛⎞α = ⎜⎟⎜⎟"⎜⎟ mm!!" m !nn n 12 d ⎝⎠⎝⎠12⎝⎠ki

α n! ki ! ⎛⎞⎛⎞αα ⎛⎞ = ⎜⎟⎜⎟"⎜⎟, k ! mm!!" m !nn n i 12 d ⎝⎠⎝⎠12⎝⎠ki k ! where i is the number of permutations of nn,,," nof every possible order. 12 ki mm12!!" md !

ki ⎛⎞n j When sum Pnis (), ⎜⎟∏α over the same value of ki , say, k, then, ⎝⎠j=1

ki ⎛⎞n j ∑ Pnis (), ⎜⎟∏α kki = ⎝⎠j=1 ⎡⎤α n! ki ! ⎛⎞⎛⎞αα ⎛⎞ = ⎢⎥⎜⎟⎜⎟"⎜⎟ ∑ kmm!!!" m !nn n kki = ⎣⎦⎢⎥id12 ⎝⎠⎝⎠12⎝⎠ki n! ⎡⎤k! ⎛⎞⎛⎞αα ⎛⎞α = ⎢⎥⎜⎟⎜⎟"⎜⎟ k! ∑ mm!!" m !nn n kki = ⎣⎦12 d ⎝⎠⎝⎠12⎝⎠k which equalsCnk(),,α , the C-numbers defined by Charalambides (1977). Note that the summation is over all the permutations of n with ki=k. ki kk Using the equality()−=−1 iiccki () (Goldman, Joichi, Reiner and White, 1976), ⎡⎤c Ey⎣⎦n n Pn() ki −1 ⎡⎤k ⎛⎞n () −1 i j ki = ∑ ⎢⎥αα()−Γ1,Pnis ( )⎜⎟∏ () cc Γ()n i=1 ⎣⎦⎢⎥⎝⎠j=1 n ⎧⎫Pn()⎡⎤ki ()−1 −1 ⎪⎪⎛⎞n j ki = ααΓ−()cPnic⎨⎬∑ ⎢⎥s (, )⎜⎟∏ () Γ()n ⎩⎭⎪⎪i=1 ⎣⎦⎢⎥⎝⎠j=1 n ⎧⎫n ⎡⎤⎛⎞ki ()−1 ⎪⎪⎛⎞n j ki = αα−1Γ−cPnic⎢⎥, ()⎨⎬∑∑⎜⎟s ( )⎜⎟∏ () Γ()n kkk==1 j=1 ⎩⎭⎪⎪⎣⎦⎢⎥i ⎝⎠⎝⎠ n n ()−1 k = αα−1Γ−cCnkc⎡⎤,, ()∑ ⎣⎦ ( )( ) Γ()n k =1 n ()−1 n = αα−1Γ−()(cc ) (using equation 1.3 by Charalambides, 1977) Γ()n n ()−1 nn = αα−1Γ−()()(cc1 ) (using the formula by Goldman et al. 1976) Γ()n 1 = ()()()()ααcc++12" α cnc +−Γ+ 11. Γ()n

Therefore, the general moment of x12,,,xx" n is ⎡⎤ii12" in Exx⎣⎦12 xn

ii12αα iin−1α ii12 n ⎡⎤⎡⎤inα ++" ii i γγ γγγγ 12nn 1 2 −1 γ 12 n = λλ12"" λnnEy⎢⎥⎢⎥ 1 y 2 y−− 1()1−−−− y 1 y 2 " y nn 1n Ey ⎣⎦⎣⎦⎢⎥⎢⎥

⎛⎞⎛⎞iiαα⎛⎞i α ΓΓ()n ⎜⎟⎜⎟12 +Γ11 +" Γ⎜⎟n + 1 rr r ii12 in ⎝⎠⎝⎠12⎝⎠n = λ12λλ" n ⎡⎤⎛⎞ii i Γ++++⎢⎥α ⎜⎟12" n n ⎣⎦⎝⎠γγ12 γn 1 ⎡⎤⎡⎤⎛⎞⎛⎞iiii ii .12⎢⎥⎢⎥αα⎜⎟⎜⎟12++++""nn 12 ++++… Γ()n ⎣⎦⎣⎦⎝⎠⎝⎠γγ12 γnn γγ 12 γ ⎡⎤⎛⎞ii i ⎡ ⎛⎞ii i ⎤ ⎢⎥α ⎜⎟12+++" n +−()n 1 Γ ⎢α ⎜⎟12+++" n +1⎥ ⎣⎦⎝⎠γγ12 γn ⎣ ⎝⎠γγ12 γn ⎦

⎛⎞⎛⎞iiαα⎛⎞⎛iiα ⎡ ii ⎞⎤ ii12 in 12nn 12 λλ12""" λn Γ+Γ+Γ+Γ++++⎜⎟⎜⎟11⎜⎟⎜ 1⎢ ⎟ 1⎥ ⎝⎠⎝⎠rr12⎝⎠⎝ rnnγγ 12 γ ⎠ = ⎣ ⎦ (9) ⎡⎤⎛⎞ii12 in Γ++++⎢⎥α ⎜⎟" 1 ⎣⎦⎝⎠γγ12 γn

From the general moment, the expectation and the variance of any random variable, and the covariance and the correlation coefficient of any number of random variables can be derived.

5. Application We analyze the data published by Mantel, Bohidar and Ciminear (1997). The data (Table 1) contains 50 litters of female rats with one drug-treated and two control rats in each letter. The same data was also analyses by Hougaard (1986) using a bivariate Weibull distribution. We assume the time to the appearance of tumor of the treatment group, the control group 1 and the control group 2 are Weibull distributed. Table 2 displays the counts of combinations of censoring status of litters. The results of parameter estimates and standard errors based on the second derivatives valued at the maximized log- likelihood function are in Table 3. The estimates for the 3 shape parameters are significantly greater than 1 with significant level equal to 0.05 indicating that the 3 group have a monotonically increasing hazard function. The estimate of the association parameter α is not significantly different from 1 indicating that the time to the tumor occurrence among the 3 groups are not associated. Without considering the standard errors, by equation (9), the correlation coefficient of the treatment group and control group 1 is 0.163. The correlation coefficient of the treatment group and the control group 2 is 0.159. The correlation coefficient of the two control groups is 0.157.

Table 1. Time to tumor appearance in weeks of treatment group (T) and control groups (C1, C2).

Litter T C1 C2 Litter T C1 C2 1 101+ 49+ 104+ 26 104+ 102+ 104+ 2 104+ 104+ 104+ 27 77+ 97+ 79+ 3 89+ 104+ 104+ 28 88 96 104 4 104 94 77 29 96 104 104 5 82+ 77+ 104+ 30 70 104 77 6 89 91 90 31 91+ 70+ 92+ 7 39 45 50 32 103 69 91 8 93+ 104+ 103+ 33 85+ 72+ 104+ 9 104+ 63+ 104+ 34 104+ 104+ 74+ 10 81+ 104+ 69+ 35 67 104 68 11 104+ 104+ 104+ 36 104+ 104+ 104+ 12 104+ 83+ 40+ 37 87+ 104+ 104+ 13 104+ 104+ 104+ 38 89+ 104+ 104+ 14 78+ 104+ 104+ 39 104+ 81+ 64+ 15 86 55 94 40 34 104 54 16 76+ 87+ 74+ 41 103 73 84 17 102 104 80 42 80 104 73 18 45 79 104 43 94 104 104 19 104+ 104+ 104+ 44 104+ 101+ 94+ 20 76+ 84+ 78+ 45 80 81 76 21 72 95 104 46 73 104 66 22 92 104 102 47 104+ 98+ 73+ 23 55+ 104+ 104+ 48 49+ 83+ 77+ 24 89 104 104 49 88+ 79+ 99+ 25 103 91 104 50 104+ 104+ 79+ + denote right censored times

Table 2. Number of litters of various censoring status combinations

Censoring Status (T, C1, C2) Number of Litters (0, 0, 0) 1 (0, 0, 1) 3 (0, 1, 0) 6 (1, 0, 0) 2 (0, 1, 1) 11 (1, 0, 1) 2 (1, 1, 0) 2 (1, 1, 1) 23 Tumor occurrence is denoted as 0 and censored is denoted as 1

Table 3. Maximum likelihood estimates and standard errors.

Parameter Estimate Standard Error α 0.900 0.101 Scale (T) 112.035 6.484 Shape (T) 4.393 0.879 Scale (C1) 157.641 28.649 Shape (C1) 3.568 1.150 Scale (C2) 154.119 25.667 Shape (C2) 2.890 0.792

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