A Multivariate Weibull Distribution David D. Hanagal Department Of ...
A MULTIVARIATE WEIBULL DISTRIBUTION
DAVID D. HANAGAL
Department of Statistics, University of Pune,
Pune-411007, INDIA.
Visiting Professor, Colegio de Postgraduados, ISEI,
Montecillo, Texcoco, CP56230, MEXICO.
ABSTRACT
In this paper, we introduce a new multivariate Weibull (MVW) dis-
tribution with many interesting properties. We obtain the maximum
likelihood estimate (MLE) of the parameters and their asymptotic mul-
tivariate normal (AMVN) distribution in MVW model. We propose large
sample studentized tests for testing multivariate exponentiality and also
tests for independence and identical marginals of the components.
Key words and Phrases: Independence, Maximum likelihood esti-
mate, Multivariate weibull model, Multivariate exponentiality, Symme-
try.
1. INTRODUCTION
In reliability theory and life testing experiments, weibull distribution
plays an important role. It reduces to exponential distribution when the
shape parameter equals one. Weibull distribution has increasing failure
rate (IFR) when the shape parameter is greater than one and has de-
creasing failure rate (DFR) when shape parameter is less than one. The
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extension of univariate Weibull distribution to multivariate case is desir-
able in view of the crucial role that Weibull distribution plays in reliabil-
ity as well as building models for various failure or life time distribution.
However, in the exponential case, there does not exist a unique natural
extension of the univariate exponential distribution to bivariate or mul-
tivariate case. So, we have many bivariate or multivariate extensions of
univariate exponential distribution. [See Weinmann(1966), Block(1975)
and Hanagal(1993a,1993b)]. In the similar way, we have many bivariate
or multivariate Weibull distributions based on these bivariate or multi-
variate exponential distributions. In this paper we consider the multi-
variate Weibull (MVW) distribution which can be obtained from mul-
tivariate exponential (MVE) model of Marshall - Olkin (1967). This is
the only MVE having the marginals as exponentials and this is the main
reason to choose this particular MVE model to obtain MVW model.
If Y = (Y1, ..., Yn) is (k + 1) parameter version of MVE distribution
of Marshall - Olkin (1967) as stated in Proschan - Sullo (1976) and
Hanagal (1991), then by taking the transformation Xi = Y 1/c
i
, c > 0, i =
1, ..., k, we have X = (X1, ..., Xk) follow MVW model which contains
singularities. The above transformation can also be done to (2k − 1)
parameter version of MVE of Marshall - Olkin (1967) and from that we
obtain 2k parameter version of MVW model. But we are not interested
in 2k parameter version of MVW and so, we study only (k + 2) prameter
version of MVW model.
In Section 2, we obtain MVW model and present some interesting
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properties. In Section 3, we obtian MLEs of the parameters of MVW.
In the last Section, we develop large sample studentized test for testing
multivariate exponentiality and also test for independence and symmetry
or identical marginals of the components.
2. MULTIVARIATE WEIBULL MODEL AND ITS
PROPERTIES
The survival function of Y of MVE of Marshall - Olkin (1967) is
¯
FY (y) = P [Y1 > y1, ..., Yk > yk]
= exp[−λ1y1 − ... − λkyk − λ0M ax(y1, ..., yk)]
where λ0, ..., λk > 0. Taking the transformation Xi = Y 1/c
i
, c > 0, i =
1, ..., k, we get the corresponding survival function of X of MVW which
is given by
¯
FX(x) = P [X1 > x1, ..., Xk > xk]
= exp{−λ1xc − λ
− ... − λ
− λ
1
2xc2
kxck
0{M ax(x1, ..., xk)}c}.
The above MVW model is not absolutely continuous with respect
to Lebesgue measure on Rk. As MVW distribution is failure time dis-
tribution and derived from MVE of Marshall-Olkin(1967), all real life
applications of MVE of Marshall-Olkin(1967) will become real life appli-
cations of this proposed MVW. For e.g., simultaneous failure of nuclear
power stations, simultaneous failure of hydroelectric pumps in aeroplane
etc.
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The marginal of Xi, i = 1, ..., k are obtained as
P [Xi > xi] = ¯
Fx(0, ..., xi, 0, ..., 0)
= exp{−(λi + λ0)xc}, i = 1, ..., k
i
which is the survival function of Weibull with parameters (λi + λ0, c),
i=1,...,k.
The distribution of M in(X1, ..., Xk) is obtained by
P [M in(X1, ..., Xk) > x] = F X(x, ..., x)
= exp{−λxc}, λ = λ0 + λ1 + .... + λk
which is the survival function of Weibull with parameters (λ, c).
The random variables Xi, i = 1, ..., k are independent iff λ0 = 0 and
Xi, i = 1, ..., k are identically distributed iff λ1 = λ2 = ... = λk. The
probability that all Xi, i = 1, ..., k are equal to each other is P [X1 =
X2 = ... = Xk] = λ0/λ. This MVW model has IFR when c > 1 and
DFR when c < 1.
3. ESTIMATION OF THE PARAMETERS
In this section, we obtain the MLEs of the parameters of MVW model.
Let (x1j, x2j, ..., xkj), j = 1, ..., n be i.i.d. random observations of sample
of size n. Now we see that there are some similarities in writing the
likelihood of MVW and MVE of Marshall - Olkin [See Proschan - Sullo
(1976)]. The likelihood of the sample of size n is
k
n
k
−
L = cpλn
(r−1)(c−1)
0
0
λni
i (λi + λ0)ni(e)
x(c−1)
ij
x(k)j
i=1
j=1
r=2 j Sr
k
n
n
exp{−
λi
xc − λ
xc
}
ij
0
(k)j
i=1
j=1
j=1
4
where p = [nk −
k
(r − 1)n
r=2
0(r)], n0(r) = number of observations with
r of X s, (i = 1, ..., k) are equal, n
n
i
0 =
k
r=2
0(r), ni = number of ob-
servations in which the random variable Xi < X(k), ni(e) = number of
observations with Xi is strictly the maximum of the (X1, ..., Xk),
Sr = {Xi = X = ... = X = X
1
i2
ir
(k), i1 = i2 = ... = ir = 1, ..., k}
and X(k) = M ax(X1, ..., Xk).
The loglikelihood of the sample of size n is given by
k
logL = plogc + n0logλ0 +
λilogλi
i=1
k
k
n
+
ni(e)log(λi + λ0) +
(c − 1)
logxij
i=1
i=1
j=1
k
k
n
n
−
(r − 1)(c − 1)
logx(k)j −
λi
xc − λ
xc
ij
0
(k)j
r=2
j Sr
i=1
j=1
j=1
The expected values of ni, n0, ni(e) and n0(r) are
E(ni) = nλi(1 − φi)/(λi + λ0), i = 1, ..., k
E(n0) = n(1 − k φ
i=1
i)
E(ni(e)) = nφi, i = 1, ..., k
and
k
λi , ..., λi
λ0
E[n
1
k−r
0(r)] =
...
,
i
(λi
+ ... + λi + λ0)...(λ)
1=
=ik=1
k−r+1
k
i1 = ... = ik = 1, ..., k
where
φi
= P [X > M ax(X , ..., X )], i
1
i1
i2
ik
1 = ... = ik = 1, ..., k
k
λi λi ...λi
=
....
2
3
k
.
i
(λi + λi + λ0)...(λ)
1=
=ik=1
1
2
5
The likelihood equations with respect to the parameters (λ0, λ1, ..., λk, c)
are
k
n
n0/λ0 +
ni(e)/(λi + λ0) −
xc
= 0
(k)j
i=1
j=1
n
ni/λi + ni(e)/(λi + λ0) −
xc = 0, i = 1, ..., k
ij
j=1
p
k
n
k
+
logxij −
(r − 1)
logx
c
(k)j
i=1 j=1
r=2
j Sr
k
n
n
−
λi
xc logx
xc logx
ij
ij − λ0
(k)
(k)j = 0.
i=1
j=1
j=1
The likelihood equations are not easy to solve. So one can generate
some consistent estimators say (u0, ..., uk+1) of λ = (λ0, ..., λk, c) and
use (u0, ..., uk+1) as a trial solution in Newton - Raphson procedure or
Fisher’s method of scoring to obtain MLEs ˆ
λ = (ˆ
λ0, ˆ
λ1, ..., ˆ
λk, ˆ
c).
So, we choose the consistent estimators (u0, ..., uk+1) of λ = (λ0, ...,
λk, c) as
n
ui = ri/
xc
, i = 0, 1, ..., k
(1)j
j=1
π
1 n
u
√
k+1 =
[
(logx(1)j − logx(1))2]−1/2
6 n j=1
where ri, i = 1, ..., k be the number of observations with xij < M in =i(x ),
j
= i = 1, ..., k and r0 be the number of observations with x1j = ... = xkj
in the sample of size n,
k
r
i=0
i
= n, x(1)j = M in(x1j, ..., xkj), and
logx
n
(1) = 1
logx
n
j=1
(1)j.
The distribution of (r1, ..., rk) is multinomial
with parameters (n, λ1/λ, ..., λk/λ) and the distribution of xc
is ex-
(1)j
ponential with failure rate λ and it is easy to check that u
P
i → λi, i =
0, 1, .., k, u
P
k+1 → c.
Here the initial estimator uk+1 is obtained by the
expression V ar(logX(1)) = (π2/6)c−2.
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The Fisher information matrix is nI(λ) = n(Iij)) where
k
k
I00 = [1 −
φi]/λ2 +
φ
0
i/(λi + λ0)2
i=1
i=1
Iii = [1 − φi]/[λi(λi + λ0)] + φi/(λi + λ0)2, i = 1, ..., k,
Ii0 = φi/(λi + λ0)2, i = 1, ..., k,
Iij = 0, i = j = 1, ..., k
k
Icc = E(p)/[nc2] +
λiE[xc (logx
(logx
)2]
ij
ij)2] + λ0E[xc(k)j
(k)j
i=1
Iic = E[xc logx
logx
ij
ij], i = 1, ..., k,
I0c = E[xc(k)j
(k)j]
where
k
E(p) = nk −
(r − 1)E[n0(r)],
r=2
E[xc (logx
ij
ij)2] = [[log(λi + λ0) − ψ(2)]2 + ψ (1) − 1]/[(λi + λ0)c2], i = 1, ..., k
E[xc logx
ij
ij] = [ψ(2) − log(λi + λ0)]/[(λi + λ0)c], i = 1, ..., k
E[xc
(logx
(k)j
(k)j)]
1 k
[ψ(2) − log(λ
i + .. + λi + λ0)]
=
1
r
(−1)r+1
...
c r=1
i
(λ + ... + λ + λ
1<
<i
i
i
0)
r =1:k
1
r
E[xc
(logx
(k)j
(k)j)2]
1 k
[[log(λ
i + ... + λi + λ0) − ψ(2)]2 + ψ (1) − 1]
=
1
r
(−1)r+1
...
c2 r=1
i
(λ + ... + λ + λ
1<
<i
i
i
0)
r =1:k
1
r
where ψ(·) and ψ (·) are digamma and trigamma functions and are given
by
∂logΓz
ψ(2) =
|z=2 = 0.422785,
∂z
∂2logΓz
ψ (1) =
|z=1 = π2/6 = 1.64474.
∂z2
The above Fisher information matrix nI(λ) is positive definite and using
√
multivariate central limit theorem,
n(ˆ
λ − λ) has an asymptotic multi-
variate normal (MVN) with mean vector zero and variance - covariance
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matrix I−1(λ) = ((Iij)), i, j = 0, 1, ..., k + 1 where Iij are the elements of
the inverse of I(λ).
4. LARGE SAMPLE TESTS
TEST FOR MVE :
First we develop large sample studentized test for testing multivariate
exponentiality (MVE) based on the MLE of c i.e., ˆ
c. The hypothesis of
the test for MVE is H0 : c = 1. The asymptotic distribution of ˆ
c is
√
AN (c, Ik+1,k+1/n). One can obtain studentized test statistic
n(ˆ
c −
1)/( ˆ
Ik+1,k+1)1/2 which is AN (0, 1) under H0 where ˆ
Ik+1,k+1 is estimated
using MLEs of the parameters under (H0 U H1). For the alternatives
H1 : c = 1, we reject H0 if n(ˆ
c − 1)2/( ˆ
Ik+1,k+1) > χ2
where χ2
is
1,1−α
1,1−α
100(1 − α)% point of the chisquare with 1 d.f.
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TEST FOR INDEPENDENCE
We next consider the hypothesis of the test for independence of (X1, .., Xk)
i.e., H0 : λ0 = 0 . Here the proposed test is based on the MLE ˆ
λ0 which
√
is AN (λ0, I00/n). The studentized test statistic is
n(ˆ
λ0/( ˆ
I00)1/2 which
is AN (0, 1) under H0 where ˆ
I00 is the estimate of the variance of ˆ
λ0. For
√
the alternatives H1 : λ0 > 0, we reject H0 if
nˆ
λ0/( ˆ
I00)1/2 > ξ1−α where
ξ1−α is 100(1 − α)% point of the standard normal variate.
TEST FOR SYMMETRY :
We next consider the hypothesis of the test for symmetry or identical
marginals or exchangebility of (X1, ..., Xk) i.e., H0 : λ1 = λ2 = ... = λk
or µ = 0 where µ = (λ2 − λ1, λ3 − λ2, ..., λk − λk−1) . We develop a test
based on MLEs i.e. ˆ
µ = (ˆ
λ2 − ˆ
λ1, ..., ˆ
λk − ˆ
λk−1) and the studentized test
statistic is ˆ
µ ˆ
Σ−1ˆ
µ which is χ2
under H
k−1
0 where ˆ
Σ−1 is the estimate of
variance - covariance matrix of ˆ
µ. For the alternatives H1 : µ = 0, we
reject H0 in favour of H1 if ˆ
µ ˆ
Σ−1ˆ
µ > χ2
. Under the alternatives
k−1,1−α
H1 : µ = 0, ˆ
µ ˆ
Σ−1ˆ
µ is non-cental χ2
with non-centrality parameter
k−1
µ ˆ
Σ−1µ.
ACKNOWLEDGEMENTS
I thank the referee for the constructive suggestions and comments.
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Hanagal, D.D. (1991). Large sample tests for independence
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Hanagal, D.D.(1993b). Some inference results in several sym-
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