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The generalized inverse Weibull distribution
Article in Statistical Papers · August 2011
DOI: 10.1007/s00362-009-0271-3
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Stat Papers
DOI 10.1007/s00362-009-0271-3
REGULAR ARTICLE
Felipe R. S. de Gusmão · Edwin M. M. Ortega ·
Gauss M. Cordeiro
Received: 2 February 2009 / Revised: 21 July 2005
© Springer-Verlag 2009
Abstract The inverse Weibull distribution has the ability to model failure rates
which are quite common in reliability and biological studies. A three-parameter gen-
eralized inverse Weibull distribution with decreasing and unimodal failure rate is
introduced and studied. We provide a comprehensive treatment of the mathematical
properties of the new distribution including expressions for the moment generating
function and the rth generalized moment. The mixture model of two generalized
inverse Weibull distributions is investigated. The identiﬁability property of the mix-
ture model is demonstrated. For the ﬁrst time, we propose a location-scale regression
model based on the log-generalized inverse Weibull distribution for modeling lifetime
data. In addition, we develop some diagnostic tools for sensitivity analysis. Two appli-
cations of real data are given to illustrate the potentiality of the proposed regression
model.
Keywords Censored data · Data analysis · Inverse Weibull distribution · Maximum
likelihood estimation · Moment · Weibull regression model
F. R. S. de Gusmão · G. M. Cordeiro
DEINFO, Universidade Federal Rural de Pernambuco, Recife, Brazil
F. R. S. de Gusmão
e-mail: felipe556@gmail.com
G. M. Cordeiro
e-mail: gausscordeiro@uol.com.br
E. M. M. Ortega (B)
Departamento de Ciências Exatas, ESALQ, Universidade de São Paulo, Av. Pádua Dias 11,
Caixa Postal 9, Piracicaba, SP 13418-900, Brazil
e-mail: edwin@esalq.usp.br
123

F. R. S. de Gusmão et al.
1 Introduction
The inverse Weibull (IW) distribution has received some attention in the literature.
Keller and Kamath (1982) study the shapes of the density and failure rate functions
for the basic inverse model. The random variable Y has an inverse Weibull distribution
if its cumulative distribution function (cdf) takes the form
G(t) = exp −
α
t
β
, t > 0, (1)
where α > 0 and β > 0. The corresponding probability density function (pdf) is
g(t) = βαβ
t−(β+1)
exp −
α
t
β
. (2)
The IW distribution is also a limiting distribution of the largest order statistics. Drapella
(1993) and Mudholkar and Kollia (1994) suggested the names complementary Weibull
and reciprocal Weibull for the distribution (2). The corresponding survival and hazard
functions are
SG(t) = 1 − G(t) = 1 − exp −
α
t
β
and
hG(t) = βαβ
t−(β+1)
exp −
α
t
β
1 − exp −
α
t
β −1
,
respectively. The kth moment about zero of (2) is
E(T k
) = αk
(1 − kβ−1
),
where (·) is the gamma function.
The rest of the paper is organized as follows. In Sect. 2, we define the general-
ized inverse Weibull (GIW) distribution. Section 3 provides a general formula for its
moments. General expansions for the moments of order statistics of the new distri-
bution are given in Sect. 4. We discuss in Sect. 5 maximum likelihood estimation
with censored data. Section 6 provides some properties of a mixture of two GIW
distributions. Sections 7 and 8 discuss the log-generalized inverse Weibull distribu-
tion and the corresponding regression model, respectively. Some diagnostic measures
for the log-generalized inverse Weibull model are proposed in Sect. 9. Section 10
defines two type of residuals. Section 11 illustrates two applications to real data
sets including the model fitting and influence analysis. Section 12 ends with some
conclusions.
123

2 The generalized inverse Weibull distribution
Let G(t) be the cdf of the inverse Weibull distribution discussed by Drapella (1993),
Mudholkar and Kollia (1994) and Jiang et al. (1999), among others. The cdf of the
GIW distribution can be defined by elevating G(t) to the power of γ > 0, say F(t) =
G(t)γ = exp −γ α
t
β
. Hence, the GIW density function with three parameters
α > 0, β > 0 and γ > 0 is given by
f (t) = γβαβ
t−(β+1)
exp −γ
α
t
β
, t > 0. (3)
We can easily prove that (3) is a density function by substituting u = −γ αβt−β. The
inverse Weibull distribution is a special case of (3) when γ = 1. If T is a random
variable with density (3), we write T ∼ GIW(α, β, γ ). The corresponding survival
and hazard functions are
S(t) = 1 − F(t) = 1 − exp −γ
α
t
β
and
h(t) = γβαβ
t−(β+1)
exp −γ
α
t
β
1 − exp −γ
α
t
β −1
,
respectively.
We can simulate the GIW distribution using the nonlinear equation
t = α
− log(u)
γ
−1/β
, (4)
where u has the uniform U(0, 1) distribution. Plots of the GIW density and survival
function for selected parameter values are given in Figs. 1 and 2, respectively.
2.1 Characterizing the failure rate function
The first derivative h (t) = dh(t)/dt to study the hazard function shape is given by
h (t) = h(t)t−(β+1)
γβαβ
1 − exp −γ
α
t
β −1
− (β + 1)tβ
.
We can easily show that h(t) is unimodal with a maximum value at t∗, where t∗
satisfies the nonlinear equation
γ
α
t∗
1 − exp −γ
α
t∗
β −1
= 1 + β−1
.
123

Fig. 1 Plots of the GIW density for some values of α with β = 5 and γ = 0.5
Fig. 2 Plots of the GIW survival functions for some values of α with β = 5 and γ = 0.5
Evidently, lim
t→0
h(t) = lim
t→∞
h(t) = 0.
3 A general formula for the moments
We hardly need to emphasize the necessity and importance of the moments in any
statistical analysis especially in applied work. Some of the most important features
and characteristics of a distribution can be studied through moments (e.g., tendency,
dispersion, skewness and kurtosis). If the random variable T has the GIW density
function (3), the kth moment about zero is
123

E(T k
) = γ
k
β αk
(1 − kβ−1
). (5)
The moment generating function M(z) of T for |z| < 1 is
M(z) = E(ezT
) =
n
k=0
zk
k!
γ
k
β αk
1 − kβ−1
.
Hence, for |z| < 1, the cumulant generating function of T is
K(z) = log[M(z)] = log
n
k=0
zk
k!
γ
k
β αk
1 − kβ−1
. (6)
The ﬁrst four cumulants obtained from (6) are
κ1 = γ
1
β α 1(β), κ2 = γ
2
β α2
{ 2(β) − 1(β)2
},
κ3 = γ
3
β α3
{ 3(β) − 3 2(β) 1(β) + 2 1(β)3
}
and
κ4 = γ
4
β α4
{ 4(β) − 4 3(β) 1(β) − 3 1(β)2
+ 12 2(β) 1(β)2
− 6 1(β)4
},
where k(β) = (1 − kβ−1) for k = 1, . . . , 4. The skewness and kurtosis are then
ρ3 = κ3/κ
3
2
2 and ρ4 = κ4/κ2
2 , respectively.
The Shannon entropy of a random variable T with density f (t) is a measure of the
uncertainty deﬁned by E{− log[ f (T )]}. The Shannon entropy for the GIWdistribution
can be expressed as
E{− log[ f (T )]} = − log(αβγ ) + (β + 1){log(α) + β−1
[log(γ ) + 0.577216]} + 1,
where 0.577216 is the approximate value for Euler’s constant.
4 Moments of order statistics
Order statistics make their appearance in many areas of statistical theory and practice.
Let the random variable Tr:n be the rth order statistic (T1:n ≤ T2:n ≤ · · · ≤ Tn:n) in a
sample of size n from the GIW distribution, for r = 1, . . . , n. The pdf of Tr:n is given
by
fr:n(t) = Cr:n f (t)F(t)r−1
[1 − F(t)]n−r
, t > 0,
where f (t) comes from (3), F(t) = exp −γ α
t
β
and Cr:n = n!/[(r −1)!(n −r)!].
123

The kth moment about zero of the rth order statistic is
µ(k)
r:n = E(T k
r:n) = Cr:n
∞
0
f (t)F(t)r−1
[1 − F(t)]n−r
dt.
After some algebra, we can obtain
µ(k)
r:n = Cr:nαk
γ k/β
[1 − (k/β)]
n−r
i=0
n − r
i
(−1)i
(r + i)
k
β −1
. (7)
An alternative formula for this moment (Barakat and Abdelkader 2004) is
E(T k
r:n) = k
n
j=n−i+1
(−1)j−n+i−1 j − 1
n − i
n
j
Ij (k), (8)
where Ij (k) denotes the integral
Ij (k) =
∞
0
tk−1
[1 − F(t)]j
dt. (9)
From Eqs. 8 and 9, we have
E(T k
r:n)=k(γ αβ
)
k
β
k(β)
n
j=n−i+1
j
l=0
(−1)j−n+i−1 j − 1
n − i
n
j
(−1)l j
l
l
k
β −1
.
(10)
For γ = 1, we obtain the moments of order statistics of the IW distribution. Equations
7 and 10 are the main results of this section.
5 Maximum likelihood estimation with censored data
Let Ti be a random variable distributed as (3) with the vector of parameters θ =
(α, β, γ )T . The data encountered in survival analysis and reliability studies are often
censored. A very simple random censoring mechanism that is often realistic is one
in which each individual i is assumed to have a lifetime Ti and a censoring time Ci ,
where Ti and Ci are independent random variables. Suppose that the data consist of
n independent observations ti = min(Ti , Ci ) for i = 1, . . . , n. The distribution of Ci
does not depend on any of the unknown parameters of Ti . Parametric inference for
such model is usually based on likelihood methods and their asymptotic theory. The
censored log-likelihood l(θ) for the model parameters reduces to
123

l(θ) = r log(γ ) + log(β) + β log(α) − (β + 1)
i∈F
log(ti ) − γ αβ
i∈F
t
−β
i
+
i∈C
log 1 − exp −γ
α
ti
β
,
where r is the number of failures and F and C denote the sets of uncensored and
censored observations, respectively.
The score functions for the parameters α, β and γ are given by
Uα(θ) =
rβ
α
− γβαβ−1
i∈F
t
−β
i + γβαβ−1
i∈C
t
−β
i
1 − ui
ui
,
Uβ(θ) =
r
β
+ r log(α) −
i∈F
log(ti ) − γ αβ
i∈F
t
−β
i log
α
ti
+γ αβ
i∈C
t
−β
i log
α
ti
1 − ui
ui
and
Uγ (θ) =
r
γ
− αβ
i∈F
t
−β
i + αβ
i∈C
t
−β
i
1 − ui
ui
,
where ui = 1 − exp −γ α
ti
β
is the ith transformed observation.
The MLE θ of θ is obtained by solving the nonlinear likelihood equations Uα(θ) =
0, Uβ(θ) = 0 and Uγ (θ) = 0 using iterative numerical techniques such as the
Newton–Raphson algorithm. We employ the programming matrix language Ox
(Doornik 2007).
For interval estimation of (α, β, γ ) and hypothesis tests on these parameters, we
derive the observed information matrix since the expected information matrix is very
complicated and requires numerical integration. The 3×3 observed information matrix
J(θ) is
J(θ) = −
⎛
⎝
Lαα Lαβ Lαγ
· Lββ Lβγ
· · Lγ γ
⎞
⎠ ,
whose elements are given in Appendix 1. Under conditions that are fulﬁlled for param-
eters in the interior of the parameter space but not on the boundary, the asymptotic
distribution of
√
n(θ − θ) is N3(0, I (θ)−1
),
where I (θ) is the expected information matrix. This asymptotic behavior is valid if
I (θ) is replaced by J(θ), i.e., the observed information matrix evaluated at θ. The
multivariate normal N3(0, J(θ)−1) distribution can be used to construct approximate
conﬁdence regions for the parameters and for the hazard and survival functions. The
123

asymptotic normality is also useful for testing goodness of ﬁt of the GIW distribution
and for comparing this distribution with some of its special sub-models using one of
the three well-known asymptotically equivalent test statistics—namely, the likelihood
ratio (LR) statistic, Wald and Rao score statistics.
6 Mixture of two GIW distributions
Mixture distributions have been considered extensively by many authors. For an excel-
lent survey of estimation techniques, discussion and applications, see Everitt and Hand
(1981), Maclachlan and Krishnan (1997) and Maclachlan and Peel (2000). Recently,
AL-Hussaini and Sultan (2001), Jiang et al. (2001) and Sultan et al. (2007) reviewed
some properties and estimation techniques of ﬁnite mixtures for some lifetime models.
The mixture of two generalized inverse Weibull (MGIW) distributions has density
given by
f (t; θ) =
2
i=1
pi fi (t; θi ),
where 2
i=1 pi = 1, θ = (θT
1 , θT
2 )T , θ1 = (α1, β1, γ1)T , θ2 = (α2, β2, γ2)T and
fi (t; θi ) is the density function of the ith component given by
fi (t; θi ) = γi βi α
βi
i t−(βi +1)
exp −γi
αi
t
βi
, t, αi , βi , γi > 0, i = 1, 2.
Different shapes of the MGIW density are shown in Figs. 3 and 4. The cdf of the
MGIW distribution can be expressed as
F(t; θ) =
2
i=1
pi Fi (t; θi ),
where Fi (t; θi ) = exp −γi
αi
t
βi
is the cdf of the ith GIW distribution for i = 1, 2.
6.1 Properties
Here, we derive some properties of the MGIW distribution by extending the corre-
sponding results for the GIW distribution. The kth moment of the random variable T
following the MGIW distribution (5) is given by
E(T k
) =
2
i=1
pi γ
k
βi
i αk
i (1 − kβ−1
i ).
123

Fig. 3 Plots of the MGIW density for some values of γ2 with α1 = 1, β1 = 2, γ1 = 3, α2 = 5 and β2 = 6
Fig. 4 Plots of the MGIW density for some values of γ1 and γ2 with α1 = 1, β1 = 2, α2 = 5 and β2 = 6
The corresponding survival and the hazard functions are
S(t) =
2
i=1
pi 1 − exp −γi
αi
t
βi
123

and
h(t) =
2
i=1 pi γi βi α
βi
i t−(βi +1) exp −γi
αi
t
βi
2
i=1 pi 1 − exp −γi
αi
t
βi
,
respectively. If γi = 1 and αi = 1/δi , the results derived by Sultan et al. (2007) are
obtained as a special case.
6.2 Identifiability
Let φ be a transform associated with each Fi ∈ having the domain of definition
Dφi with linear map M : Fi → φi . If there exists a total ordering ( ) of such that
(i) F1 F2 (F1, F2 ∈ ) ⇒ Dφ1 ⊆ Dφ2 and
(ii) for each F1 ∈ , there is some s1 ∈ Dφ1 , φ1(s) = 0 such that lims→s1 φ2(s)/
φ1(s) = 0 for F1 < F2 (F1, F2 ∈ ),
the class of all finite mixing distributions is identifiable relative to (Chandra
1977).
Using Chandra’s approach (assumptions (i) and (ii)) and the results by Sultan et al.
(2007), we now prove the following proposition.
Proposition The class of all finite mixing distributions relative to the GIW distribution
is identifiable.
Proof If Ti is a random variable with the GI W distribution, the sth moment is
φi (s) = E(T s
i ) = γ
s/βi
i α−s
i (1 − sβ−1
i ), i = 1, 2.
From the cdf of Ti we have
F1 < F2 when β1 = β2, γ1 = γ2 and α1 < α2 (11)
and
F1 < F2 when α1 = α2 >
1
t
, γ1 = γ2 and β1 < β2. (12)
Let Dφ1 (s) = (−∞, β1), Dφ2 (s) = (−∞, β2) and s = β1. From the facts (11) and
(12), we have
lim
s→β1
φ1(s) = γ
β1/β1
1 α
−β1
1 (1 − β1β−1
1 ) = (0+) = ∞. (13)
When α1 = α2 > 1/t,γ1 = γ2 and β1 < β2, we obtain
lim
s→β1
φ2(s) = γ
β1/β2
1 α
−β1
1 (1 − β1β−1
2 ) > 0. (14)
123

Finally, the results (13) and (14) lead to
lim
s→β1
φ2(s)
φ1(s)
= 0,
and then the identifiability is proved.
It is a much more difficult problem to study the identifiability of a mixture of n ≥ 3
distributions. Lack of identifiability gives no guarantee of convergence to the true
values of parameters and therefore usually gives rise to confusing results.
7 The log-generalized inverse Weibull distribution
Let T be a random variable having the GIW density function (3). The random variable
Y = log(T ) has a log-generalized inverse Weibull (LGIW) distribution whose density
function, parameterized in terms of σ = 1/β and µ = log(α), is given by
f (y; γ, σ, µ)=
γ
σ
exp −
y − µ
σ
− γ exp −
y − µ
σ
, −∞ < y < ∞
(15)
where γ > 0, σ > 0 and −∞ < µ < ∞. Four density curves (15) for selected
parameter values are given in Fig. 5 to show great flexibility of the new parameter γ .
The corresponding survival function to (15) is
S(y) = 1 − exp −γ exp −
y − µ
σ
.
We now define the standardized random variable Z = (Y −µ)/σ with density function
π(z; γ ) = γ exp[−z − γ exp(−z)], −∞ < z < ∞ and γ > 0. (16)
The inverse extreme value distribution follows as a special case when γ = 1.
The kth ordinary moment of the standardized distribution (16), say µk = E(Zk),
is
µk =
∞
−∞
zk
γ exp −z − γ exp(−z) dz. (17)
Substituting u = exp(−z), we can rewrite Eq. 17 as
µk = (−1)k
γ
∞
0
log(u)
k
exp(−γ u)du. (18)
123

Fig. 5 Plots of the LGIW density for some parameter values
The integral in (18) follows from Prudnikov et al. (1986) and Nadarajah (2006) as
∞
0
[log(u)]k
exp(−γ u)du =
∂k[γ −a (a)]
∂ak
a=1
.
Inserting the last equation in (18), yields the kth moment of Z
µk = (−1)k
γ
∂k[γ −a (a)]
∂ak
a=1
. (19)
The ordinary moments of Y can be obtained via binomial expansion and Eq. 19.
8 The log-generalized inverse Weibull regression model
In many practical applications, the lifetimes ti are affected by explanatory variables
such as the cholesterol level, blood pressure and many others. Let xi = (xi1, . . . , xip)T
be the explanatory variable vector associated with the ith response variable yi for
i = 1, . . . , n. Consider a sample (y1, x1), . . . , (yn, xn) of n independent observations,
where each random response is deﬁned by yi = min{log(ti ), log(ci )}. We assume
non-informative censoring and that the observed lifetimes and censoring times are
independent.
For the ﬁrst time, we construct a linear regression model for the response variable
yi based on the LGIW density given by
yi = xT
i β + σzi , i = 1, . . . , n, (20)
123

where the random error zi follows the distribution (16), β = (β1, . . . , βp)T , σ > 0
and γ > 0 are unknown scalar parameters and xi is the explanatory variable vector
modeling the location parameter µi = xT
i β. Hence, the location parameter vector
µ = (µ1, . . . , µn)T of the LGIW model has a linear structure µ = Xβ, where
X = (x1, . . . , xn)T is a known model matrix. The log-inverse Weibull (LIW) (or the
inverse extreme value) regression model is defined by Eq. 20 with γ = 1.
Let F and C be the sets of individuals for which yi is the log-lifetime or log-
censoring, respectively. The total log-likelihood function for the model parameters
θ = (γ, σ, βT
)T can be written from Eqs. 16 and 20 as
l(θ)=r log(γ )−log(σ) −
i∈F
yi − xT
i β
σ
−γ
i∈F
exp −
yi − xT
i β
σ
+
i∈C
log 1 − exp −γ exp −
yi − xT
i β
σ
, (21)
where r is the observed number of failures. The MLE θ of θ can be obtained by maxi-
mizing the log-likelihood function (21). We use the matrix programming language Ox
(MAXBFGS function) (see Doornik 2007) to compute this estimate. From the fitted
model (20), the survival function for yi can be estimated by
S(yi ; ˆγ , ˆσ, β
T
) = 1 − exp − ˆγ exp −
yi − xT
i β
ˆσ
.
Under general regularity conditions, the asymptotic distribution of
√
n(θ − θ)
is multivariate normal Np+2(0, K(θ)−1), where K(θ) is the expected information
matrix. The asymptotic covariance matrix K(θ)−1 of θ can be approximated by the
inverse of the (p+2)×(p+2) observed information matrix J(θ) and then the asymp-
totic inference for the parameter vector θ can be based on the normal approximation
Np+2(0, J(θ)−1) for θ. The observed information matrix is
J(θ) = {− ¨Lr,s} =
⎛
⎝
−Lγ γ −Lγ σ −Lγβj
· −Lσσ −Lσβj
· · −Lβj βs
⎞
⎠ ,
whose elements are given in Appendix 2.
The asymptotic multivariate normal Np+2(0, J(θ)−1) distribution can be used to
construct approximate confidence regions for some parameters in θ and for the hazard
and survival functions. In fact, an 100(1 − α)% asymptotic confidence interval for
each parameter θr is given by
AC Ir = ˆθr − zα/2 − ¨L
r,r
, ˆθr + zα/2 − ¨L
r,r
,
123

where − ¨L
r,r
denotes the rth diagonal element of the inverse of the estimated observed
information matrix J(θ)−1 and zα/2 is the quantile 1 − α/2 of the standard nor-
mal distribution. The asymptotic normality is also useful for testing goodness of
fit of some sub-models and for comparing some special sub-models using the LR
statistic.
The interpretation of the estimated coefficients could be based on the ratio of median
times (see Hosmer and Lemeshow 1999) which holds for continuous or categorical
explanatory variables. When the explanatory variable is binary (0 or 1), and consider-
ing the ratio of median times with x = 1 in the numerator, if ˆβ is negative (positive),
it implies that the individuals with x = 1 present reduced (increased) median survival
time in [exp( ˆβ) × 100%] as compared to those individuals in the group with x = 0,
assuming the other explanatory variables fixed.
We are also interested to investigate if the LIW model is a good model to fit the
data under investigation. Clearly, the LR statistic can be used to discriminate between
the LIW and LGIW models since they are nested models. The hypotheses to be tested
in this case are H0 : γ = 1 versus H1 : γ = 1 and the LR statistic reduces to
w = 2{l(θ) − l(θ)}, where θ is the MLE of θ under H0. The null hypothesis is
rejected if w > χ2
1−α(1), where χ2
1−α(1) is the quantile of the chi-square distribution
with one degree of freedom.
9 Sensitivity analysis
9.1 Global influence
A first tool to perform sensitivity analysis is by means of the global influence starting
from case-deletion. Case-deletion is a common approach to study the effect of drop-
ping the ith observation from the data set. The case-deletion model corresponding to
model (20) is given by
yl = xT
l β + σzl, l = 1, . . . , n, and for l = i. (22)
From now on, a quantity with subscript (i) refers to the original quantity with the ith
observation deleted. Let l(i)(θ) be the log-likelihood function for θ under model (22)
and θ(i) = ( ˆγ(i), ˆσ(i), β
T
(i))T be the MLE of θ obtained by maximizingl(i)(θ). To assess
the influence of the ith observation on the unrestricted estimate θ = ( ˆγ , ˆσ, β
T
)T , the
basic idea is to compare the difference between θ(i) and θ. If deletion of an observa-
tion seriously affect the estimates, more attention should be paid to that observation.
Hence, if θ(i) is far from θ, then the ith case is regarded as an influential observation.
A first measure of the global influence can be expressed as the standardized norm of
θ(i) − θ (so-called generalized Cook distance) given by
GDi (θ) = (θ(i) − θ)T
J(θ)−1
(θ(i) − θ).
Another alternative is to assess the values GDi (β) and GDi (γ, σ) which reveal the
impact of the ith observation on the estimates of β and (γ, σ), respectively. Another
123

well-know measure of the difference between θ(i) and θ is the likelihood distance
L Di (θ) = 2{l(θ) − l(θ(i))}.
Further,wecanalsocompute ˆβj − ˆβj(i)( j = 1, . . . , p)toobtainthedifferencebetween
β and β(i). Alternative global influence measures are possible. We can develop the
behavior of a test statistic, such as the Wald test for explanatory variable or censoring
effect, under a case deletion scheme.
9.2 Local influence
As a second tool for sensitivity analysis, we now describe the local influence method
for the LGIW regression model with censored data. Local influence calculation can be
carried out for the model (20). If likelihood displacement L D(ω) = 2{l(θ) − l(θω)}
is used, where θω is the MLE under the perturbed model, the normal curvature for
θ at the direction d, d = 1, is given by Cd(θ) = 2|dT T
J(θ)−1 d| (see Cook
1986). Here, is a (p + 2) × n matrix depending on the perturbation scheme, whose
elements are given by ji = ∂2l(θ|ω)/∂θj ∂ωi , i = 1, . . . , n and j = 1, . . . , p + 2
evaluated at θ and ω0, and ω0 is the no perturbation vector. For the LGIW model, the
elements of J(θ) are given in Appendix 2. We can calculate normal curvatures Cd(θ),
Cd(γ ), Cd(σ) and Cd(β) to perform various index plots, for instance, the index plot
of dmax, the eigenvector corresponding to Cdmax , the largest eigenvalue of the matrix
B = T
J(θ)−1 and the index plots of Cdi (θ), Cdi (γ ), Cdi (σ) and Cdi (β), so-called
the total local influence (see, for example, Lesaffre and Verbeke 1998), where di is an
n×1 vector of zeros with one at the ith position. Hence, the curvature at direction di
has the form Ci = 2| T
i J(θ)−1
i |, where T
i is the ith row of . It is usual to point
out those cases such that
Ci ≥ 2 ¯C, ¯C =
1
n
n
i=1
Ci .
9.3 Curvature calculations
Here, under three perturbation schemes, we calculate the matrix
= ( ji )(p+2)×n =
∂2l(θ|ω)
∂θi ωj (p+2)×n
, j = 1, . . . , p + 2 and i = 1, . . . , n,
from the log-likelihood function (21). Let ω = (ω1, . . . , ωn)T be the vector of weights.
123

9.3.1 Case-weight perturbation
In this case, the log-likelihood function has the form
l(θ|ω) = log(γ ) − log(σ)
i∈F
ωi −
i∈F
ωi zi − γ
i∈F
ωi exp(−zi )
+
i∈C
ωi log {1 − exp[−γ exp(−zi )]} ,
where zi = (yi − xT
i β)/σ, 0 ≤ ωi ≤ 1 and ω = (1, . . . , 1)T . From now on, we take
= ( 1, . . . , p+2)T , ˆzi = (yi − xT
i
ˆβ)/ˆσ and ˆhi = exp[− ˆγ exp(−ˆzi )].
The elements of the vector 1 have the form
1i =
ˆγ −1 − exp(−ˆzi ) if i ∈ F
ˆhi exp(−ˆzi )(1 − ˆhi )−1 if i ∈ C.
On the other hand, the elements of the vector 2 are
2i =
ˆσ−1{−1 + ˆzi [−1 + ˆγ exp(−ˆzi )]} if i ∈ F
ˆγ ˆσ−1 ˆzi ˆhi exp(−ˆzi )(1 − ˆhi )−1 if i ∈ C.
The elements of the vector j for j = 3, . . . , p + 2 can be expressed as
ji =
ˆσ−1xi j 1 − ˆγ exp(ˆzi ) if i ∈ F
ˆγ ˆσ−1xi j ˆhi exp(−ˆzi )(1 − ˆhi )−1 if i ∈ C.
9.3.2 Response perturbation
We assume that each yi is perturbed as yiw = yi + ωi Sy, where Sy is a scale factor
such as the standard deviation of Y and ωi ∈ . The perturbed log-likelihood function
becomes
l(θ|ω) = r log(γ ) − log(σ) −
i∈F
z∗
i − γ
i∈F
exp(−z∗
i )
+
i∈C
log 1 − exp[−γ exp(−z∗
i )] ,
where z∗
i = (yi + ωi Sy) − xT
i β /σ. In addition, the elements of the vector 1 take
the form
1i =
ˆσ−1Sy exp(ˆzi ) if i ∈ F
ˆσ−1Sy ˆhi exp(−ˆzi ){[ ˆγ exp(−ˆzi ) − 1] 1 − ˆhi + ˆhi exp(−ˆzi )}(1 − ˆhi )−2 if i ∈ C.
123

On the other hand, the elements of the vector 2 are
2i =
⎧
⎨
⎩
ˆσ−2Sy[1 + ˆγ ˆzi exp(−ˆzi ) − exp(−ˆzi )] if i ∈ F
ˆσ−2 ˆγ Sy ˆhi exp(−ˆzi ){ˆzi ˆγ exp(−ˆzi ) − 1 + 1}(1 − ˆhi )−1
+ˆσ−2 ˆγ Sy ˆzi ˆh2
i exp(−2ˆzi )(1 − ˆhi )−2 if i ∈ C.
The elements of the vector j , for j = 3, . . . , p + 2, can be expressed as
ji =
⎧
⎨
⎩
ˆγ ˆσ−2Sy xi j exp(−ˆzi ) if i ∈ F
− ˆγ ˆhi exp(−ˆzi ){−ˆσ−2Sy xi j [ ˆγ exp(−ˆzi ) − 1]}(1 − ˆhi )−1
+ ˆγ ˆσ−2Sy xi j ˆh2
9.3.3 Explanatory variable perturbation
We now consider an additive perturbation on a particular continuous explanatory vari-
able, say Xt , by making xitω = xit +ωi Sx , where Sx is a scale factor and ωi ∈ . This
perturbation scheme leads to the following expressions for the log-likelihood function
and for the elements of the matrix :
l(θ|ω) = r log(γ ) − log(σ) −
i∈F
z∗∗
i − γ
i∈F
exp(−z∗∗
i )
+
i∈C
log 1 − exp[−γ exp(−z∗∗
i )] ,
where z∗∗
i = (yi −x∗T
i β)/σ andx∗T
i β = β1+β2xi2+· · ·+βt (xit +ωi Sx )+· · ·+βpxip.
In addition, the elements of the vector 1 can be expressed as
1i =
−ˆσ−1Sx ˆβt exp(−ˆzi ) if i ∈ F
−ˆσ−1Sx ˆβt ˆhi exp(−ˆzi ){[ ˆγ exp(−ˆzi ) − 1](1 − ˆhi ) + ˆhi exp(−ˆzi )}(1 − ˆhi )−2 if i ∈ C,
the elements of the vector 2 are
2i =
⎧
⎨
⎩
ˆσ−2Sx ˆβt {−1 − ˆγ ˆzi exp(−ˆzi ) + exp(−ˆzi )} if i ∈ F
− ˆγ ˆσ−2Sx ˆβt ˆhi exp(−ˆzi ){ˆzi [ ˆγ exp(−ˆzi ) − 1] + 1}(1 − ˆhi )−1
− ˆγ ˆσ−2Sx ˆβt ˆzi ˆh2
i exp(−2ˆzi )(1 − ˆhi )−2 if i ∈ C,
the elements of the vector j , for j = 3, . . . , p + 2 and j = t, have the forms
ji =
⎧
⎨
⎩
− ˆγ ˆσ−2Sx ˆβt xi j exp(−ˆzi ) if i ∈ F
− ˆγ ˆσ−2Sx ˆβt xi j ˆhi exp(ˆzi )[γ exp(−ˆzi ) + 1](1 − ˆhi )−1
− ˆγ ˆσ−2Sx ˆβt xi j ˆh2
i exp(−2ˆzi )(1 − ˆhi )−2 if i ∈ C,
and the elements of the vector t are given by
123

ti =
⎧
⎪⎨
⎪⎩
ˆσ−1Sx 1 − ˆγ ˆσ−1 ˆβt xit exp(−ˆzi ) − exp(−ˆzi ) if i ∈ F
− ˆγ ˆσ−1Sx ˆhi exp(−ˆzi ){ˆσ−1 ˆβt xit [ ˆγ exp(−ˆzi ) − 1] − 1}(1 − ˆhi )−1
− ˆγ ˆσ−2Sx ˆβt xi j ˆh2
10 Residual analysis
In order to study departures from the error assumption as well as the presence of outli-
ers, we consider the martingale residual proposed by Barlow and Prentice (1988) and
some transformations of this residual.
10.1 Martingale residual
This residual was introduced in the counting process (see Fleming and Harrington
1991). For LGIW regression models can be written as
rMi =
1 + log 1 − exp − ˆγ exp(−ˆzi ) if i ∈ F
log 1 − exp − ˆγ exp(−ˆzi ) if i ∈ C,
where ˆzi = (yi − xT
i β)/ˆσ. The distribution of the residual rMi is skewed with a max-
imum value +1 and minimum value −∞. Transformations to achieve a more normal
shaped form would be more adequate for residual analysis.
10.2 Martingale-type residual
Another possibility is to use a transformation of the martingale residual, based on
the deviance residuals for the Cox model with no time-dependent covariates as intro-
duced by Therneau et al. (1990). We use this transformation of the martingale residual
in order to have a new residual symmetrically distributed around zero (see Ortega
et al. 2008). Hence, a martingale-type residual for the LGIW regression model can be
expressed as
rDi = sgn(rMi ) −2 rMi + δi log(δi − rMi )
1/2
, (23)
where rMi is the martingale residual.
11 Applications
11.1 Vitamin A
We illustrate the proposed model using a data set from a randomized community trial
that was designed to evaluate the effect of vitamin A supplementation on diarrheal
episodes in 1,207 pre-school children, aged 6–48 months at the baseline, who were
123

0
0,2
0,4
0,6
0,8
1
0,4 0,80 0,2 0,6 1
i / n
TTT
Fig. 6 TTT-plot on vitamin A data
assigned to receive either placebo or vitamin A in a small city of the Northeast of
Brazil from December 1990 to December 1991 (see, for example, Barreto et al. 1994).
The vitamin A dosage was 100,000 IU for children younger than 12 months and
200,000 IU for older children, which is the highest dosage guideline established by
the World Health Organization (WHO) for the prevention of vitamin A deficiency.
The total time was defined as the time from the first dose of vitamin A until the
occurrence of an episode of diarrhea. An episode of diarrhea was defined as a sequence
of days with diarrhea and a day with diarrhea was defined when three or more liquid
or semi-liquid motions were reported in a 24-h period. The information on the occur-
rence of diarrhea collected at each visit corresponds to a recall period of 48–72h. The
number of liquid and semi-liquid motions per 24h was recorded. In many applica-
tions, there is a qualitative information about the failure rate function shape which
can help in selecting a particular model. In this context, a device called the total
time on test (TTT) plot (Aarset 1987) is useful. The TTT plot is obtained by plotting
G(r/n) = [( r
i=1 Ti:n) + (n − r)Tr:n]/( n
i=1 Ti:n) for r = 1, . . . , n against r/n
(Mudholkar et al. 1996), where Ti:n, for i = 1, . . . , n, are the order statistics of the
sample.
Figure 6 shows the TTT-plot for the current data with the convex and then concave
shape, which indicates that the hazard function has a unimodal shape. Hence, the GIW
distribution could be an appropriate model for fitting such data. The MLEs (approxi-
mate standard errors in parentheses) are: ˆα = 3.3422 (52.08), ˆβ = 0.9696 (0.0245)
and ˆγ = 8.6448 (130.66). In order to assess if the model is appropriate, Fig. 7a
plots the empirical survival function and the estimated survival function for the GIW
distribution. This distribution fits well to the data under analysis. Additionally, the
estimated hazard function in Fig. 7b has an upside-down bathtub-shaped curve.
11.2 Multiple myeloma
As a second application, we consider the data set given in Krall et al. (1975) and
reported in Lawless (2003, p. 334) connected with survival analysis. This data set has
123

(a) (b)
0
0,25
0,5
0,75
1
100 110 120 130
Time
EstimatedSurvivorFunction
Survival Fitted GIW
Kaplan-Meier estimate
Lower and Upper 95% C.L.
0
0,005
0,01
0,015
0,02
0,025
0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
TiTime
FittedHazardRateFunction
Fig. 7 a Estimated survival function and the empirical survival for vitamin A data. b Estimated hazard rate
function for vitamin A data
recently been analyzed by Jin et al. (2003) and Ghosh and Ghosal (2006) who use a
semi-parametric accelerated failure time model. The aim of the study is to relate the
logarithm of the survival time (y) for multiple myeloma with a number of prognostic
variables for censored data. The data consist of the survival times, in months, of 65
patients who were treated with alkylating agents, of which 48 died during the study
and 17 survived. They also include measurements on each patient for the following
five predictors: logarithm of a blood urea nitrogen measurement at diagnosis (urea)
(x1); hemoglobin measurement at diagnosis (hemoglobin) (x2); age at diagnosis (age)
(x3); sex (sex) (x4, 0 for male and 1 for female) and serum calcium measurement at
diagnosis (serum) (x5).
We fit the LGIW no-intercept regression model
yi = β1xi1 + β2xi2 + β3xi3 + β4xi4 + β5xi5 + σzi , (24)
where the errors z1, . . . , z65 are independent random variables with density function
(16).
11.2.1 Maximum likelihood results
Table 1 provides the MLEs of the parameters for the LGIW and LIW regression models
fitted to the current data using the subroutine MAXBFGS in Ox. The LR statistic for
testing the hypotheses H0: γ = 1 versus H1: γ = 1, i.e. to compare the LIW and LGIW
regression models, is w = 2{−76.87 − (−82.69)} = 11.64 (P-value <0.01) which
yields favorable indications toward to the LGIW model. The LGIW model involves an
extra parameter which gives it more flexibility to fit the data, although some parame-
ters may be estimated with less precision, as is the case of β3 in this example. We note
from the fitted LGIW regression model that the age at diagnostic and the sex do not
seem to be significant. We expect that the survival time increases as the blood urea
nitrogen and the serum calcium measurements at diagnostic decrease, but it increases
123

Table 1 MLEs of the parameters for the LGIW and LIW regression model fitted to the myeloma data
LGIW LIW
Parameter Estimate SE P-value Estimate SE P-value
γ 130.25 180.25 – 1 – –
σ 0.9179 0.1025 – 1.0525 0.1162 –
β1 –1.5459 0.3605 <0.0001 –1.1777 0.3885 0.0035
β2 0.1604 0.0456 0.0008 0.2421 0.0502 <0.0001
β3 0.0006 0.0119 0.9612 0.0309 0.0099 0.0027
β4 0.3093 0.2604 0.2393 0.8197 0.2659 0.0030
β5 – 0.1618 0.0607 0.0097 – 0.0605 0.0549 0.2745
Table 2 MLEs of the parameters for the log-Weibull model fitted to the myeloma data
Parameter Estimate SE P-value
σ 0.8822 0.0943 –
β0 4.5642 1.4489 0.0016
β1 −1.6257 0.5180 0.0017
β2 0.1181 0.0538 0.0282
β3 0.0187 0.0140 0.1815
β4 0.0352 0.2741 0.8979
β5 −0.1215 0.0883 0.1691
Table 3 Statistics AIC, BIC and CAIC for comparing the LGIW and LIW models
Model AIC BIC CAIC
LGIW 167.8 183.0 169.7
LIW 177.4 190.4 178.8
LW 174.6 189.8 176.6
as the hemoglobin measurement at diagnosis increases. As an alternative analysis, we
fit the log-Weibull (LW) regression model (see, for example, Lawless 2003) to these
data. The MLEs of the parameters for this model are given in Table 2. Further, we
provide in Table 3 the Akaike Information Criterion (AIC), the Bayesian Information
Criterion (BIC) and the Consistent Akaike Information Criterion (CAIC) to compare
the LGIW, LIW and log-Weibull regression models. The LGIW regression model out-
performs the other two models irrespective of the criteria and can be used effectively
in the analysis of these data.
123

(a) (b)
0
10
20
30
40
50
60
10 15 20 25 30 35 40 45 50 55 60 65
Index
GeneralizedCookDistance
5
0
2
4
6
10 15 20 25 30 35 40 45 50 55 60 65
Index
LikelihoodDistance
5
40
12
0 5 0 5
Fig. 8 a Index plot of GDi (θ) (Generalized Cook’s distance) on the myeloma data. b Index plot of L Di (θ)
(Likelihood distance) on the myeloma data
(a) (b)
-0,8
-0,7
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
0,3
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Index
dmaxi(θ)
5 0
0,12
0,24
0,36
0,48
0,6
0,72
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Index
5
Ci(θ)
12
Fig. 9 Index plot for θ on the myeloma data (case-weight perturbation): a dmax and b total local influence
11.2.2 Sensitivity analysis
Global influence analysis. In this section, we compute using Ox the case-deletion
measures GDi (θ) and L Di (θ) introduced in Sect. 9.1. The influence measure index
plots are given in Fig. 8, where we note that the cases 5, 12 and 40 are possible
influential observations.
Local and total influence analysis. In this section, we perform an analysis of the
local influence for the Golden shiner data set by fitting LGIW regression models.
Case-weight perturbation. We apply the local influence theory developed in
Sect. 9.2, where the case-weight perturbation is used and the value Cdmax = 1.27
was obtained as a maximum curvature. Figure 9a plots the eigenvector corresponding
to dmax, whereas Fig. 9b plots the total influence Ci . The observations 5 and 12 are
very distinguished in relation to the others.
123

(a) (b)
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Index
dmaxi(θ)
12
5
0
1
2
3
4
5
6
7
8
0 5 10 15 20 25 30 35 40 45 50 55 60 65
Index
5
Ci(θ)
12
Fig. 10 Index plot for θ on the myeloma data (response perturbation): a dmax and b total local influence
Response variable perturbation. The influence of perturbations on the observed
survival times is now analyzed. The value of the maximum curvature was Cdmax =
14.37. Figure 10a shows the plot of dmax versus the observation index, thus indicating
that the observations 5 and 12 are far way from the others. Figure 10b plots the total
local influence (Ci ) versus the observation index, thus suggesting that the observations
5 and 12 again stand out.
Explanatory variable perturbation. The perturbation of vectors for each continuous
explanatory variable (x1, x2, x3 and x5) are now investigated. The values for the max-
imum curvature were Cdmax = 2.05, Cdmax = 1.75, Cdmax = 1.05 and Cdmax = 1.75
for x1, x2, x3 and x5, respectively. Then plots of dmax and the total local influence Ci
against the index of the observations are shown in Figs. 11a–d and 12a–d. We note
that the observations 5 and 12 are very distinguished in relation to the others.
11.2.3 Residual analysis
In order to detect possible outliers in the fitted LGIW regression model, Fig. 13 plots
the martingale-type residualsrDmodi versus fitted values. We note that almost all obser-
vations are within the interval [−2, 2], although a few observations ( 5 and 12) appear
as possible outliers, thus indicating that the model is well fitted.
11.2.4 Impact of the detected influential observations
From the previous sections we can consider the observations 5 and 12 as possi-
ble influential. The observation 5 corresponds to one of the lowest survival times
and shows the extent of hemoglobin. The observation 12 is also one of the low-
est stroke survivors and also shows the extent of the serum calcium. Table 4 gives
the relative changes (in percentages) of the parameter estimates defined by RCθ j
=
[(θ j − θ j (I))/θ j ]×100, and the corresponding p-values, where θ j (I) denotes the
123

(a) (b)
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
10 15 20 25 30 35 40 45 50 55 60 65
Index
dmaxi(θ)
5
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
10 15 20 25 30 35 40 45 50 55 60 65
Index
5
Ci(θ)
(c) (d)
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
10 15 20 25 30 35 40 45 50 55 60 65
Index
dmaxi(θ)
1212
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
0 5
0 5
0 5 0 5 10 15 20 25 30 35 40 45 50 55 60 65
Index
dmaxi(θ)
12
Fig. 11 Index plots for θ on the myeloma data (explanatory variable perturbation): a dmax (urea), b total
local influence (urea), c dmax (hemoglobin) and d total local influence (hemoglobin)
MLE of θ j after removing the “set I” of observations in the LGIW model fitted to the
current data. We consider the following sets: I1 = { 5}, I2 = { 12} and I3 = { 5, 12}.
Table 4 shows that the estimates of the LGIW regression model are not highly sensi-
tive under deletion of the outstanding observations. In general, the significance of the
parameter estimates does not change (at the significance level of 5%) after removing
the set I. Hence, we do not have inferential changes after removing the observations
handed out in the diagnostic plots. The large variations in the parameter estimates
occur for the estimates which are not significant, and then they could be removed
from the model.
12 Conclusions
We introduce a three parameter lifetime distribution, so-called the generalized inverse
Weibull (GIW) distribution, to extend some well-known distributions in the lifetime
123

(a) (b)
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
10 15 20 25 30 35 40 45 50 55 60 65
Index
dmaxi(θ)
12
5
0
0,05
0,1
0,15
10 15 20 25 30 35 40 45 50 55 60 65
Index
5
Ci(θ)
12
(c) (d)
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
10 15 20 25 30 35 40 45 50 55 60 65
Index
12
dmaxi(θ)
5
-0,5
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
0 5
0 5
0 5 0 5 10 15 20 25 30 35 40 45 50 55 60 65
Index
12
dmaxi(θ)
5
Fig. 12 Index plot for θ on the myeloma data (explanatory variable perturbation): a dmax (age), b total
local influence (age), c dmax (serum) and d Total local influence (serum)
0,250 0,5
Fitted Value
Martingale-typeResidual(rDi)
5 12
3
2
1
0
-1
-2
-3
Fig. 13 Plot of the martingale-type residuals (rDi
) versus fitted values for the LGIW model fitted to
myeloma data
123

Table 4 Relative changes [-RC- in %], parameter estimates and their P-values in parentheses for the
indicated set
Dropping ˆγ ˆσ ˆβ1 ˆβ2 ˆβ3 ˆβ4 ˆβ5
– – – – – – –
All observations 130.3 0.918 −1.546 0.160 0.001 0.309 −0.162
(–) (–) (0.000) (0.000) (0.961) (0.239) (0.010)
[313] [3] [−4] [30] [808] [65] [−9]
Set I1 537.7 0.892 −1.603 0.112 −0.004 0.108 −0.177
(–) (–) (0.000) (0.030) (0.720) (0.695) (0.003)
[13] [2] [3] [−7] [−714] [−48] [−13]
Set I2 112.8 0.896 −1.504 0.172 0.005 0.459 −0.182
(–) (–) (0.000) (0.000) (0.694) (0.082) (0.003)
[−219] [5] [−1] [20] [117] [14] [−21]
Set I3 414.9 0.876 −1.555 0.128 0.000 0.265 −0.195
(–) (–) (0.000) (0.009) (0.993) (0.337) (0.001)
literature. The new distribution is much more flexible than the inverse Weibull distri-
bution and could have increasing, decreasing and unimodal hazard rates. We derive
explicit algebraic formula for the rth moment which holds in generality for any param-
eter values. We discuss maximum likelihood estimation and hypothesis tests for the
parameters.
We also propose a log-generalized inverse Weibull (LGIW) regression model for
analysis of lifetime censored data. We show how to obtain the maximum likelihood
estimates (MLEs) and develop asymptotic tests for the parameters based on the asymp-
totic distribution of the estimates. We provide applications of influence diagnostics
(global, local and total influence) in LGIW regression models to censored data. Further,
we discuss the robustness aspects of the MLEs from the fitted LGIW regression model
through residual and sensitivity analysis. The practical relevance and applicability of
the proposed regression model are demonstrated in two real data analysis.
Appendix 1: The observed information matrix for the GIW model
The elements of the observed information matrix J(θ) for the model parameters
(α, β, γ ) under censored data are given by
Lαα(θ) = −
rβ
α2
− γβ(β − 1)αβ−2
i∈F
t
−β
i
+ γβαβ−2
i∈C
t
−β
i
1 − ui
ui
β − 1 −
γβ
ui
α
ti
β
;
123

Lαβ(θ) =
r
α
− γ αβ−1
i∈F
t
−β
i 1 + β log
α
ti
+ γ αβ−1
i∈C
t
−β
i
1 − ui
ui
1 + β log
α
ti
1 −
γ
ui
α
ti
β
;
Lγβ(θ)=−αβ
i∈F
t
−β
i log
α
ti
+αβ
i∈C
t
−β
i log
α
ti
1−ui
ui
1−
γ
ui
α
ti
β
;
Lαγ (θ) = −βαβ−1
i∈F
t
−β
i + βαβ−1
i∈C
t
−β
i
1 − ui
ui
1 −
γ
ui
α
ti
β
;
Lγ γ (θ) = −
r
γ 2
− α2β
i∈C
t
−2β
i
1 − ui
u2
i
;
and
Lββ(θ) = −
r
β2
− γ αβ
i∈F
log
α
ti
log(α) − t
−β
i log(ti )
+ γ αβ
i∈C
log
α
ti
1 − ui
ui
log(α) − t
−β
i log(ti )
−
t
−β
i γ
ui
α
ti
β
log
α
ti
.
Here, ui = 1 − exp −γ α
ti
β
.
Appendix 2: The observed information matrix for the LGIW model
We derive the second-order partial derivatives of the log-likelihood function. After
some algebraic manipulations, we obtain
Lγ γ = −
r
σ2
−
i∈C
exp(−2zi )hi (1 − hi )−2
;
Lγ σ =
i∈F
exp(−zi )(˙zi )σ +
i∈C
hi exp(−zi )(˙zi )σ [γ exp(−zi )](1 − hi )−1
+
i∈C
h2
i γ exp(−2zi )(˙zi )σ (1 − h−2
i );
123

Lγβj =
i∈F
exp(−zi )(˙zi )βj +
i∈C
hi exp(−zi )(˙zi )βj [γ exp(−zi )](1 − hi )−1
+
i∈C
h2
i γ exp(−2zi )(˙zi )βj (1 − h−2
i );
Lσσ =
r
σ2
+
i∈F
−(¨zi )σσ + γ exp(−zi ) − [(˙zi )σ ]2
+ (¨zi )σσ
−γ
i∈C
hi exp(−zi ) [(˙zi )σ ]2
[γ exp(−zi ) − 1] + (¨zi )σσ (1 − hi )−1
−γ
i∈C
h2
i exp(−2zi ) [(˙zi )σ ]2
(1 − hi )−2
;
Lσβj = −
i∈F
(¨zi )βj σ + γ
i∈F
− exp(−zi )(˙zi )βj (˙zi )σ + exp(−zi )(¨zi )βj σ
−γ
i∈C
hi exp(−zi ) (˙zi )βj (˙zi )σ [γ exp(−zi ) − 1] + (¨zi )βj σ (1 − hi )−1
+ γ
i∈C
h2
i exp(−2zi )(˙zi )β− j (˙zi )σ (1 − hi )−2
;
and
Lβj βs =
i∈F
−γ exp(−zi )(˙zi )βj (˙zi )βs
−γ
i∈C
hi exp(−zi ) (˙zi )βj (˙zi )βs [γ exp(−zi ) − 1] (1 − hi )−1
−γ
i∈C
h2
i exp(−2zi )(˙zi )βj (˙zi )βs (1 − hi )−2
;
where hi = exp[−γ exp(−zi )], (˙zi )σ = −zi /σ, (˙zi )βj = −xi j /σ, (˙zi )βs = −xis/σ,
(¨zi )σσ = 2zi /σ2, (¨zi )βj σ = xi j /σ2 and zi = (yi − xT
i β)/σ.
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