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On Order Statistics from the Gompertz–Makeham Distribution and the Lambert W Function

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The aim of this paper is twofold. First, we show that the expected value of the minimum order statistic from the Gompertz-Makeham distribution can be expressed in closed form in terms of the incomplete gamma function. We also give a general formula for the moments of the minimum order statistic in terms of the generalized integro-exponential function. As a consequence, the moments of all order statistics from this probability distribution can be more easily evaluated from the moments of the minimum order statistic. Second, we show that the maximum and minimum order statistics from the Gompertz-Makeham distribution are in the domains of attraction of the Gumbel and Weibull distributions, respectively. Lambert W function plays an important role in solving these problems.
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Mathematical Modelling and Analysis Publisher: Taylor&Francis and VGTU
Volume 18 Number 3, June 2013, 432–445 http://www.tandfonline.com/TMMA
http://dx.doi.org/10.3846/13926292.2013.807316 Print ISSN: 1392-6292
c
Vilnius Gediminas Technical University, 2013 Online ISSN: 1648-3510
On Order Statistics from
the Gompertz–Makeham Distribution and
the Lambert W Function
Pedro Jodr´a
Universidad de Zaragoza
Mar´ıa de Luna 3, 50018 Zaragoza, Spain
E-mail: pjodra@unizar.es
Received September 12, 2012; revised May 16, 2013; published online June 1, 2013
Abstract. The aim of this paper is twofold. First, we show that the expected
value of the minimum order statistic from the Gompertz–Makeham distribution can
be expressed in closed form in terms of the incomplete gamma function. We also
give a general formula for the moments of the minimum order statistic in terms of
the generalized integro-exponential function. As a consequence, the moments of all
order statistics from this probability distribution can be more easily evaluated from
the moments of the minimum order statistic. Second, we show that the maximum
and minimum order statistics from the Gompertz–Makeham distribution are in the
domains of attraction of the Gumbel and Weibull distributions, respectively. Lambert
W function plays an important role in solving these problems.
Keywords: Gompertz–Makeham distribution, order statistics, moments, domains of at-
traction, Lambert W function.
AMS Subject Classification: 33B30; 60E05; 33F10.
1 Introduction
The Gompertz–Makeham distribution was introduced by the British actuary
William M. Makeham [16] in the second half of the 19th century as an ex-
tension of the Gompertz distribution. Since then, this probability model has
been successfully used in actuarial science, biology and demography to des-
cribe mortality patterns in numerous species, including humans. Marshall and
Olkin [17, Chap. 10] provides a comprehensive review of the history and theory
of this probability distribution and its practical importance is strongly high-
lighted in Golubev [9]. Some more recent mathematical contributions can also
be found in Feng et al. [7], Jodr´a [12], Lager˚as [14] and Teimouri and Gupta [20].
This work has been partially supported by Diputaci´on General de Arag´on (Grupo conso-
lidado PDIE) and by Universidad de Zaragoza, research project UZ2012-CIE-09.
On Order Statistics from the Gompertz–Makeham Distribution 433
To be more precise, let Xbe a continuous random variable having a Gom-
pertz–Makeham distribution with positive real parameters α,βand λ, that is,
the cumulative distribution function F(x;α, β, λ) := P(Xx) is given by
F(x;α, β, λ)=1expλx (α/β)eβ x 1, x > 0, α, β, λ > 0.(1.1)
The parameters αand βare usually interpreted as the initial mortality and the
mortality increase when advancing age, respectively, whereas the parameter λ
represents the risk of death due to causes which do not depend on age such as
accidents and acute infections, among others.
Jodr´a [12, Theorem 2] expresses the inverse of the cumulative distribution
function of the Gompertz–Makeham distribution in closed form, more specifi-
cally,
F1(u) = α
βλ 1
λlog (1 u)1
βW0α
λeα/λ(1 u)β ,0<u<1,(1.2)
where log denotes the natural logarithm and W0denotes the principal branch of
the Lambert W function, which is briefly described in the following paragraph.
Throughout this paper, for notational simplicity we use F1(u) instead of the
more cumbersome notation F1(u;α, β, λ).
For the sake of completeness, we recall that the Lambert W function is a
multivalued complex function whose defining equation is W(z)eW(z)=z, where
zis a complex number. This function has attracted a great deal of attention
after the seminal paper by Corless et al. [6], which summarizes a wide variety
of applications in mathematics and physics together with the main properties
of W. In this regard, if zis a real number such that z≥ −1/e then the Lambert
W function has only two real branches. The real branch taking on values in
[1,) (resp. (−∞,1]) is called the principal (resp. negative) branch and
denoted in the literature by W0(resp. W1). Both real branches are depicted
in Figure 1. In this paper we use only the principal branch W0and it is
known that W0(z) is increasing as zincreases, W0(1/e) = 1, W0(0) = 0,
W0(zez) = zand, in addition, the first derivative of W0is given by
W0
0(z) := dW0(z)
dz =W0(z)
z(1 + W0(z)),for z6= 0.(1.3)
This paper is mainly motivated by the observation in Marshall and Olkin
[17, p. 380] that the moments of the Gompertz–Makeham distribution, that
is, E[Xk] := R
0xkdF (x;α, β, λ), k= 1,2, . . . , cannot be given in closed form.
Here, we show that the expected value of the minimum order statistic from the
Gompertz–Makeham distribution can be expressed in closed form in terms of
the incomplete gamma function. This result can be derived using Eq. (1.2) in
order to highlight the importance of the Lambert W function. We also see that
the kth moment of the minimum order statistic can be expressed explicitly in
terms of the generalized integro-exponential function for any k= 1,2. . . . Then,
as a particular case, we have a closed-form expression for the moments of the
Gompertz–Makeham distribution, E[Xk], for any k= 1,2, . . . . Additionally,
using Eq. (1.2), we also determine the limit distributions of the maximum and
minimum order statistics from this probability distribution.
Math. Model. Anal., 18(3):432–445, 2013.
434 P. Jodr´a
1
−1
−2
−3
−1 1 2 30
−1/e
z
W(z)
Figure 1. The two real branches of W(z): , W1(z); −−−, W0(z).
The remainder of this paper is organized as follows. In Section 2, we pro-
vide a closed-form expression for an integral involving the principal branch
of the Lambert W function, specifically, R1
0xpW0(s/xq)dx, where p,qand s
are positive real numbers. In Section 3, we see that the above integral arises
when we compute the expected value of the minimum order statistic from the
Gompertz–Makeham distribution and, as a consequence, the expected value
can be expressed in closed form in terms of the incomplete gamma function.
In this section, we also see that the moments of the minimum order statis-
tic can be expressed in terms of the generalized integro-exponential function.
These explicit expressions corresponding to the minimum are useful to evaluate
the moments of all order statistics from the Gompertz–Makeham distribution.
A numerical example illustrates the results. Finally, in Section 4, we iden-
tify the asymptotic behavior of the maximum and minimum order statistics
from the Gompertz–Makeham distribution. More precisely, we establish that
the domains of attraction corresponding to the maximum and minimum order
statistics are the Gumbel and Weibull distributions, respectively.
2 An Integral Involving the Principal Branch of the Lam-
bert W Function
In the following result, we provide a closed-form expression for an integral in-
volving the principal branch of the Lambert W function. Denote by Γ(a, z) the
(upper) incomplete gamma function (cf. Abramowitz and Stegun [1, p. 260]),
that is,
Γ(a, z) := Z
z
ta1etdt, z C\R, a C.(2.1)
Lemma 1. For any positive real number q, we have
Z1
0
W01/xqdx =q1/q Γ1
q,W0(1)
q+ W0(1) + q. (2.2)
On Order Statistics from the Gompertz–Makeham Distribution 435
Proof. First, we make the change of variable u= 1/xqin the integral in
Eq. (2.2) and we obtain
Z1
0
W01/xqdx =1
qZ
1
u(q+1)/qW0(u)du.
By setting w= W0(u) on the right-hand side in the above equation, which
implies u=w ewand also du = (1 + w)ewdw by virtue of Eq. (1.3), we get
Z
1
u(q+1)/qW0(u)du =Z
W0(1)
(1 + w)w1/qew/q dw. (2.3)
Now, on the right-hand side in Eq. (2.3) we take into account Eq. (2.1) together
with the fact that Γ(a, ) = 0, and thus we obtain
Z
W0(1)
(1 + w)w1/qew/q dw
=q(q1)/qq Γ 21
q,W0(1)
q+Γ11
q,W0(1)
q.(2.4)
It is well-known that the incomplete gamma function satisfies the recurrence
formula Γ(a+1, z) = (a, z)+zaez(see Abramowitz and Stegun [1, pp. 260–
262]). According to this, Eq. (2.4) becomes
Z
W0(1)
(1 + w)w1/qew/q dw
=q(q1)/q Γ1
q,W0(1)
q+qW0(1) + qW0(1)eW0(1)1/q
=q(q1)/q Γ1
q,W0(1)
q+qW0(1) + q,
where in the last equality we have used the fact that W0(z)eW0(z)=z. This
completes the proof. ut
Essentially the same reasoning as in Lemma 1 leads to the result stated in
Lemma 2 below, so we omit the proof here.
Lemma 2. For any positive real numbers p,qand s, we have
Z1
0
xpW0(s/xq)dx =1
1 + ps(1 + p)
q(1+p)/q
Γ1 + p
q,1 + p
qW0(s)
+ W0(s) + q
(1 + p).
In the next section, we apply Lemma 2 to obtain a closed-form expression
for the expected value of the minimum order statistic from the Gompertz–
Makeham distribution.
Math. Model. Anal., 18(3):432–445, 2013.
436 P. Jodr´a
3 Expected Value of Order Statistics from the Gompertz
– Makeham Distribution
We first introduce some notation and terminology. Denote by Nthe set of
non-negative integers and by N:= N r {0}. For any nN, let X1, . . . , Xn
be a random sample of size nfrom the Gompertz–Makeham distribution with
parameters α,βand λ, and let X1:n, X2:n, . . . , Xn:nbe the corresponding or-
der statistics obtained by arranging Xi,i= 1, . . . , n, in non-decreasing order
of magnitude. The rth element of this sequence, Xr:n, is called the rth or-
der statistic and, in particular, the minimum and maximum order statistics,
that is, X1:n= min{X1, . . . , Xn}and Xn:n= max{X1, . . . , Xn}, the so-called
extremes, are very important in practical applications.
For any nNand kN, it is known that the kth moment of Xr:n,
r= 1, . . . , n, can be computed as follows (cf. Balakrishnan and Rao [3, p. 7])
EXk
r:n=rn
rZ
0
xkF(x;α, β, λ)r11F(x;α, β , λ)nrdF (x;α, β, λ),
(3.1)
where Fis given by Eq. (1.1). For the Gompertz–Makeham distribution,
even the problem of computing the expected value E[Xr:n] by direct numerical
integration of Eq. (3.1) is not efficient from a computational viewpoint, and, in
particular, the problem becomes intractable when the sample size nis relatively
large (see the example at the end of this section).
Our first aim in this section is to compute the expected value E[Xr:n] in a
more efficient manner, that is, avoiding the numerical integration of Eq. (3.1),
and, to this end, the minimum order statistic X1:nplays a crucial role. More
precisely, the expected value of X1:ncan also be obtained by means of the
following formula involving F1(cf. Arnold et al. [2, Chapter 5] for further
details)
E[X1:n] = nZ1
0
(1 u)n1F1(u)du, n N,(3.2)
where in our case F1is given by Eq. (1.2). By using Eq. (3.2) together
with Lemma 2, in the following result we give an analytical expression for the
expected value of the minimum order statistic X1:n, for n= 1,2, . . . . The
special case n= 1 corresponds to the expected value of X.
Corollary 1. Let X1, . . . , Xnbe a random sample of size nfrom a Gompertz–
Makeham distribution with parameters α,βand λ. Then, the expected value
of X1:nis given by
E[X1:n] = enα/β
β
βnλ/β
Γ
β,
β, n N.
Proof. For any nN, from Eq. (3.2) together with Eq. (1.2) we have
E[X1:n] = α
βλ +1
n
βZ1
0
wn1W0α
λeα/λwβdw.
On Order Statistics from the Gompertz–Makeham Distribution 437
Now, by taking into account Lemma 2 in the above equation, we get
E[X1:n] = α
βλ 1
βW0α
λeα/λ
+1
β
βeα/λnλ/β
Γ
β,
βW0α
λeα/λ
=enα/β
β
βnλ/β
Γ
β,
β,
where in the last equality we have used the fact that W0(zez) = z. The proof
is completed. ut
Here, we point out the importance of the closed-form expression given in Co-
rollary 1. On the one hand, from a theoretical point of view, the expected value
of the minimum order statistic can be used to determine if two distributions
with finite expected values are identical (cf. Chan [5] and also Huang [11]). On
the other hand, from a numerical point of view, in order to calculate E[X1:n] by
virtue of Corollary 1, computer algebra systems such as Maple or Mathematica,
among others, can be used to directly evaluate the incomplete gamma function.
Moreover, the analytical expression of E[X1:n] in Corollary 1 can also be used
to compute E[Xr:n], for r= 2, . . . , n, avoiding the numerical integration of
Eq. (3.1). To this end, for any nNand kNwe can use the following
relation (cf. Balakrishnan and Rao [3, p. 156])
EXk
r:n=
n
X
j=nr+1
(1)j(nr+1)n
jj1
nrEXk
1:j, r = 2, . . . , n. (3.3)
By setting k= 1 in Eq. (3.3) and in view of Corollary 1, we get the following
result.
Corollary 2. Let X1, . . . , Xnbe a random sample of size nfrom a Gompertz–
Makeham distribution with parameters α,βand λ. Then, for r= 2, . . . , n, we
have
E[Xr:n] =
n
X
j=nr+1
(1)j(nr+1)n
jj1
nrejα/β
β
βjλ/β
Γ
β,
β.
Unfortunately, we have not found simple analytical expressions for the mo-
ments E[Xk
1:n] for integers k2. However, we provide a general formula for
the moments E[Xk
1:n], which can be used for computing these moments for
integers k2 in a more efficient manner than using Eq. (3.1). We first give
the following preliminary result.
Lemma 3. Let X1, . . . , Xnbe a random sample of size nfrom a Gompertz–
Makeham distribution with parameters α,βand λ. Then, the minimum order
statistic X1:nhas a Gompertz–Makeham distribution with parameters ,β
and .
Math. Model. Anal., 18(3):432–445, 2013.
438 P. Jodr´a
Proof. For any nN, recall that the cumulative distribution function of
X1:n,F1:n(x;α, β, λ) := P(X1:nx), can be computed as follows
F1:n(x;α, β, λ)=11F(x;α, β, λ)n, x > 0.
Then, by using Eq. (1.1) in the above formula we obtain
F1:n(x;α, β, λ)=1expnλx (nα/β)eβ x 1, x > 0,
that is, F1:n(x;α, β, λ) = F(x;nα, β , nλ) which implies the result. ut
Now, we are in a position to express the moments E[Xk
1:n] in terms of the
generalized integro-exponential function. We recall that the generalized inte-
gro-exponential function can be defined by the following integral representation
(cf. Milgram [18] for further details)
Em
s(z) := 1
Γ(m+ 1) Z
1
(log u)musezu du, z > 0,(3.4)
where m= 0,1, . . . and sis a real number.
Proposition 1. Let X1, . . . , Xnbe a random sample of size nfrom a Gompertz
– Makeham distribution with parameters α,βand λ. Then, for the minimum
order statistic X1:nwe have
EXk
1:n=Γ(k+ 1)enα/β
βkEk1
(
β+1)
β, k N.(3.5)
Proof. By virtue of Lemma 3, together with Eq. (1.1), we get
EXk
1:n=Z
0
xkdF (x;nα, β, nλ)
=Z
0
xk+nαeβx expnλx (nα/β)eβ x 1dx.
Now, the change of variable u=eβx leads to the following
EXk
1:n=enα/β
βkZ
1
β+
βu(log u)kenαu/βu(nλ/β)1du.
From the above equation and the definition in Eq. (3.4), for any kNwe
have
EXk
1:n=Γ(k+ 1)enα/β
βk
βEk
(
β+1)
β+
βEk
β
β.
Finally, by taking into account the following recursion formula (cf. Milgram [18,
Eq. (2.4)])
(1 s)Em
s(z) = zEm
s1(z)Em1
s(z), z > 0, s 6= 1, m = 0,1,...,
On Order Statistics from the Gompertz–Makeham Distribution 439
defining E1
s(z) := ez, we achieve the desired result. ut
It is interesting to note that if we take k= 1 in Proposition 1, and using
the fact that E0
s(z) = zs1Γ(1 s, z) (cf. Milgram [18, Eq. (2.2)]), we have an
alternative derivation of Corollary 1. Moreover, it is clear that if we take n= 1
in Proposition 1, then we have an explicit expression for the moments of the
Gompertz–Makeham distribution, E[Xk], for any k= 1,2, . . . . We highlight
that Lenart [15] has recently given explicit expressions for the moments of the
Gompertz distribution in terms of the generalized integro-exponential function.
In fact, if we take n= 1 and λ= 0 in Eq. (3.5), then we obtain the same
expression provided by Lenart for the moments of the Gompertz distribution.
As a consequence of Proposition 1 together with Eq. (3.3), we give a gene-
ral formula for the moments of order statistics from the Gompertz–Makeham
distribution, as stated below.
Corollary 3. Let X1, . . . , Xnbe a random sample of size nfrom a Gompertz–
Makeham distribution with parameters α,βand λ. Then, for any kNand
for r= 2, . . . , n, we have
EXk
r:n=Γ(k+ 1)
βk
n
X
j=nr+1
(1)j(nr+1)n
jj1
nrejα/β Ek1
(
β+1)jα
β.
To complete this section, we provide an example in which we compute the
two first moments corresponding to order statistics from the Gompertz–Make-
ham distribution. Unless differently specified, all of the computations in the
example below were performed with the software Mathematica 8.0, on an Intel
Core2 Quad Q8200 at 2.33GHz with 4GB RAM.
Numerical Example. Promislow and Haselkorn [19] have studied the
aging process for different species of flies. In particular, it is considered a sample
of size n= 990 from a Gompertz–Makeham distribution Xwith parameters
α= 0.00049, β= 0.071 and λ= 0.00092. In this case, the random variable X
represents the lifetime (in days) of females of Drosophila melanogaster (fruit
fly).
With the help of the computer algebra system Mathematica and applying
Corollary 1, in a preprocessing step we have first calculated and stored the
expected values E[X1:j], for j= 1,...,990. Then, from Corollary 2 and ta-
king into account the stored values E[X1:j], we have obtained E[Xr:990], for
r= 2,...,990. Table 1 below displays the results for several values of r. We
remark that the expected values E[Xr:990] could not be obtained by numerical
integration of Eq. (3.1) for integers r275. We also note that these compu-
tational results could not be improved using the software package Maple.
Finally, in Table 1, we also show several values of E[X2
r:990] together with the
standard deviation of Xr:990, that is, Stdv(Xr:990):=(E[X2
r:900]E2[Xr:900 ])1/2.
Based on Proposition 1, in a preprocessing step we have computed and stored
the values E[X2
1:j], for j= 1,...,990, and, then, the values of E[X2
r:990] have
been calculated by means of Corollary 3.
Math. Model. Anal., 18(3):432–445, 2013.
440 P. Jodr´a
Table 1. The two first moments of Xr:990 together with the standard deviation.
r E[Xr:990 ]E[X2
r:990] Stdv(Xr:990 )
1 0.7037690414 0.9732338953537 0.4779430317
100 34.671246464 1204.6301869524 1.5921231108
200 46.617592929 2174.4393369591 1.1132681618
300 53.923823454 2908.5744034664 0.8920022421
400 59.414516903 3530.6685856898 0.7640471189
500 64.018193614 4098.7930277133 0.6811130596
600 68.189517484 4650.2009359519 0.6250135998
700 72.248250231 5220.1564414941 0.5888802934
800 76.560537704 5861.8453664336 0.5739625423
900 81.951502261 6716.4130816494 0.6036215701
990 98.064118070 9621.9189792750 2.3125150810
4 Domains of Attraction of the Extreme Order Statistics
Let X1, . . . , Xnbe a random sample of size nfrom a Gompertz–Makeham
distribution with parameters α,βand λ. Let us consider the maximum and
minimum order statistics, Xn:nand X1:n, respectively, and denote by Fn:nand
F1:nthe corresponding cumulative distribution functions. It is known that
Fn:n(x) = Fn(x) and F1:n(x)=1(1 F(x))n,x > 0, where Fis given by
Eq. (1.1). For notational simplicity, in this section we use F(x) instead of
F(x;α, β, λ) (similarly for F1:nand Fn:n).
In order to identify the asymptotic distributions of Xn:nand X1:nwhen the
sample size nincreases to , it is well-known that the limits of Fn:nand F1:n
take only values 0 and 1, which means that the limit distributions are degene-
rate. To avoid degeneracy, it is necessary to look for linear transformations to
find the asymptotic non-degenerate distributions. If there exist non-degenerate
cumulative distribution functions Hand Lsuch that
lim
n→∞ Fn(an+bnx) = H(x),lim
n→∞ 11F(cn+dnx)n=L(x),(4.1)
where an,bn>0, cnand dn>0 are normalization constants depending on the
sample size n, then it is said that Fbelongs to the maximum –resp. minimum–
domain of attraction of the limit distribution H–resp. L–. Moreover, it is well-
established in the statistical literature that the limit distributions Hand Lcan
only be either a Fr´echet, a Gumbel or a Weibull distribution. For further
details on the asymptotic theory of extremes see, for instance, Castillo at al. [4,
Chap. 9], Kotz and Nadarajah [13, Chap. 1] and also Galambos [8].
In this section, we determine the limit distribution of the extremes of the
Gompertz–Makeham distribution, that is, we identify Hand Lin Eqs. (4.1)
together with the normalizing constants. With the previous notations, below
we state the result and the proof is given in the remainder of this section.
Theorem 1. The Gompertz–Makeham distribution belongs to:
(i) the Gumbel maximum domain of attraction,
(ii) the Weibull minimum domain of attraction.
On Order Statistics from the Gompertz–Makeham Distribution 441
The normalization constants in Eqs. (4.1)
an=F111
n, bn=F111
neF111
n(4.2)
cn= 0, dn=F1(1/n),(4.3)
where F1is given by Eq. (1.2).
The proof of Theorem 1 is based on applying Theorems 9.5 and 9.6 given
in Castillo et al. [4, pp. 203–205]. Both theorems require that the inverse of
the cumulative distribution function corresponding to the continuous random
variable involved can be expressed analytically and, by virtue of Eq. (1.2), this
is the case for the Gompertz–Makeham distribution.
Before proceeding with the proof of Theorem 1, we need some auxiliary
results concerning asymptotic properties of the principal branch of the Lambert
W function. Denote by
LW0(x) := log xlog log x+1
2
log log x
log x, x e,
UW0(x) := log xlog log x+e
e1
log log x
log x, x e,
where log denotes the natural logarithm. Hoorfar and Hassani [10, Theo-
rem 2.7] have shown that LW0(x) and UW0(x) are bounds for W0(x), more
specifically,
LW0(x)W0(x)UW0(x), x e, (4.4)
where the equality is attained only in the case x=e. Now, we establish the
following results.
Lemma 4. For any a > 0and b > 0, we have
lim
x→∞W0(ax)W0(bx)= log(a/b).
Proof. For any positive real numbers aand b, from Eq. (4.4) we can bound
the difference W0(ax)W0(bx) as follows
LW0(ax)UW0(bx)W0(ax)W0(bx)UW0(ax)LW0(bx),(4.5)
which holds for xmax{e/a, e/b}. Now, it can be checked that the upper
bound in Eq. (4.5) can be written as below
UW0(ax)LW0(bx)= loga
b+ loglog(bx)
log(ax)+e
e1
log log(ax)
log(ax)1
2
log log(bx)
log(bx)
where xmax{e/a, e/b}. Then, from the above equality it can be checked
that
lim
x→∞UW0(ax)LW0(bx)= log(a/b).
By similar arguments, for the lower bound in Eq. (4.5) we obtain the following
lim
x→∞LW0(ax)UW0(bx)= log(a/b).
Math. Model. Anal., 18(3):432–445, 2013.
442 P. Jodr´a
As a consequence, the statement of this lemma is obtained by applying the
squeeze theorem to Eq. (4.5). ut
Lemma 5. For any a > 0, we have
lim
x→∞
1+W0(ax)
1+W0(x)= 1.
Proof. For any positive real number a, we can bound (1+W0(ax))/(1+W0(x))
by using Eq. (4.4) as follows
1 + LW0(ax)
1 + UW0(x)1+W0(ax)
1+W0(x)1 + UW0(ax)
1 + LW0(x), x e/a. (4.6)
It can be seen that limx→∞ UW0(ax) = and also that limx→∞ LW0(ax) = .
Then, we can compute limx→∞(1 + UW0(ax))/(1 + LW0(x)) via l’Hˆopital’s
rule. Denote by U0
W0(resp. L0
W0) the first derivative of UW0(resp. LW0). In
addition, denote by
h(x) := 2 log2x2 log xlog log x+ 1, x > 0.(4.7)
After a bit of algebra, U0
W0(ax)/L0
W0(x) can be written as below
U0
W0(ax)
L0
W0(x)=2 log2x
h(x)2 log2x
log(ax)h(x)+2e(log log(ax)1) log x
(e1) log2(ax)h(x), x e/a,
where h(x) is given by Eq. (4.7). Moreover, it can be checked the following
lim
x→∞
2 log2x
h(x)= 1,lim
x→∞
2 log2x
log(ax)h(x)= 0,
lim
x→∞
2e(log log(ax)1) log x
(e1) log2(ax)h(x)= 0.
Thereby, we have
lim
x→∞
1 + UW0(ax)
1 + LW0(x)= lim
x→∞
U0
W0(ax)
L0
W0(x)= 1.(4.8)
On the other hand, by similar arguments, it can also be verified that
lim
x→∞
1 + LW0(ax)
1 + UW0(x)= 1.(4.9)
Finally, from Eqs. (4.8) and (4.9) the statement of this lemma is obtained by
applying the squeeze theorem to Eq. (4.6). ut
With the previous results, we are now in a position to prove Theorem 1.
On Order Statistics from the Gompertz–Makeham Distribution 443
Proof of Theorem 1. (i) By Theorem 9.5 in Castillo et al. [4, pp. 203–
204], a necessary and sufficient condition to determine whether the Gompertz–
Makeham distribution belongs to the maximum domain of attraction of the
Gumbel distribution is the following
lim
ε0
F1(1 ε)F1(1 2ε)
F1(1 2ε)F1(1 4ε)= 1,(4.10)
where F1is given by Eq. (1.2). Below, we show that Eq. (4.10) holds.
Denote by Hk(ε) := F1(1 )F1(1 2), k= 1,2. Then, we need
to compute limε0H1(ε)/H2(ε). From Eq. (1.2), we have
Hk(ε)= 1
λlog 2+ 1
βW0α
λeα/λ(2kε)β/λW0α
λeα/λ(kε)β/λ,(4.11)
where α,βand λare positive real parameters. By virtue of Lemma 4, since
limε0εβ/λ =, for k= 1,2 we have
lim
ε0W0α
λeα/λ(2kε)β/λW0α
λeα/λ(kε)β/λ =β
λlog 2.(4.12)
Therefore, from Eqs. (4.11) and (4.12) we get limε0Hk(ε) = 0 for k= 1,2.
We can now apply l’Hˆopital’s rule to obtain limε0H1(ε)/H2(ε). Denote by
H0
ithe first derivative of Hi,i= 1,2. Then, we have
lim
ε0
H0
1(ε)
H0
2(ε)= lim
ε0
W0
0((α/λ)eα/λ(2ε)β )W0
0((α/λ)eα/λ(ε)β )
W0
0((α/λ)eα/λ(4ε)β )W0
0((α/λ)eα/λ(2ε)β )
= lim
ε0
W0((α/λ)eα/λ(2ε)β )W0((α/λ)eα/λ (ε)β/λ )
W0((α/λ)eα/λ(4ε)β )W0((α/λ)eα/λ (2ε)β/λ )
×lim
ε0
1+W0((α/λ)eα/λ(4ε)β)
1+W0((α/λ)eα/λ(ε)β),(4.13)
where W0
0denotes the first derivative of W0with respect to εand the second
equality in Eq. (4.13) is obtained by taking into account Eq. (1.3). Finally,
we apply Lemmas 4 and 5 to calculate the limit in Eq. (4.13) and then we get
limε0H0
1(ε)/H0
2(ε) = 1, which implies that Eq. (4.10) holds. The normalizing
constants anand bngiven in Eq. (4.2) are chosen by virtue of Castillo et al. [4,
Theorem 9.5]. The proof of part (i) is completed.
(ii) By Theorem 9.6 in Castillo et al. [4, pp. 204–205], a necessary and
sufficient condition for part (ii) is that there exists a constant c > 0 such that
lim
ε0
F1(ε)F1(2ε)
F1(2ε)F1(4ε)= 2c,(4.14)
where F1is given by Eq. (1.2). We claim that c= 1 and, if this is the case,
then the Gompertz–Makeham distribution belongs to the minimum domain
of attraction of the Weibull distribution. To see this, we denote by Lk(ε) :=
Math. Model. Anal., 18(3):432–445, 2013.
444 P. Jodr´a
F1()F1(2), k= 1,2. We need to compute limε0L1(ε)/L2(ε). From
Eq. (1.2), we have
Lk(ε) = 1
λlog12
1
+1
βW0α
λeα/λ(1 2kε)β/λW0α
λeα/λ(1 kε)β/λ,
where α,βand λare positive real parameters. Now, by taking into account
that W0(xex) = xit is easy to check that limε0Lk(ε) = 0 for k= 1,2. Then,
we can apply l’Hˆopital’s rule to compute limε0L1(ε)/L2(ε), so that, denoting
by L0
ithe first derivative of Li,i= 1,2, we have to compute
lim
ε0
L0
1(ε)
L0
2(ε)= lim
ε0
(F1)0(ε)(F1)0(2ε)
(F1)0(2ε)(F1)0(4ε),
where (F1)0denotes the first derivative of F1with respect to ε. For k=
1,2,4, it can be checked the following
F10() = k
λ(1 )(1 + W0((α/λ)eα/λ(1 )β/λ )) .(4.15)
In view of Eq. (4.15) together with the fact that W0(xex) = x, we get
lim
ε0F10() = k
α+λ, k = 1,2,4.(4.16)
Now, by using Eq. (4.16) we obtain
lim
ε0
(F1)0(ε)(F1)0(2ε)
(F1)0(2ε)(F1)0(4ε)=1
2,
which implies that c= 1 in Eq. (4.14), as it was claimed. Finally, the nor-
malizing constants cnand dnin Eq. (4.3) are chosen according to Castillo
et al. [4, Theorem 9.6]. This concludes the proof of part (ii). The proof of
Theorem 1 is completed. ut
Acknowledgements
The author wishes to express his sincere thanks to the anonymous referees
for their helpful comments and suggestions, which led to an improvement of
the paper. In particular, one of the referees pointed out the recent paper by
Lenart [15], which led to Proposition 1.
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