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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(X≤x) is given by

F(x;α, β, λ)=1−exp−λx −(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,

F−1(u) = α

βλ −1

λlog (1 −u)−1

βW0α

λeα/λ(1 −u)−β /λ,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 F−1(u) instead of the

more cumbersome notation F−1(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. W−1). 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): −−−−, W−1(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

ta−1e−tdt, z ∈C\R−, a ∈C.(2.1)

Lemma 1. For any positive real number q, we have

Z1

0

W01/xqdx =−q−1/q Γ−1

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)w−1/qe−w/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)w−1/qe−w/q dw

=q(q−1)/qq Γ 2−1

q,W0(1)

q+Γ1−1

q,W0(1)

q.(2.4)

It is well-known that the incomplete gamma function satisﬁes the recurrence

formula Γ(a+1, z) = aΓ (a, z)+zae−z(see Abramowitz and Stegun [1, pp. 260–

262]). According to this, Eq. (2.4) becomes

Z∞

W0(1)

(1 + w)w−1/qe−w/q dw

=−q(q−1)/q Γ−1

q,W0(1)

q+qW0(1) + qW0(1)eW0(1)−1/q

=−q(q−1)/q Γ−1

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 + p−s(1 + p)

q(1+p)/q

Γ−1 + 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 n∈N∗, 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 n∈N∗and k∈N∗, 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)r−11−F(x;α, β , λ)n−rdF (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 F−1(cf. Arnold et al. [2, Chapter 5] for further

details)

E[X1:n] = nZ1

0

(1 −u)n−1F−1(u)du, n ∈N∗,(3.2)

where in our case F−1is 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

βnα

βnλ/β

Γ−nλ

β,nα

β, n ∈N∗.

Proof. For any n∈N∗, from Eq. (3.2) together with Eq. (1.2) we have

E[X1:n] = α

βλ +1

nλ −n

βZ1

0

wn−1W0α

λeα/λw−β/λdw.

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

βnα

βeα/λnλ/β

Γ−nλ

β,nλ

βW0α

λeα/λ

=enα/β

βnα

βnλ/β

Γ−nλ

β,nα

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 n∈N∗and k∈N∗we can use the following

relation (cf. Balakrishnan and Rao [3, p. 156])

EXk

r:n=

n

X

j=n−r+1

(−1)j−(n−r+1)n

jj−1

n−rEXk

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=n−r+1

(−1)j−(n−r+1)n

jj−1

n−rejα/β

βjα

βjλ/β

Γ−jλ

β,jα

β.

Unfortunately, we have not found simple analytical expressions for the mo-

ments E[Xk

1:n] for integers k≥2. However, we provide a general formula for

the moments E[Xk

1:n], which can be used for computing these moments for

integers k≥2 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 nα,β

and nλ.

Math. Model. Anal., 18(3):432–445, 2013.

438 P. Jodr´a

Proof. For any n∈N∗, recall that the cumulative distribution function of

X1:n,F1:n(x;α, β, λ) := P(X1:n≤x), can be computed as follows

F1:n(x;α, β, λ)=1−1−F(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)=1−exp−nλx −(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)mu−se−zu 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

βkEk−1

(nλ

β+1)nα

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

xknλ +nαeβx exp−nλx −(nα/β)eβ x −1dx.

Now, the change of variable u=eβx leads to the following

EXk

1:n=enα/β

βkZ∞

1nλ

β+nα

βu(log u)ke−nαu/βu−(nλ/β)−1du.

From the above equation and the deﬁnition in Eq. (3.4), for any k∈N∗we

have

EXk

1:n=Γ(k+ 1)enα/β

βknλ

βEk

(nλ

β+1)nα

β+nα

βEk

nλ

βnα

β.

Finally, by taking into account the following recursion formula (cf. Milgram [18,

Eq. (2.4)])

(1 −s)Em

s(z) = zEm

s−1(z)−Em−1

s(z), z > 0, s 6= 1, m = 0,1,...,

On Order Statistics from the Gompertz–Makeham Distribution 439

deﬁning E−1

s(z) := e−z, 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) = zs−1Γ(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 k∈N∗and

for r= 2, . . . , n, we have

EXk

r:n=Γ(k+ 1)

βk

n

X

j=n−r+1

(−1)j−(n−r+1)n

jj−1

n−rejα/β Ek−1

(jλ

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 r≥275. 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→∞ 1−1−F(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=F−11−1

n, bn=F−11−1

ne−F−11−1

n(4.2)

cn= 0, dn=F−1(1/n),(4.3)

where F−1is 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 x−log log x+1

2

log log x

log x, x ≥e,

UW0(x) := log x−log log x+e

e−1

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 x≥max{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

e−1

log log(ax)

log(ax)−1

2

log log(bx)

log(bx)

where x≥max{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 log2x−2 log x−log 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

(e−1) 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

(e−1) 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

ε→0

F−1(1 −ε)−F−1(1 −2ε)

F−1(1 −2ε)−F−1(1 −4ε)= 1,(4.10)

where F−1is given by Eq. (1.2). Below, we show that Eq. (4.10) holds.

Denote by Hk(ε) := F−1(1 −kε)−F−1(1 −2kε), k= 1,2. Then, we need

to compute limε→0H1(ε)/H2(ε). From Eq. (1.2), we have

Hk(ε)= 1

λlog 2+ 1

βW0α

λeα/λ(2kε)−β/λ−W0α

λeα/λ(kε)−β/λ,(4.11)

where α,βand λare positive real parameters. By virtue of Lemma 4, since

limε→0ε−β/λ =∞, for k= 1,2 we have

lim

ε→0W0α

λeα/λ(2kε)−β/λ−W0α

λeα/λ(kε)−β/λ =−β

λlog 2.(4.12)

Therefore, from Eqs. (4.11) and (4.12) we get limε→0Hk(ε) = 0 for k= 1,2.

We can now apply l’Hˆopital’s rule to obtain limε→0H1(ε)/H2(ε). Denote by

H0

ithe ﬁrst derivative of Hi,i= 1,2. Then, we have

lim

ε→0

H0

1(ε)

H0

2(ε)= lim

ε→0

W0

0((α/λ)eα/λ(2ε)−β /λ)−W0

0((α/λ)eα/λ(ε)−β /λ)

W0

0((α/λ)eα/λ(4ε)−β /λ)−W0

0((α/λ)eα/λ(2ε)−β /λ)

= lim

ε→0

W0((α/λ)eα/λ(2ε)−β /λ)−W0((α/λ)eα/λ (ε)−β/λ )

W0((α/λ)eα/λ(4ε)−β /λ)−W0((α/λ)eα/λ (2ε)−β/λ )

×lim

ε→0

1+W0((α/λ)eα/λ(4ε)−β/λ)

1+W0((α/λ)eα/λ(ε)−β/λ),(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

limε→0H0

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

ε→0

F−1(ε)−F−1(2ε)

F−1(2ε)−F−1(4ε)= 2−c,(4.14)

where F−1is 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

F−1(kε)−F−1(2kε), k= 1,2. We need to compute limε→0L1(ε)/L2(ε). From

Eq. (1.2), we have

Lk(ε) = 1

λlog1−2kε

1−kε

+1

βW0α

λeα/λ(1 −2kε)−β/λ−W0α

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 limε→0Lk(ε) = 0 for k= 1,2. Then,

we can apply l’Hˆopital’s rule to compute limε→0L1(ε)/L2(ε), so that, denoting

by L0

ithe ﬁrst derivative of Li,i= 1,2, we have to compute

lim

ε→0

L0

1(ε)

L0

2(ε)= lim

ε→0

(F−1)0(ε)−(F−1)0(2ε)

(F−1)0(2ε)−(F−1)0(4ε),

where (F−1)0denotes the ﬁrst derivative of F−1with respect to ε. For k=

1,2,4, it can be checked the following

F−10(kε) = k

λ(1 −kε)(1 + W0((α/λ)eα/λ(1 −kε)−β/λ )) .(4.15)

In view of Eq. (4.15) together with the fact that W0(xex) = x, we get

lim

ε→0F−10(kε) = k

α+λ, k = 1,2,4.(4.16)

Now, by using Eq. (4.16) we obtain

lim

ε→0

(F−1)0(ε)−(F−1)0(2ε)

(F−1)0(2ε)−(F−1)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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