Journal of Statistical Theory and Applications

Volume 17, Issue 1, March 2018, Pages 59 - 76

The Kumaraswamy Marshall-Olkin Log-Logistic Distribution with Application

Authors
Selen Cakmakyapanselencakmakyapan@hacettepe.edu.tr
Department of Statistics, Hacettepe University, Ankara, Turkey
Gamze Ozelgamzeozl@hacettepe.edu.tr
Department of Statistics, Hacettepe University, Ankara, Turkey
Yehia Mousa Hussein El Gebalyyehia1958@hotmail.com
Department of Statistics, Mathematics and Insurance, Benha University, Egypt
G. G. Hamedanigholamhoss.hamedani@marquette.edu
Department of Mathematics, Statistics and Computer Science, Marquette University, USA
Received 1 November 2016, Accepted 20 March 2017, Available Online 31 March 2018.
DOI
https://doi.org/10.2991/jsta.2018.17.1.5How to use a DOI?
Keywords
Kumaraswamy-G, Maximum likelihood, Log-Logistic, Order statistic
Abstract

In this paper, we define and study a new lifetime model called the Kumaraswamy Marshall-Olkin log-logistic distribution. The new model has the advantage of being capable of modeling various shapes of aging and failure criteria. The new model contains some well-known distributions as special cases such as the Marshall-Olkin log-logistic, log-logistic, lomax, Pareto type II and Burr XII distributions. Some of its mathematical properties including explicit expressions for the quantile and generating functions, ordinary moments, skewness, kurtosis are derived. The maximum likelihood estimators of the unknown parameters are obtained. The importance and flexibility of the new model is proved empirically using a real data set.

Copyright
Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

There has been an increased interest in defining new generated classes of univariate continuous distributions by introducing additional shape parameter(s) to a baseline model. The extended distributions have attracted several statisticians to develop new models. The addition of parameters has been proven to be useful in exploring skewness and tail properties, and also for improving the goodness-of-fit of the generated family. The well-known generators are the following: the Marshall-Olkin distribution family by Marshall and Olkin (1997), the beta-G by Eugene et al. (2002), the Kumaraswamy-G (Kw-G) by Cordeiro and de Castro (2011), the Logistic-G by Torabi and Montazari (2014), the transformed-transformer (T-X) by Alzaatreh et al. (2013), the odd exponentiated generalized by Cordeiro et al. (2013), the Weibull-G by Bourguignon et al. (2014), the Kumaraswamy Marshal-Olkin distribution family by Alizadeh et al. (2015), the transmuted geometric-G by Afify et al. (2016a) and the beta transmuted-H by Afify et al. (2017).

Marshall and Olkin (1997) proposed a flexible family of distributions and introduced an interesting method of adding a new parameter to an existing distribution. The resulting new distribution includes the original distribution as a special case and gives more flexibility to model various types of data. For further information about the Marshall–Olkin family of distributions, see Barreto-Souza et al. (2013). The log-logistic (LL) distribution (known as the Fisk distribution in economics) has been widely used particularly in survival and reliability over the last few decades. It is the probability distribution of a random variable whose logarithm has a logistic distribution, an alternative to the log-normal distribution since it presents a failure rate function that increases initially and decreases later. The cumulative distribution function (cdf) and probability density function (pdf) of the LL distribution are given (for x > 0) by

GLL(x;γ,α)=1[1+(xα)γ]1andgLL(x;γ,α)=γxγ1αγ[1+(xα)γ]2,
respectively, where α > 0 is the scale parameter and γ > 0 is the shape parameter.

Searching a more flexible LL distribution, many authors defined generalizations and modified forms of the LL distribution, with different number of parameters. For example, the Kumaraswamy log-logistic (de Santana et al., 2012), Marshall-Olkin LL (MOLL) (Gui, 2013), Lomax log-logistic (Cordeiro et al., 2014), McDonald log-logistic (Tahir et al., 2014), beta log-logistic (Lemonte, 2014), transmuted log-logistic (Granzotto and Louzada, 2015), Kumaraswamy transmuted log-logistic (Afify et al., 2016b) and generalized transmuted log-logistic (GTLL) (Nofal et al., 2017) distributions.

Gui (2013) defined the cdf and pdf of the MOLL distribution (for x > 0) by

GMOLL(x;γ,α,β)=1βαγ(xγ+βαγ)andgMOLL(x;γ,α,β)=βγαγxγ1(xγ+βαγ)2,
respectively, where α, γ, β > 0. For β = 1, we obtain the LL distribution.

The goal of this paper is to define and study a new lifetime model called the Kumaraswamy Marshall-Olkin Log-Logistic (“KMOLL” for short) distribution. The main feature of this model is that two additional shape parameters are inserted in (2) to give more flexibility in the form of the generated distribution. Based on the Kumaraswamy-generalized (K-G) family proposed by Cordeiro and de Castro (2011), we construct the new five-parameter KMOLL distribution. We give some mathematical properties of the new distribution with the hope that it will attract wider applications in engineering, reliability, life testing and other research. In fact, the KMOLL distribution can provide better fits than other models.

Let g(x) and G(x) denote the pdf and cdf of the baseline model. Cordeiro and de Castro (2011) defined the cdf of the K-G family by

F(x)=1{1G(x)a}b.
The corresponding pdf of (1.3) is given by
f(x)=abg(x)G(x)a1{1G(x)a}b1,
where a > 0 and b > 0 are two extra shape parameters whose role are to govern skewness and tail weights. Clearly, for a = b = 1, we obtain the baseline distribution.

To this end, we start from the MOLL distribution to define the new KMOLL distribution by inserting (1.2) in equations (1.3) and (1.4). Then, the cdf (for x > 0) of the KMOLL distribution is given by

F(x;a,b,γ,α,β)=1[1(xγxγ+βαγ)a]b.
The corresponding pdf of (1.5) is
f(x;a,b,γ,α,β)=abβγαγxγa1(xγ+βαγ)a+1[1(xγxγ+βαγ)a]b1.
where α is a scale parameter and the shape parameters a,b,γ and β govern the skewness of (1.6).

A random variable X with the pdf (1.6) is denoted by X ~ KMOLL(a,b,α,γ,β). The survival function, hazard rate function (hrf) and cumulative hazard rate function (chrf) of X are, respectively, given by

F¯(x;a,b,γ,α,β)=[1(xγxγ+βαγ)a]b,
h(x;a,b,γ,α,β)=abβγαγxγ1(xγ+βαγ)2(xγxγ+βαγ)a1[1(xγxγ+βαγ)a]1
and
H(x;a,b,γ,α,β)=blog1(xγxγ+βαγ)a.
Some of the possible shapes of the pdf (1.6) for selected parameter values are illustrated in Figure 1. As seen from Figure 1, the density function can take various forms depending on the parameter values. It is evident that the KMOLL distribution is much more flexible than the MOLL distribution, i.e. the additional shape parameters a and b allow for a high degree of flexibility of the KMOLL distribution. Both unimodal and monotonically decreasing shapes appear to be possible.

Fig. 1.

Plots for the pdf of the KMOLL distribution for several parameter values.

Plots for the hrf of the KMOLL distribution for several parameter values are displayed in Figure 2. Figure 2 shows that the hrf of the KMOLL distribution can be bathtub, upside down bathtub (unimodal), increasing or decreasing. This attractive flexibility makes the hrf of the KMOLL useful and suitable for non-monotone empirical hazard behaviors which are more likely to be encountered or observed in real life situations.

Fig. 2.

Plots of the hrf of the KMOLL distribution for several parameter values.

We now state a useful expansion for the KMOLL density. Using the binomial expansion, the pdf of the KMOLL reduces to

f(x)=g(x)i=1wiGa(i+1)1(x)
where wi=(1)iab(b1i)

The importance of the KMOLL distribution is that it contains as special sub-models several well-known distributions. Table 1 lists the special distributions related to KMOLL distribution.

Reduced Model a b γ α β
KLL a b γ α 1
MOLL 1 1 γ α β
EMOLL a 1 γ α β
GMOLL 1 b γ α β
ELL a 1 γ α 1
GLL 1 b γ α 1
LL 1 1 γ α 1
Dagum a 1 γ - αβ1/γ
BurrXII 1 b 1 - αβ
EBurrXII a b 1 - αβ
ParetoII 1 1/ b 0 - αβ/b
EParetoII a 1/ b 0 - αβ/b
Lomax a 1/ b - - αβ/b
Table 1.

Sub-models of the KMOLL distribution

The rest of the article is outlined as follows. In Section 2, we obtain the quantile function, shapes, skewness, kurtosis, moments, moment generating functions, Rényi entropies, reliability function and order statistics of X. Certain characterizations are presented in Section 3. The maximum likelihood estimates (MLEs) of the model parameters are obtained in Section 4. An application to real data set is considered in Section 5. Finally, Section 6 provides some concluding remarks.

2. The KMOLL Properties

In this section, we investigate mathematical properties of the KMOLL distribution including quantile function, skewness, kurtosis, shapes of functions, moments, the Rényi and Shannon entropies, reliability and order statistics.

2.1. Quantile function

Quantile functions are in widespread use in statistics and often find representations in terms of lookup tables for key percentiles. Let X ~ KMOLL(a,b,α,γ,β). The quantile function say Q(u) is defined by inverting F(x) in (1.5) as

x=Q(u)=αβ1/γ[[1(1u)1/b]1/a1]1/γ.
The effect of the shape parameters a,b,α, γ, β, on the skewness and kurtosis can be considered based on quantile measures. There are many heavy tailed distributions for which this measure is infinite. So, it becomes uninformative precisely when it needs to be. The Bowley’s skewness is based on quartiles:
S=Q(3/4)2Q(1/2)+Q(1/4)Q(3/4)Q(1/4)
and the Moors’ kurtosis is based on octiles:
K=Q(7/8)Q(5/8)+Q(3/8)Q(1/8)Q(6/8)Q(2/8),
where Q(.) represents the quantile function of X. These measures are less sensitive to outliers and they exist even for distributions without moments. Skewness measures the degree of the long tail and kurtosis is a measure of the degree of tail heaviness. When the distribution is symmetric, S = 0 and the when the distribution is right (or left) skewed, S > 0 or (S < 0). As K increases, the tail of the distribution becomes heavier.

2.2. Moments and moment generating function

Some of the most important features and characteristics of a distribution can be studied through moments (e.g. tendency, dispersion, skewness and kurtosis). Now we obtain ordinary moments and the moment generating function of the KMOLL distribution. The ordinary moments E(X)n = μ′n, n = 1,2,..., of the KMOLL distribution can be obtained, using (1.7), as

μn=i=0wi0xngMOLL(x)GMOLLa(i+1)1(x)dx,
where wi=(1)iab(b1i). Here, gMOLL(.) and GMOLL(.) are the pdf and cdf of the MOLL distribution, respectively. Then, the integral part in (2.1) is defined as
0xngMOLL(x)GMOLLa(i+1)1(x)dx=01QMOLLn(u)ua(i+1)1du,
where QMOLL(.) is the quantile function of the MOLL distribution for 0 < u< 1. Then, we obtain
μn=βn/γαni=0wiB(nγ+a(i+1),1nγ),
where B(.,.) is the beta function.

Further, the central moments (μn) and cumulants (κn), n = 1, 2,..., of the KMOLL distribution can be obtained from

μn=k=0n(1)k(nk)μ1nμnkandκn=μnk=0n(1)k(nk)μ1kμnk.
Here, κ1 = µ′1, κ2=μ2μ12, κ3=μ33μ2μ1+2μ13, κ4=μ44μ3μ13μ22+12μ2μ126μ14 etc.

The skewness γ1=κ3/κ23/2 and kurtosis γ2=κ4/κ22 are also computed from the second, third and fourth cumulants. Table 2 gives moments, skewness, and kurtosis of the KMOLL distribution for some parameter values.

KMOLL(a, b, γ, α, β) μ′1 μ′2 μ′3 μ′4 S K
(0.1,1.98,16,18,20) 11.806 167.936 2656.865 45090.060 0.000 −0.811
(5,10,2,3,8) 11.362 133.290 1612.749 20108.035 0.335 0.357
(2,1,5,4,3) 6.228 47.418 526.448 185560.800 4.883 2403.251
(4,9,1.2,1.5,8) 10.148 119.066 1601.209 24581.778 1.028 2.180
(0.4,15,0.5,1.2,5) 0.018 0.002 0.000 0.000 5.511 53.883
(7,2,3.5,3,0.2) 0.565 0.356 0.258 0.230 2.200 13.764
(1,4,1,8,1) 0.807 0.668 0.567 0.491 −0.207 0.262
(0.5,9,16,1,13) 4.126 86.379 4204.147 381961.900 5.671 63.640
(1.5,2.5,2,7,4) 4.163 17.806 78.167 352.098 0.303 0.765
(25,10,0.5,1.2,1) 3.660 15.332 73.762 409.965 1.287 3.412
Table 2.

Moments, skewness and kurtosis of the KMOLL distribution for some parameter values

Table 2 indicates that the skewness value can be positive and negative, also close to zero. Hence, the KMOLL distribution can be right-skewed, left-skewed or symmetric.

Figure 3 also depicts plots for the skewness and kurtosis coefficients related to additional parameters. In the figure, a parameter decreases while other parameters are kept fixed. These plots indicate that both measures can be very sensitive on these shape parameters. Thus, indicating the importance of the proposed distribution.

Fig. 3.

Skewness and kurtosis plots of the KMOLL distribution for some parameter values

The moment generating function (mgf) is widely used as an alternative way to analytical results compared with working directly with pdf and cdf. The mgf of X is

M(t)=E(etx)=n=0μnn!tn,
or another representation for M(t) can be obtained using (1.7)
M(t)=i=0wi0etxgMOLL(x)GMOLLa(i+1)1(x)dx
where wi=(1)iab(b1i). Then, the integral part in (2.2) is given as 0etxgMOLL(x)GMOLLa(i+1)1(x)dx=01etQMOLL(u)ua(i+1)1du where QMOLL(.) is the quantile function of the MOLL distribution for 0 < u < 1. From the Maclaurin expansion, we obtain etQMOLL(u)=j=0[tα(βu1u)1/γ]j/j!.

Then we obtain

M(t)=j=0(tα)jβj/γj!01uj/γ+a(i+1)1(1u)j/γdu=i=0j=0wi(tα)jβj/γj!B(jγ+a(i+1),1jγ)=j=0wi(tα)jβj/γj!B(jγ+a(i+1),1jγ).

2.3. Unimodality

The pdf of the KMOLL model is decreasing or unimodal. In order to investigate the critical points of its density function, its first derivative with respect to x is

ddxf(x)=b(b1)a2β2γ2α2γx2γ2(xγxγ+βαγ)2a2((xγxγ+βαγ)a+1)b2(x2γ+2β(xα)γ+β2α2γ)2+ab(a1)β2γ2α2γx2γ2(xγxγ+βαγ)a2((xγxγ+βαγ)a+1)b1(x2γ+2β(xα)γ+β2α2γ)22abβγ2αγx2γ2(xγxγ+βαγ)a1((xγxγ+βαγ)a+1)b1(xγ+βαγ)3+abβγ(γ1)αγxγ2(xγxγ+βαγ)a1((xγxγ+βαγ)a+1)b1(xγ+βαγ)2=0.
There may be more than one root to (2.4). If x= x0 is a root of (2.4), then it corresponds to a local maximum If df(x) / dx > 0 for all x < x0 and df (x)/dx < 0 for all x > x0. It corresponds to a local minimum if df (x)/dx < 0 for all x < x0 and df (x)/dx > 0 for all x> x0. It corresponds to a point of inflexion if either df (x)/dx > 0 for all xx0 or df (x) /dx < 0 for all xx0.

2.4. Entropies

The entropy of a random variable X with density function f(x) is a measure of variation of the uncertainty. Two popular entropy measures are the Rényi and Shannon entropies (Rényi (1961). Shannon (1951)). Here. we derive expressions for the Rényi and the Shannon entropies of the KMOLL distribution. The Rényi entropy of a random variable with pdf f(x) is defined as

IR(δ)=11δlogfδ(x)dx
for δ > 0 and δ ≠ 1. Then, we can write
0fδ(x)dx=0[gMOLL(x)i=0wiGMOLLa(i+1)1(x)]δdx=0gMOLLδ(x)GMOLL(a1)δ(x)[i=0wiGMOLLai(x)]δdx=i=0cδ,i0gMOLLδ(x)GMOLL(a1)δ+ai(x)dx,
where wi=(1)iab(b1i) and cδ,i=(iw0)1m=1i[m(δ+1)i]wmcδ,im.

The integral part of (2.5) is

0gMOLLδ(x)GMOLL(a1)δ+ai(x)dx=01uai+δaδgMOLLδ1QMOLL(u)du=(γα1β1/γ)δ101uai+δaδ+(γ1)(δ1)γ(1u)(γ+1)(δ1)γdu=(γα1β1/γ)δ1B(ai+δaδ+(γ1)(δ1)γ+1,(γ+1)(δ1)γ+1).
Then the Rényi entropy of the KMOLL distribution is given by
IR(δ)=11δlog(γα1β1/γ)δ1B(ai+δaδ+(γ1)(δ1)γ+1,(γ+1)(δ1)γ+1).
The Shannon entropy plays a similar role as the kurtosis measure in comparing the shapes of various densities and measuring heaviness of tails. The Shannon entropy of a random variable X is defined by. E⌊−log f(x)⌋. It is the special case of Rényi entropy when δ > 1. The Shannon entropy of the KMOLL distribution is
Elogf(x)=log(ab)E[loggMOLL(x)](a1)E[logGMOLL(x)](b1)E{log[1GMOLLa(x)]}.
To obtain three expectations terms given above. We define and compute
E[loggMOLL(x)]=ab0log(gMOLL(x))gMOLL(x)GMOLLa1(x)[1GMOLLa(x)]b1dx=k=0wkIk.
Here we have
Ik=0log[gMOLL(x)]gMOLL(x)GMOLLa(k+1)1(x)dx=01log{gMOLL[QMOLL(u)]}ua(k+1)1du=01log[γα1β1/γuγ1γ(1u)γ+1γ]ua(k+1)1du=1ak+a(logγlogαlogβγ)+log12j=1(1)j12jj(ak+a+1)F12(j,ak+a,ak+a+1,2)
where 2F1 is the generalized hypergeometric function defined by
F12(j,ak+a,ak+a+1,2)=k=0(a)k(b)k(c)kxkk!
and (a)k = a(a + 1)…(a + k − 1) denotes ascending factorial.

Similarly, the following expectations are defined for (12) as E[1GMOLLa(x)]=b01loguub1du=1b and E[logGMOLL(x)]=ba01logu(1u)b1du=baB(α+1,b)α|α=0=C+ψ(b+1)a. Here. C is Euler’s constant (Nadarajah et al. 2012).

2.5. Reliability

In the context of reliability. the stress-strength model describes the life of a component which has a random strength X1 that is subjected to a random stress X2 The component fails at the instant that the stress applied to it exceeds the strength. and the component will function satisfactorily whenever X1 > X2. Hence, R = Pr(X2 < X1 ) is a measure of component reliability. Here. we obtain the reliability R when X1 ~ KMOLL(a1,b1,α,γ,β) and X2 ~ KMOLL(a2,b2,α,γ,β) are independent random variables. Probabilities of this form have many applications especially in engineering concepts.

Let fi and Fi denote the pdf and cdf Xi for i = 1,2,…,. Then, the reliability function for the KMOLL distribution is given by

R=0f1(x)F2(x)dx.
The cdf of X2 and the pdf of X1 are obtained as
F2(x)=1[1GMOLLa2(x)]b2=1k=0skGMOLLa2k(x)andf1(x)=i=0wiGMOLLa1(i+1)1(x)gMOLL(x),
After some algebra, we arrive at
R=i=0wia1(i+1)i=0k=0wiska1(i+1)+a2k.
where wi=(1)iab(b1i) and sk=(1)k(b2k).

2.6. Order statistics

Order statistics make their appearance in many areas of statistical theory and practice. They enter in the problems of estimation and hypotheses testing in a variety of ways. Therefore, we now discuss some properties of the order statistics for the proposed class of distributions. Let Xi:n denote the ith order statistic. Nadarajah et al. (2012) obtained the general results for the Kumaraswamy-G distribution. We use the results about the pdf fi:n(x) of the ith order statistic. Then. we can give the pdf fi:n(x) for a random sample X1, X2,…,Xn from the KMOLL distribution. It is well-known that

fi:n(x)=f(x)B(i,ni+1)F(x)i1[1F(x)]ni
for i = 1, 2,..., n. Using the binomial expansion in the last equation. We obtain
fi:n(x)=f(x)B(i,ni+1)j=0ni(1)j(nij)Fi+j1(x).
The pdf in (2.7) can also be defined as
fi:n(x)=r,k=0qr,kGMOLLa(k+1)+r1(x)gMOLL(x),
where qr,k=j=0ni(1)j(nij)wkpr,i+j1B(i,ni+1) and pr,u(a,b)=k=0u(1)k(uk)m=0l=r(1)mr+l(kbm)(mal)(lr).

Several mathematical properties for the KMOLL order statistics (mgf, ordinary moments) can be derived from the mixture form in (2.9). Thus, from (2.9). the sth ordinary moment of Xi:n is given by

E(Xi:ns)=r,k=0qr,k01QMOLLs(u)ua(k+1)+r1du=βs/γαsB(sγ+ak+a+r,sγ+1).
Then the mgf of Xi:n is obtained as
Mi:n(t)=r,k=0qr,k01etQMOLL(u)uadu=r,k=0n=0qr,ktnαnβn/γn!B(nγ+a(k+1)+r,1nγ),
where qr,k=j=0ni(1)j(nij)wkpr,i+j1B(i,ni+1) and B(.,.) is the beta function.

3. Characterizations

This section deals with various characterizations of KMOLL distribution. These characterizations are based on: (i) the ratio of two truncated moments; (ii) the hazard function and (iii) certain functions of the random variable. It should be mentioned that for characterization (i) the cdf need not have a closed form. We present our characterizations (i) − (iii) in three subsections.

3.1. Characterizations based on ratio of two truncated moments

In this subsection, we present characterizations of KMOLL distribution in terms of a simple relationship between two truncated moments. This characterization result employs a theorem due to (Glänzel. 1987). see Theorem 3.1 below. Note that the result holds also when the interval H is not closed. Moreover, as mentioned above. It could be also applied when the cdf F does not have a closed form. As shown in (Glänzel, 1990). This characterization is stable in the sense of weak convergence.

Theorem 3.1.

Let (Ω, F, P) Ω, be a given probability space and let H = [d,e] be an interval for some d < e (d = −∞, e = ∞ might as well be allowed). Let X: Ω → H be a continuous random variable with the distribution function F and let g and h be two real functions defined on H such that

E[g(X)|Xx]=E[h(X)|Xx]ξ(x),xH,
is defined with some real function ξ. Assume that the equation g, hC1(H), ξC2 (H) and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation ξh = g has no real solution in the interior of H. Then, F is uniquely determined by the functions g, h and ξ particularly
F(x)=axC|ξ(u)ξ(u)h(u)g(u)|exp(s(u))du,
where the function s is a solution of the differential s=ξhξhg and C is the normalization constant. such that HdF=1.

Proposition 3.1.

Let X : Ω → (0, ∞) be a continuous random variable and let h(x)=xγ(1α)[1(xγxγ+βαγ)α]1b and g(x) = h(x)(xγ + βαγ) −1 for x > 0. The random variable X has pdf (1.6) if and only if the function ξ defined in Theorem 3.1 has the form

ξ(x)=αα+1(xγ+βαγ)1,x>0.

Proof.

Let X be a random variable with pdf (1.6). Then,

(1F(x))E[h(x)|Xx]=bβαγ(xγ+βαγ)a,x>0,
and
(1F(x))E[g(x)|Xx]=(aa+1)bβαγ(xγ+βαγ)a1,x>0,
and finally
ξ(x)h(x)g(x)=1a+1h(x)(xγ+βαγ)1,forx>0.
Conversely, if ξ is given as above. Then
s(x)=ξ(x)h(x)ξ(x)h(x)g(x)=aγxγ1(xγ+βαγ)1,x>0,
and hence
s(x)=log{(xγ+βαγ)a},x>0.
Now, in view of Theorem 3.1. X has density (1.6).

Corollary 3.1.

Let X : Ω → (0, ∞) be a continuous random variable and let h(x) be as in Proposition 3.1. The pdf of X is (6) if and only if there exist functions g and ξ defined in Therorem 3.1 satisfying the differential equation

ξ(x)h(x)ξ(x)h(x)g(x)=aγxγ1(xγ+βαγ)1,x>0.
The general solution of the differential equation in corollary 3.1 is
ξ(x)=(xγ+βαγ)aaγxγ1(xγ+βαγ)a1(h(x))1g(x)+D,
where D is a constant. Note that a set of function satisfying the above differential equation is given in Proposition 3.1 with D = 0 However, it should be also noted that there are other triplets (h,g,ξ) satisfying the conditions of Theorem 3.1.

Remark 3.1.

For b = 1, (Mendoza et al., 2016). we let h(x) = g(x)[xβ + αβ] −1 with g(x) = xβ(a−1). Then ξ(x)=a+1a[xβ+αβ] for x > 0.

The differential equation and general solution in this case are. respectively.

ξ(x)h(x)ξ(x)h(x)g(x)=β(a+1)xβ1xβ+αβ,x>0,
and
ξ(x)=[xβ+αβ]a[β(a+1)xβ1[xβ+αβ](a+1)(h(x))1g(x)dx+D].

3.2. Characterization based on hazard function

It is known that the hazard function. hF. a twice differentiable distribution function, F, satisfies the first order differential equation

f(x)f(x)=hF(x)hFhF(x).
For many univariate continuous distributions. this is the only characterization available in terms of the hazard function. The following characterization establishes a characterization of KMOLL distribution which is not of the above trivial form.

Proposition 3.2.

Let X : Ω → (0, ∞) be a continuous random variable. The pdf of X is (1.6) if and only if its hazard function hF (x) satisfies the differential equation

hF(x)(aγ1)x1hF(x)=abβγαγxaγ1{γxγ1(xγ+βαγ)2[aβαγxγ(a1)(a+1)(xγ+βαγ)a][(xγ+βαγ)axaγ]},
x > 0. with the initial condition hF(0) = 0 for > 1.

Proof.

If X has pdf (1.6) then clearly the above differential equation holds. Now, if it holds, then

ddx{x1aγhF(x)}=abβγαγddx{(xγ+βαγ)(a+1)[1(xγxγ+βαγ)a]1},
or
hF(x)=abβγαγxaγ1(xγ+βαγ)(a+1)1(xγxγ+βαγ)a
which is the hazard function of the KMOLL distribution.

Remark 3.2.

For a = b = 1 (special case of (1.4)). we have the following simple differential equation

hF(x)(β1)x1hF(x)=β2x2(β1)[xβ+αβ]2,x>0.

3.3. Characterization based on certain functions of the random variable

The following propositions have already appeared in (Hamedani, 2013). so we will just state them here which can be used to characterize KMOLL distribution.

Proposition 3.3.

Let X : Ω → (d, e) be a continuous random variable with cdf F. Let ψ(x) be a differentiable function on (d, e) with limxd+ ψ(x) = 1. Then for δ ≠ 1.

E[ψ(x)|Xx]=δψ(x),x(d,e),
If and only if
ψ(x)=(1F(x))1δ1,x(d,e).

Remark 3.3.

It is easy to see that for certain functions. e.g., ψ(x)=1(xγxγ+βαγ)a, δ=bb+1 and (d, e) = (0,∞).

Proposition 3.3 provides a characterization of KMOLL distribution. Clearly there are other suitable functions ψ. We chose the above one for simplicity.

4. Maximum Likelihood Estimation

Several approaches for parameter estimation have been proposed in the literature but the maximum likelihood method is the most commonly employed. Here we consider estimation of the unknown parameters of the KMOLL distribution by the method of maximum likelihood. Let x1, x2,..., xn be observed values from the KMOLL distribution with parameters a,b,γ,α and β. The log-likelihood function for (a,b,γ,α,β) is given by

logL=loga+logb+logβ+logγ+γlogα+(γa1)i=1nlogxi(a+1)i=1nlog(xiγ+βαγ)+(b1)log(1sia),
where si=xiγ/(xiγ+βαγ).

The derivatives of the log-likelihood function with respect to the parameters a,b,γ,α and β are given by respectively.

logLa=1a+γi=1nlogxii=1nlog(xiγ+βαγ)(b1)i=1n(1sia)1sialog(si),logLb=1b+i=1nlog(1sia),
logLγ=1γ+logα+ai=1nlogxia(b1)i=1nβlogxi(xiα)γβlogα(xiα)γ(xi2γ+2β(xiα)γ+β2α2γ)(1sia)+ai=1n[logxiβαγlogα+xiγlogxixiγ+βαγ]i=1n(βαγlogα+xiγlogxi)xiγ+βαγ,
logLα=aβγ(b1)αγ1i=1nxiγsia1(1sia)1(xiγ+βαγ)2[βγ(a1)αγ1+2βγαγ1]i=1n(xiγ+βαγ)1+γα,logLβ=1β+a(b1)i=1n(xiα)γsia1(xiγ+βαγ)2(1sia)(a1)αγ2αγi=1n(xiγ+βαγ)1.
The MLEs of (a, b, γ, α, β), say (â, b^, γ^, α^, β^), are the simultaneous solutions of the equations logLa=0, logLb=0, logLγ=0, logLα=0 and logLβ=0. Maximization of the likelihood function can be performed by using nlm or optimize in R statistical package.

5. An Illustrative Application

In this section, we use a real data set to compare the fits of the KMOLL distribution with MOLL, LL and Weibull Fréchet (WFr) (Afify et al., 2016c) distributions. We will use a data set consists of 63 observations of the strengths of 1.5 cm glass fibres (Smith and Naylor, 1987), originally obtained by workers at the UK National Physical Laboratory. Unfortunately, the measurement units are not given in their paper. We estimate the unknown parameters of the distributions by the maximum likelihood. Then, we provide the values of the following statistics: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC) and Bayesian Information Criterion (BIC).

In general, the smaller the values of these statistics, the better the fit to the data. Table 3 lists the MLEs of the parameters and the values of AIC, CAIC and BIC statistics.

Distribution Estimated Parameters (Standard Error) AIC CAIC BIC
KMOLL(a, b, γ, α, β) 1.8355 (0.096) 0.0028 (0.002) 47.4236 (13.307) 0.0588 (0.030) 0.2786 (0.095) 28.0861 29.1387 38.8018
MOLL(γ, α, β) 2.3267 (1.289) 0.0353 (0.154) 7.9260 (0.873) 51.5799 51.9867 58.0093
LL(γ, α) 1.5262 (0.041) 7.9260 (0.873) 49.5799 49.7799 53.8662
WFr(α, β, a, b) 0.3865 (0.799) 0.2436 (0.285) 1.4762 (4.782) 16.8561 (20.485) 39.0 47.6 42.4
Table 3.

MLEs and the values of AIC, CAIC and BIC statistics

Based on Table 3, it is clear that KMOLL distribution provides the overall best fit and therefore could be chosen as the more adequate model than other models for explaining the data set. Table 4 gives Cramer-von Misses (W) and Anderson Darling statistics (A) for the three models which are the KMOLL, MOLL and LL distributions. More information is provided by a histogram of the data given in Figure 4. Fitted lines in Figure 4 represent the KMOLL, MOLL, LL and WFr distributions. Figure 5 shows empirical cdf and the fitted cdfs. Finally, we give Q-Q plots for all fitted models. The figures also reveals that the KMOLL fits the data very well.

A W
KMOLL(a, b, γ, α, β) 0.0181403 0.127219
MOLL(γ, α, β) 0.4969404 2.748973
LL(γ, α) 0.4969402 2.748972
Table 4.

Cramer-von Misses and Anderson Darling statistics

Fig. 4.

Fitted densities of the KMOLL, MOLL, LL and WFr distributions for the data set.

Fig. 5.

Estimated and KMOLL, MOLL, LL and WFr cdfs for the data set.

6. Conclusion

In this paper. we introduce a five-parameter distribution called the Kumaraswamy Marshal-Olkin log-logistic (KMOLL) distribution. Interestingly. our proposed model has increasing. upside-down bathtub and bathtub shaped hazard rate function. A study on the mathematical properties of the new distribution is presented. We obtain the moment generating function. ordinary moments. skewness. kurtosis. hazard and survival functions. The estimation of the model parameters is done via maximum likelihood method. We also provide a numerical example of our findings. We hope that the proposed model may attract applications in survival analysis and customer lifetime duration etc.

Fig. 6.

Q-Q plots for KMOLL, MOLL, LL and WFr distributions.

References

[1]AZ Afify, M Alizadeh, HM Yousof, G Aryal, and M Ahmad, The transmuted geometric-G family of distributions: theory and applications, In: Pak. J. Statist, Vol. 32, 2016a, pp. 139-160.
[2]AZ Afify, GM Cordeiro, HM Yousof, A Alzaatreh, and ZM Nofal, The Kumaraswamy transmuted-G family of distributions: Properties and Applications, In: J. Data Sci, Vol. 14, No. 2, 2016b, pp. 245-270.
[6]M Alizadeh, MH Tahir, GM Cordeiro, M Mansoor, M Zubair, and GG Hamedani, The Kumaraswamy Marshal-Olkin family of distributions, In: J. Egyptian Math. Soc, 2015. http://dx.doi.org/10.1016/j.joems.2014.12.002
[8]M Bourguignon, RB Silva, and GM Cordeiro, The Weibull-G family of probability distributions, In: J. Data Sci, Vol. 12, 2014, pp. 53-68.
[11]TVF de Santana, EMM Ortega, GM Cordeiro, and GO Silva, The Kumaraswamy-log-logistic distribution, In: J. Stat. Theory Appl, Vol. 11, No. 3, 2012, pp. 265-291.
[17]GG Hamedani, In: On certain generalized gamma convolution distributions II, SCS, Arquette University, 2013. Technical Report No. 484,
[23]A Rényi, On measures of entropy and information, University of California Press, Berkeley, in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, 1961, pp. 547-561.
Journal
Journal of Statistical Theory and Applications
Volume-Issue
17 - 1
Pages
59 - 76
Publication Date
2018/03
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
https://doi.org/10.2991/jsta.2018.17.1.5How to use a DOI?
Copyright
Copyright © 2018, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Selen Cakmakyapan
AU  - Gamze Ozel
AU  - Yehia Mousa Hussein El Gebaly
AU  - G. G. Hamedani
PY  - 2018
DA  - 2018/03
TI  - The Kumaraswamy Marshall-Olkin Log-Logistic Distribution with Application
JO  - Journal of Statistical Theory and Applications
SP  - 59
EP  - 76
VL  - 17
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.1.5
DO  - https://doi.org/10.2991/jsta.2018.17.1.5
ID  - Cakmakyapan2018
ER  -