Hacettepe Journal of Mathematics and StatisticsVolume 48 (3) (2019), 897 – 930RESEARCH ARTICLE

Method for generating distributions and classes ofprobability distributions: the univariate case

Cıcero Ramos de Brito∗ , Leandro Chaves Rego † , Wilson Rosa de Oliveira ‡ andFrank Gomes-Silva §¶

Abstract

In this work, we present a method to generate probability distributions andclasses of probability distributions, which broadens the process of probabilitydistribution construction. In this method, distribution classes are built from pre-defined monotonic functions and from known distributions. With its use, wecan obtain different classes of probability distributions described in literature.Beside these results, we obtain results on the support and nature of the generateddistributions.

Keywords: Probabilistic distributions generating method, Distribution classes genera-ting functions, probabilistic distributions classes, probabilistic distribution support.

Mathematics Subject Classification (2010): 60E05; 62N05.

Received : 15.12.2017 Accepted : 28.06.2018 Doi : 10.15672/HJMS.2018.619

∗Federal Institute of Pernambuco, Recife, Brazil. Email: cicerocarlosbrito@yahoo.com.br†Department of Statistics and Applied Mathematics, Federal University of Ceara, Fortaleza, Brazil. Email:

leandrochavesrego@gmail.com‡Department of Statistics and Informatics, Federal Rural University of Pernambuco, Recife, Brazil. Email:

wilson.rosa@gmail.com§Department of Statistics and Informatics, Federal Rural University of Pernambuco, Recife, Brazil. Email:

franksinatrags@gmail.com¶Corresponding Author.

898

1. IntroductionThe amount of data available for analysis is growing increasingly faster, requiring new proba-

bilistic distributions to better describe each phenomenon or experiment studied. Computer basedtools allow the use of more complex distributions with a larger number of parameters to better studysizeable masses of data.

The literature in the field describes several generalizations and extensions of symmetric, asym-metric, discrete and continuous distributions. It is worth quoting Lee et al. (2013) regarding themain methods of generating distributions and classes of probability distributions:

Generally speaking, the methods developed prior to 1980s may be summarizedinto three categories: (1) method of differential equation, (2) method of trans-formation, and (3) quantile method. Techniques developed since 1980s may becategorized as methods of combination for the reason that these methods attemptto combine existing distributions into new distributions or adding new parame-ters to an existing distribution. (Lee et al., 2013, 219)

The relevance of these new models is that, according to the situation, each one of them canbetter fit the mass of data. Table 1 presents several classes of distributions described in literature,their nomenclature and the title of the work where they have been presented. For a comprehensivediscussion about the classes of probability distributions see three excellent articles written by Leeet al. (2013), Tahir and Nadarajah (2015) and Tahir and Cordeiro (2016).

Table 1. Some classes of distributions described in literatura

Distribution classes Nomenclature

F (x) = Ga(x), where a > 0 defined by Mudholkar et al.(1995)

F (x) = 1B(a,b)

∫G(x)0 ta−1(1− t)b−1dt, beta-G type 1 defined by

where a > 0, b > 0 and 0 < t < 1 Eugene et al. (2002)F (x) = 1

B(a,b)

∫G(x)0 ta−1(1 + t)−(a+b)dt, beta-G type 3 defined by

where a > 0, b > 0 and t > 0 Thair and Nadarajah (2015)F (x) = 1

B(a,b)

∫Gc(x)0 ta−1(1− t)b−1dt, Mc-G type 1 defined by

where a > 0, b > 0, c > 0 and 0 < t < 1 McDonald (1984)F (x) = 1

B(a,b)

∫Gc(x)0 ta−1(1 + t)−(a+b)dt, Mc-G type 3 defined by

where a > 0, b > 0, c > 0 and t > 0 Thair and Nadarajah (2015)F (x) = 1− (1−Ga(x))b, where Kumaraswamy-G defined by

a > 0 and b > 0 Cordeiro and Castro (2011)Kumaraswamy-G type 2

F (x) = 1− (1− (1−G(x))a)b, where defined bya > 0 and b > 0 Thair and Nadarajah (2015)

F (x) = 1−G(x)

G(x) + b(1−G(x), where b > 0 Marshall-Olkin-G defined by

Marshall and Olkin (1997)

F (x) = 1−(

b(1−G(x))G(x)+b(1−G(x)

)θ, where b > 0 and Marshall-Olkin-G

θ > 0 defined by Jayakumarand Mathew (2008)

F (x) =(

G(x)G(x)+b(1−G(x)

)θ, where b > 0 and θ > 0 Marshall-Olkin-G defined by

Thair and Nadarajah (2015)gamma-G defined by

Continues on the next page

899

Table 1. Continued from the previous page

Distribution classes Nomenclature

F (x) =βα

Γ(α)

∫ − log[1−G(x)]

0tα−1e−βtdt, Zografos and Balakrishnan

where α > 0 and β > 0 (2009)

F (x) = 1−βα

Γ(α)

∫ 1−G(x)G(x)

0tα−1 e−β tdt, gamma-G defined by

where α > 0 and β > 0 Brito et al. (2017)

F (x) = 1−βα

Γ(α)

∫ − log(G(x))

0tα−1 e−β tdt, gamma-G defined by

where α > 0 and β > 0 Cordeiro et al. (2017)

F (x) = 1− a+[1−G(x)](1+a) [1−G(x)]

exp{−a G(x)

1−G(x)

}, Odd Lindley-G defined by

where a > 0 Gomes-Silva et al. (2017)

F (x) = 1− C(θe−αH(x))C(θ)

, where x > 0, θ > 0 and Extended Weibull distributionC(θ) =

∑∞n=1 anθ

n defined by Silva et al. (2013)

F (x) =1− exp[−λG(x)]

1− e−λKumaraswamy-G Poisson defi-

ned by Ramos et al. (2014)F (x) = {1− [1−Ga(x)]b}c, where a > 0, b > 0 exponentiated

and c > 0 Kumaraswamy-G defined bySilva (2019)

F (x) =eλe

−βxa − eλ

1− eλ, x > 0 beta Weibull Poisson Family

defined by Percontini (2013)F (x) =

∫G(x)0 Kta−1(1− t)b−1 exp(−ct)dt, where beta Kummer generalized

a > 0, b > 0 and c ∈ R defined by Pescim et al.(2012)

F (x) = e−λaW (−a e−a)−e

−λaW (ψ(x))

e−λaW (−a e−a)−1

, where Weibull Generalized Poisson

W (x) =∑∞

n=1(−1)n−1nn−z

(n−1)!xn and distribution defined

ψ(x) = −a e−a−bxa by Percontini (2014)

F (x) =(1− β)s − {1− β[1−G(x)]}−s

(1− β)s − 1, G-Negative Binomial family

where β ∈ (0, 1) and s > 0 defined by Percontini (2014)F (x) =

ζ(s)−Lis[1−G(x)](s)

, where Zeta-G defined by Percontini

Lis(z) =∑∞

k=1zk

ksand ζ(s) =

∑∞k=1

1ks

(2014)Power Series Distributions

F (x) =

x∑k=0

C(k)(a)

k!C(λ)(λ− a)k Family by Consul and Famoye,

(2006)

F (x) =1∑

k=1

1

k!

[(C(0))k

](k−1)Basic Lagrangian defined by

Consul and Famoye (2006)

F (x) =x∑

k=n

n

(k − n)!k

[(C(0))k

](k−n)Lagrangian Delta defined by

Continues on the next page

900

Table 1. Continued from the previous page

Distribution classes Nomenclature

Consul and Famoye (2006)F (x) =

(∑xk=0 P (X = k)

)δ, Generalized Lagrangian

P (X = k) =

{w(0), k = 0[

(C(0))kw(1)(0)](k−1)

, k = 1, 2, ...defined by Consul and

Famoye (2006)Generalized Pearson in

F (x) =

∫ x

−∞e∫ a0+a1t+···+astsb0+b1t+···+brtr

dtdt Ordinary Differential

Equation form defined byShakil et al. (2010)Generalized Pearson in

F (x) =

∫ x

−∞e∫ a0+a1t+···+astsb0+b1t+···+brtr

(f(t))βdtdt, β ≥ 0 Ordinary Differential

Equation form defined byShakil et al. (2010)

Generalized Family in

F (x) =

∫ x

−∞

∫ y

−∞

(2∑

i=1

ai(t)fβi (t)

)dt dy Ordinary Differential

Equation form defined byVoda (2009)

F (x) =

∫ W [G(x)]

ar(t) dt, T-X class

where W [G(x)] ∈ [a, b], with by Alzaatreh et al. (2013)W [G(x)] differentiable and monotonically

non-decreasing

The aim of this work is to propose a method to create distributions and probabilistic distributionclasses that could unify the various methods to generate distribution classes already described inliterature. The idea of this method is to generate classes from already known distributions, usingmonotonic functions and a cumulative distribution function.

We show that the proposed method has high power of generality. The well-known T-X classgeneralizes most of the classes presented in Table 1. To get an idea of its power we will show thatthe T-X class appears as a sub-case of a simple sub-model of the proposed method that we willdenote it by 3S1C1.2 (see Table 2, page 907). In addition to generalizing existing classes, the newmethod provides a source of new probability distribution classes.

This paper is organized in the following way: in Section 2, we describe two methods to generateprobability distributions, establishing the conditions that must be satisfied by the used monotonicfunctions and probability distribution to guarantee that the proposed method indeed generates aprobability distribution. In Section 3, we analyze a special case of the methods described in theprevious section for the case where the monotonic functions are compositions of known probabilitydistribution functions. Still in Section 3 we present several specific cases of these methods thatmay be easily used to obtain new probability distributions. At the end of this section, we demons-trate that all methods presented in Sections 2 and 3 are equivalent. In Section 4, we analyze thesupport and nature of the distributions generated by the methods proposed in Section 3. Section 5

901

presents our conclusions and directions for further works. As an application of the proposed meth-ods, the appendix A to the article contains a table (Table 4) showing how to obtain several classesof probability distributions described in the literature using our proposed methods.

2. The methodThe method we suggest to create distribution classes uses monotonic functions, υ : R → R,√

: R → R, Uj : R → R ∪ {±∞}, Lj : R → R ∪ {±∞}, Mj : R → R ∪ {±∞} andVj : R → R ∪ {±∞}, and a cumulative distribution function (cdf) F . The idea of this method isto generate a probability distribution integrating F from Lj(x) to Uj(x) and from Mj(x) to Vj(x)for any x ∈ R and j = 1, 2, 3, . . . , n. Theorem 1 that follows shows sufficient conditions that thefunctions υ(x),

√(x), Lj(x), Uj(x), Mj(x) and Vj(x) must satisfy to guarantee that the method

generates a probability distribution function.

1. Theorem (T1). Method to generate distributions and classes of probability distributions.Let F : R → R, υ: R → R,

√: R → R, Uj : R → R ∪ {±∞}, Lj : R → R ∪ {±∞},

Mj : R → R ∪ {±∞} and Vj : R → R ∪ {±∞}, for j = 1, 2, 3, . . . , n, be monotonic and rightcontinuous functions such that:

[c1] F is a cdf andυ and√

are non-negative;

[c2] υ(x), Uj(x) and Mj(x) are non-decreasing and√(x), Vj(x) and Lj(x) are non-increasing

∀ j = 1, 2, 3, . . . , n;

[c3] If limx→−∞

υ(x) = limx→−∞

√(x), then lim

x→−∞υ(x) = 0 or lim

x→−∞Uj(x) = lim

x→−∞Lj(x) ∀

j = 1, 2, 3, . . . , n, and limx→−∞

√(x) = 0 or lim

x→−∞Mj(x) = lim

x→−∞Vj(x), ∀ j = 1, 2, 3, . . . , n;

[c4] If limx→−∞

υ(x) = limx→−∞

√(x) = 0, then lim

x→−∞Uj(x) = lim

x→−∞Vj(x) and lim

x→−∞Mj(x) =

limx→−∞

Lj(x), ∀ j = 1, 2, 3, . . . , n;

[c5] limx→−∞

Lj(x) ≤ limx→−∞

Uj(x) and if limx→−∞

√(x) = 0, then lim

x→+∞Mj(x) ≤ lim

x→+∞Vj(x),

∀ j = 1, 2, 3, . . . , n;

[c6] limx→+∞

Un(x) ≥ sup{x ∈ R : F (x) < 1} and limx→+∞

L1(x) ≤ inf{x ∈ R : F (x) > 0};

[c7] limx→+∞

υ(x) = 1;

[c8] limx→+∞

√(x) = 0 or lim

x→+∞Mj(x) = lim

x→+∞Vj(x), ∀ j = 1, 2, 3, . . . , n and n ≥ 1;

[c9] limx→+∞

Uj(x) = limx→+∞

Lj + 1(x), ∀ j = 1, 2, 3, . . . , n− 1 and n ≥ 2;

[c10] F is a cdf without points of discontinuity or all functions Lj(x) and Vj(x) are constantat the right of the vicinity of points whose image are points of discontinuity of F , being also contin-uous in that points. Moreover, F does not have any point of discontinuity in the set { lim

x→±∞Lj(x),

limx→±∞

Uj(x), limx→±∞

Mj(x), limx→±∞

Vj(x), for some j = 1, 2, . . . , n}. Then,

H(x) = υ(x)

n∑j=1

∫ Uj(x)

Lj(x)

dF (t)−√(x)

n∑j=1

∫ Vj(x)

Mj(x)

dF (t)(2.1)

is a cdf .

902

Proof. (i) limx→−∞

H(x) = 0.

limx→−∞

H(x) = limx→−∞

(υ(x)

n∑j=1

∫ Uj(x)

Lj(x)

dF (t)

)

− limx→−∞

(√(x)

n∑j=1

∫ Vj(x)

Mj(x)

dF (t)

)

=

(lim

x→−∞υ(x)

) n∑j=1

∫ limx→−∞

Uj(x)

limx→−∞

Lj(x)

dF (t)

−(

limx→−∞

√(x)

) n∑j=1

∫ limx→−∞

Vj(x)

limx→−∞

Mj(x)

dF (t),

where the last equality holds because F is continuous in{lim

x→−∞Uj(x), lim

x→−∞Lj(x), lim

x→−∞Vj(x), lim

x→−∞Mj(x)

}.

Conditions [c3] and [c4] guarantee that:

limx→−∞

H(x) =

(lim

x→−∞υ(x)

) n∑j=1

∫ limx→−∞

Uj(x)

limx→−∞

Lj(x)

dF (t)

−(

limx→−∞

√(x)

) n∑j=1

∫ limx→−∞

Vj(x)

limx→−∞

Mj(x)

dF (t) = 0.

(ii) limx→+∞

H(x) = 1.

limx→+∞

H(x) = limx→+∞

(υ(x)

n∑j=1

∫ Uj(x)

Lj(x)

dF (t)

)

− limx→+∞

(√(x)

n∑j=1

∫ Vj(x)

Mj(x)

dF (t)

)

=

(lim

x→+∞υ(x)

) n∑j=1

∫ limx→+∞

Uj(x)

limx→+∞

Lj(x)

dF (t)

−(

limx→+∞

√(x)

) n∑j=1

∫ limx→+∞

Vj(x)

limx→+∞

Mj(x)

dF (t),

where the last equality holds because F is continuous in{lim

x→+∞Uj(x), lim

x→+∞Lj(x), lim

x→+∞Vj(x), lim

x→+∞Mj(x)

}.

Thus, conditions [c1], [c6], [c7], [c8] and [c9] guarantee that

limx→+∞

H(x) = 1.

(iii) If x1 ≤ x2, then H(x1) ≤ H(x2).Let x1 ≤ x2, then [c2] implies that: Uj(x1) ≤ Uj(x2), Lj(x1) ≥ Lj(x2), Mj(x1) ≤

Mj(x2), Vj(x1) ≥ Vj(x2), υ(x1) ≤ υ(x2) and√(x1) ≥

√(x2). Beside this, [c2] and [c5]

imply,∑n

j=1

∫ Uj(x1)

Lj(x1)dF (t) ≥ 0,

∑nj=1

∫ Vj(x1)

Mj(x1)dF (t) ≥ 0,

∑nj=1

∫ Uj(x2)

Lj(x2)dF (t) ≥ 0 and∑n

j=1

∫ Vj(x2)

Mj(x2)dF (t) ≥ 0.

903

Thus, since, by [c1],υ and√

are non-negative, we haveH(x1) =υ(x1)

∑nj=1

∫ Uj(x1)

Lj(x1)dF (t)−

√(x1)

∑nj=1

∫ Vj(x1)

Mj(x1)dF (t)

≤ υ(x2)

n∑j=1

∫ Uj(x2)

Lj(x2)

dF (t)−√(x2)

n∑j=1

∫ Vj(x2)

Mj(x2)

dF (t) = H(x2).

(iv) limx→x+

0

H(x) = H(x0).

limx→x+

0

H(x) = limx→x+

0

υ(x)

n∑j=1

∫ Uj(x)

Lj(x)

dF (t)− limx→x+

0

√(x)

n∑j=1

∫ Vj(x)

Mj(x)

dF (t)

=

(lim

x→x+0

υ(x)

)n∑

j=1

∫ limx→x

+0

Uj(x)

limx→x

+0

Lj(x)

dF (t)−

(lim

x→x+0

√(x)

)n∑

j=1

∫ limx→x

+0

Vj(x)

limx→x

+0

Mj(x)

dF (t)

= υ(x0)

n∑j=1

∫ Uj(x0)

Lj(x0)

dF (t)−√(x0)

n∑j=1

∫ Vj(x0)

Mj(x0)

dF (t) = H(x0).

The above equalities hold due to [c10] and because υ(x), Uj(x), Mj(x),√(x), Vj(x) and

Lj(x) are right continuous.

From the facts (i), (ii), (iii) and (iv), we may conclude that (2.1) is a cdf . �

Corollary 1.1 presents an alternative method to generate distributions and classes of probabilitydistributions.

1.1. Corollary (C1.1). Complementary method to generate distributions and classes of probabilitydistributions.Let φ : R → R, 0 : R → R, W : R → R, U : R → R ∪ {±∞}, Mj : R → R ∪ {±∞} andVj : R → R ∪ {±∞}, ∀ j = 1, 2, 3, . . . , n, be monotonic and right continuous functions suchthat:[cc1] F is a cdf and 0 and W are non-negative;

[cc2]0(x), Uj(x) and Mj(x) are non-decreasing and W (x), Vj(x) and Lj(x) are non-increasing∀ j = 1, 2, 3, . . . , n;

[cc3] If limx→+∞

W (x) = limx→+∞

0(x), then limx→+∞

0(x) = 0 or limx→+∞

Lj(x) = limx→+∞

Uj(x), ∀ j = 1, 2, 3, . . . , n, and limx→+∞

W (x) = 0 or limx→+∞

Mj(x) = limx→+∞

Vj(x), ∀j = 1, 2, 3, . . . , n;

[cc4] If limx→+∞

W (x) = limx→+∞

0(x) = 0, then limx→+∞

Uj(x) = limx→+∞

Vj(x) and limx→+∞

Mj(x) = limx→+∞

Lj(x), ∀ j = 1, 2, 3, . . . , n;

[cc5] limx→+∞

Mj(x) ≤ limx→+∞

Vj(x) and if limx→+∞

0(x) = 0, then limx→−∞

Lj(x) ≤ limx→−∞

Uj(x),

∀ j = 1, 2, 3, . . . , n;

[cc6] limx→−∞

Vn(x) ≥ sup{x ∈ R : φ(x) < 1} and limx→−∞

L1(x) ≤ inf{x ∈ R : φ(x) > 0};

[cc7] limx→−∞

W (x) = 1;

[cc8] limx→−∞

0(x) = 0 or limx→−∞

Lj(x) = limx→−∞

Uj(x), ∀ j = 1, 2, 3, . . . , n and n ≥ 1;

904

[cc9] limx→−∞

Vj(x) = limx→−∞

Mj + 1(x), ∀ j = 1, 2, 3, . . . , n− 1 and n ≥ 2;

[cc10] φ is a cdf without points of discontinuity or all functions Lj(x) and Vj(x) are constant atthe right of the vicinity of points whose image are points of discontinuity of φ, being also continuousin that points. Moreover, φ does not have any point of discontinuity in the set { lim

x→±∞Lj(x)(x),

limx→±∞

Uj(x), limx→±∞

Mj(x), limx→±∞

Vj(x), for some j = 1, 2, . . . , n}.

Then,

H(x) = 1−W (x)

n∑j=1

∫ Vj(x)

Mj(x)dφ(t) + 0(x)

n∑j=1

∫ Uj(x)

Lj(x)dφ(t)

is a cdf.

Proof. In Theorem 1, consider n = 1, υ(x) = 1,√(x) = 0, U1(x) = 1 and L1(x) =

W (x)∑n

j=1

∫ Vj(x)Mj(x)

dφ(t)− 0(x)∑n

j=1

∫ Uj(x)Lj(x)

dφ(t), ∀x ∈ R, and F a cdf of the uni-form[0,1] distribution. Note that U1(x) and L1(x) satisfy the hypotheses of Theorem 1, since[cc1], [cc2] and [cc5] guarantee that L1(x) = W (x)

∑nj=1

∫ Vj(x)Mj(x)

dφ(t)−0(x)∑n

j=1

∫ Uj(x)Lj(x)

dφ(t)

is non-increasing and U1(x) = 1 is non-decreasing. Thus, conditions [cc2] and [cc5] are satisfied.Moreover, conditions [cc3] and [cc4] guarantee that:

limx→−∞

U1(x) = limx→−∞

L1(x) = 1, limx→+∞

U1(x) = sup{x∈R : F (x) < 1} = 1, limx→+∞

L1(x) =

inf{x ∈ R : F (x) > 0} = 0, that both L1(x) and U1(x) are right continuous and that F is a cdfwithout points of discontinuity.

As all conditions of Theorem 1 are satisfied, it follows that

H(x) =

∫ U1(x)

L1(x)

dF (s) =

∫ 1

W (x)∑nj=1

∫ Vj(x)Mj(x)

dφ(t)−0(x)∑nj=1

∫Uj(x)Lj(x)

dφ(t)

ds

= 1−W (x)

n∑j=1

∫ Vj(x)

Mj(x)dφ(t) + 0(x)

n∑j=1

∫ Uj(x)

Lj(x)dφ(t)

is a cdf . �

In the next section, we present some corollaries of Theorem 1 where the monotonic functionsυ(x),

√(x), Uj(x), Lj(x), Mj(x) and Vj(x) are compositions of monotonic functions of known

probability distributions.

3. Monotonic functions involving probabilities distributionsIn this section, we show how to generate classes of probability distributions using monotonicfunctions which are compositions known probability distributions. Formally, consider that u:[0, 1]m → R, ϑ : [0, 1]m → R, µj : [0, 1]m → R ∪ {±∞}, ℓ : [0, 1]m → R ∪ {±∞},vj : [0, 1]m → R ∪ {±∞} and mj : [0, 1]m → R ∪ {±∞} are monotonic and right continuousfunctions. The results of this section are achieved considering that: υ(x) = u(G1, . . . , Gm)(x),√(x) = ϑ(G1, . . . , Gm)(x), Uj(x) = µj(G1, . . . , Gm)(x), Lj(x) = ℓ(G1, . . . , Gm)(x),Mj

(x) = mj(G1, . . . , Gm)(x) and Vj(x) = vj(G1, . . . , Gm)(x).We use the abbreviation (·)(x) = (G1, . . . , Gm)(x) = (G1(x), . . . , Gm(x)) to represent the

vector formed by the cdf ′s calculated on the same point of the domain x.

905

1.2. Corollary (C1.2). Method to generate classes of probability distributions.Let F : R → R, µj : [0, 1]m → R ∪ {±∞}, ℓj : [0, 1]m → R ∪ {±∞}, u: [0, 1]m → R,vj : [0, 1]m → R∪{±∞}, mj : [0, 1]m → R∪{±∞} and ϑ : [0, 1]m → R, ∀ j = 1, 2, 3, . . . , n,be monotonic and right continuous functions such that:

[d1] F is a cdf andu and ϑ are non-negative;

[d2] µj , mj and u are non-decreasing and ℓj , vj and ϑ are non-increasing, ∀ j = 1, 2, 3, . . . , n,in all of its variables;

[d3] If u(0, . . . , 0) = ϑ(0, . . . , 0), then u(0, . . . , 0) = 0 or µj(0, . . . , 0) = ℓj(0, . . . , 0), ∀j = 1, 2, 3, . . . , n, and ϑ(0, . . . , 0) = 0 or mj(0, . . . , 0) = vj(0, . . . , 0), ∀ j = 1, 2, 3, . . . , n;

[d4] Ifu(0, . . . , 0) = ϑ(0, . . . , 0) = 0, then µj(0, . . . , 0) = vj(0, . . . , 0) and mj(0, . . . , 0)= ℓj(0, . . . , 0), ∀ j = 1, 2, 3, . . . , n;

[d5] ℓj(0, . . . , 0) ≤ µj(0, . . . , 0) and if ϑ(0, . . . , 0) = 0, then mj(1, . . . , 1) ≤ vj(1, . . . , 1),∀ j = 1, 2, 3, . . . , n;

[d6] µn(1, . . . , 1) ≥ sup{x ∈ R : F (x) < 1} and ℓ1(1, . . . , 1) ≤ inf{x ∈ R : F (x) > 0};

[d7]u(1, . . . , 1) = 1;

[d8] ϑ(1, . . . , 1) = 0 or vj(1, . . . , 1) = mj(1, . . . , 1), ∀ j = 1, 2, 3, . . . , n− 1 and n ≥ 2;

[d9] µj(1, . . . , 1) = ℓj+1(1, . . . , 1), ∀ j = 1, 2, 3, . . . , n− 1 and n ≥ 2;

[d10] F is a cdf without points of discontinuity or the functions ℓj(·)(x) and vj(·)(x) are constantat the right of the vicinity of points whose image are points of discontinuity of F , being also contin-uous in that points. Moreover, F does not have any point of discontinuity in the set {ℓj(0, . . . , 0),µj(0, . . . , 0), mj(0, . . . , 0), vj(0, . . . , 0), ℓj(1, . . . , 1), µj(1, . . . , 1), mj(1, . . . , 1), vj(1, . . . , 1),for some j = 1, 2, . . . , n}.

Then,

HG1,...,Gm(x) = u(·)(x)n∑

j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)− ϑ(·)(x)

∫ vj(·)(x)

mj(·)(x)dF (t)

is a functional generator of classes of probability distributions where (·)(x) = (G1, . . . ,Gm)(x).

Proof. In Theorem 1, set υ(x) = u(·)(x),√(x) = ϑ(·)(x), Uj(x) = µj(·)(x), Lj(x) =

ℓj(·)(x), Mj(x) = mj(·)(x) and Vj(x) = vj(·)(x), and observe that condition [di] implies con-dition [ci] of Theorem 1, for i = 1, 2, . . . , 10. �

Let us now consider a special case of Corollary 1.2 that is a functional constructor of classes ofprobability distributions that can be easily used.

1st special case of Corollary 1.2 (1C1.2). Easy to use method for the construction of classes ofprobability distributions.Let ui : [0, 1]m → [0, 1] and νi : [0, 1]m → [0, 1] be monotonic and right continuous func-tions such that ui’s are non-decreasing and νi’s are non-increasing in each one of its variables,with ui(0, . . . , 0) = 0, ui(1, . . . , 1) = 1, νi(0, . . . , 0) = 1 and νi(1, . . . , 1) = 0, for all

906

i = 1, . . . , k. If, in Corollary 1.2, u(·)(x) =∏k

i=1

((i − θi)ui(·)(x) + θi

)αiand ϑ(·)(x) =∏k

i=1 (θiνi(·)(x))αi , with αi ≥ 0 and 0 ≤ θi ≤ 1, then

HG1,...,Gm(x) =

k∏i=1

((1− θi)ui(·)(x) + θi)αi

n∑j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)

−k∏

i=1

(θiνi(·)(x))αin∑

j=1

∫ vj(·)(x)

mj(·)(x)dF (t),(3.1)

is a functional generator of classes of probability distributions, where (·)(x) = (G1, . . . ,Gm)(x).

Table 2 shows some particular cases of the functional constructor of classes of probability dis-tributions, given by Equation (3.1), that may be more easily used for the generation of classesof distribution. Consider the following functions in the expressions from 15S1C1.2 to 20S1C1.2:µ : [0, 1] → R∪{±∞}, ℓ : [0, 1] → R∪{±∞}, v : [0, 1] → R∪{±∞}, m : [0, 1] → R∪{±∞}such that µ and m are non-decreasing and right continuous, and v and ℓ are non-increasing and rightcontinuous.

907

Tabl

e2.

Som

efu

nctio

nalc

onst

ruct

ors

ofcl

asse

sof

prob

abili

tydi

stri

butio

nsob

tain

edfr

om1C

1.2

Som

esu

b-ca

ses

of1C

1.2

Spec

ialc

ondi

tions

ofm

onot

onic

func

tions

and

para

met

ers

Func

tiona

lcon

stru

ctor

obta

ined

1S1C

1.2

k=

1,a

1=

0an

dvj(1,...,1)=mj(1,...,1)

HG

1,...,Gm

(x)=

n ∑ j=

1

∫ µ j(·)

(x)

ℓj(·)

(x)

dF(t)

2S1C

1.2

n=

1,θi=

0an

dvj(1,...,1)=mj(1,...,1)

HG

1,...,Gm

(x)=

k ∏ i=

1

uαii

(·)(x)

∫ µ j(·)

(x)

ℓj(·)

(x)

dF(t)

3S1C

1.2

n=

1,k=

1,a

1=

0an

d

v1(1,...,1)=m

1(1,...,1)

HG

1,...,Gm

(x)=

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dF(t)

4S1C

1.2

n=

1,k=

1,a

1=

0,

vj(1,...,1)=mj(1,...,1)

and

f(t)=

1

µ1(1,...,1)−ℓ 1

(1,...,1)

,fort

inHG

1,...,Gm

(x)=

µ1(·)(x)−ℓ 1

(·)(x)

µ1(1,...,1)−ℓ 1

(1,...,1)

[ℓ1(1,...,1),µ1(1,...,1)]

5S1C

1.2

n=

1,k=

1,a

1=

0,

ℓ 1(·)(x)=µ1(0,...,0),

v1(1,...,1)=m

1(1,...,1)

and

f(t)=

1

µ1(1,...,1)−µ1(0,...,0)

,fort

inHG

1,...,Gm

(x)=

µ1(·)(x)−µ1(0,...,0)

µ1(1,...,1)−µ1(0,...,0)

[µ1(0,...,0),µ1(1,...,1)]

Con

tinue

son

the

next

page

908

Tabl

e2.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Som

esu

b-ca

ses

of1C

1.2

Spec

ialc

ondi

tions

ofm

onot

onic

func

tions

and

para

met

ers

Func

tiona

lcon

stru

ctor

obta

ined

6S1C

1.2

n=

1,k=

1,a

1=

0,

µ1(·)(x)=ℓ 1

(0,...,0),

v1(1,...,1)=m

1(1,...,1)

and

HG

1,...,Gm

(x)=

ℓ 1(·)(x)−ℓ 1

(0,...,0)

ℓ 1(1,...,1)−ℓ 1

(0,...,0)

f(t)=

1

ℓ 1(1,...,1)−ℓ 1

(0,...,0)

,fort

in

[ℓ1(1,...,1),ℓ 1

(0,...,0)]

7S1C

1.2

k=

1,a

1=

0an

dn ∑ j=

1

∫ µ j(·)

(x)

ℓj(·)

(x)

dF(t)=

1HG

1,...,Gm

(x)=

1−

n ∑ j=

1

∫ v j(·)

(x)

mj(·)

(x)

dF(t)

8S1C

1.2

n=

1,θ

1=

1,

µ1(·)(x)=

+∞

and

HG

1,...,Gm

(x)=

1−

k ∏ i=

1

(νi(·)(x))αi

∫ v 1(·)

(x)

m1(·)

(x)

dF(t)

ℓ 1(·)(x)=

−∞

9S1C

1.2

n=

1,k

=1a1=

0,

µ1(·)(x)=

+∞

andℓ 1

(·)(x)=

−∞

HG

1,...,Gm

(x)=

1−∫ v 1

(·)

(x)

m1(·)

(x)

dF(t)

10S1

C1.

2n

=1

,k=

1a1=

0,

µ1(·)(x)=

+∞

,ℓ1(·)(x)=

−∞

and

f(t)=

1

v1(0,...,0)−m

1(0,...,0)

,fort

inHG

1,...,Gm

(x)=

1−

v1(·)(x)−m

1(·)(x)

v1(0,...,0)−m

1(0,...,0)

[m1(0,...,0),v1(0,...,0)]

11S1

C1.

2n

=1

,k=

1a1=

0,

m1(·)(x)=v1(1,...,1),

µ1(·)(x)=

+∞

,ℓ1(·)(x)=

−∞

and

f(t)=

1

v1(0,...,0)−v1(1,...,1)

,fort

inHG

1,...,Gm

(x)=

v1(·)(x)−v1(0,...,0)

v1(1,...,1)−v1(0,...,0)

[v1(1,...,1),v1(0,...,0)]

Con

tinue

son

the

next

page

909

Tabl

e2.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Som

esu

b-ca

ses

of1C

1.2

Spec

ialc

ondi

tions

ofm

onot

onic

func

tions

and

para

met

ers

Func

tiona

lcon

stru

ctor

obta

ined

12S1

C1.

2n

=1

,k=

1a1=

0,

v1(·)(x)=m

1(1,...,1),

µ1(·)(x)=

+∞

,ℓ1(·)(x)=

−∞

and

f(t)=

1

m1(0,...,0)−m

1(1,...,1)

,fort

inHG

1,...,Gm

(x)=

m1(·)(x)−m

1(0,...,0)

m1(1,...,1)−m

1(0,...,0)

[m1(0,...,0),m

1(1,...,1)]

13S1

C1.

2n

=1

,k=

1a1=

0

HG

1,...,Gm

(x)=

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dF(t)−∫ v 1

(·)

(x)

m1(·)

(x)

dF(t)

14S1

C1.

2n

=1

,k=

1a1=

0an

df(t)=

1µ1(1,...,1

)−ℓ1(1,...,1

)−v1(1,...,1

)+m

1(1,...,1

),

HG

1,...,Gm

(x)=

µ1(·)

(x)−ℓ1(·)

(x)−v1(·)

(x)+m

1(·)

(x)

µ1(1,...,1

)−ℓ1(1,...,1

)−v1(1,...,1

)+m

1(1,...,1

)

fort

in[ℓ

1(1,...,1)+v1(1,...,1),m

1(1,...,1)+µ1(1,...,1)]

.

15S1

C1.

2µ1(·)(x)=µ((1−γ)uk+

1(·)(x)+γ),

ℓ 1(·)(x)=ℓ((1−γ)uk+

1(·)(x)+γ),

HG

1,...,Gm

(x)=

k ∏ i=

1

((1−θi)ui(·)(x)+θi)αi

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dF(t)

v1(·)(x)=µ(γνk+

1(·)(x))

,

m1(·)(x)=ℓ(γνk+

1(·)(x))

,−

n+k ∏

i=n+

1

(θiνi(·)(x))αi

∫ v 1(·)

(x)

m1(·)

(x)

dF(t)

n=

1,αi>

0an

d0≤γ≤

1.

Con

tinue

son

the

next

page

910

Tabl

e2.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Som

esu

b-ca

ses

of1C

1.2

Spec

ialc

ondi

tions

ofm

onot

onic

func

tions

and

para

met

ers

Func

tiona

lcon

stru

ctor

obta

ined

16S1

C1.

2µ1(·)(x)=µ((1−γ)uk+

1(·)(x)+γ),

ℓ 1(·)(x)=

−∞

,HG

1,...,Gm

(x)=

k ∏ i=

1

((1−θi)ui(·)(x)+θi)αi

∫ µ 1(·)

(x)

−∞

dF(t)

v1(·)(x)=µ(γν1(·)(x))

,−

k ∏ i=

1

(θiνi(·)(x))αi

∫ v 1(·)

(x)

−∞

dF(t)

m1(·)(x)=

−∞

,n

=1

and0≤γ≤

1.

17S1

C1.

2µ1(·)(x)=

+∞

,

ℓ 1(·)(x)=ℓ((1−γ)uk+

1(·)(x)+γ),

HG

1,...,Gm

(x)=

k ∏ i=

1

((1−θi)ui(·)(x)+θi)αi

∫ +∞

ℓ1(·)

(x)

dF(t)

v1(·)(x)=

+∞

,−

k ∏ i=

1

(θiνi(·)(x))αi

∫ +∞

m1(·)

(x)

dF(t)

m1(·)(x)=ℓ(γνk+

1(·)(x))

,n

=1

and0≤γ≤

1.

18S1

C1.

2µ1(·)(x)=v(γνk+

1(·)(x))

,HG

1,...,Gm

(x)=

k ∏ i=

1

((1−θi)ui(·)(x)+θi)αi

∫ ℓ 1(·)

(x)

µ1(·)

(x)

dF(t)

ℓ 1(·)(x)=m

(γνk+

1(·)(x))

,

v1(·)(x)=v((1−γ)uk+

1(·)(x)+γ),

−n+k ∏

i=n+

1

(θiνi(·)(x))αi

∫ v 1(·)

(x)

m1(·)

(x)

dF(t)

m1(·)(x)=m

((1−γ)uk+

1(·)(x)+γ),

n=

1,αi>

0an

d0≤γ≤

1.

19S1

C1.

2µ1(·)(x)=v(γνk+

1(·)(x))

,

ℓ 1(·)(x)=

−∞

,HG

1,...,Gm

(x)=

k ∏ i=

1

((1−θi)ui(·)(x)+θi)αi

∫ µ 1(·)

(x)

−∞

dF(t)

v1(·)(x)=v((1−γ)uk+

1(·)(x)+γ),

−k ∏ i=

1

(θiνi(·)(x))αi

∫ +∞

m1(·)

(x)

dF(t)

m1(·)(x)=

−∞

,n

=1

and0≤γ≤

1.

Con

tinue

son

the

next

page

911

Tabl

e2.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Som

esu

b-ca

ses

of1C

1.2

Spec

ialc

ondi

tions

ofm

onot

onic

func

tions

and

para

met

ers

Func

tiona

lcon

stru

ctor

obta

ined

20S1

C1.

2µ1(·)(x)=

+∞

,

ℓ 1(·)(x)=m

(γνk+

1(·)(x))

,HG

1,...,Gm

(x)=

k ∏ i=

1

((1−θi)ui(·)(x)+θi)αi

∫ +∞

ℓ1(·)

(x)

dF(t)

v1(·)(x)=

+∞

,−

k ∏ i=

1

(θiνi(·)(x))αi

∫ +∞

m1(·)

(x)

dF(t)

m1(·)(x)=m

((1−γ)uk+

1(·)(x)+γ),

n=

1an

d0≤γ≤

1.

21S1

C1.

2n

=1

.HG

1,...,Gm

(x)=

k ∏ i=

1

((1−θi)ui(·)(x)+θi)αi

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dF(t)

−k ∏ i=

1

(θiνi(·)(x))αi

∫ v 1(·)

(x)

m1(·)

(x)

dF(t)

22S1

C1.

2µ1(·)(x)=

+∞

,

ℓ 1(·)(x)=

−∞

,HG

1,...,Gm

(x)=

k ∏ i=

1

((1−θi)ui(·)(x)+θi)αi−

k ∏ i=

1

(θiνi(·)(x))αi

v1(·)(x)=

+∞

,m

1(·)(x)=

−∞

,n

=1

andαi>

0.

912

Corollary 1.3 shows an alternative method to obtain classes of probability distributions fromCorollary 1.1. It shows what hypothesesu, ϑ, µj , ℓj , vj and mj must satisfy so that the functions0(x), W (x), Uj(x), Lj(x), Mj(x) and Vj(x) satisfy the conditions of Corollary 1.1 and classesof probability distributions can be obtained.

1.3. Corollary (C1.3). Complementary method to generate classes of probability distributions.Let φ : R → R, µj : [0, 1]m → R ∪ {±∞}, ℓj : [0, 1]m → R ∪ {±∞}, u: [0, 1]m → R,vj : [0, 1]m → R∪{±∞}, mj : [0, 1]m → R∪{±∞} and ϑ : [0, 1]m → R, ∀ j = 1, 2, 3, . . . , η,be monotonic and right continuous functions such that:

[cd1] φ is a cdf andu and ϑ are non-negative;

[cd2] µj , mj andu are non-decreasing and ℓj , vj and ϑ are non-increasing, ∀ j = 1, 2, 3, . . . , η,in all of its variables;

[cd3] If u(1, . . . , 1) = ϑ(1, . . . , 1), then ϑ(1, . . . , 1) = 0 or mj(1, . . . , 1) = vj(1, . . . , 1), ∀j = 1, 2, 3, . . . , η, and u(1, . . . , 1) = 0 or ℓj(1, . . . , 1) = µj(1, . . . , 1), ∀ j = 1, 2, 3, . . . , η;

[cd4] If u(1, . . . , 1) = ϑ(1, . . . , 1) = 0, then µj(1, . . . , 1) = vj(1, . . . , 1), ∀ j = 1, 2, 3, . . . , η,and mj(1, . . . , 1) = ℓj(1, . . . , 1), ∀ j = 1, 2, 3, . . . , η;

[cd5] ℓj(0, . . . , 0) ≤ µj(0, . . . , 0) and if ϑ(1, . . . , 1) = 0, then mj(1, . . . , 1) ≤ vj(1, . . . ,1), ∀ j = 1, 2, 3, . . . , η;

[cd6] vn(0, . . . , 0) ≥ sup{x ∈ R : F (x) < 1} and m1(0, . . . , 0) ≤ inf{x ∈ R : F (x) > 0};

[cd7] ϑ(0, . . . , 0) = 1;

[cd8]u(0, . . . , 0) = 0 or ℓj(0, . . . , 0) = µj(0, . . . , 0), ∀ j = 1, 2, 3, . . . , η − 1 and η ≥ 1;

[cd9] vj(0, . . . , 0) = mj+1(0, . . . , 0), ∀ j = 1, 2, 3, . . . , η − 1 and η ≥ 2;

[cd10] φ is a cdf without points of discontinuity or the functions ℓj(·)(x) and vj(·)(x) are constantat the right of the vicinity of points whose image are points of discontinuity of φ, being also contin-uous in that points. Moreover, φ does not have any point of discontinuity in the set {ℓj(0, . . . , 0),µj(0, . . . , 0), mj(0, . . . , 0), vj(0, . . . , 0), ℓj(1, . . . , 1), µj(1, . . . , 1), mj(1, . . . , 1), vj(1, . . . , 1),for some j = 1, 2, . . . , η}.

Then,

HG1,...,Gm(x) = 1− ϑ(·)(x)η∑

j=1

∫ vj(·)(x)

mj(·)(x)dφ(t) + u(·)(x)

η∑j=1

∫ µj(·)(x)

ℓj(·)(x)dφ(t),

is a functional generator of classes of probability distributions, where (·)(x) = (G1, . . . ,Gm) (x).

Proof. In Corollary 1.1, set 0(x) = u(·)(x), W (x) = ϑ(·)(x), Uj(x) = µj(·)(x), Lj(x) =ℓj(·)(x), Mj(x) = mj(·)(x) and Vj(x) = vj(·)(x) and observe that condition [cdi] impliescondition [cci] of Corollary 1.1, for i = 1, 2, . . . , 10. �

Let us now consider a special case of Corollary 1.3 that is a functional constructor of classes ofprobability distributions that can be easily used.

913

1st special case of Corollary 1.3 (1C1.3). Easy to use complementary method for the constructionof classes of probability distributions.Let ui : [0, 1]m → [0, 1] and νi : [0, 1]m → [0, 1] be monotonic and right continuous func-tions such that ui’s are non-decreasing and νi’s are non-increasing in each one of its variables,with ui(0, . . . , 0) = 0, ui(1, . . . , 1) = 1, νi(0, . . . , 0) = 1 and νi(1, . . . , 1) = 0, for all

i = 1, . . . , k. If, in Corollary 1.3, ϑ(·)(x) =∏k

i=1

((i − θi)νi(·)(x) + θi

)αiand u(·)(x) =∏k

i=1 (θiui(·)(x))αi , with αi ≥ 0 and 0 ≤ θi ≤ 1, then

HG1,...,Gm(x) = 1−k∏

i=1

((1− θi)νi(·)(x) + θi)αi

n∑j=1

∫ vj(·)(x)

mj(·)(x)dφ(t)

+

k∏i=1

(θiui(·)(x))αin∑

j=1

∫ µj(·)(x)

ℓj(·)(x)dφ(t),(3.2)

is a functional generator of classes of probability distributions, where (·)(x) = (G1, . . . ,Gm)(x).

Table 3 shows how to obtain some special cases of the function given by Equation (3.2), thatmay be more easily used to generate classes of distributions. It is important to emphasize that wecan obtain the same constructors from 1S1C1.2 to 12S1C1.2 using 1C1.3, we omit the details hereshowing only how to obtain different constructors from those of Table 2. Consider the followingfunctions in the expressions from 15S1C1.3 to 20S1C1.3: µ : [0, 1] → R ∪ {±∞}, ℓ : [0, 1] →R∪{±∞}, v : [0, 1] → R∪{±∞}, m : [0, 1] → R∪{±∞} such that µ and m are non-decreasingand right continuous, and v and ℓ are non-increasing and right continuous.

914

Tabl

e3.

Som

efu

nctio

nalc

onst

ruct

ors

ofcl

asse

sof

prob

abili

tydi

stri

butio

nsob

tain

edfr

om1C

1.3

Som

esu

b-ca

ses

of1C

1.3

Spec

ialc

ondi

tions

over

mon

oton

icfu

nctio

nsan

dpa

ram

eter

sC

onst

ruct

orfu

nctio

nsob

tain

ed

13S1

C1.

3η=

1,k

=1

,a1=

0.

HG

1,...,Gm

( x)=

1−∫ v 1

(·)

(x)

m1(·)

(x)

dF(t)+

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dF(t)

14S1

C1.

3η=

1,k

=1

,a1=

0an

df(t)=

1v1(0,...,0

)−m

1(0,...,0

)−µ1(0,...,0

)+ℓ1(0,...,0

),

HG

1,...,Gm

(x)=

1−

v1(·)

(x)−m

1(·)

(x)−µ1(·)

(x)+ℓ1(·)

(x)

v1(0,...,0

)−m

1(0,...,0

)−µ1(0,...,0

)+ℓ1(0,...,0

)

fort

in[m

1(0,...,0)+µ1(0,...,0),ℓ 1

(0,...,0)+v1(0,...,0)]

.

15S1

C1.

3µ1(·)(x)=µ(γuk+

1(·)(x))

,

ℓ 1(·)(x)=ℓ(γuk+

1(·)(x))

,HG

1,...,Gm

(x)=

1−

k ∏ i=

1

((1−θi)νi(·)(x)+θi)αi

∫ v 1(·)

(x)

m1(·)

(x)

dφ(t)

v1(·)(x)=µ((1−γ)νk+

1(·)(x)+γ),

+

k ∏ i=

1

(θiui(·)(x))αi

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dφ(t)

m1(·)(x)=ℓ((1−γ)νk+

1(·)(x)+γ),

η=

1an

d0≤γ≤

1.

16S1

C1.

3µ1(·)(x)=µ(γuk+

1(·)(x))

,

ℓ 1(·)(x)=

−∞

,HG

1,...,Gm

(x)=

1−

k ∏ i=

1

((1−θi)νi(·)(x)+θi)αi

∫ v 1(·)

(x)

m1(·)

(x)

dφ(t)

v1(·)(x)=µ((1−γ)νk+

1(·)(x)+γ),

m1(·)(x)=

−∞

,+

k ∏ i=

1

(θiui(·)(x))αi

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dφ(t)

η=

1an

d0≤γ≤

1.

17S1

C1.

3µ1(·)(x)=

+∞,ℓ 1

(·)(x)=ℓ(γuk+

1(·)(x))

,

v1(·)(x)=

+∞,η=

1,0≤γ≤

1,a

ndHG

1,...,Gm

(x)=

1−

k ∏ i=

1

((1−θi)νi(·)(x)+θi)αi

∫ +∞

m1(·)

(x)

dφ(t)

m1(·)(x)=ℓ((1−γ)νk+

1(·)(x)+γ).

+

k ∏ i=

1

(θiui(·)(x))αi

∫ +∞

ℓ1(·)

(x)

dφ(t)

Con

tinue

son

the

next

page

915

Tabl

e3.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Som

esu

b-ca

ses

of1C

1.3

Spec

ialc

ondi

tions

over

mon

oton

icfu

nctio

nsan

dpa

ram

eter

sC

onst

ruct

orfu

nctio

nsob

tain

ed

18S1

C1.

3µ1(·)(x)=m

(γuk+

1(·)(x))

,

ℓ 1(·)(x)=v(γuk+

1(·)(x))

,HG

1,...,Gm

(x)=

1−

k ∏ i=

1

((1−θi)νi(·)(x)+θi)αi

∫ v 1(·)

(x)

m1(·)

(x)

dφ(t)

v1(·)(x)=m

((1−γ)νk+

1(·)(x)+γ),

+

k ∏ i=

1

(θiui(·)(x))αi

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dφ(t)

m1(·)(x)=v((1−γ)νk+

1(·)(x)+γ),

η=

1an

d0≤γ≤

1.

19S1

C1.

3µ1(·)(x)=m

(γuk+

1(·)(x))

,

ℓ 1(·)(x)=

−∞

,HG

1,...,Gm

(x)=

1−

k ∏ i=

1

((1−θi)νi(·)(x)+θi)αi

∫ v 1(·)

(x)

−∞

dφ(t)

v1(·)(x)=m

((1−γ)νk+

1(·)(x)+γ),

m1(·)(x)=

−∞

,+

k ∏ i=

1

(θiui(·)(x))αi

∫ µ 1(·)

(x)

−∞

dφ(t)

η=

1an

d0≤γ≤

1.

20S1

C1.

3µ1(·)(x)=

+∞

,

ℓ 1(·)(x)=v(γuk+

1(·)(x))

,HG

1,...,Gm

(x)=

1−

k ∏ i=

1

((1−θi)νi(·)(x)+θi)αi

∫ +∞

m1(·)

(x)

dφ(t)

v1(·)(x)=

+∞

,+

k ∏ i=

1

(θiui(·)(x))αi

∫ +∞

ℓ1(·)

(x)

dφ(t)

m1(·)(x)=v((1−γ)νk+

1(·)(x)+γ),

η=

1an

d0≤γ≤

1.

21S1

C1.

3η=

1.

HG

1,...,Gm

(x)=

1−

k ∏ i=

1

((1−θi)νi(·)(x)+θi)αi

∫ v 1(·)

(x)

m1(·)

(x)

dφ(t)

+k ∏ i=

1

(θiui(·)(x))αi

∫ µ 1(·)

(x)

ℓ1(·)

(x)

dφ(t)

22S1

C1.

3µ1(·)(x)=

+∞

,ℓ1(·)(x)=

−∞

,v1(·)(x)=

+∞

,m1(·)(x)=

−∞

,

η=

1an

dαi>

0.

HG

1,...,Gm

(x)=

1−

k ∏ i=

1

((1−θi)νi(·)(x)+θi)αi+

k ∏ i=

1

(θiui(·)(x))αi

916

The following theorem shows that Theorem 1 and of its corollaries are equivalent. In otherwords, Theorem 1 and all of its corollaries generate the same probabilistic distributions.

2. Theorem (T2). Equivalence among Theorem 1 and its corollaries.Theorem 1 and all of its corollaries generate exactly the same probabilistic distributions

Proof. To demonstrate Theorem 2 we show that C1.1 is a corollary of T1, that C1.2 is a corollaryof C1.1, that C1.3 is a corollary of C1.2, and finally that T1 is a corollary of C1.3.

(1) C1.1 is a corollary of T1: it is obvious, as it has been already demonstrated.(2) C1.2 is a corollary of C1.1: In Corollary 1.1, W (x) = 1, η = 1, V1(x) = 1, M1(x) =

u(·)(x)∑n

j=1

∫ µj(·)(x)ℓj(·)(x)

dF (t)−ϑ(·)(x)∑n

j=1

∫ vj(·)(x)mj(·)(x)

dF (t), 0(x) = 0, ∀x ∈ R and φ(t)

the cdf of the uniform[0,1].(3) C1.3 is a corollary of C1.2: In Corollary 1.2, set n = 1, u(·)(x) = 1, µ1(·)(x) = 1,

ℓ1(·)(x) = 0, ϑ(·)(x) = 1, m1(·)(x) = 0, v1(·)(x) = ϑ(·)(x)∑η

j=1

∫ vj(·)(x)mj(·)(x)

dφ(t) −

u(·)(x)∑η

j=1

∫ µj(·)(x)ℓj(·)(x)

dφ(t), ∀x ∈ R and F (t) as the cdf of the uniform[0,1].(4) T1 is a corollary of C1.3: In Corollary 1.3, set η = 1,u(·)(x) = 1, ϑ(·)(x) = 1, v1(·)(x) = 1,

m1(·)(x) = 0, ℓ1(·)(x) = 0, µ1(·)(x) = υ(x)∑n

j=1

∫ Uj(x)

Lj(x)dF (t)−

√(x)∑n

j=1

∫ Vj(x)

Mj(x)dF (t)

∀x ∈ R, and φ(t) cdf of the uniform[0,1].

From (1) to (4), we conclude Theorem 2. �

Several classes of probability distributions existing in the literature can be obtained as specialcases of the functional constructors of classes of probability distributions proposed here. Table 4,in the Appendix A, shows how to obtain such classes using some corollaries of Theorem 1.

4. Support of the classes of probability distributionsIn this section, we provide an analysis about the support and nature of the probability distributionsgenerated through the methods described in Corollaries 1.2 and 1.3. These results are important togain a deeper understanding about the proposed method, especially considering the fact that thereis little work on this theme in the literature.

In order to state the results, we remind the reader that, by the definition, the support of a cumula-tive distribution function F is given by SF = {x ∈ R : (x)− F (x− ε) > 0, ∀ε > 0}. Theorem 3shows that the support of the generated distribution is contained in the union of the supports of thebaseline distributions Gi’s.

3. Theorem (T3). General theorem of the supports.Let HG1,...,Gm(x) be a cumulative distribution function generated from Corollary 1.2 (respectively,1.3). Then,

SHG1,...,Gm⊂ ∪m

j=1SGj .

Proof. Consider that HG1,...,Gm(x) has the functional form of Corollary 1.2:

HG1,...,Gm(x) = u(·)(x)∑n

j=1

∫ µj(·)(x)ℓj(·)(x)

dF (t)− ϑ(·)(x)∑n

j=1

∫ vj(·)(x)mj(·)(x)

dF (t).

917

Thus, it follows that

HG1,...,Gm(x)−HG1,...,Gm(x− ε) = u(·)(x)n∑

j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)

− ϑ(·)(x)n∑

j=1

∫ vj(·)(x)

mj(·)(x)dF (t)

− u(·)(x− ε)

n∑j=1

∫ µj(·)(x−ε)

ℓj(·)(x−ε)

dF (t)

+ ϑ(·)(x− ε)

n∑j=1

∫ vj(·)(x−ε)

mj(·)(x−ε)

dF (t).

Suppose that x /∈ ∪mj=1 SGj . Then, there exists ε > 0 such that Gj(x) − Gj(x − ε) = 0, for

all j = 1, 2, . . . ,m. Let us show that HG1,...,Gm(x)−HG1,...,Gm(x− ε) = 0.

First, note that

HG1,...,Gm(x) − HG1,...,Gm(x− ε) = (u(·)(x)−u(·)(x− ε))

n∑j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)

− (ϑ(·)(x)− ϑ(·)(x− ε))

∫ vj(·)(x)

mj(·)(x)dF (t)

+ u(·)(x− ε)

n∑j=1

{∫ µj(·)(x)

µj(·)(x−ε)

dF (t)−∫ ℓj(·)(x)

ℓj(·)(x−ε)

dF (t)

}

− ϑ(·)(x− ε)

n∑j=1

{∫ vj(·)(x)

vj(·)(x−ε)

dF (t)−∫ mj(·)(x)

mj(·)(x−ε)

dF (t)

}.

Since u(·)(x) = u(·)(x − ε), ϑ(·)(x) = ϑ(·)(x − ε), µj(·)(x) = µj(·)(x − ε), ℓj(·)(x) =ℓj(·)(x− ε), mj(·)(x) = mj(·)(x− ε), vj(·)(x) = vj(·)(x− ε), it follows that

HG1,...,Gm(x)−HG1,...,Gm(x− ε) = 0.

Thus, we have x /∈ SHG1,...,Gm. Therefore, SHG1,...,Gm

⊂ ∪mj=1SGj . A similar argument

works for the case where HG1,...,Gm(x) has the functional form of Corollary 1.3. �

Corollary 3.1 shows a special case where the distribution HG1,...,Gm(x) is discrete.

3.1. Corollary (C3.1). Discrete baselines generate discrete distributions.If all Gj’s are discrete in Corollary 1.2 (respectively, 1.3), then HG1,...,Gm(x) is discrete.

Proof. Being all Gj’s discrete, then ∪mj=1SGj has a countable number of values. Since, by Theo-

rem 3, SHG1,...,Gm⊂ ∪m

j=1SGj , then SHG1,...,Gmhas a countable number of values and, for this

reason, HG1,...,Gm(x) is acdf of a discrete random variable. �

Theorem 4 shows conditions when SHG1,...,Gm⊂ ∪m

j=1SGj .

4. Theorem (T4). The support of distribution is a union of the supports of the baselines.Assume, in Corollary 1.2, (respectively, 1.3) that:[f1] SF is a convex set;[f2] µn(1, . . . , 1) = sup{x ∈R : φ(x) < 1}, ℓ1(1, . . . , 1) = inf{x ∈R : φ(x) < 1},u(·)(x) >0, ∀ x ∈ R, and that µj(·)(x) or ℓj(·)(x), for some j = 1, 2, . . . , n, are strictly monotonic orvn(0, . . . , 0) = sup{x ∈ R : φ(x) < 1}, m1(0, . . . , 0) = inf{x ∈ R : φ(x) > 0}, ϑ(·)(x) > 0,∀ x ∈ R, and that vj(·)(x) or mj(·)(x), for some j = 1, 2, . . . , n, are strictly monotonic.

918

Then,

SHG1,...,Gm⊂ ∪m

j=1SGj .

Proof. As a consequence of Theorem 3, we only need to show that

SHG1,...,Gm⊂ ∪m

j=1SGj .

Consider that HG1,...,Gm(x) has the functional form of Corollary 1.2:HG1,...,Gm(x) = u(·)(x)

∑nj=1

∫ µj(·)(x)ℓj(·)(x)

dF (t)− ϑ(·)(x)∑n

j=1

∫ vj(·)(x)mj(·)(x)

dF (t).Thus, using an argument identical to that of the proof of Theorem 3, we have

HG1,...,Gm(x) − HG1,...,Gm(x− ε) = (u(·)(x)−u(·)(x− ε))

n∑j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)

− (ϑ(·)(x)− ϑ(·)(x− ε))

∫ vj(·)(x)

mj(·)(x)dF (t)

+ u(·)(x− ε)

n∑j=1

{∫ µj(·)(x)

µj(·)(x−ε)

dF (t)−∫ ℓj(·)(x)

ℓj(·)(x−ε)

dF (t)

}

− ϑ(·)(x− ε)

n∑j=1

{∫ vj(·)(x)

vj(·)(x−ε)

dF (t)−∫ mj(·)(x)

mj(·)(x−ε)

dF (t)

}.

Suppose that x ∈ ∪mj=1SGj . Then, there exists ε > 0 such that Gj(x) − Gj(x − ε) > 0 for

some j = 1, 2, . . . ,m. Conditions [f1] and [f2] imply that at least one of the integrals of theform

∫ h(·)(x)h(·)(x−ε)

dF (t) is different from zero for h = µj , h = ℓj , h = vj or h = mj , for somej = 1, 2, . . . , n. This fact together with the fact thatu or ϑ are strictly monotonic imply that:

HG1,...,Gm(x) − HG1,...,Gm(x − ε) > 0. Thus, x ∈ SHG1,...,Gm, as desired. A similar argu-

ment works for the case where HG1,...,Gm(x) has the functional form of Corollary 1.3. �

Theorem 5 shows some conditions that guarantee that HG1,...,Gm(x) is a continuous cdf .

5. Theorem (T5). Generating continuous cumulative distribution functions.Suppose that F (x), G1, . . . , Gm are continuous cdf ′s and that µj , ℓj , u, vj , mj and ϑ are con-tinuous functions in Corollary 1.2 (respectively, 1.3). Then, HG1,...,Gm(x) is a continuous cdf .

Proof. Consider that HG1,...,Gm(x) has the functional form of Corollary 1.2

HG1,...,Gm(x) = u(·)(x)∑n

j=1

∫ µj(·)(x)ℓj(·)(x)

dF (t)− ϑ(·)(x)∑n

j=1

∫ vj(·)(x)mj(·)(x)

dF (t).

Thus, it follows that

HG1,...,Gm(x)−HG1,...,Gm(x−) = u(·)(x)n∑

j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)

− ϑ(·)(x)n∑

j=1

∫ vj(·)(x)

mj(·)(x)dF (t)

− u(·)(x−)

n∑j=1

∫ µj(·)(x−)

ℓj(·)(x−)

dF (t)

+ ϑ(·)(x)n∑

j=1

∫ vj(·)(x−)

mj(·)(x−)

dF (t).

919

Using a similar approach as that used in the proof of Theorem 3, we obtain:

HG1,...,Gm(x)−HG1,...,Gm(x−) =(u(·)(x)− u(·)(x−)

) n∑j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)

−(ϑ(·)(x)− ϑ(·)(x−)

) n∑j=1

∫ vj(·)(x)

mj(·)(x)dF (t)

+u(·)(x−)

n∑j=1

{∫ µj(·)(x)

µj(·)(x−)

dF (t)−∫ ℓj(·)(x)

ℓj(·)(x−)

dF (t)

}

− ϑ(·)(x−)

{n∑

j=1

∫ vj(·)(x)

vj(·)(x−)

dF (t)−∫ mj(·)(x)

mj(·)(x−)

dF (t)

}.

Since all functions included in the previous expression are continuous, we have:

HG1,...,Gm(x)−HG1,...,Gm(x−) = 0.

Therefore, we shall conclude that HG1,...,Gm(x) is a continuous function. A similar argumentworks for the case where HG1,...,Gm(x) has the functional form of Corollary 1.3. �

Theorem 6 shows conditions where distribution HG1,...,Gm(x) will be a continuous cdf of ran-dom variables.

6. Theorem (T6). Generating cumulative distribution functions of continuous random variables.Suppose that F (x), G1, . . . , Gm are cdf ′s of continuous random variables and that µj , ℓj ,u, vj ,mj and ϑ are continuous and differentiable functions in Corollary 1.2 (respectively, 1.3). Then,HG1,...,Gm(x) is a cdf of a continuous random variable.

Proof. Consider that HG1,...,Gm(x) has the functional form of Corollary 1.2HG1,...,Gm(x) = u(·)(x)

∑nj=1

∫ µj(·)(x)ℓj(·)(x)

dF (t)− ϑ(·)(x)∑n

j=1

∫ vj(·)(x)mj(·)(x)

dF (t).

Since F (x), G1, . . . , Gm are cdf ′s of continuous random variables and µj , ℓj , u, vj , mj andϑ are continuous and differentiable functions , then HG1,...,Gm(x) is a cdf of a continuous randomvariable with density given by:

HG1,...,Gm(x) =

(m∑

z=1

∂u(·)(x)∂Gz

gz(x)

)(n∑

j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)

)

+u(·)(x)n∑

j=1

{F (µj(·)(x)

m∑z=1

∂µj(·)(x)∂Gz

gz(x)

− F (ℓj(·)(x))m∑

z=1

∂ℓj(·)(x)∂Gz

gz(x)

}

−

(m∑

z=1

∂ϑ(·)(x)∂Gz

gz(x)

)(n∑

j=1

∫ vj(·)(x)

mj(·)(x)dF (t)

)

− ϑ(·)(x)n∑

j=1

{F (vj(·)(x))

m∑z=1

∂vj(·)(x)∂Gz

gz(x)

− F (mj(·)(x))m∑

z=1

∂mj(·)(x)∂Gz

gz(x)

}

920

where (·)(x) = (G1, . . . , Gm)(x). A similar argument works for the case whereHG1,...,Gm(x) has the functional form of Corollary 1.3. �

The next theorem shows an alternative way of generating discrete distributions.

7. Theorem (T7). Integrals with respect to discrete distributions generate discrete distributions.Suppose that the probability distribution F (x) is discrete and that u(·)(x) = ϑ(·)(x) = 1, inCorollary 1.2 (respectively, 1.3). Then, HG1,...,Gm(x) is discrete independent from the monotonicfunctions that are used as limits of the integration.

Proof. We can write the following

HG1,...,Gm(x) =

n∑j=1

∫ µj(·)(x)

ℓj(·)(x)dF (t)−

n∑j=1

∫ vj(·)(x)

mj(·)(x)dF (t)

=

n∑j=1

[F (µj(·)(x))− F (ℓj(·)(x))]

−n∑

j=1

[F (vj(·)(x))− F (mj(·)(x))] .

Since F (x) is a cdf of a discrete random variable, then F (x) assumes a countable number ofdifferent values. Thus, as HG1,...,Gm(x) is given by the sum of differences of F (x) evaluatedin at most 4n distinct points, it also can only assume a countable number of different values and,therefore, it is a cdf of a discrete random variable. �

5. ConclusionsThe method to generate distributions and classes of probability distributions that we presented in thispaper combines several methods to generate classes of distribution that have already been describedin the literature. By this unification, we could draw conclusions on the supports of generated classes.Using the proposed method, we can generate any probability distribution in different ways. The onlynecessity is to modify the monotonic functions involved in the method.

As a further step, we aim to explore several classes of distribution which may be generated usingthis method, developing their properties and applying them to model several datasets. In a parallelwork, we are proposing a method for generating multivariate distributions.

It is important to stress that a model can better describe a phenomenon by increasing its numberof parameters, providing higher flexibility. On the other hand, we should not forget that increasingthe number of parameters may cause identifiability and computational problems in the estimation ofthe parameters. Moreover, a large number of parameters increase the chance of overfitting, which isa problem particularly in forecasting and prediction studies. Thus, the best approach is to choose amethod that best describes the analyzed phenomenon or experiment with the lowest possible numberof parameters.

Appendix A. how to obtain generalizations of already existing class modelsIn this appendix we will show some applications to obtain some very special examples of functionsgenerating classes of probabilistic distributions by finding probability distribution classes that havealready been described in literature.

Table 4 shows how to obtain classes of probability distributions already existing in the literatureby the use of some corollaries of Theorem 1.

921

Tabl

e4.

Gen

eral

izat

ions

ofal

read

yex

istin

gcl

asse

sof

prob

abili

tydi

stri

butio

ns

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

θ=

0,m

=1

and

beta

-Gty

pe1

ℓ 1(·)(x)=θ

m ∏ i=

1

( 1−Gαii

(x)) δ i

β1=

1de

fined

byE

ugen

eet

3S1C

1.2

1

B(a,b)ta

−1(1

−t)b−

1al

.(20

02)

µ1(·)(x)=

(1−θ)

m ∏ j=

1

Gβjj

(x)+θ

Mc-G

type

1

θ=

0an

dm

=1

defin

edby

McD

onal

d(1

984)

θ=

0,m

=1

and

beta

-Gty

pe1

m1(·)(x)=θ

m ∏ j=

1

Gβjj

(x)

β1=

1de

fined

byE

ugen

eet

9S1C

1.2

1

B(a,b)ta

−1(1

−t)b−

1al

.(20

02)

v1(·)(x)=

(1−θ)

m ∏ i=

1

( 1−Gαii

(x)) δ i

+θ

Mc-G

type

1

θ=

0an

dm

=1

defin

edby

McD

onal

d(1

984)

3S1C

1.2

btb

−1

ℓ 1(·)(x)=θ

m ∏ i=

1

( 1−Gαii

(x)) δ i

θ=

0,m

=1

and

expo

nent

iate

d

β1=

1ge

nera

lized

defin

edby

µ1(·)(x)=

(1−θ)

m ∏ j=

1

Gβjj

(x)+θ

Mud

holk

aret

al.

(199

5)ex

pone

ntia

ted

9S1C

1.2

btb

−1

m1(·)(x)=θ

m ∏ j=

1

Gβjj

(x)

θ=

0,m

=1

and

gene

raliz

edde

fined

by

v1(·)(x)=

(1−θ)

m ∏ i=

1

( 1−Gαii

(x)) δ i

+θ

β1=

1M

udho

lkar

etal

.

(199

5)

Con

tinue

son

the

next

page

922

Tabl

e4.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

expo

nent

iate

d

3S1C

1.2

btb

−1

ℓ 1(·)(x)=θ

m ∏ i=

1

( 1−Gαii

(x)) δ i

θ=

0an

dm

=1

gene

raliz

edde

fined

by

µ1(·)(x)=

(1−θ)

m ∏ j=

1

Gβjj

(x)+θ

Mud

holk

aret

al.

(199

5)ex

pone

ntia

ted

5S1C

1.2

——

—µ1(·)(x)=

m ∏ i=

1

( b i+Gαii

(x)) β i

θ=

0an

dβ

=1

gene

raliz

edde

fined

by

Mud

holk

aret

al.

(199

5)K

umar

asw

amy-G

3S1C

1.2

abta

−1(1

−ta

)b−

1ℓ 1

(·)(x)=θ

m ∏ i=

1

( 1−Gαii

(x)) δ i

θ=

0an

dm

=1

and

defin

edby

µ1(·)(x)=

(1−θ)

m ∏ j=

1

Gβjj

(x)+θ

β1=

1C

orde

iro

and

Cas

tro

(201

1)

Kum

aras

wam

y-G

6S1C

1.2

——

—µ1(·)(x)=

m ∏ i=

1

( b i−Gαii

(x)) β i

m=

1,b

1=

1de

fined

by

β1=β

andα

1=α

Cor

deir

oan

dC

astr

o(2

011)

beta

-Gty

pe3

θ=

0,m

=1

and

defin

edby

Tha

iran

d

3S1C

1.2

ta−

1(1

−t)b−

1

B(a,b)(1+t)a+b

ℓ 1(·)(x)=θm ∏ i=

1

( 1−Gαii

(x)) δ i

β1=

1N

adar

ajah

(201

5)[2

2]

µ1(·)(x)=

(1−θ)m ∏ j=

1

Gβjj

(x)+θ

Mc-G

type

3

θ=

0an

dm

=1

defin

edby

Tha

iran

dN

adar

ajah

(201

5)

Con

tinue

son

the

next

page

923

Tabl

e4.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

beta

-Gty

pe1

θ=

0,m

=1

and

defin

edby

Eug

ene

et

m1(·)(x)=θ

m ∏ j=

1

Gβjj

(x)

β1=

1al

.(20

02)

9S1C

1.2

ta−

1(1

−t)b−

1

B(a,b)(1+t)a+b

v1(·)(x)=

(1−θ)

m ∏ i=

1

( 1−Gαii

(x)) δ i

+θ

Mc-G

type

3

θ=

0an

dm

=1

defin

edby

Tha

iran

dN

adar

ajah

(201

5)

ℓ 1(·)(x)=θ

m ∏ i=

1

( 1−Gαii

(x)) δ i

θ=

0,m

=1

and

Kum

mer

beta

gene

ra-

3S1C

1.2

ta−

1(1

−t)b−

1exp(−ct)

B(a,b)

β1=

1liz

edde

fined

byPe

s-

µ1(·)(x)=

(1−θ)

m ∏ j=

1

Gβjj

(x)+θ

cim

etal

.(20

12)

Kum

mer

beta

gene

ra-

9S1C

1.2

ta−

1(1

−t)b−

1exp(−ct)

B(a,b)

m1(·)(x)=θ

m ∏ j=

1

Gβjj

(x)

θ=

1,m

=1

and

lized

defin

edby

Pes-

v1(·)(x)=

(1−θ)

m ∏ i=

1

( 1−Gαii

(x)) δ i

+θ

β1=

1ci

met

al.(

2012

)

θ=

0,m

=1

,ga

mm

a-G

defin

edby

3S1C

1.2

ba

Γ(a

)ta

−1e−bt

ℓ 1(·)(x)=θm ∏ i=

1

( 1−Gβjj

(x)) γ j

α1=

1,λ

=1

,r=

1Z

ogra

fos

and

Bal

a-

µ1(·)(x)=θ+

( −log

( m ∏ i=

1

(1−Gβjj

(x))δ1

) λ)r

andδ1=

1kr

ishn

an(2

009)

Con

tinue

son

the

next

page

924

Tabl

e4.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

ℓ 1(·)(x)=ρ

m ∏ i=

1

( 1−Gωij

(x)) s i

ρ=

0,m

=1

,ga

mm

a-G

defin

edby

3S1C

1.2

ba

Γ(a

)ta

−1e−bt

α=

0an

dλ1=

1Z

ogra

fos

and

Bal

a-

µ1(·)(x)=ρ−

log

( 1−

m ∏ i=

1

Gλll

(x))

kris

hnan

(200

9)

ℓ 1(·)(x)=ρ

m ∏ i=

1

( 1−Gλij

(x)) s i

m=

1,α

1=

1,

gam

ma-G

defin

edby

3S1C

1.2

ba

Γ(a

)ta

−1e−bt

ρ=

0,ω

1=

1,λ

=1

Zog

rafo

san

dB

ala-

µ1(·)(x)=ρ+

( −log

( m ∏ i=

1

( 1−Gαii

(x)) ω i

) λ)r

andr=

1kr

ishn

an(2

009)

m1(·)(x)=ρ

m ∏ i=

1

( 1−Gωij

(x)) s i

ρ=

0,m

=1

and

gam

ma-G

defin

edby

9S1C

1.2

ba

Γ(a

)ta

−1e−bt

λ1=

1Z

ogra

fos

and

Bal

a-

v1(·)(x)=ρ−

log

( 1−

m ∏ i=

1

Gλll

(x))

kris

hnan

(200

9)

m1(·)(x)=ρm ∏ i=

1

( 1−Gλij

(x)) s i

m=

1,α

1=

0,

gam

ma-G

defin

edby

9S1C

1.2

ba

Γ(a

)ta

−1e−bt

ρ=

0an

dβ1=

1Z

ogra

fos

and

Bal

a-

v1(·)(x)=ρ+

( −log

( m ∏ i=

1

( 1−Gαii

(x)) β i

) λ)r

kris

hnan

(200

9)

Con

tinue

son

the

next

page

925

Tabl

e4.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

9S1C

1.2

ba

Γ(a

)ta

−1e−bt

m1(·)(x)=θ

m ∏ j=

1

( 1−Gβjj

(x)) γ j

θ=

0,m

=1

,ga

mm

a-G

defin

ed

v1(·)(x)=θ+

( −log

( m ∏ i=

1

Gαii

(x))) δ

α1=

1an

dδ=

1by

Cor

deir

oet

al.

(201

7)

9S1C

1.2

ba

Γ(a

)ta

−1e−bt

m1(·)(x)=θ

1−m ∏ j=

1

Gαjj

(x) λ

θ=

0,m

=1

,ga

mm

a-G

defin

ed

v1(·)(x)=θ+

( −log

( 1−

m ∏ i=

1

( 1−Gβii

(x)) γ i

) r)s

β1=

1,γ

1=

1,r

=1

byC

orde

iro

etal

.

ands=

1(2

017)

3S1C

1.2

ba

Γ(a

)ta

−1e−bt

ℓ 1(·)(x)=ρ

m ∏ i=

1

Gωii

(x)

ρ=

0,m

=1

,ga

mm

a-G

defin

ed

µ1(·)(x)=ρ−

log

( m ∏ l=1

Gλll

(x))

andλ1=

1by

Cor

deir

oet

al.

(201

7)

3S1C

1.2

ba

Γ(a

)ta

−1e−bt

ℓ 1(·)(x)=ρ

m ∏ i=

1

Gλii

(x)

m=

1, ρ

=0

,ga

mm

a-G

defin

ed

µ1(·)(x)=ρ+

( −log

( 1−

m ∏ i=

1

( 1−Gαii

(x)) ω i

) λ)r

α1=

1,δ

1=

1,

byC

orde

iro

etal

.

λ=

1an

dr=

1(2

017)

9S1C

1.2

ba

Γ(a

)ta

−1e−bt

m1(·)(x)=ρm ∏ i=

1

Gωii

(x)

ρ=

0,m

=1

,ga

mm

a-G

defin

ed

v1(·)(x)=ρ−

log

( m ∏ l=1

Gλll

(x))

α=

0an

dλ1=

1by

Cor

deir

oet

al.

(201

7)

Con

tinue

son

the

next

page

926

Tabl

e4.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

9S1C

1.2

ba

Γ(a

)ta

−1e−bt

m1(·)(x)=ρ

m ∏ i=

1

Gλii

(x)

m=

1,ρ

=0

,ga

mm

a-G

defin

ed

v1(·)(x)=ρ+

( −log

( 1−

m ∏ i=

1

( 1−Gαii

(x)) ω i

) λ)r

α1=

1,ω

1=

1,

byC

orde

iro

etal

.

λ=

1an

dr=

1(2

017)

G1(x

)=G

2(x

),M

arsh

all-

Olk

in-G

α=

1,β

=1

and

defin

edby

Mar

shal

lθ=

1an

dO

lkin

(199

7)

12S1

C1.

2—

——

——

—m

1(·)(x)=

b( 1

−Gβ 2(x

)) γGα 1(x

)+b( 1

−Gβ 2(x

)) γ θ

G1(x

)=G

2(x

),M

arsh

all-

Olk

in-G

α=

1an

dβ

=1

defin

edby

Jaya

kum

aran

dM

athe

w(2

008)

G1(x

)=G

2(x

),M

arsh

all-

Olk

in-G

α=

1,β

=1

and

defin

edby

Mar

shal

lθ=

1an

dO

lkin

(199

7)

5S1C

1.2

——

——

——

µ1(·)(x)=

Gα 1(x

)

Gα 1(x

)+b( 1

−Gβ 2(x

)) γ θ

G1(x

)=G

2(x

),M

arsh

all-

Olk

in-G

α=

1an

dβ

=1

defin

edby

Tha

iran

dN

adar

ajah

(201

5)

Kum

aras

wam

y-G

12S1

C1.

2—

——

——

—m

1(·)(x)=

exp

( −λ

m ∏ l=1

Gαll

(x))

m=

1an

dα

1=

1Po

isso

nde

fined

by

Ram

oset

al.

(201

4)

Con

tinue

son

the

next

page

927

Tabl

e4.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

Kum

aras

wam

y-G

12S1

C1.

2—

——

——

—m

1(·)(x)=

exp

( −λ−λ

m ∏ l=1

( 1−Gαll

(x)) β l

)m

=1

andα

1=

1Po

isso

nde

fined

by

andβ1=

1R

amos

etal

.(2

014)

Kum

aras

wam

y-G

12S1

C1.

2—

——

——

—m

1(·)(x)=

exp

( −λ−λ

( 1−

m ∏ l=1

Gαll

(x)) β)

m=

1an

dα

1=

1Po

isso

nde

fined

by

andβ1=

1R

amos

etal

.(2

014)

u1(·)(x)=

eλe−βxa

−eλ

1−

eλ

k=

2,α

1=

1B

eta

Wei

bull

Pois

son

α2=

1,θ

=0

and

Fam

ilyde

fined

by

u2(·)(x)=

e−a λW

(−a

e−a)−

e−a λW

(ψ(x))

e−a λW

(−a

e−a)−

1δ=

0Pe

rcon

tini(

2013

)

k=

2,α

1=

0,

Wei

bull

Gen

eral

ized

W(x

)=∑ ∞ n=

1(−

1)n−

1nn−

2

(n−

1)!

xn

α2=

1,θ

=0

and

Pois

son

defin

edby

ψ(x

)=

−αe−α−bxa

δ=

0Pe

rcon

tini(

2014

)

2S1C

1.2

γtγ−

1ℓ 1

(·)(x)=θ

( 1−

(1−β)−s−

{1−β[1

−G

(x)]}

−s

(1−β)−s−

1

)k=

2,α

1=

0,

G-N

egat

ive

Bin

omia

l

α2=

0an

dθ=

1de

fined

byPe

rcon

-

µ1(·)(x)=

(1−θ)( ζ(s

)−Lis[1

−G

(x)]

ζ(s)

) δ +θ

,tin

ieta

l.(2

013)

LIs(Z

)=∑ ∞ j=

1zj

js

k=

2,α

1=

0,

Zet

a-G

defin

edby

ζ(s)=∑ ∞ j=

11 js

α2=

0an

dθ=

0Pe

rcon

tini(

2014

)

Con

tinue

son

the

next

page

928

Tabl

e4.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

k=

2,α

1=

1,

Pow

erSe

ries

defin

ed

u1(·)(x)=

x ∑ j=

0

C(j)(a

)

j!C

(λ)(λ

−a)j

α2=

0,θ

=0

and

byC

onsu

land

Fam

oye

δ=

0(2

006)

u2(·)(x)=

x ∑ j=

1

1 j!

[ (C(0

))j] (j−

1)

k=

2,α

1=

0,

Bas

icL

agra

ngia

n

α2=

1,θ

=0

and

defin

edby

Con

sula

nd

δ=

0Fa

moy

e(2

006)

2S1C

1.2

btb

−1

ℓ 1(·)(x)=θ

1−x ∑ j=n

n

(j−n)!j

[ (C(0

))j] (j−

n)

k=

2,α

1=

0,

Lag

rang

ian

Del

ta

α2=

0an

dθ=

1de

fined

byC

onsu

land

µ1(·)(x)=

(1−θ)

x ∑ j=

0

P(X

=j) δ

+θ

Fam

oye

(200

6)

P(X

=j)=

{w(0

),j=

0[ (C

(0))jw

(1)(0

)] (j−1),

j=

1,2,3,...

Gen

eral

ized

k=

2,α

1=

0,

Lag

rang

ian

fam

ily

α2=

0an

dθ=

0by

Con

sula

nd

Fam

oye

(200

6)

Con

tinue

son

the

next

page

929

Tabl

e4.

Con

tinue

dfr

omth

epr

evio

uspa

ge

Sub-

case

Use

ddi

stri

butio

nsf(t)

Mon

oton

icfu

nctio

nsV

alue

sfo

rthe

para

met

ers

Obt

aine

dcl

ass

Gen

eral

ized

Pear

son

in

k=

1,α

1=

0,θ

=0

Ord

inar

yD

iffer

entia

l

andδ=

0E

quat

ion

form

bySh

a-

u1(·)(x)=

∫ x −∞

e

∫ a0+a1t+

···+asts

b0+b1+t+

···+brtrf(t)dtdt

kile

tal.

(201

0)

Gen

eral

ized

Pear

son

2S1C

1.2

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