# General Results for the Transmuted Family of Distributions and New Models

Authors: Marcelo Bourguignon; Indranil Ghosh; Gauss M. Cordeiro
Journal: Journal of Probability and Statistics
 (2016)

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## Abstract

The transmuted family of distributions has been receiving increased attention over the last few years. For a baselineG distribution, we derive a simple representation for the transmuted-G family density function as a linear mixture of theG and exponentiated-G densities. We investigate the asymptotes and shapes and obtain explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean deviations, Rényi and Shannon entropies, and order statistics and their moments. We estimate the model parameters of the family by the method of maximum likelihood. We prove empirically the flexibility of the proposed model by means of an application to a real data set.


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## Body

## 1. Introduction

Adding parameters to a well-established distribution is a time honored device for obtaining more flexible new families of distributions. Shaw and Buckley [1] pioneered an interesting method of adding a new parameter to an existing distribution that would offer more distributional flexibility. They used the quadratic rank transmutation map (QRTM) in order to generate a flexible family of distributions. The generated family, also called the transmuted extended distribution, includes the parent distribution as a special case and gives more flexibility to model various types of data.In the last three years, there has been a growing interest in transmuted distributions and several of them have been investigated. A significant amount of work has been attributed towards developing a new transmuted model and subsequently discussing its utilities as enhanced flexibility in modeling various types of real life data, where the parent model does not provide a good fit. Aryal and Tsokos [2] defined the transmuted generalized extreme value distribution and studied some basic mathematical characteristics of the transmuted Gumbel distribution and its applications to climate data. Aryal and Tsokos [3] presented a new generalized Weibull distribution called the transmuted Weibull distribution. Recently, Aryal [4] proposed and studied various structural properties of the transmuted log-logistic distribution. Khan and King [5] introduced the transmuted modified Weibull distribution, which extends the transmuted Weibull distribution [3], and studied its mathematical properties and maximum likelihood estimation of the model parameters. Elbatal [6] proposed the transmuted modified inverse Weibull distribution. Elbatal and Aryal [7] explored the transmuted additive Weibull model, which extends the additive Weibull distribution and some other distributions using the QRTM method [1]. However, several published works did not investigate many properties such as finite mixture of the density function, Rényi and Shannon entropies, extreme values, probability weighted moments (PWMs), and bivariate and multivariate generalization. This paper aims to fill out this gap in the existing literature and contribute with general properties of the transmuted family.This vast amount of literature merits for a detailed study for the most general transmuted family of distributions, which is our major motivation to carry out this work. In this paper, we derive general mathematical properties for the transmuted family, which hold for any baseline distribution, such as the ordinary, central, and incomplete moments, quantile and generating functions, mean deviations, Rényi and Shannon entropies, extreme values, PWMs, order statistics and their moments, and bivariate and multivariate generalizations. We provide a comprehensive description of these properties with the hope that the transmuted family will attract wider applications in biology, medicine, economics, reliability, and engineering and in other areas of research. We also introduce new distributions based on the transmuted construction.The rest of the paper is organized as follows. In Section2, we discuss the general theory behind the transmuted distribution and present useful representations for the density and cumulative functions. In Section 3, we investigate its asymptotes and shapes. In Section 4, we provide an algorithm for generating samples from the transmuted family based on its quantile function (qf). In Section 5, we derive expressions for the moments and generating function. In Section 6, we obtain mean deviations and provide some examples. In Section 7, we present two special transmuted models. In Section 8, we discuss the limiting behavior of the extreme statistics. In Section 9, we derive the PWMs. In Section 10, we obtain the order statistics. We derive expressions for the Shannon and Rényi entropies and Kullback-Leibler divergence measure in Section 11. We introduce in Section 12 the bivariate and multivariate extensions of the univariate transmuted family. In Section 13, we use the maximum likelihood method to estimate the model parameters. In Section 14, we fit some special models of the transmuted family to a real data set to prove empirically its usefulness. In Section 15, we offer some concluding remarks.


## 2. Distribution and Density Functions

LetF1 and F2 be the cumulative distribution functions (cdfs) of two models with a common sample space. The general rank transmutation as given in Shaw and Buckley [1] is defined as GR12(u)=F2(F1-1(u)) and GR21(u)=F1(F2-1(u)). Note that the qf is defined by F-1(y)=infx∈R{F(x)≥y} for y∈[0,1]. Functions GR12(u) and GR21(u) both map the unit interval I=[0,1] into itself and, under suitable assumptions, are mutual inverses and satisfy GRij(0)=0 and GRij(1)=1 (for i=1,2). The QRTM is defined by GR12(u)=u+λu(1-u),λ≤1, from which it follows that F2(x)=(1+λ)F1(x)-λF1(x)2. Differentiating gives f2(x)=f1(x)[1+λ-2λF1(x)], where f1(x) and f2(x) are the probability density functions (pdfs) corresponding to the cdfs F1(x) and F2(x), respectively. For more details about the QRTM approach, see Shaw and Buckley [1].A random variableX has thetransmuted-G (TG) family if the pdf and cdf are defined through the QRTM method by (for λ∈[-1,1])(1)fx=fx;ξ,λ=1+λ-2λGx;ξgx;ξ,x∈D⊆R,(2)Fx=Fx;ξ,λ=1+λGx;ξ-λGx;ξ2,where G(x;ξ) is the parent cdf and g(x;ξ) is the parent pdf. Both functions depend on the parameter vector ξ. For λ=0, it reduces to the parent model. Hereafter, the random variable X following (1) with parameter λ and baseline vector of parameters ξ is denoted by X~TG(λ,ξ). The computations for fitting family (1) to real data in practical problems can be easily performed using theAdequacyModel script in the R software.For an arbitrary baseline cdfG(x;ξ), a random variable is said to have the Exp-G distribution with power parameter α>0, say Y~Exp-G(ξ;α), if its pdf and cdf are given by (3)πx;ξ,α=αGx;ξα-1gx;ξ,Πx;ξ,α=Gx;ξα,respectively. Note that π(x;ξ,1)=g(x;ξ). The properties of exponentiated distributions have been studied by many authors in recent years. See, for example, Mudholkar and Srivastava [8] for exponentiated Weibull, Gupta et al. [9] for exponentiated Pareto, Gupta and Kundu [10] for exponentiated exponential, Nadarajah [11] for exponentiated Gumbel, Kakde and Shirke [12] for exponentiated lognormal, and Nadarajah and Gupta [13] for exponentiated gamma distributions.Theorem 1.
The density function ofX can be expressed as the linear mixture (4)fx;ξ,λ=cgx;ξ+1-cπx;ξ,2,where c=1+λ.Corollary 2.
Ifλ=-1, then X~Exp-G(ξ;2).Theorem1 is important to obtain some measures of X from those of exponentiated distributions. This result plays an important role in the paper, since we can obtain, for example, the moments, generating function, and mean deviations of X. Established explicit expressions for these measures can be simpler than using numerical integration.The hazard rate function (hrf) ofX is given by(5)hx;ξ,λ=1+λ-2λGx;ξgx1-Gx;ξ1-λGx;ξ=hGx1+λ-2λGx;ξ1-λGx;ξ,x>0,where hG(x) is the baseline hrf. The multiplying quantity [1+λ-2λG(x;ξ)]/[1-λG(x;ξ)] is a kind of correction factor for the baseline hrf.Equation (5) can deal with general situations for modeling survival data with various hrf shapes. From this equation, we note that h(x;ξ,λ)/hG(x;ξ) is decreasing in x for λ≥0 and it is increasing in x for λ≤0. Additionally, we have (6)hGx≤hx≤1+λhGx,1+λhGt≤hx≤hGx,for λ≥0 and λ≤0, respectively.Equation (5) can be expressed as (7)hx;ξ,λ=wthGx;ξ+1-wthExp-Gx;ξ,2,where w(t)={c[1-G(x;ξ)]}/{1-(1+λ)G(x;ξ)+λ[G(x;ξ)]2}, c=1+λ, and hG(x;ξ) and hExp-G(x;ξ,2) are the hrfs of the G and Exp-G distributions, respectively.


## 3. Asymptotes and Shapes

Proposition 3.
The asymptotics of (1) and (2) as G(x)→0 are(i)
F(x)~(1+λ)G(x),(ii)
f(x)~(1+λ)g(x),(iii)
h(x)~(1+λ)g(x)/G¯(x), where G¯(x)=1-G(x).Proposition 4.
The asymptotics of (1) and (2) as x→∞ are(i)
1-F(x)~λG(x)G(x)-1,(ii)
f(x)~(1-λ)g(x),(iii)
h(x)~(1-λ)g(x)/G¯(x).The shapes of the density and hazard functions ofX can be described analytically. The critical points of the TG pdf are the roots of the equation(8)g′xgx-2λgx1+λ-2λGx=0.There may be more than one root to (8). Let δ(x)=d2/dx2log⁡[f(x)]. We have(9)δx=g′′xgx-g′x2gx2-2λ1+λg′x-2λg′xGx+2λgx21+λ-2λGx2.If x=x0 is a root of (9), then it corresponds to a local maximum if δ(x)>0 for all x<x0 and δ(x)<0 for all x>x0. It corresponds to a local minimum if δ(x)<0 for all x<x0 and δ(x)>0 for all x>x0. It refers to a point of inflection if either δ(x)>0 for all x≠x0 or δ(x)<0 for all x≠x0.The critical points of the hrf ofX are obtained from(10)g′xgx-2λgx1+λ-2λGx+gx1-Gx+λgx1-λGx.Again, there may be two roots to (10). Let θ(x)=d2/dx2log⁡h(x). We have(11)θx=g′′xgx-g′x2gx2+1-Gxg′x+g2x1-Gx2+λ1-λGxg′x+λg2x1-λGx2-2λ1+λ-2λGxg′x+2λg2x1+λ-2λGx2.If x=x0 is a root of (11), then it corresponds to a local maximum if θ(x)>0 for all x<x0 and θ(x)<0 for all x>x0. It corresponds to a local minimum if θ(x)<0 for all x<x0 and θ(x)>0 for all x>x0. It refers to a point of inflection if either θ(x)>0 for all x≠x0 or θ(x)<0 for all x≠x0.


## 4. Quantile Function and Simulation

The qf of theTG family is given by(12)Qu;ξ,λ=QG1+λ-1+λ2-4λu2λ;ξ,ifλ≠0;QGu;ξ,ifλ=0,where QG(u;ξ)=G-1(u;ξ) is the inverse of the baseline cdf. The TG family is easily simulated by Algorithm 1.Algorithm 1:Random number generator for the TG distribution.
(1)  Generate a random number u from U~U(0,1);
(2)  If λ≠0 then compute a random number x=QG(1+λ-(1+λ)2-4λU)/2λ;ξ; Otherwise x=QG(U;ξ);
(3) Repeat steps (1) to (2) until the required amount of random numbers to be completed.Table1 gives QG(u;ξ) and the corresponding parameters for some special distributions.Table 1
Distributions and corresponding qfs.
Distribution
Q
G
u
;
ξ
ξ
Uniform (0<x<θ)
θ
u
θ
Exponential (x>0)
α
-
1
l
o
g
(
1
-
u
)
α
Weibull (x>0)
[
α
-
1
l
o
g
(
1
-
u
)
]
1
/
β
α
,
β
Fréchet (x>0)
σ
[
-
l
o
g
(
u
)
]
-
1
/
λ
λ
,
σ
Half-logistic (x>0)
-
l
o
g
[
(
1
-
u
)
/
(
1
+
u
)
]
⌀
Power function (0<x<1/θ)
u
1
/
k
/
θ
θ
,
k
Pareto (x≥θ)
θ
(
1
-
u
)
-
1
/
k
θ
,
k
Burr XII (x>0)
s
[
(
1
-
u
)
-
1
/
k
-
1
]
1
/
c
s
,
k
,
c
Logistic (x∈R)
-
l
o
g
(
u
-
1
-
1
)
⌀
Log-logistic (x>0)
s
[
(
1
-
u
)
-
1
-
1
]
1
/
c
s
,
c
Lomax (x>0)
s
[
(
1
-
u
)
-
1
/
k
-
1
]
s
,
k
Gumbel (x∈R)
μ
-
σ
l
o
g
-
l
o
g
(
u
)
μ
,
σ
Kumaraswamy (0<x<1)
[
1
-
(
1
-
u
)
1
/
b
]
1
/
a
a
,
b
Normal (x∈R)
Φ
-
1
(
(
u
-
μ
)
/
σ
)
μ
,
σ


## 5. Moments and Generating Function

Many of the important characteristics and features of a distribution are determined through the ordinary moments. Therth ordinary moment of X is obtained from Theorem 1 as(13)μr′=EXr=cEY1r+1-cEY2r,where Yi~Exp-G(ξ;i) for i=1,2. Some moments obtained from (13) are reported in Table 2.Table 2
Distributions and their moments.
Distribution
μ
r
′
Reference
Transmuted Weibull
Γ
1
+
r
/
β
α
1
/
β
1
-
λ
+
2
-
r
/
β
λ
Aryal and Tsokos [3]
Transmuted Lindley
r
!
θ
r
(
θ
+
1
)
1
-
λ
θ
+
r
+
1
+
λ
θ
2
r
-
1
θ
+
1
2
θ
+
3
+
r
Merovci [14]
Transmuted Fréchet
σ
r
Γ
1
-
r
λ
1
+
λ
-
2
r
/
λ
λ, r<λ
Mahmoud and Mandouh [15]
Transmuted log-logistic
s
r
r
π
/
c
sin
⁡
r
π
/
c
1
+
λ
-
λ
s
c
1
+
r
c
s
r
r
π
/
c
sin
⁡
r
π
/
c
Aryal [4]
Transmuted Pareto
k
β
2
k
-
r
1
+
λ
k
-
r
2
k
-
r, r<k
Merovci and Puka [16]The central moments (μr) and cumulants (κr) of X follow from (13) as(14)μr=∑k=0rrk-1kμ1′kμr-k′,κr=μr′-∑k=1r-1r-1k-1κkμr-k′,respectively, where κ1=μ1′. Further, the skewness and kurtosis are obtained from the third and fourth standardized cumulants ζ1=κ3/κ23/2 and ζ2=κ4/κ22, respectively.The moment generating function (mgf) ofX, say M(t)=EetX, can be expressed from Theorem 1 as (15)Mt=cI0ξ+21-cI1ξ,where Ij-1=Ij-1(ξ)=∫01‍exp⁡tQG(u;ξ)uj-1du for j=1,2. The integrals I0(ξ) and I1(ξ) can be evaluated numerically for most parent distributions.Three closed forms forIj-1 (for j=1,2) follow by selecting from Table 1 the exponential (with parameter α), standard logistic, and Fréchet as baseline distributions, for which Ij-1=B(j,1-α/t) (for t>α), Ij-1=B(j,1-t) (for t<1), and Ij-1=(tσ/λ+j)-1, respectively.The characteristic function (chf) has many useful and important properties and plays a central role in statistical theory. It is particularly useful in analysis of linear combination of independent random variables. Clearly, a simple representation for the chfϕ(t)=M(it) of X, where i=-1, is given by (16)ϕt=∫0∞cos⁡txfx;ξ,λdx+i∫0∞sin⁡txfx;ξ,λdx.From expansions cos⁡(tx)=∑r=0∞‍((-1)r/(2r)!)(tx)2r and sin⁡(tx)=∑r=0∞‍((-1)r/(2r+1)!)(tx)2r+1, we obtain (17)ϕt=∑r=0∞-1rt2r2r!μ2r′+i∑r=0∞-1rt2r+12r+1!μ2r+1′.


## 6. Mean Deviations

Thenth incomplete moment of X, say mn(y)=∫-∞y‍xnf(x;ξ,λ)dx, is expressed as(18)mny=cvn,1y+1-cvn,2y,where(19)vn,jy=j∫0Gy;ξQGu;ξruj-1dun=1,2.The integral vn,j(y) can be determined analytically for some special models with closed form expressions for QG(u;ξ) or evaluated at least numerically for most baseline distributions. It can also be obtained for several baseline G distributions using power series methods. These methods are at the heart of many aspects of applied mathematics and statistics. If this function does not have a closed form expression, it can be expressed as a power series:(20)QGu;ξ=∑i=0∞aiui,where coefficients ai are suitably chosen real numbers. For some important distributions, such as the normal, Student t, gamma, and beta distributions, QG(u;ξ) does not have closed form but it can be expanded as in (20). For example, for the standard normal distribution, coefficients ai′s are given by (21)ai=2πi/2∑m=i∞-12m-jmipi,where pi=0 (for i=0,2,4,…) and pi=q(i-1)/2 (for i=1,3,5,…), and the qk′s are determined recursively from (22)qk+1=122k+3∑r=0k2r+12k-2r+1qrqk-rr+12r+1.Then, q0=1, q1=1/6, q2=7/120, and q3=127/7560,…We consider a result by Gradshteyn and Ryzhik [17] for a power series raised to a positive integer n:(23)QGu;ξn=∑i=0∞aiuin=∑i=0∞cn,iui,where coefficients cn,i (for i=1,2,…) are determined from the recurrence equation (24)cn,i=ia0-1∑m=1imn+1-iamcn,i-m,and cn,0=a0r. Coefficient cn,i can be obtained from quantities a0,…,ai in any analytical or numerical software. Hence, quantity vn,j(y) (for j=1,2) in (19) is given by (25)vn,jy=j∑i=0∞cn,ii+jGy;ξi+j.An important application of the first incomplete moment ofX in (18) is related to the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance, and medicine. For a given probability π, they are given by B(π)=m1(q)/(πμ1′) and L(π)=m1(q)/μ1′, where q=Q(π;ξ,λ) comes from (12).The magnitude of dispersion associated with the population can be measured by the totality of deviations from the mean and median. Another application refers to the the deviations about the mean (δ1=E(|X-μ1′|)) and about the median (δ2=E(|X-M|)) of X given by (26)δ1=2μ1′Fμ1′-2m1μ1′,δ2=μ1′-2m1M,respectively, where M is the median of X, μ1′=E(X) is determined from (13), F(μ1′) is easily evaluated from (2), and m1(y)=cv1,1(y)+(1-c)v1,2(y) is obtained from (19) with n=1.Next, we provide two applications of (19) by taking for the baseline model the exponential (with parameter α) and standard logistic distributions listed in Table 1. By using the generalized binomial expansion, we obtain v1,j(y) (for j=1 and 2) for the transmuted-exponential (TE) (with parameter α) and transmuted-standard logistic (TSL) as (27)v1,jy=jα∑m=0∞-1m1-e-mαym+1jm,v1,jy=1Γy∑m=0∞-1mΓm+jm+1!1-e-mz,respectively.


## 7. Special Transmuted Models

The pdf and cdf ofX in (1) and (2) will be most tractable when G(x) and g(x) have simple analytic expressions. In this section, we present two special TG models.


### 7.1. The Transmuted Burr XII (TBXII) Distribution

We consider the parent Burr XII distribution, where the pdf and cdf (forx>0) are g(x)=cks-cxc-1[1+(x/s)c]-k-1, s,k,c>0, and G(x)=1-[1+(x/s)c]-k, respectively. Then, the TBXII density function is given by (28)Fx;k,s,c,λ=1-1+xsc-k1+λ-λ1-1+xsc-k,where λ∈[-1,1]. The corresponding pdf is given by (29)fx;k,s,c,λ=kcsxsc-11+xsc-k-11-λ+2λ1+xsc-k.The TBXII distribution includes an important special case whenk=1: the transmuted-log-logistic [4] distribution. Further, we obtain the transmuted Lomax [18] distribution when c=1. Some plots of the TBXII density function are displayed in Figure 1.Figure 1
Plots of the TBXII density function for some parameter values.
(a)
k
=
2, c=5
(b)
k
=
5, c=2Therth ordinary moment of the TBXII model can be obtained from (13) as (30)EXr=1-λBk-rc-1,rc-1+1+λB2k-rc-1,rc-1+1,ck>r,where B(·,·) is the beta function.


### 7.2. The Transmuted Kumaraswamy (TKw) Distribution

The baseline Kumaraswamy (Kw) distribution has pdf and cdf, forx∈(0,1) and a,b>0, given by g(x)=abxa-1(1-xa)b-1 and G(x)=1-(1-xa)b, respectively. Then, the TKw cdf is given by (31)Fx;a,b,λ=1-1-xab1+λ1-xab,where λ∈[-1,1]. The corresponding pdf is given by (32)fx;a,b,λ=abxa-11-xab-11-λ+2λ1-xab.Ahmad et al. [19] proposed the transmuted Kumaraswamy (TKw) distribution as an extension of the Kw distribution and obtained the density and cumulative functions. However, they did not investigate an application to real data and explore the qf. In Figure 2, we plot the TKw density function for some parameter values.Figure 2
Plots of the TKw density function for some parameter values.
(a)
a
=
2, b=5
(b)
a
=
5, b=2Therth moment of TKw can be obtained from (13) as (33)EXr=1-λbB1+ra,b+2λbB1+ra,2b,r>-a,and the qf is given by (34)F-1u;a,b,λ=1-1-1+λ-1+λ2-4λu2λ1/b1/a,λ≠0.


## 7.1. The Transmuted Burr XII (TBXII) Distribution

We consider the parent Burr XII distribution, where the pdf and cdf (forx>0) are g(x)=cks-cxc-1[1+(x/s)c]-k-1, s,k,c>0, and G(x)=1-[1+(x/s)c]-k, respectively. Then, the TBXII density function is given by (28)Fx;k,s,c,λ=1-1+xsc-k1+λ-λ1-1+xsc-k,where λ∈[-1,1]. The corresponding pdf is given by (29)fx;k,s,c,λ=kcsxsc-11+xsc-k-11-λ+2λ1+xsc-k.The TBXII distribution includes an important special case whenk=1: the transmuted-log-logistic [4] distribution. Further, we obtain the transmuted Lomax [18] distribution when c=1. Some plots of the TBXII density function are displayed in Figure 1.Figure 1
Plots of the TBXII density function for some parameter values.
(a)
k
=
2, c=5
(b)
k
=
5, c=2Therth ordinary moment of the TBXII model can be obtained from (13) as (30)EXr=1-λBk-rc-1,rc-1+1+λB2k-rc-1,rc-1+1,ck>r,where B(·,·) is the beta function.


## 7.2. The Transmuted Kumaraswamy (TKw) Distribution

The baseline Kumaraswamy (Kw) distribution has pdf and cdf, forx∈(0,1) and a,b>0, given by g(x)=abxa-1(1-xa)b-1 and G(x)=1-(1-xa)b, respectively. Then, the TKw cdf is given by (31)Fx;a,b,λ=1-1-xab1+λ1-xab,where λ∈[-1,1]. The corresponding pdf is given by (32)fx;a,b,λ=abxa-11-xab-11-λ+2λ1-xab.Ahmad et al. [19] proposed the transmuted Kumaraswamy (TKw) distribution as an extension of the Kw distribution and obtained the density and cumulative functions. However, they did not investigate an application to real data and explore the qf. In Figure 2, we plot the TKw density function for some parameter values.Figure 2
Plots of the TKw density function for some parameter values.
(a)
a
=
2, b=5
(b)
a
=
5, b=2Therth moment of TKw can be obtained from (13) as (33)EXr=1-λbB1+ra,b+2λbB1+ra,2b,r>-a,and the qf is given by (34)F-1u;a,b,λ=1-1-1+λ-1+λ2-4λu2λ1/b1/a,λ≠0.


## 8. Extreme Values

IfX¯=n-1(X1+X2+⋯+Xn) denotes the sample mean from iid random variables following (2), then by standard central limit theorem n(X¯-E(X))/Var(X) converges in distribution to the standard normal as n→∞ under suitable conditions. However, one might be interested in the asymptotics of the extreme values Xmax=max⁡(X1,…,Xn) and Xmin=min⁡(X1,…,Xn). We consider the following:(i)
Suppose thatG belongs to the max. domain of attraction of the Gumbel extreme value distribution. Then, by Leadbetter et al. [20], there must be a strictly positive function, say h(t), such that (35)limt→∞⁡1-Gt+xht1-Gt=exp⁡-x,for everyx∈(-∞,∞). In our case, we have (36)limt→∞⁡1-Ft+xht1-Ft=limt→∞⁡1-Gt+xht;ξ1-λGt+xht;ξ1-Gt;ξ1-λGt;ξ=limt→∞⁡1-Gt+xht;ξ1-Gt;ξlimt→∞⁡1-λGt;ξ1-λGt;ξ=λexp⁡-x,for everyx∈(-∞,∞). Hence, it follows by Leadbetter et al. [20] that F also belongs to the max. domain of attraction of the Gumbel extreme value distribution with (37)limn→∞⁡PanXmax-bn≤x=exp⁡-λexp⁡-xfor some suitable norming constantsan>0 and bn.(ii)
Again, suppose thatG belongs to the max. domain of attraction of the Fréchet extreme value distribution. By Leadbetter et al. [20], there must exist a β>0, such that (38)limt→∞⁡1-Gtx1-Gt=xβ,for everyx>0. In our case, (39)limt→∞⁡1-Ftx1-Ft=limt→∞⁡1-Gtx;ξ1-Gt;ξlimt→∞⁡1-λGtx;ξ1-λGt;ξ=λxβ,for everyx>0. Hence, it follows by Leadbetter et al. [20] that F also belongs to the max. domain attraction of the Fréchet extreme value distribution with (40)limn→∞⁡PanXmax-bn≤x=exp⁡-λxβfor some suitable norming constantsan>0 and bn.(iii)
Also, suppose thatG belongs to the max. domain of attraction of the Weibull extreme value distribution. By Leadbetter et al. [20], there must be an α>0 such that (41)limt→0⁡1-Gt+xht1-Gt=xα,for everyx<0. In our case, we have(42)limt→0⁡FtxFt=limt→0⁡1+λGtx;ξ-λG2tx;ξ1+λGt;ξ-λG2t;ξ=xα+1,for everyx<0. Hence, it follows by Leadbetter et al. [20, Chapter  1] that F also belongs to the max. domain of attraction of the Weibull extreme value distribution with (43)limn→∞⁡PanXmax-bn≤x=exp⁡--xα+1for some suitable norming constantsan>0 and bn.Similar arguments apply to min. domains of attraction. That is,F belongs to the same min. domain of attraction as that of G.


## 9. Probability Weighted Moments

The PWMs of a baseline model can be very useful to determine the moments of more complex distributions. Distributions that can be expressed in inverse form may present problems in estimating their parameters as functions of ordinary moments. For these distributions, the relations between the PWMs and the parameters have simpler analytical structure than those between the ordinary moments and the parameters. The PWMs are also widely used for estimating parameters of distributions from complete or censored samples.We demonstrate that the(n,s)th PWM of X, say δn,s=E{XnF(X;ξ,λ)s} (for n,s=0,1,…), can be expressed as linear combinations of the baseline PWMs defined by τn,s=∫-∞∞‍xnG(X;ξ)sdx. First, we can write τn,s from (20) and (23) by interchanging the sum and the integral(44)τn,s=∑i=0∞cn,ii+s+1.Further, we have(45)δn,r=∫-∞+∞xn1+λGx;ξ-λGx;ξ2dx,and then using the binomial expansion, we can express the PWMs of X as (46)δn,r=∑j=0r-λj1+λr-jrj1+λτn,r-j-2λτn,r+1-j,where τn,r-j and τn,r+1-j are obtained from (44).


## 10. Order Statistics

Order statistics are required in many fields, such as climatology, engineering, and industry. Further, they play an important role in Statistical Inference and Nonparametric Statistics. In this section, we present some results with respect to the order statistics. We obtain an expression for the density of therth order statistic and the large sample distribution of the minimum and maximum when a random sample of size n is drawn from the TG family. The density function of the rth order statistic, say Xr:n, from a random sample of size n drawn from (5) is given by (47)fXr:nx=1Br,n-r+1Fxr-11-Fxn-rfx=1Br,n-r+1∑j=0n-r-1jn-rj1+λGx;ξ-λGx;ξ2r-1+j1+λ-2λGx;ξgx;ξ.Thekth order moment of Xr:n is obtained from (20) as (48)EXr:nk=1Br,n-r+1∑j=0n-r-1jn-rj∫0∞xk1+λGx;ξ-λGx;ξ2r+j-11+λ-2λGx;ξgx;ξdx=1Br,n-r+1∑j=0n-r-1jn-rjJ,where J=∫01‍G-1(t;ξ)1+λ-2λt(1+λ)t-λt2r+j-1dt can be evaluated numerically for most parent distributions using statistical software.


## 11. Information Theory

### 11.1. Entropies

An entropy is a measure of variation or uncertainty of a random variableX. Two well-known entropy measures are the Rényi and Shannon entropies. The Rényi entropy of a random variable X with pdf f(x) is defined by (49)IRρ=11-ρlog⁡∫0∞fρxdx,for ρ>1 and ρ≠1. The Rényi entropy for the TG family should be evaluated numerically.The Shannon entropy of a random variableX is defined by E-log⁡[f(X)]. It is the special case of the Rényi entropy when ρ→1. For the proposed model in (1), the Shannon entropy reduces to(50)E-log⁡fX=-Elog⁡gX;ξ-Elog1+λ-2λGX;ξ=-I1+I2,where (51)I1=EloggX;ξ=∫0∞loggx;ξ1+λ-2λGx;ξ⁡gx;ξdx=12λ∫1-λ1+λtlog⁡gG-11+λ-t2λ;ξdt=A1λ;ξ,I2=Elog⁡1+λ-2λGX;ξ=121+λ2log⁡1+λ-λ-12log⁡1-λ-2λ.By substituting the last two expressions in (50), the Shannon entropy becomes (52)E-logfX=-A1λ;ξ-121+λ2log⁡1+λ-λ-12log⁡1-λ-2λ.Next, we consider the Rényi entropy. From (1), we have (53)∫0∞fρxdx=12λ∫1-λ1+λtρ+1gG-11+λ-t2λ;ξρdt=A2λ;ξ,which can be computed numerically.


### 11.2. Kullback-Leibler (KL) Divergence

Consider two distributions from the same family (but with different parametric configuration). To be more specific, letP(x)~TG(λ1,ξ1) and Q(x)~TG(λ2,ξ2). Then, the KL divergence measure between P(x) and Q(x), say KLP∣Q, is given by (54)KLP∣Q=∫Pxlog⁡PxQxdx=∫Pxlog⁡1+λ1-2λ1Gx;ξ1gx;ξ11+λ2-2λ2Gx;ξ2gx;ξ2dx=∫Pxlog⁡1+λ1-2λ1Gx;ξ1dx+∫Pxlog⁡gx;ξ1dx-∫Pxlog⁡1+λ2-2λ2Gx;ξ2dx-∫Pxlog⁡gx;ξ2dx=J1+J2-J3-J4,where (55)J1=∫Pxlog⁡1+λ1-2λ1Gx;ξ1dx=12-2λ1-λ1-12log⁡1-λ1+1+λ12log⁡1+λ1,J2=A1λ;ξ,J3=12λ1∫1-λ11+λ1tlog⁡1+λ2-2λ2GG-11+λ1-t2λ1;ξ2;ξ1dt.Special Case. If ξ1=ξ2, then J3 reduces to (56)A3λ;ξ=∫1-λ11+λ1tlog⁡1+λ2-2λ2GG-11+λ1-t2λ1;ξ2dt=4λ12λ22-1-λ1λ21+λ1-2+-2+3λ1+3λ12λ2-4λ12λ22-1λ1λ2λ1-1-2+-2+3λ1+λ12λ2+24λ12λ22-11+λ2-1+-1+2λ1+2λ12λ2log⁡1+λ2-21-21+λ1+λ12λ2+λ221-2λ1-2λ12+4λ13log⁡1+λ2-2λ12λ2.Thus,J3=(1/2λ1)A3(λ;ξ). Also, in this case, J2=J4. Hence, the KL divergence is given by (57)KLP∣Q=12-2λ1-λ1-12log⁡1-λ1+1+λ12log⁡1+λ1-2A1λ;ξ-12λ1A3λ;ξ.


## 11.1. Entropies

An entropy is a measure of variation or uncertainty of a random variableX. Two well-known entropy measures are the Rényi and Shannon entropies. The Rényi entropy of a random variable X with pdf f(x) is defined by (49)IRρ=11-ρlog⁡∫0∞fρxdx,for ρ>1 and ρ≠1. The Rényi entropy for the TG family should be evaluated numerically.The Shannon entropy of a random variableX is defined by E-log⁡[f(X)]. It is the special case of the Rényi entropy when ρ→1. For the proposed model in (1), the Shannon entropy reduces to(50)E-log⁡fX=-Elog⁡gX;ξ-Elog1+λ-2λGX;ξ=-I1+I2,where (51)I1=EloggX;ξ=∫0∞loggx;ξ1+λ-2λGx;ξ⁡gx;ξdx=12λ∫1-λ1+λtlog⁡gG-11+λ-t2λ;ξdt=A1λ;ξ,I2=Elog⁡1+λ-2λGX;ξ=121+λ2log⁡1+λ-λ-12log⁡1-λ-2λ.By substituting the last two expressions in (50), the Shannon entropy becomes (52)E-logfX=-A1λ;ξ-121+λ2log⁡1+λ-λ-12log⁡1-λ-2λ.Next, we consider the Rényi entropy. From (1), we have (53)∫0∞fρxdx=12λ∫1-λ1+λtρ+1gG-11+λ-t2λ;ξρdt=A2λ;ξ,which can be computed numerically.


## 11.2. Kullback-Leibler (KL) Divergence

Consider two distributions from the same family (but with different parametric configuration). To be more specific, letP(x)~TG(λ1,ξ1) and Q(x)~TG(λ2,ξ2). Then, the KL divergence measure between P(x) and Q(x), say KLP∣Q, is given by (54)KLP∣Q=∫Pxlog⁡PxQxdx=∫Pxlog⁡1+λ1-2λ1Gx;ξ1gx;ξ11+λ2-2λ2Gx;ξ2gx;ξ2dx=∫Pxlog⁡1+λ1-2λ1Gx;ξ1dx+∫Pxlog⁡gx;ξ1dx-∫Pxlog⁡1+λ2-2λ2Gx;ξ2dx-∫Pxlog⁡gx;ξ2dx=J1+J2-J3-J4,where (55)J1=∫Pxlog⁡1+λ1-2λ1Gx;ξ1dx=12-2λ1-λ1-12log⁡1-λ1+1+λ12log⁡1+λ1,J2=A1λ;ξ,J3=12λ1∫1-λ11+λ1tlog⁡1+λ2-2λ2GG-11+λ1-t2λ1;ξ2;ξ1dt.Special Case. If ξ1=ξ2, then J3 reduces to (56)A3λ;ξ=∫1-λ11+λ1tlog⁡1+λ2-2λ2GG-11+λ1-t2λ1;ξ2dt=4λ12λ22-1-λ1λ21+λ1-2+-2+3λ1+3λ12λ2-4λ12λ22-1λ1λ2λ1-1-2+-2+3λ1+λ12λ2+24λ12λ22-11+λ2-1+-1+2λ1+2λ12λ2log⁡1+λ2-21-21+λ1+λ12λ2+λ221-2λ1-2λ12+4λ13log⁡1+λ2-2λ12λ2.Thus,J3=(1/2λ1)A3(λ;ξ). Also, in this case, J2=J4. Hence, the KL divergence is given by (57)KLP∣Q=12-2λ1-λ1-12log⁡1-λ1+1+λ12log⁡1+λ1-2A1λ;ξ-12λ1A3λ;ξ.


## 12. Bivariate and Multivariate Generalization

First, we consider a bivariate extension of the new model. The joint cdf is expressed as(58)Fx,y;ξ,λ=1+λGx,y;ξ-λGx,y;ξ2,x,y∈D⊆R,λ∈-1,1,where G(x,y;ξ) is a bivariate continuous distribution with marginal cdfs G1(x;ξ) and G2(x;ξ).The corresponding joint pdf is given by(59)fx,y=∂2∂x∂yFx,y=λ+Gx,y∂2∂x∂yGx,y-2λ∂∂xGx,y∂∂yGx,y.The marginal cdfs are given by(i)
F(x)=(1+λ)G1(x)-λG1(x)2,(ii)
F(y)=(1+λ)G2(y)-λG2(y)2.The marginal pdfs are given by(i)
f(x)=(1+λ)g1(x)-2λG1(x)g1(x),(ii)
f(y)=(1+λ)g2(y)-2λG2(y)g2(y).A natural multivariate extension (say,m-variate, m≥2) of the above bivariate (for λ∈[-1,1]) is given by (60)Fx1,…,xm;ξ,λ=1+λGx1,…,xm;ξ-λGx1,…,xm;ξ2,x1,…,xm∈D⊆R,where G(x1,…,xm;ξ) is a multivariate continuous distribution with marginal cdfs Gi(xi;ξ), for i=1,2,…,m.


## 13. Maximum Likelihood Estimation

Several methods for parameter estimation have been proposed in the literature but the maximum likelihood method is the most commonly employed. The maximum likelihood estimators (MLEs) enjoy desirable properties and can be used to obtain confidence intervals for the model parameters. In this section, we consider the estimation of the unknown parameters of theTG family from complete samples only by maximum likelihood. Let x1,…,xn be observed values from this family with parameters λ and ξ.Letθ=(λ,ξ⊤)⊤ be the p×1 parameter vector. The total log-likelihood function for θ is given by (61)lθ=∑i=1nlog⁡gxi;ξ+∑i=1nlog⁡1+λ-2λGxi;ξ.The components of the score functionU(θ)=Uλ,Uξ⊤ are (for k=1,…,p) (62)Uλ=-2∑i=1nGxi;ξ1+λ-2λGxi;ξ,Uξk=∑i=1n∂gxi;ξ/∂ξkgxi;ξ-2λ∑i=1n∂Gxi;ξ/∂ξk1+λ-2λGxi;ξ.We can obtain the MLEθ^=(λ^,ξ^⊤)⊤ of θ=(λ,ξ⊤)⊤ by maximizing the log-likelihood either directly by using the R (optim function), SAS (PROC NLMIXED), and Ox (subroutine MaxBFGS) programs or by setting Uλ and Uξ equal to zero and solving the equations simultaneously. These equations can be solved numerically using iterative methods such as the Newton-Raphson type algorithms.For interval estimation on the model parameters, we require the observed information matrix(63)Jθ=-Uλλ∣Uλξ⊤------Uλξ∣Uξξ,whose elements are (64)Uλλ=-2∑i=1nGxi;ξ21+λ-2λGxi;ξ2,Uλξk=-21+λ∑i=1n∂Gxi;ξ/∂ξk1+λ-2λGxi;ξ2,Uξkξl=∑i=1ngkl′′xi;ξgxi;ξ-gk′xi;ξgl′xi;ξg2xi;ξ-2λ∑i=1nGkl′′xi;ξ1+λ-2λGxi;ξ-4λ∑i=1nGk′xi;ξGl′xi;ξ1+λ-2λGxi;ξ2,where tk′(·;ξ)=∂t(·;ξ)/∂ξk and tkl′′(·;ξ)=∂2t(·;ξ)/∂ξk∂ξl.We can easily check if the fit using theTG model is statistically “superior” to a fit using the G model by testing the null hypothesis H0:λ=0 against H1:λ≠0. For testing H0:λ=0, the likelihood ratio (LR) statistic is given by w=2{l(λ^,ξ^)-l(0,ξ~)}, where λ^ and ξ^ and ξ~ are the unrestricted and restricted estimates obtained from the maximization of l(θ) under H1 and H0, respectively. The limiting distribution of this statistic is χ12 under the null hypothesis. The null hypothesis is rejected if the test statistic exceeds the upper 100(1-γ)% quantile of the χ12 distribution.


## 14. An Application

In this section, we compare the results of fitting the TKw and Kw distributions to a real data set. We estimate the unknown parameters of the models by the maximum likelihood (as discussed in Section13) and all the computations are performed using the NLMixed subroutine of the SAS software. The data are from a study on anxiety performed in a group of 166 “normal” women, that is, outside of a pathological clinical picture (Townsville, Queensland, Australia). The data originally reported by Smithson and Verkuilen [21] are as follows: 0.01, 0.17, 0.01, 0.05, 0.09, 0.41, 0.05, 0.01, 0.13, 0.01, 0.05, 0.17, 0.01, 0.09, 0.01, 0.05, 0.09, 0.09, 0.05, 0.01, 0.01, 0.01, 0.29, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.09, 0.37, 0.05, 0.01, 0.05, 0.29, 0.09, 0.01, 0.25, 0.01, 0.09, 0.01, 0.05, 0.21, 0.01, 0.01, 0.01, 0.13, 0.17, 0.37, 0.01, 0.01, 0.09, 0.57, 0.01, 0.01, 0.13, 0.05, 0.01, 0.01, 0.01, 0.01, 0.09, 0.13, 0.01, 0.01, 0.09, 0.09, 0.37, 0.01, 0.05, 0.01, 0.01, 0.13, 0.01, 0.57, 0.01, 0.01, 0.09, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.05, 0.01, 0.01, 0.01, 0.13, 0.01, 0.25, 0.01, 0.01, 0.09, 0.13, 0.01, 0.01, 0.05, 0.13, 0.01, 0.09, 0.01, 0.05, 0.01, 0.05, 0.01, 0.09, 0.01, 0.37, 0.25, 0.05, 0.05, 0.25, 0.05, 0.05, 0.01, 0.05, 0.01, 0.01, 0.01, 0.17, 0.29, 0.57, 0.01, 0.05, 0.01, 0.09, 0.01, 0.09, 0.49, 0.45, 0.01, 0.01, 0.01, 0.05, 0.01, 0.17, 0.01, 0.13, 0.01, 0.21, 0.13, 0.01, 0.01, 0.17, 0.01, 0.01, 0.21, 0.13, 0.69, 0.25, 0.01, 0.01, 0.09, 0.13, 0.01, 0.05, 0.01, 0.01, 0.29, 0.25, 0.49, 0.01, and 0.01.Table3 lists some descriptive statistics of these data. Table 4 gives the MLEs (with corresponding standard errors in parentheses) and some goodness of fit measures for the fitted TKw and Kw distributions. Since the values of the Akaike information criterion (AIC), Bayesian information criterion (BIC), and consistent Akaike information criterion (CAIC) are smaller for the TKw distribution compared to those values of the Kw model, the new distribution is a very competitive model to these data.Table 3
Descriptive statistics for the data set.
n
Min.
Q
1
Q
2
Mean
Q
3
Max.
Var.
166
0.0100
0.0100
0.0300
0.0912
0.1300
0.6900
0.0176Table 4
Estimates of the parameters (standard errors in parentheses), goodness-of-fit statistics, and confidence intervals with significance level at5% for the data set.
Model
Estimates
Confidence intervals
AIC
BIC
CAIC
TKw
a
^ = 0.7037 (0.0483) b^ = 3.8089 (0.6890) λ^ = 0.5680 (0.1784)
(0.6083, 0.7991) (2.4485, 5.1693)  (0.2159, 0.9202)
−489.8
−480.5
−489.7
Kw
a
^ = 0.6425 (0.0457) b^ = 4.4733 (0.5827)
(0.5523, 0.7326) (3.3228, 5.6239)
−484.6
−478.4
−484.6In order to test the null hypothesisH0:λ=0 against the alternative hypothesis H1:λ≠0, we obtain a confidence interval (with 5% significance level). The 95% confidence interval for λ is (0.2159,0.9202) and since it does not contain zero, we can reject the null hypothesis in favor of the alternative hypothesis that the data set is generated from a TKw model with λ≠0. Further, the LR statistic (see Section 13) to test H0:Kw against H1:TKw is 7.2 (p value < 0.01). Thus, using any usual significance level, we reject the null hypothesis in favor of the TKw distribution, that is, this distribution is significantly better than the Kw distribution to explain the current data.Plots of the pdf and cdf of the fitted TKw and Kw models to these data are displayed in Figure3. They indicate that the TKw model is superior to the Kw model in terms of model fitting.Figure 3
Estimated pdf and cdf from the fitted TKw and Kw models for the data.


## 15. Conclusions

In this paper, we explore general properties of the transmuted-G family of distributions. This family is obtained by adding shape parameters to an existing well-known distribution by using the transmutation map approach [1]. We investigate some of its general mathematical properties including ordinary and incomplete moments, generating function, probability weighted moments, Shannon and Rényi entropies, Kullback-Leibler divergence, and limiting behavior of the extremum values. The existing literature does not include these general properties. We also provide a real life application to prove empirically the usefulness of the transmuted-G family. The application indicates that the transmuted model performs better compared to the parent model. Consequently, it merits further a thorough study in terms of its extension to a bivariate and subsequently multivariate set-up (we have discussed briefly this topic in Section 12). Needless to say, inferential procedures, especially under the Bayesian paradigm, will not be an easy task. We plan to work on it as a future project and will report our findings elsewhere.



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