Universidade Federal de Santa Maria
Ci. e Nat., Santa Maria, v.42, e111, 2020
DOI:10.5902/2179460X39914
ISSN 2179460X
Received: 09/09/19 Accepted: 16/04/20 Published: 23/12/20
Statistics
Classe ZetaG: Alguns Aspectos Computacionais e Analíticos
The ZetaG Class: Some Computational and Analytical Aspects
Ana Carla Percontini ^{I}
Frank GomesSilva ^{II}
Gauss Moutinho Cordeiro ^{III}
Pedro Rafael Marinho ^{IV}
^{I }Universidade Estadual de Feira de Santana, Feira de Santana, BA  anappaixao@gmail.com
^{II }Universidade Federal Rural de Pernambuco, Recife, PE  franksinatrags@gmail.com
^{III} Universidade Federal de Pernambuco, Recife, PE  gausscordeiro@gmail.com
^{III} Universidade Federal da Paraíba, João Pessoa, PB  pedro.rafael.marinho@gmail.com
ABSTRACT
We define a new class of distributions with one extra shape parameter including some special cases. We provide numerical and computational aspects of the new class. We propose functions using the R language to fit any distribution in this family to a data set. In addition, such functions are implemented efficiently using the library Rcpp that enables the incorporation of the codes C++ in R automatically. Some examples are presented for using the implemented routines in practice. We derive some mathematical properties of this class including explicit expressions for the moments, generating function and mean deviations. We discuss the estimation of the model parameters by maximum likelihood and provide an application to a real data set.
Keywords: Computational aspects, Generating function, Mean deviations; Moment, Zeta distribution
RESUMO
Definimos uma nova classe de distribuições com um parâmetro de forma extra incluindo alguns casos especiais. Estabelecemos aspectos numéricos e computacionais da nova classe. Propomos funções usando a linguagem R para ajustar qualquer distribuição nesta família a conjuntos de dados. Tais funções são eficientemente implementadas utilizando a biblioteca Rcpp que permite incorporação automática de códigos C++ em R. Alguns exemplos são apresentados para o uso das rotinas implementadas na prática. Derivamos algumas propriedades matemáticas dessa classe, incluindo expressões explícitas para os momentos, função geradora e desvios médios. Discutimos a estimação dos parâmetros do modelo por máxima verossimilhança e fornecemos uma aplicação a dados reais.
Palavraschave: Aspectos computacionais, Função geradora, Desvios médios, Momentos, Distribuição Zeta
1 INTRODUCTION
Recently, new families have been proposed by compounding any continuous baseline G distribution with a discrete distribution supported on integers n ≥ 1. By this method, we can obtain new classes with additional parameters to govern skewness and generate densities with heavier or ligther tails. These parameters are sought as a manner to furnish a more flexible distribution for modeling the hazard rate function (hrf). Another important method for generating continuous distributions was proposed by Alzaatreh et al. (2013). Accordingly, several new distributions have been published as, for example, an extended Weibull distribution (Cordeiro e Lemonte, 2013) that includes the Weibull as a special case and gives more flexibility to model various types of data.
We propose a general family of continuous distributions called the ZetaG class with one additional shape parameter. The ZetaG is a generated class from the Riemann Zeta distribution studied by Lin and Hu (2001). They proved that the Riemann Zeta random variable can be represented by a linear function of infinitely many independent geometric random variables. Gupta et al. (2008) studied the Hurwitz Lerch Zeta (HLZ) distribution and investigated its structural properties. In recent years, the HLZ distribution and its variants have been studied by various authors including Vilaplana (1988), Zörnig e Altmann (1995), Doray e Luong (1995), Doray e Luong (1997) and Gut (2005). Zörnig e Altmann (1995) have shown that many wellknown discrete distribution are special cases of the HLZ family. The Riemman Zeta distribution is one of them, which has been used for describing ranking problems in linguistics.
The ZetaG class can generate new distributions from specified baseline distributions. We demonstrate that the ZetaG class density is a linear combination of exponentiatedG (“expG” for short) density functions.
Let W_{1}, ···, W_{Z} be a random sample from a continuous cumulative distribution function (cdf) G(·) with positive support, where Z is an unknown positive integer number. We assume that the random variable Z has a Zeta probability mass function (pmf)


where ζ(s) is the Riemann Zeta function. All real zeros of the function are on the negative real axis, located in points ζ(2s) = 0, s = 1,2, .... In the particular case of the origin, we have ζ(0) = − 1/2. The Riemann Zeta function is undefined for s = 1 and ζ(s) < 0 for s ∈ (0,1). Further, ζ(s) > 1 for s > 1 and ζ(s) → 1 when s → ∞. For details and other analytic properties of the Riemann Zeta function, see Lin e Hu (2001) and Gut (2006).
Let Z and W_{i}’s be independent random variables and X = min{W_{1}, ···, W_{Z}}. Then, the conditional cdf of X given Z is


The unconditional cdf of X has the form (for x > 0)


where s > 1 is a shape parameter. After some algebra, the cdf of X can be expressed as

(1) 
where Li_{s}(x) is the polylogarithm function [K. Oldham e Spainer (2009), Section 25:12] given by the power series

(2) 
and z < 1. Equation (1) defines the cdf of the ZetaG class.
The polylogarithm function can be represented by more general functions, for example, using the generalized hypergeometric function, the Lerch transcendent function and the Meijer Gfunction given in Wolfram website (http://functions.wolfram.com/10.08.26.0008.01  Accessed 13/06/2018.).
We provide two motivations for the ZetaG class. Let Z have a Zeta distribution. First, suppose the failure of a device occurs due to the presence of an unknown number Z of initial defects of same kind, which can be identifiable only after causing failure and are repaired perfectly. Define by W_{i} the time to the failure of the device due to the ith defect, for i ≥ 1. Under the assumptions that the W_{i}’s are independent and identically distributed (iid) random variables with cdf G(x) independent of Z, equation (1) is appropriate for modeling the time to the first failure. Secondly, suppose that an individual in the population is susceptible to a certain type of cancer. Let Z be the number of carcinogenic cells for that individual left active after the initial treatment and denote by W_{i} the time spent for the ith carcinogenic cell to produce a detectable cancer mass, for i ≥ 1. Under the assumptions that {W_{i}}_{i}_{≥1} is a sequence of iid random variables independent of Z having the cdf G(x), where Z has a Zeta distribution, the time to relapse of cancer of a susceptible individual is defined by X = min {Wi}^{Z}_{i}_{=1}, which follows (1).
The probability density function (pdf) corresponding to (1) is

(3) 
where g(x) = dG(x)/dx. We can verify using Mathematica that . The density function (3) will be most tractable when G(x) and g(x) have simple analytic expressions.
A positive point of the ZetaG model is that it includes as a special case the G distribution when s → ∞. Hereafter, a random variable X having density (3) is denoted by X ∼ ZetaG(τ ,s), where τ is the parameter vector associated with G. The survival function and hrf of X are given by


and


respectively.
The rest of the paper is organized as follows. In Section 2, we present some new distributions in the ZetaG class. Some of its numerical and computational aspects are addressed in Section 3. We obtain a useful linear representation for its density, explicit expressions for the ordinary and incomplete moments, moment generating function (mgf) and mean deviations in Sections 4 to 7. The estimation of the model parameters using the method of maximum likelihood is presented in Section 8. An application to a real data set is performed in Section 9. Finally, some conclusions are offered in Section 10.
2 SPECIAL ZETAG DISTRIBUTIONS
The ZetaG class of density functions (3) allows for greater flexibility of its tails and can be widely applied in many areas of engineering and biology. This class extends several widelyknown distributions in the literature. Next, we present four special cases.
2.1 ZetaWeibull (ZW) distribution
If G(x) is the Weibull cdf with scale parameter β > 0 and shape parameter α > 0, say G(x) = 1 − exp(−βx^{α} ), the pdf (for x > 0) and cdf of the ZW distribution are, respectively,


Figure 1 displays some possible shapes of the ZW density function.
2.2 ZetaFréchet (ZFr) distribution
Consider the Fréchet distribution (for x,σ,λ > 0) with cdf and pdf G(x) = e^{−(σ/x) λ} and g(x) = λ σ^{λ} x^{−λ−1} e^{−(σ/x) λ} , respectively. The pdf and cdf of the ZFr distribution, for x > 0, are

(4) 
Figura 1  The ZW density functions for: (a) s = 3 and β = 1; (b) s = 5 and α = 0.7
(a) 
(b) 
and


respectively, where σ > 0 is scale parameter and λ > 0 is a shape parameter. Plots of (4) for selected parameter values are displayed in Figure 2.
Figura 2  The ZFr density function for: (a) s = 2 and σ = 0.5; (b) s = 2 and λ = 1.5
(a) 
(b) 
2.3 ZetaFréchet (ZFr) distribution
Let G(x) be the Burr XII distribution with cdf G(x) = 1 − (1 + x^{c})^{−k} and pdf g(x) = ckx^{c}^{−1}/(1 + x^{c})^{k}+1, where c > 0 is a shape parameter and k > 0 is a scale parameter. The pdf and cdf of the ZBXII distribution, for x > 0, are

(5) 
and


respectively. Plots of (5) for some parameter values are displayed in Figure (3).
Figura 3  The ZBXII density function for some parameter values: (a) s = 3 and k = 0.5; (b) s = 3 and c = 2
(a) 
(b) 
2.4 ZetaLomax (ZLo) distribution
The pdf and cdf of the Lomax distribution are (for x ≥ 0 and α, λ > 0) g(x) = α/λ (1 + x/λ)^{−α−1} and G(x) = 1 − (1 + x/λ)^{−α}, respectively. The pdf and cdf of the ZLo distribution, for x ≥ 0, are


and


respectively. Some plots of the ZLo density function are displayed in Figure 4.
Figura 4  The ZLo density function for: (a) s = 3 and α = 1.5; (b) s = 3 and λ = 1
(a) 
(b) 
3 NUMERICAL AND COMPUTATIONAL ASPECTS
The use and acceptance of a family of distributions is closely related to the ease of use and implementation of their particular models. Many of these families have an analytical approach in closedform but can require the use of infinite sums and numerical approximations which are tiring to be implemented in a computationally efficient form. We facilitate the implementation of this family using some simple functions
The functions are implemented using the language C++ and can easily be invoked by the language R. The programmer of R does not need to spend time understanding the C++ code nor even configuring the R language to compile the C++ code. For configuration, the programer of R only need to run the config_cpp(dir) defined over the front. This function is responsible for installing the necessary dependencies for compiling and linking of C++ code with the R language, in which the argument dir is the directory in which the user will save the C++ code.
Once the ZetaG family involves the functions Li_{S}(x) and ζ(s), polylogarithm and Rimman Zeta functions, we should obtain a numerical approximation considering a large number of sums, which can be computationally intensive depending on the problem in which these functions are applied. This fact justifies writing these functions in a computationally efficient language such as the case of the C++ language. The steps for communicating the C++ code with R are summarized in three simple steps described below. Soon after, a diagram helps in the explanation.
1. Create, in some directory, the fast.cpp file and save it with the C++ Code 1 (Appendix A). This is the file with C++ code that will be compiled in R;
2. Run, in R, the code of the function of the name config_cpp (see Code 2, Appendix B);
3. After the previous step, run config_cpp (dir = "path"), where “path” is the path of the directory the user saved the file fast.cpp in step 1.
The Code 1 refers to C++ code that should be saved in a file named fast.cpp. The user must save this file into a directory of free choice. The code makes use of the Rcpp library which provides a clean API that allows you to write highperformance R code using C++. The code presents the derivationcpp(), polylogcpp() and riemann_zetacpp() functions that are used to obtain numerical derivatives and approximation of Riemann’s polylogarithm and zeta functions, respectively.
Appendix B lists the R code for the config_cpp() function that is responsible for compiling the C++ code, making the functions implemented in fast.cpp available in R. The user should pass as argument to the config_cpp() function the path of the directory containing the file fast.cpp. If no argument is passed, the function assumes that the fast.cpp file is in the default working directory that R considers in the system. To find out what this path is, open a section of R and run getwd(), that is, if the dir parameter of the config_cpp() function is not given, the fast.cpp code should be saved in the path given by getwd().
After the previous steps, the user can, finally, make use of the functions cdf_zeta() and pdf_zeta() implemented in R. These functions use the compiled functions of the C++. Given any G function (baseline cdf), the function cdf_zeta(G) provides the ZetaG cdf. Consider the example below:
Example (Implementing the ZW cdf): At the end, the ZW cdf (α = 0.5, β = 0.5, s = 2) is evaluated at x = 0 and x = 10000, returning 0 and 1, respectively.
# Zeta−G class of distributions.
cdf_zeta <− function (G){
# Using the concept of closures.
function (par , x ){
s <− tail (par , n = 1)
stopifnot (s > 1 )
zeta_s <− riemann_zetacpp(s)
w <− polylogcpp (z = 1 − G(par = par[−length (par)], x), s = s )
return ((zeta_s − w) / zeta_s)
}
}
# Weibull distribution.
cdf_weibull <− function (par , x){
alpha <− par[1]
beta <− par[2]
1 − exp (−(x/beta)^alpha)
}
# Zeta−Weibull distribution.
cdf_zetag_weibull <− cdf_zeta (G = cdf_weibull)
cdf_zetag_weibull (par = c(0.5, 0.5, 2), 10)
#> 0
cdf_zetag_weibull (par = c(0.5, 0.5, 2), 1e4)
#> 1
It is important to note that making use of the cdf_zeta() function, the user simply need to implement the G function. In the example above, it is only need to implement the G cdf. The way it was implemented in cdf_weibull distribution(), other G cdfs could be implemented to generate new ZetaG distributions. It is also important to note that the parameter s will always be the last of the vector par of the function obtained by cdf_zeta(). In this example, cdf_zetag_weibull(par = c(0.5,0.5,2), x = 1e4) we set s = 2. Considering what was exemplified above, we can easily obtain the ZW density, as it can seen in the next example.
Example (Obtaining the ZW density by means of the pdf_zeta() function): Note that we take as an argument to pdf_zeta() the ZW cdf obtained in the previous example. At the end of the example, note that the integral of the ZW density (α = 0.5, β = 0.5, s = 2) is one.
pdf_zeta <− function(cdf_zeta){
# Using the concept of closures.
function (par , x){
derivationcpp(cdf_zeta , par = par , x)
}
}
pdf_zetag_weibull <− pdf_zeta(cdf_zeta = cdf_zetag_weibull)
integrate (f = pdf_zetag_weibull , par = c(0.5, 0.5, 2), lower = 0,
upper = Inf)
#> 1
The cdf_zeta() and pdf_zeta() functions are interesting, since they create new functions (cdf and pdf of the ZetaG model) in a layout accepted by the AdequacyModel package, version 2.0.0, developed by Diniz Marinho et al. (2016). This package is widely used in the area of distributions to obtain goodnessoffits statistics, being one of the most cited packages in the literature for this purpose. Details regarding the AdequacyModel package can be seen in https://cran.rproject.org/package=AdequacyModel. Automatically, we obtain cdfs and pdfs of ZetaG distributions in the forms accepted by the AdequacyModel package, which facilitates the implementation of distributions in the proposed class.
The cdf_zeta() and pdf_zeta() functions make use of what is known in the computation of lambda functions, which basically refer to the anonymous functions that can be applied for various purposes. These are used in the concept of programming known as “closures”. Closure refers to any function that closes in the environment in which it was defined, being able to access variables that are not in its parameter list. The introduction of these concepts into the cdf_zeta() and pdf_zeta() functions is what allows them to construct new functions by passing G and the ZetaG cdfs, respectively, in the form they are accepted by the AdequacyModel package. It is a metaprogramming technique that can be very well explored in this class of problems.
4 USEFUL REPRESENTATIONS
Some useful expansions for (1) and (3) can be derived using the concept of exponentiated distributions. For an arbitrary baseline cdf G(x), a random variable is said to have the exponentiatedG (“expG”) distribution with parameter r > 0, say Yr ∼ expG(r), if its pdf and cdf are


respectively. The properties of exponentiated distributions have been studied by several authors in recent years. See Nadarajah et al. (2013) for exponentiated Weibull, Kundu e Gupta (1999) for exponentiated exponential, Nadarajah e Kotz (2006) for exponentiated Fréchet and Nadarajah e Gupta (2007) for exponentiated gamma distributions.
Using expansion (2), we can write (3) as


Expanding the binomial term in this equation, we have

(6) 
where h_{r}(x) denotes the expG(r) density function and (for r = 1,2, . . .)


We can prove, for example, using Mathematica that . Equation (6) reveals that the ZetaG density function is a linear combination of expG densities. So, several mathematical properties of the ZetaG class can be obtained by knowing those of the expG distribution, see, for example, Nadarajah et al. (2013), Nadarajah e Kotz (2006), among others.
By integrating (6), we can express F(x) as


where H_{r}(x) denotes the expG(r) cdf.
5 MOMENTS
A first formula for the nth moment of X, say µ’_{n} = E(X_{n}), can be obtained from (6) (let Yr ∼ expG(r) for r ≥ 1) as

(7) 
Expressions for moments of some exponentiated distributions are given by Nadarajah e Kotz (2006), which can be used to obtain E(X_{n}). We now provide an application of (7) by taking the baseline Weibull introduced in Section 2.1. The expWeibull density with power parameter r is h_{r}(x) = r α β x^{α}^{−1} e^{−βxα} (1 − e^{−βxα})^{r−1}, which gives the nth moment of the ZW distribution


Plots of the skewness and kurtosis of the ZW distribution for some choices of α and β as functions of s are displayed in Figure 5.
A second formula for µ’_{n} can be derived from (7) in terms of the baseline quantile function Q_{G}(u) = G^{−1} (u)

(8) 
where

(9) 
The ordinary moments of several ZetaG distributions can be determined directly from equations (8) and (9).
For empirical purposes, the shapes of many distributions can be described by the incomplete moments. These types of moments play an important role for measuring inequality, for example, mean deviations and Lorenz and Bonferroni curves, which depend upon the incomplete moment of a distribution. The nth incomplete moment of X can be determined from (6) as

(10) 
The last integral can be computed for most baseline G distributions at least numerically.
The symbolic computational softwares Maple, Mathematica and Matlab can automate all previous formulae since they have currently the ability to deal with analytic recurrence equations and sums of formidable size and complexity. In practical terms, we can substitute ∞ in the sums by a large number such as 20 for most practical applications. Equations (7)–(10) are the main results of this section.
Figura 5  Skewness and kurtosis measures of the ZW distribution for some parameter values
(a) 
(b) 
6 GENERATING FUNCTION
The mgf M(t) = E(e^{tX}) of X follows from (6) as

(10) 
where M_{r}(t) is the mgf of Y_{r}. Hence, M(t) can be immediately determined from the generating function of the expG distribution.
Another formula for M(t) follows from (6) as

(11) 
where γ_{r}(t) can be expressed in terms of Q_{G}(u) as

(12) 
We can obtain the mgfs of several ZetaG distributions directly from equations (11) and (12). For example, the mgfs of the ZetaExponential (with parameter λ and for t < λ^{−1}) and ZetaStandard Logistic (for t < 1) distributions are


respectively.
7 MEAN DEVIATIONS
The mean deviations about the mean (δ_{1}(X) = E(X − µ’_{n})) and about the median (δ_{2}(X) = E(X − M)) of X can be expressed as

(13) 
respectively, where µ’_{n} = E(X), M = Median(X) is the median, F(µ’_{n}) is easily calculated from (1) and m_{1}(z) is the first incomplete moment given by (10) with n = 1.
We have two alternative ways to compute δ_{1}(X) and δ_{2}(X). A general equation for m_{1}(z) can be derived from (6) as

(13) 
where

(14) 
Equation (14) is the basic quantity to compute the mean deviations of the expG distributions. Hence, the mean deviations in (13) depend only on the mean deviations of the expG distribution. So, alternative representations for δ_{1}(X) and δ_{2}(X) are


In a similar manner, the mean deviations of the ZetaG class can be determined from equation (10) with n = 1.
8 ESTIMATION
We calculate the maximum likelihood estimates (MLEs) of the parameters of the ZetaG class from complete samples only. Let x_{1}, ···,x_{n} be a observed sample of size n from the ZetaG(s,τ) distribution, where τ is a p × 1 vector of unknown parameters in the baseline distribution G(x; τ). The loglikelihood function for the vector of parameters θ = (s,τ^{T})^{T} is



(15) 
The loglikelihood can be maximized by using well established routines like nlm or optimize in the R statistical package or by solving the nonlinear likelihood equations obtained by differentiating (15). The components of the score vector U(θ) are






for j = 1, ... p, and ζ (1,s) = ζ (s)’.
For interval estimation and hypothesis tests on the model parameters, we require the (p+1)×(p+1) observed information matrix J = J(θ) given in Appendix C. Under standard regularity conditions, the asymptotic distribution of is , where I(θ) is the Fisher information matrix. In practice, we can replace I(θ) by the observed information matrix evaluated at , say , to construct approximate confidence intervals for the parameters based on the multivariate normal distribution for .
9 APPLICATION
In this section, we apply the ZetaG class to a real data set. We compare the ZW distribution with the Exponentiated Weibull (ExpW), Modified Weibull (MW), Kumaraswamyinverse Weibull (KwIW), Exponentiated NadarajahHaghighi (ExpNH), Fréchet (Fr) and Chen distributions.
We use the AdequacyModel package, see Diniz Marinho et al. (2016), available in the Comprehensive R Archive Network  CRAN, currently in its stable version 2.0.0. The AdequacyModel project is maintained in GitHub at https://github.com/ prdm0/AdequacyModel under the terms of the GNU General Public License, GNUGPL (≥ 3), where improvements can be made and sent to https://github.com/prdm0/AdequacyModel/issues. The package can also be installed directly of the GitHub, which enables developing versions to be tested and used before they are even available in CRAN. For the installation of the developing version, after removing preinstalled versions of the package that can be made with the remove.packages() function, the R programmer must have installed the devtools package and run the code below:
devtools : : install_github (repo = "prdm0 / AdequacyModel" , ref = "development" ,
force = TRUE, dependencies = TRUE,
type = "source")
The data refer to the observations of the failure time of secondary pumps of a reactor installed in a RSGGSA, see Barlow e Davis (1977). The primary pump is a single phase centrifugal type and uses mechanical seals. The primary pump design parameters are: 1570 m^{3}/h, engine power of 160 kW and total discharge height of 27 m. The secondary pump design parameters are pump flow of 1950 m^{3}/h, engine power of 200 kW, and total discharge height of 29 m.
The data composed of 23 observations (Suprawhardana e Prayoto, 1999) are: 2.1600, 0.7460, 0.4040, 0.9540, 0.4910, 6.5600, 4.9920, 0.3470, 0.1500, 0.3580, 0.1010, 1.3590, 3.4650, 1.0600, 0.6140, 1.9210, 4.0820, 0.1990, 0.6050, 0.2730, 0.0700, 0.0620, 5.3200. Table 1 highlights some descriptive statistics for the current data and may provide important descriptions for understanding the observations. Figure 6 displays the histogram of the data.
Tabela 1  Descriptive statistics
Statistics 
Real data sets 
Time between failures (hours) 

Mean 
1.5779 
Median 
0.6140 
Mode 
0.5000 
Variance 
3.7275 
Skewness 
1.3643 
Kurtosis 
0.5445 
Minimun 
0.0620 
Maximun 
6.5600 
n 
23 
Figura 6  Failure time of secondary pumps of a reactor installed in an RSGGSA
Table 2 gives the MLEs and corresponding standard errors (SEs) and the values of the Cramérvon Mises (W^{*}) and AndersonDarling (A^{*}) statistics (Chen e Balakrishnan, 1995). In general, the smaller the values of these statistics, the better the fit to the data. To obtain the statistics, one can proceed as follows: (i) compute and , where the are in ascending order, is an estimate of , is the standard normal cumulative function and denotes its inverse; (ii) compute , where is the sample mean of and is the sample variance; (iii) compute and , and then and . According to Chen e Balakrishnan (1995), these steps provide an approximation to the statistics below:






Table 2 lists the MLEs of the parameters and their SEs (in parentheses) for the compared models using the BFGS method. The goodnessoffit statistics are also presented, i.e., A*, W*, AIC (Akaike’s Information Criterion) by Akaike (1981), AICc (Consistent Akaike’s Information Criterion) by Hurvich e Tsai (1989), BIC (Bayesian Information Criterion) by Schwarz (1978) and HQIC (HannanQuinn Information Criterion) by Hannan e Quinn (1979).
Using AdequacyModel, these statistics can be easily obtained through the goodness.fit() function. The figures in Table 2 indicate that the ZW distribution provides the best fit to these data, according to the A* and W* statistics. Figure 7 displays the plots of the fitted densities.
Tabela 2  MLEs, SEs and the adequacy statistics (AIC, AICc, BIC and HQIC)
Distributions 
Estimates 
A* 
W* 
AIC 
AIC_{c} 
BIC 
HQIC 

ZW(α, β, s) 
0.0033 
0.4800 
1.8012 
0.1910 
0.0194 
95.2608 
96.5240 
98.6673 
96.1176 

(0.0032 
0.2863 
0.0001) 






ExpW(α, β, a) 
2.9924 
0.3030 
9.9812 
0.2129 
0.0234 
69.6642 
70.9273 
73.0707 
70.5209 

(2.8568 
0.2752 
27.6582) 






MW(β, γ, λ) 
0.7523 
0.7924 
0.0090 
0.4443 
0.0682 
71.0165 
72.2796 
74.4229 
71.8732 

(0.2199 
0.1925 
0.0849) 






KwIW(α, β, θ) 
3.1975 
21.5102 
0.2416 
0.2170 
0.0244 
69.6465 
70.9097 
73.053 
70.5032 

(4.6991 
99.5367 
0.3046) 






ExpNH(α, λ, β) 
0.3239 
19.7504 
2.4286 
0.2027 
0.0214 
69.4723 
70.7354 
72.8788 
70.3290 

(0.1514 
55.9603 
2.9442) 






Fr(a, b) 
0.3569 
0.7832 

0.3732 
0.0448 
69.8834 
70.4834 
72.1544 
70.4546 

(0.1007 
0.1234) 







Chen(β, λ) 
0.4569 
0.4045 

0.7408 
0.1235 
71.6804 
72.2804 
73.9514 
72.2515 

(0.0628 
0.1026) 







Figura 7  Failure time of secondary pumps of a reactor installed in an RSGGSA
10 CONCLUDING REMARKS
We propose a general family of continuous distributions called the ZetaG class. It extends several common distributions such as the Weibull, Fréchet, Burr XII and Lomax distributions. In fact, for each distribution G, we can define a new ZetaG distribution using a simple equation. We demonstrate that some mathematical properties of the ZetaG distribution can be readily obtained from those of the exponentiatedG distribution. The ordinary and incomplete moments, generating function and mean deviations of the ZetaG class can be expressed explicitly in terms of the baseline quantile function. We discuss maximum likelihood estimation and inference on the parameters based on the Cramérvon Mises and AndersonDarling statistics. An example to real data illustrates the potentiality of the new class.
Appendix A: Code 1
#include <Rcpp.h>
#include <math.h>
using namespace Rcpp;
// http://www.rcpp.org/
// [[Rcpp :: export]]
NumericVector derivationcpp (Function f, NumericVector par, NumericVector x, long double h = 1 . 0 e−8)
if (h == 0  h > 1){
stop("h should assume values in the interval (0, 1).");
}
NumericVector result , a , b;
a = as <NumericVector>(f(par, x + h));
b = as <NumericVector>(f(par, x));
return (a − b)/h;
}
// [[Rcpp :: export]]
NumericVector polylogcpp(NumericVector z, long int s = 2,
long int n = 1e4){
NumericVector result(z.size ( ));
for (int k = 1; k <= n; ++k){
result = result + pow(z, k)/pow(k, s);
}
return result;
}
// [[Rcpp :: export]]
double riemann_zetacpp(long int s, long int n = 1e4){
double result;
if(s <= 1) stop("s > 1 is not TRUE");
for (long int k = 1; k <= n; ++k){
result = result + 1/pow(k, s);
}
return result;
}
Code 1: C++ code that implements the derivationcpp(), polylogcpp() and riemann_zetacpp() functions
Appendix B: Code 2
config_cpp <− function (dir = NULL){
ifelse(is . null(dir) , base :: setwd(base :: getwd( )) ,
base :: setwd(dir))
if(!("fast.cpp" %in% base :: dir( ))){
stop("The file fast.cpp is not found in the directory: ",
base :: getwd( ))
}
depend <− function (...){
return("Rcpp" %in% utils :: installed.packages (...)[ , "Package"])
}
if(depend( )){
Rcpp :: sourceCpp("fast.cpp")
return(message("All ready. Setup completed!"))
} else {
message ("== > The Rcpp package will need to be installed.")
install <− function(...){
tryCatch(expr = utils :: install.packages(...),
warning = function(w) NA)
}
pkg <− NULL
pkg <− install("Rcpp")
if(!(is.null(pkg))){
stop(" ==> Check your internet connection!")
}
Rcpp :: sourceCpp("fast.cpp")
return(message("All ready. Setup completed!"))
}
}
Code 2: Setup code and compilation of fast.cpp file
Appendix C: Information Matrix
The elements of the observed information matrix J(θ) for the model parameters (s,τ) of the ZetaG class are given by










where , , , , , and .
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