A new generalized Weibull family of distributions: mathematical properties and applications
 Gauss M. Cordeiro^{1},
 Edwin M. M. Ortega^{2}Email author and
 Thiago G. Ramires^{2}
DOI: 10.1186/s4048801500366
© Cordeiro et al. 2015
Received: 1 April 2015
Accepted: 11 November 2015
Published: 26 November 2015
Abstract
We propose a generalized Weibull family of distributions with two extra positive parameters to extend the normal, gamma, Gumbel and inverse Gausssian distributions, among several other wellknown distributions. We provide a comprehensive treatment of its general mathematical properties including quantile and generating functions, ordinary and incomplete moments and other properties. We introduce the loggeneralized Weibullloglogistic, this is new regression model represents a parametric family of models that includes as submodels several widely known regression models that can be applied to censored survival data. We discuss estimation of the model parameters by maximum likelihood and provide two applications to real data.
Keywords
Estimation Generating function Mean deviation Moment Weibull distributionMathematics Subject Classification
47N30 97K70 97K80Introduction
Hereafter, a random variable X having the called generalized WeibullG (GWG) density function (2) is denoted by X∼ GWG (α,β,η). The aim of this paper is to derive some mathematical properties of X in explicit forms.
We provide explicit expressions for the quantile function (qf), ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, Rényi entropy, Shannon entropy, reliability and some properties of the order statistics.
The paper is outlined as follows. Section 2 provides some special distributions in the GW family. In Section 3, we derive useful expansions for the pdf and cdf of X. We can easily apply these expansions for all GWG distributions. In Section 4, we obtain the quantile function (qf) of X. In Section 5, we derive explicit expressions for the ordinary and incomplete moments. The moment generating function (mgf) of X is determined in Section 6. Mean deviations, probability weighted moments (PWMs), entropies and reliability are investigated in Sections 7, 8, 9 and 10. In Section 11, we derive an expansion for the density function of the GW order statistics. Some inferential tools are discussed in Section 12. In Section 13, we present a generalization of regression models based on the GW family. The performance of the maximum likelihood estimators (MLEs) are also investigated by a simulation study in this section. In Section 14, we fit some GWG distributions to two real data sets to demonstrate the potentiality of this family. Finally, Section 15 ends with some conclusions.
Special WeibullG distributions
The GW family density function (2) allows for greater flexibility of its tails and can be widely applied in many areas of engineering and biology. Here, we present and study some special cases of this family because it extends several widelyknown distributions in the literature. The density (2) will be most tractable when the cdf G(x;η) and pdf g(x;η) have simple analytic expressions.
2.1 The generalized Weibullnormal (GWN) distribution
2.2 The generalized WeibullGumbel (GWGu) distribution
2.3 The generalized Weibulllognormal (GWLN) distribution
2.4 The generalized Weibullloglogistic (GWLL) distribution
Useful expansions
where p _{ i }(c) are Stirling polynomials. The first six polynomials are p _{0}(w)=1/2, p _{1}(w)=(2+3w)/24, p _{2}(w)=(w+w ^{2})/48, p _{3}(w)=(−8−10w+15w ^{2}+15w ^{3})/5760, p _{4}(w)=(−6w−7w ^{2}+2w ^{3}+3w ^{4})/11520 and p _{5}(w)=(96+140w−224w ^{2}−315w ^{2}+63w ^{5})/2903040. These coefficients are related to the Stirling polynomials^{1} by p _{ n−1}(w)=S _{ n }(w)/[n!(w + 1)] for n≥1, where S _{0}(w) = 1,S _{1}(w)=(w+1)/2, etc. The proof of the expansion (7) is given in details by Flajonet and Odlyzko (1990) (see Theorem 3A, page 227) and Flajonet and Sedgewick (2009) (see Theorem VI.2, page 385). In this paper, we adopt the polynomials p _{ i }(w) in accordance with Nielson (1906) and Ward (1934).
respectively. The properties of exponentiated distributions have been studied by many authors in recent years, see Mudholkar and Srivastava (1993) for exponentiated Weibull, Gupta et al. (1998) for exponentiated Pareto, Gupta and Kundu (1999) for exponentiated exponential, Nadarajah (2005) for exponentiated Gumbel, Kakde and Shirke (2006) for exponentiated lognormal, and Nadarajah and Gupta (2007) for exponentiated gamma distributions.
where h _{ k+1}(x) denotes the pdf of the expG (k+1) distribution and v _{ k }=w _{ k+1}. So, several properties of the GWG distribution can be obtained by knowing those of the expG distribution, see, for example, Mudholkar et al. (1995), Gupta and Kundu (2001) and Nadarajah and Kotz (2006a), among others.
Quantile function
where v _{ k }=(−1)^{ k+1}/(k α). Hence, the last equation reveals that the GWG qf can be expressed as the G qf applied a power series.
Moments
Explicit expressions for moments of several exponentiated distributions are given by Nadarajah and Kotz (2006a). They can be used to produce \(\mu _{n}^{\prime }\).
where \(B(\textit {a,b}) = {\int _{0}^{1}} t^{a1}\,(1t)^{b1} \textit {dt}\) is the beta function.
The last integral can be computed for most baseline G distributions.
is the Stirling number of the first kind which counts the number of ways to permute a list of r items into k cycles. So, we can obtain the factorial moments from the ordinary moments given before.
Generating function
where M _{ k+1}(t) is the mgf of Y _{ k+1}. Hence, M(t) can be determined from the generating function of the expG (k+1) distribution.
Mean deviations
respectively, where \(\mu ^{\prime }_{1}=E(X)\), M=M e d i a n(X) is the median given in Section 3, \(F(\mu ^{\prime }_{1})\) is easily calculated from the cdf (1) and \(m_{1}(z)=\int _{\infty }^{z} x\,f(x) dx\) is the first incomplete moment.
Equation (18) is the basic quantity to compute the mean deviations of the expG distributions. Hence, the mean deviations in (17) depend only on the mean deviations of the expG distribution.
is a simple integral defined from the baseline qf Q _{ G }(u).
Applications of the first incomplete moment can be addressed to obtain Bonferroni and Lorenz curves defined for a given probability π by \(B(\pi)= m_{1}(q)/[\pi \mu ^{\prime }_{1}]\) and \(L(\pi)=m_{1}(q)/\mu ^{\prime }_{1}\), respectively, where \(\mu ^{\prime }_{1}=E(X)\) and q=Q _{ G }(1− exp{−[−α ^{−1} log(1−π)]^{1/β }}) is the GWG qf at π, see equation.
Probability weighted moments
where τ(n,r) is given by (14).
Equation (22) can be applied for most baseline G distributions to derive explicit expressions for κ _{ n,s }, since the baseline qf can usually be expressed as a power series.
Entropies
for γ>0 and γ≠1. The Shannon entropy of a random variable X is defined by E[− logf(X)]. It is the particular case of the Rényi entropy for γ ↑1.
where \(c_{n,0}={a_{0}^{n}}\). The coefficient c _{ n,i } can be determined from c _{ n,0},…,c _{ n,i−1} and hence from the quantities a _{0},…,a _{ i }. In fact, c _{ n,i } can be given explicitly in terms of the coefficients a _{ i }, although it is not necessary for programming numerically our expansions in any algebraic or numerical software.
where the integral can be calculated for most baseline distributions using a power series expansion for Q _{ G }(u).
Reliability
where R _{ jk }= Pr(Y _{ j }<Y _{ k }) is the reliability between the independent random variables Y _{ j }∼ expG (j) and Y _{ k }∼ expG (k+1). Hence, the reliability for the GWG random variables is a linear combination of those for expG random variables. In the particular case α _{1}=α _{2} and β _{1}=β _{2}, Eq. (33) gives R=1/2.
Order statistics
Maximum likelihood estimation
Several approaches for parameter point estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The maximum likelihood estimates (MLEs) enjoy desirable properties and can be used when constructing confidence intervals and also in test statistics. Large sample theory for these estimates delivers simple approximations that work well in finite samples. Statisticians often seek to approximate quantities such as the density of a test statistic that depend on the sample size in order to obtain better approximate distributions. The resulting approximation for the MLEs in distribution theory is easily handled either analytically or numerically. The goodness of fit statistics including the Akaike information criterion (AIC), Bayesian information criterion (BIC), Consistent Akaike information criterion (CAIC), AndersonDarling (A) and Cramér–von Mises (W) are computed to compare the fitted models.
respectively.
The MLE \(\widehat {\boldsymbol {\theta }}\) of θ is obtained by solving the nonlinear likelihood equations U _{ α }(θ)=0, U _{ β }(θ)=0 and U _{ η }(θ)=0. These equations cannot be solved analytically and statistical software can be used to solve them numerically. We can use iterative techniques such as a NewtonRaphson type algorithm to obtain \(\widehat {{\boldsymbol {\theta }}}\). We employ the numerical procedure NLMixed in SAS.
Regression models
In many practical applications, the lifetimes are affected by explanatory variables such as the cholesterol level, blood pressure, weight and many others. Parametric models to estimate univariate survival functions and for censored data regression problems are widely used. A regression model that provides a good fit to lifetime data tends to yield more precise estimates of the quantities of interest.
where the functions G(·) and g(·) are defined in Section 1.
where α>0 and β>0 are shape parameters, μ∈ℜ is the location parameter and σ>0 is the scale parameter.
where the random error z _{ i } has density function (39), τ=(τ _{1},…,τ _{ p })^{ T }, σ>0, α>0 and β>0 are unknown parameters. The parameter \(v_{i}=\textbf {v}_{i}^{T} {\boldsymbol {\tau }}\) is the location of y _{ i }. The location parameter vector v=(v _{1},…,v _{ n })^{ T } is represented by a linear model v=V τ, where V=(v _{1},…,v _{ n })^{ T } is a known model matrix. The LGWLL model (40) opens new possibilities for fitted many different types of data.
where r is the number of uncensored observations (failures). The MLE \(\widehat {{\boldsymbol {\theta }}}\) of the vector of unknown parameters can be calculated by maximizing the loglikelihood (41). We use the procedure NLMixed in SAS to calculate the estimate \(\widehat {{\boldsymbol {\theta }}}\). Initial values for β and σ are taken from the fit of the logWeibull regression model with α=0 and β=1.
The elements of the (p+3)×(p+3) observed information matrix \(\ddot {\textbf {L}}({\boldsymbol {\theta }})\), namely −L _{ α α },−L _{ α β }, \(\textbf {L}_{\alpha \sigma },\textbf {L}_{{\alpha \tau }_{j}}, \textbf {L}_{\beta \beta },\textbf {L}_{\beta \sigma }, \textbf {L}_{{\beta \tau }_{j}},\textbf {L}_{\sigma \sigma },\textbf {L}_{{\sigma \tau }_{j}}\phantom {\dot {i}\!}\) and \(\textbf {L}_{\beta _{j}\beta _{s}}\phantom {\dot {i}\!}\) (for j,s=1,…,p) can be calculated numerically. Inference on θ can be conducted in the classical way based on the approximate multivariate normal \(N_{p+3}\left (0,\ddot {\textbf {L}}(\widehat {{\boldsymbol {\theta }}})^{1}\right)\) distribution for \(\widehat {{\boldsymbol {\theta }}}\). Further, we can use LR statistics for comparing the LGWLL model with some of its submodels.
13.1 Simulation
The AEs, biases and MSEs based on 1000 simulations of the GWN distribution with α=2, β=1.5, 0.5 μ=0 and σ=1, with n=50, 150 and 300
β=1.5  β=0.5  

n  Parameter  Mean  Bias  MSE  Parameter  Mean  Bias  MSE  
50  α  2.2666  0.2666  18.1264  α  1.5286  –0.4714  2.5046  
β  6.2918  4.7918  5117.984  β  0.7304  0.2304  0.3708  
μ  –1.0586  –1.0586  208.3427  μ  –0.8475  –0.8475  2.5944  
σ  3.5544  2.5544  1831.613  σ  0.9704  –0.0296  0.3156  
150  α  2.1093  0.1093  2.7102  α  1.7036  –0.2964  1.7252  
β  1.7796  0.2796  1.2247  β  0.6089  0.1089  0.0724  
μ  –0.1175  –0.1175  0.2673  μ  –0.5265  –0.5265  1.8487  
σ  1.1065  0.1065  0.3165  σ  0.9374  –0.0626  0.0423  
300  α  2.1755  0.1755  2.0205  α  1.8607  –0.1393  1.1476  
β  1.6044  0.1044  0.3035  β  0.5579  –0.4421  0.2328  
μ  –0.0329  –0.0329  0.1596  μ  –0.2802  –0.2802  1.2004  
σ  1.0379  0.0379  0.0817  σ  0.9581  –0.0419  0.0201 
Applications
In this section, we present two applications to read data. In the first, the computations were performed using the subroutine g o o d n e s s.f i t in the script AdequacyModel of the R package. In the second application for censured data the computations were done using the subroutine nlmixed of the SAS software.
14.1 Data: Strengths of glass fibers
The data (n=63) set is on the strengths of 1.5 cm glass fibers from Smith and Naylor (1987) contained in the gamlss.data library of the R software. BarretoSouza et al. (2010) fitted the beta generalized exponential (BGE) distribution to these data and proved that its fit is better than those of the beta exponential (BE) (Nadarajah and Kotz 2006b) and generalized exponential (GE) (Gupta and Kundu 1999) distributions. BarretoSouza et al. (2011) proved that the beta Fréchet (BF) distribution gives a better fit than the Fréchet and exponentiated Fréchet (EF) (Nadarajah and Kotz 2003) distributions. Alzaghal et al. (2013) fitted the exponentiated Weibullexponential (EWE) distribution to the current data and conclude that this distribution provides a better fit than the BGE and BF distributions. Recently, Bourguignon et al. (2014) fits the Weibullexponential (WE) distribution and shows that it is better than the exponentiated Weibull (EW) (Mudholkar and Srivastava 1993) and exponentiated exponential (EE) (Gupta and Kundu 1999) models.
Estimates of the model parameters for the myelogenous leukemia data, the corresponding SEs (given in parentheses) and the AIC, BIC, A and W statistics
Model  Estimates  AIC  BIC  A  W  

N (μ,σ)  1.506  0.321  39.823  44.109  1.928  0.350  
(0.040)  (0.028)  
GWN (α,β,μ,σ)  2.297  3.396  1.530  1.023  36.496  45.068  0.928  0.164 
(31.667)  (4.720)  (3.969)  (0.864)  
LL (a,b)  1.526  7.926  49.579  53.866  2.748  0.496  
(0.040)  (0.873)  
GWLL (α,β,a,b)  4.393  0.302  2.072  19.999  37.983  46.555  1.266  0.231 
(2.323)  (0.035)  (0.206)  (0.214)  
GWGu (α,β,μ,σ)  0.371  1.197  1.384  0.325  37.912  46.484  0.829  0.140 
(17.164)  (12.395)  (12.569)  (3.369)  
GWLN (α,β,μ,σ)  4.727  8.638  0.205  1.675  40.173  48.7455  1.531  0.279 
(4.927)  (6.179)  (0.432)  (1.086)  
EWE (α,γ,c)  23.614  7.249  0.003  34.654  41.083  0.960  0.170  
(3.954)  (0.994)  (0.003)  
WE (α,β,λ)  0.014  2.879  1.017  34.804  41.234  0.976  0.173  
(0.059)  (2.048)  (1.195) 
LR tests
Strengths  Hypotheses  Statistic w  pvalue 

GWN vs N  H _{0}:α=β=1 vs H _{1}:H _{0} is false  7.327  0.02 
GWLL vs LL  H _{0}:α=β=1 vs H _{1}:H _{0} is false  15.596  <0.001 
MLEs of the model parameters for the entomology data, the corresponding SEs (given in parentheses) and the statistics AIC, CAIC and BIC
Model  α  β  a  γ  AIC  CAIC  BIC  

GWLL  1.9359  0.3089  31.2742  7.1305  1276.8  1277.0  1289.4  
(0.5036)  (0.0287)  (5.6950)  (1.0640)  
LL  1  1  19.7506  2.7441  1289.1  1289.2  1295.4  
–  –  (0.9132)  (0.1877)  
β  λ  a  b  c  AIC  CAIC  BIC  
McW  0.0301  1.2154  0.6426  3.1889  2.5386  1290.5  1290.9  1306.3 
(0.0110)  (0.4149)  (0.3719)  (0.8995)  (2.2535)  
BW  0.0143  2.0779  0.8736  7.4619  1  1325.4  1325.6  1337.9 
(0.0011)  (0.3950)  (0.2642)  (0.6443)  –  
KwW  0.1055  2.0666  1  0.6934  0.1293  1323.1  1323.4  1335.7 
(0.0112)  (0.1300)  –  (0.1539)  (0.0118)  
EW  0.0449  1.5874  1.2548  1  1  1287.5  1287.6  1296.9 
(0.0075)  (0.2753)  (0.3752)  –  –  
Weibull  0.04007  1.7970  1  1  1  1286.1  1286.2  1292.4 
(0.0018)  (0.1109)  –  –  – 
14.2 Entomology data

Group 1: Control 1 (deionized water); Control 2 (acetone  5 %); aqueous extract of seeds (AES) (39 ppm); AES (225 ppm); AES (888 ppm); methanol extract of leaves (MEL) (225 ppm); MEL (888 ppm); and dichloromethane extract of branches (DMB) (39 ppm).

Group 2: MEL (39 ppm); DMB (225 ppm) and DMB (888 ppm).
For more details, see Silva et al. (2013). The response variable in the experiment is the lifetime of the adult flies in days after exposure to the treatments. The experimental period was set at 51 days, so that the numbers of larvae that survived beyond this period were considered as censored observations. The total sample size was n=72, because four cases were lost. Therefore, the variables used in this study were: x _{ i }lifetime of ceratitis capitata adults in days, δ _{ i }censoring indicator and v _{ i1}group (1 = group 1, 0 = group 2). We start the analysis of the data considering only failure (x _{ i }) and censoring (δ _{ i }) data.
Recently, Alexander et al. (2012) analyzed these data using the McDonaldWeibull (McW) distribution with scale parameter β>0 and shape parameter λ>0. We focus on this distribution since it extends various distributions previously discussed in the lifetime literature, as: beta Weibull (BW) (Lee et al. 2007), Kumaraswamy Weibull (KwW) (Cordeiro et al. 2010), exponentiated Weibull (EW) (Mudholkar et al. 1995) distributions and more.
Now, we compare the McW distribution and some of their submodels. For some fitted models, Table 4 provides the MLEs (and the corresponding standard errors in parentheses) of the parameters and the values of the AIC, BIC and CAIC statistics. The computations were performed using the NLMixed subroutine in SAS. They indicate that the GWLL model has the lowest AIC, BIC and CAIC values among those values of the fitted models, and therefore it could be chosen as the best model.
MLEs of the parameters from the fitted LGWLL regression model to the entomology data, the corresponding SEs (given in parentheses), pvalues in [ ·] and the statistics AIC, CAIC and BIC
Model  α  β  σ  θ _{0}  θ _{1}  AIC  CAIC  BIC 

LGWLL  2.1577  0.2243  0.1020  3.2690  0.3217  341.5  341.9  357.2 
(0.3215)  (0.0152)  (0.0031)  (0.0925)  (0.0612)  
[ <0.0001]  [ <0.0001]  
Logistic  1  1  0.3547  2.7392  0.3310  355.6  355.8  365.1 
(0.0241)  (0.0894)  (0.1037)  
[ <0.0001]  [0.0017] 
Conclusions
We study some mathematical properties of a new generalized Weibull family of distributions with two extra positive parameters. The family is able to generalize any continuous distribution. We provide some special models, a very useful mixture representation in terms of exponentiated distributions, explicit expressions for the ordinary and incomplete moments, generating function, mean deviations, probability weighted moments, entropies, reliability and order statistics. The model parameters are estimated by the method of maximum likelihood. We introduce a locationscale regression model based on the new family. The importance of the proposed models is illustrated by means of two real life data sets. The new models provide consistently better fits than other competitive models for these data.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Alexander, C, Cordeiro, GM, Ortega, EMM, Sarabia, JM: Generalized betagenerated distributions. Comput. Stat. Data Anal. 56, 1880–1897 (2012).MATHMathSciNetView ArticleGoogle Scholar
 Alzaatreh, A, Lee, C, Famoye, F: A new method for generating families of continuous distributions. Metron. 71, 63–79 (2013).MATHMathSciNetView ArticleGoogle Scholar
 Alzaghal, A, Famoye, F, Lee, C: Exponentiated TX Family of Distributions with Some Applications. Int. J. Stat. Probab. 2, 31 (2013).View ArticleGoogle Scholar
 BarretoSouza, W, Santos, AH, Cordeiro, GM: The beta generalized exponential distribution. J. Stat. Comput. Simul. 80, 159–172 (2010).MATHMathSciNetView ArticleGoogle Scholar
 BarretoSouza, W, Cordeiro, GM, Simas, AB: Some results for beta Fréchet distribution. Commun. Stat. Theory Methods. 40, 798–811 (2011).MATHMathSciNetView ArticleGoogle Scholar
 Bourguignon, M, Silva, RB, Cordeiro, GM: The WeibullG Family of Probability Distributions. J. Data Sci. 12, 53–68 (2014).MathSciNetGoogle Scholar
 Cordeiro, GM, Ortega, EMM, Nadarajah, S: The Kumaraswamy Weibull distribution with application to failure data. J. Frankl. Inst. 347, 1399–1429 (2010).MATHMathSciNetView ArticleGoogle Scholar
 Flajonet, P, Odlyzko, A: Singularity analysis of generating function. SIAM: SIAM J. Discr. Math. 3, 216–240 (1990).Google Scholar
 Flajonet, P, Sedgewick, R: Analytic Combinatorics. ISBN 9780521898065. Cambridge University Press (2009).
 Gradshteyn, IS, Ryzhik, IM: Table of Integrals, Series, and Products. seventh edition. Academic Press, San Diego (2007).MATHGoogle Scholar
 Gupta, RC, Gupta, PL, Gupta, RD: Modeling failure time data by Lehman alternatives. Commun. Stat. Theory Methods. 27, 887–904 (1998).MATHView ArticleGoogle Scholar
 Gupta, RD, Kundu, D: Generalized exponential distributions. Aust. N. Z. J. Stat. 41, 173–188 (1999).MATHMathSciNetView ArticleGoogle Scholar
 Gupta, RD, Kundu, D: Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biom. J. 43, 117–130 (2001).MATHMathSciNetView ArticleGoogle Scholar
 Kakde, CS, Shirke, DT: On exponentiated lognormal distribution. Int. J. Agric. Stat. Sci. 2, 319–326 (2006).Google Scholar
 Lee, C, Famoye, F, Olumolade, O: Beta Weibull distribution: some properties and applications to censored data. J. Modern Appl. Stat. Methods. 6, 173–186 (2007).Google Scholar
 Mudholkar, GS, Srivastava, DK: Exponentiated Weibull family for analyzing bathtub failurerate data. IEEE Trans. Reliab. 42, 299–302 (1993).MATHView ArticleGoogle Scholar
 Mudholkar, GS, Srivastava, DK, Freimer, M: The exponentiated Weibull family: A reanalysis of the busmotorfailure data. Technometrics. 37, 436–445 (1995).MATHView ArticleGoogle Scholar
 Nadarajah, S: The exponentiated Gumbel distribution with climate application. Environmetrics. 17, 13–23 (2005).View ArticleGoogle Scholar
 Nadarajah, S, Gupta, AK: The exponentiated gamma distribution with application to drought data. Calcutta Stat Assoc. Bull. 59, 29–54 (2007).MATHMathSciNetGoogle Scholar
 Nadarajah, S, Kotz, S: The exponentiated Frechet distribution. Interstat Electron. (2003). http://interstat.statjournals.net/YEAR/2003/abstracts/0312001.php.
 Nadarajah, S, Kotz, S: The exponentiated type distributions. Acta Applicandae Mathematicae. 92, 97–111 (2006a).MATHMathSciNetView ArticleGoogle Scholar
 Nadarajah, S, Kotz, S: The beta exponential distribution. Reliab. Eng. Syst. Saf. 91, 689–697 (2006b).MathSciNetView ArticleGoogle Scholar
 Nielson, N: Handbuch der theorie der gamma funktion. Chelsea Publ. Co., New York (1906).Google Scholar
 Prudnikov, AP, Brychkov, YA, Marichev, OI: Integrals and Series, Vol. 1, 2 and 3. Gordon and Breach Science Publishers, Amsterdam (1986).Google Scholar
 Rényi, A: On measures of entropy and information, Vol. 1. University of California Press, Berkeley (1961).Google Scholar
 Shannon, CE: Prediction and entropy of printed English. Bell Syst. Technical J. 30, 50–64 (1951).MATHView ArticleGoogle Scholar
 Silva, MA, BezerraSilva, GCD, Vendramim, JD, Mastrangelo, T: Sublethal effect of neem extract on Mediterranean fruit fly adults. Rev. Bras. Frutic. 35, 93–101 (2013).View ArticleGoogle Scholar
 Smith, RL, Naylor, JC: A comparison of maximum likelihood and Bayesian estimators for the threeparameter Weibull distribution. Appl. Stat. 36, 358–369 (1987).MathSciNetView ArticleGoogle Scholar
 Ward, M: The representation of Stirling’s numbers and Stirling’s polynomials as sums of factorials. Am. J. Math. 56, 87–95 (1934).View ArticleGoogle Scholar