 Methodology
 Open Access
The odd loglogistic logarithmic generated family of distributions with applications in different areas
 Morad Alizadeh^{1},
 S. M. T. K. MirMostafee^{2},
 Edwin M. M. Ortega^{3}Email author,
 Thiago G. Ramires^{3} and
 Gauss M. Cordeiro^{4}
https://doi.org/10.1186/s4048801700627
© The Author(s) 2017
 Received: 26 January 2017
 Accepted: 13 June 2017
 Published: 4 July 2017
Abstract
We introduce and study general mathematical properties of a new generator of continuous distributions with three extra parameters called the odd loglogistic logarithmic generated family of distributions. We present some special models and investigate the asymptotes and shapes. The new density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. Explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Shannon and Rényi entropies and order statistics, which hold for any baseline model, are determined. We discuss the estimation of the model parameters by maximum likelihood. Further, we introduce the new family in longterm survival models. We illustrate the potentiality of the proposed models by means of four applications to real data.
Keywords
 Generated family
 Maximum likelihood
 Moment
 Order statistic
 Quantile function
 Survival analysis
AMS Subject Classification
 97K50
 62N01
 62N02
Introduction
Statistical distributions are very useful in describing and predicting real world phenomena. Numerous extended distributions have been extensively used over the last decades for modeling data in several areas. Recent developments focus on defining new families that extend wellknown distributions and at the same time provide greater flexibility in modeling data in practice. Hence, several classes to generate new distributions by adding one or more parameters have been proposed in the statistical literature. Some wellknown generators are the MarshallOlkin generated (MOG) by Marshall and Olkin (1997), betaG by Eugene et al. (2002), KumaraswamyG (KwG) by Cordeiro and de Castro (2011), WeibullG by Bourguignon et al. (2014), exponentiated halflogisticG by Cordeiro et al. (2014a), LomaxG by Cordeiro et al. (2014b), among others.

The set of generalized loglogistic (GLL) transformations form an Abelian group with the binary operation of composition;

The transformation group partitions the set of all lifetime distributions into equivalence classes, so that any two distributions in an equivalence class are related through a GLL transformation;

Either every distribution in an equivalence class has a moment generating function (mgf), or none does;

Every distribution in an equivalence class has the same number of moments;

Each equivalence class is linearly ordered according to the transformation parameter, with larger values of this parameter corresponding to smaller dispersion of the distribution about the common median class; and

Within an equivalence class, the KullbackLeibler information is an increasing function of the ratio of the transformation parameters.
In addition, Gleaton and Rahman (2010, 2014) obtained asymptotic results for the maximum likelihood estimates (MLEs) of the parameters of these two distributions. They proved that for distributions generated from either a twoparameter Weibull or a twoparameter inverse Gaussian distributions by a GLL transformation, the joint MLEs of the parameters are asymptotically normal and efficient, provided the GLL transformation parameter exceeds three.
This paper is organized as follows. In Section 2, we provide a physical interpretation of the OLLLG family and define two special cases. In Section 3, two useful linear representations are derived. In Section 4, we obtain explicit expressions for the moments and generating function. In Section 5, general expressions for the Rényi and Shannon entropies and order statistics are presented. Estimation of the model parameters by maximum likelihood is investigated in Section 6. We also present the performance of the MLEs through a simulation study. In Section 7, the OLLLG model is modified for possible presence of longterm survivors in the data. Four applications to real data illustrate the performance of the proposed models in Section 8. The paper is concluded in Section 9.
Motivation and special cases
where g(x;ξ) is the baseline pdf. Hereafter, a random variable X with density function (3) is denoted by X∼OLLLG(α,β,ξ). Further, we can omit sometimes the dependence on the vector ξ of the parameters and write simply G(x)=G(x;ξ).
which is identical to (2).
has the density function (3), where Q _{ G }(u)=G ^{−1}(u) is the quantile function (qf) of the baseline G.
Remark 1
Although, we have stated that β∈(0,1), Eq. (2) is still a cdf if β<0. Hence, we can consider the OLLLG family defined for any β∈(−∞,0)∪(0,1).
In Appendix 1, we present the asymptotes and shapes of the OLLLG model.
2.1 Special OLLLG distributions
The OLLLG density function (3) allows for greater flexibility of its tails and can be widely applied in many areas of engineering and biology. It will be most tractable when G(x;ξ) and g(x;ξ) have simple analytic expressions. We now present and discuss some special cases of this family because it extends several widelyknown distributions in the literature.
2.1.1 Odd loglogistic logarithmic Weibull (OLLLW) model
2.1.2 Odd loglogistic logarithmic normal (OLLLN) model
Linear representations
Let \(A(u)=\frac {u^{\alpha }}{u^{\alpha }+(1u)^{\alpha }}\) be the cdf of the odd loglogistic uniform (Gleaton and Lynch 2004, 2006, 2010) distribution. For β∈(0,1), we have 0<β A(u)<1. Then, we can apply the power series reported in Appendix 2 by taking u=G(x)^{ α }, since they are always convergent in the interval (0,1), i.e., the power series are valid in the support of X. Henceforth, we consider that 0<β<1, which is not a restrictive assumption since it is in agreement with the logarithm distribution defined for compounding the OLLLG family.
where the coefficients \(h^{*}_{k}(\alpha,i)\) can be determined from the recursive formula given after (24).
Further, we define the exponentiatedG (“ExpG”) distribution for an arbitrary parent distribution G, say W∼Exp^{ c } G, if W has cdf and pdf given by H _{ c }(x)=G(x)^{ c } and h _{ c }(x)=c g(x) G(x)^{ c−1}, respectively. This transformed model is also called the Lehmann type I distribution, say Exp ^{ c }(G).
where \(d_{k}=\sum _{i=1}^{\infty }\frac {\beta ^{i}h^{*}_{k}(\alpha,i)}{i\,\log (1\beta)}\) and H _{ k }(x) is the ExpG cdf with power parameter k.
where h _{ k+1}(x)=(k+1) G(x)^{ k } g(x) is the ExpG density function with power parameter (k+1).
Equation (7) reveals that the OLLLG density function is a linear combination of ExpG densities. Some structural properties of the new family such as the ordinary and incomplete moments and generating function can be determined from wellestablished properties of the ExpG distribution. The properties of ExpG distributions have been studied by many authors in recent years, see Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) for exponentiated Weibull, Gupta and Kundu (1999) for exponentiated exponential and Nadarajah (2006) for exponentiated Gumbel, among others. The linear representations (6) and (7) are the main results of this section.
Moments and generating function
where \(\tau (n,k)=\int _{\infty }^{\infty } x^{n}\,G(x)^{k}\,g(x)\mathrm {d}x=\int _{0}^{1} Q_{G}(u)^{n}\,u^{k} \mathrm {d} u\). In fact, it is possible to exchange the infinite sum and the integral using the dominated convergence theorem for series.
Expressions for moments of several expG distributions are given by Nadarajah and Kotz (2006), which can be used to obtain E(X ^{ n }). Cordeiro and Nadarajah (2011) determined τ(n,k) for some wellknown distributions such as the normal, beta, gamma and Weibull distributions.
The last integral can be determined analytically or numerically for most baseline distributions. Equation (9) can be used to determine conditional moments, mean deviations and Bonferroni and Lorentz curves of X.
where M _{ k }(t) is the mgf of Y _{ k } and \(\rho (t,k)=\int _{\infty }^{\infty } {\mathrm {e}}^{t\,x}\,G(x)^{k}\,g(x) \mathrm {d}x= \int _{0}^{1} \exp [t\,Q_{G}(u)]\,u^{k} \mathrm {d} u\).
We can determine the mgfs for several OLLLG distributions directly from Eq. (10).
We present some mathematical properties of the odd loglogistic logarithmic exponential (OLLLE) distribution in Appendix 3 to illustrate the applicability of the previous results.
Other properties
We hardly need to emphasize the necessity and importance of entropies and order statistics in any statistical analysis especially in applied work.
5.1 Entropies
5.2 Order statistics
Equation (11) is the main result of this section. It reveals that the pdf of the OLLLG order statistics is a linear combination of ExpG densities. So, several mathematical quantities of the OLLLG order statistics such as ordinary, incomplete and factorial moments, mgf, mean deviations, among others, can be obtained from those quantities of the ExpG distribution.
Estimation
where h ^{(ξ)}(·) means the derivative of the function h with respect to ξ.
6.1 Simulation study
The AEs, biases and MSEs of the OLLLN distribution for μ=0, σ=1, α=0.2,0.5 and β=−0.5,0.7 varying n
α=0.2 β=−0.5  α=0.5 β=−0.5  

n  Parameter  AE  Bias  MSE  Parameter  AE  Bias  MSE 
50  μ  0.0499  0.0499  0.3033  μ  0.0602  0.0602  0.2529 
σ  0.9025  0.0975  0.2581  σ  1.1766  0.1766  42.5479  
α  0.1926  0.0074  0.0358  α  0.6543  0.1543  20.1315  
β  0.7939  0.2939  1.6520  β  1.0228  0.5228  3.4332  
150  μ  0.0518  0.0518  0.2303  μ  0.0003  0.0003  0.1117 
σ  0.8911  0.1089  0.1107  σ  0.9765  0.0235  0.1005  
α  0.1771  0.0229  0.0137  α  0.4934  0.0066  0.0487  
β  0.5812  0.0812  0.4428  β  0.9119  0.4119  2.1433  
300  μ  0.0395  0.0395  0.1105  μ  0.0037  0.0037  0.0551 
σ  0.8787  0.1213  0.0772  σ  0.9877  0.0123  0.0452  
α  0.1737  0.0263  0.0099  α  0.4973  0.0027  0.0211  
β  0.5532  0.0532  0.1873  β  0.7216  0.2216  0.9713  
α=0.2 β=0.7  α=0.5 β=0.7  
n  Parameter  AE  Bias  MSE  Parameter  AE  Bias  MSE 
50  μ  0.0436  0.0436  0.3052  μ  0.0792  0.0792  0.2650 
σ  0.9008  0.0992  0.1778  σ  1.1175  0.1175  83.1835  
α  0.1933  0.0067  0.0251  α  0.6144  0.1144  42.5964  
β  0.5943  0.1057  0.1579  β  0.4014  0.2986  0.8603  
150  μ  0.0064  0.0064  0.2759  μ  0.0152  0.0152  0.1170 
σ  0.8930  0.1070  0.1148  σ  0.9899  0.0101  0.0938  
α  0.1864  0.0136  0.0151  α  0.5029  0.0029  0.0474  
β  0.6668  0.0332  0.0500  β  0.6043  0.0957  0.1170  
300  μ  0.0624  0.0624  0.1436  μ  0.0038  0.0038  0.0616 
σ  0.8393  0.1607  0.0897  σ  0.9916  0.0084  0.0460  
α  0.1713  0.0287  0.0117  α  0.5012  0.0012  0.0227  
β  0.6797  0.0203  0.0310  β  0.6603  0.0397  0.0356 
The figures in Table 2 indicate that the MSEs and the AEs of the estimates of μ, σ, α and β decay toward zero when the sample size increases, as expected under firstorder asymptotic theory. As n increases, the AEs of the parameters tend to be closer to the true parameter values. This fact supports that the asymptotic normal distribution provides an adequate approximation to the finite sample distribution of the MLEs.
The OLLLG family with longterm survival
Models for survival analysis typically consider that every subject in the population under study is susceptible to the event of interest and will eventually experience such event if followup is sufficiently long. However, there are situations when a fraction of individuals are not expected to experience the event of interest, that is, those individuals are cured or not susceptible. Cure rate models for survival data have been used to model timetoevent data for various types of cancers, including breast cancer, nonHodgkins lymphoma, leukemia, prostate cancer and melanoma. These models have become very popular due to significant progress in treatment therapies leading to enhanced cure rates.
where f(x _{ i }N _{ i }=1) is the baseline pdf (see Section 2.1) for the susceptible individuals. Equations (13) and (14) are improper functions, since S _{ pop }(x) is not a proper survival function. We can omit sometimes the dependence on the indicator N _{ i } and write simply S(x _{ i }N _{ i }=1)=S(x), f(x _{ i }N _{ i }=1)=f(x), etc.
A random variable having density (15) is denoted by X∼OLLGcr(α,β,ξ,p). The hrf of the OLLGcr model is given by h _{ pop }(t)=f _{ pop }(t)/S _{ pop }(t).
7.1 Estimation
where r is the number of failures (uncensored observations). We can obtain the MLE \(\widehat {\boldsymbol {\theta }}\) of θ by maximizing the loglikelihood (17) either directly in R using the optim function, in SAS using the NLMixed procedure and in other statistical software or by solving the nonlinear likelihood equations obtained by differentiating (17).
Applications
In this section, we provide four applications to real data. In the first three applications, we present some results by fitting special models defined in Section 2.1. In the fourth application, we present an application using the longterm survival model defined in Section 7.
For the first three applications, the goodnessoffit statistics including the Cramérvon Mises (W ^{∗}) and AndersonDarling (A ^{∗}) test statistics are used to compare the fitted models; see Chen and Balakrishnan (1995) for more details. The smaller the values of A ^{∗} and W ^{∗}, the better the fit to the data. We also consider the KolmogrovSmirnov (KS) statistic (and its corresponding pvalue) and minus the maximized loglikelihood (\(\hat \ell _{n}\)) for the sake of comparison. For the fourth application (censored data), we adopt the AIC and BIC statistics to compare the fitted models since the A ^{∗} and W ^{∗} statistics are not suitable for censored data.

The normal distribution.

The exponentiated normal (EN) distribution.

The logarithmic normal (LN) distribution, the special case of the OLLLN distribution when α=1.

The beta normal (BN) distribution (Eugene et al. 2002) with density$$\begin{array}{@{}rcl@{}} f_{BN}(x)=\frac{1}{\sigma B(\alpha,\beta)} \left[\Phi\left(\frac{x\mu}{\sigma}\right)\right]^{\alpha1}\left[1\Phi\left(\frac{x\mu}{\sigma}\right)\right]^{\beta1} \phi\left(\frac{x\mu}{\sigma}\right). \end{array} $$

The gamma normal (GN) distribution (Alzaatreh et al. 2014) with density$$\begin{array}{@{}rcl@{}} f_{GN}(x)=\frac{\beta^{\alpha}}{\sigma\Gamma(\alpha)} \left[\log\left\{1\Phi\left(\frac{x\mu}{\sigma}\right)\right\}\right]^{\alpha1}\left[1\Phi\left(\frac{x\mu}{\sigma}\right)\right]^{\beta1} \phi\left(\frac{x\mu}{\sigma}\right). \end{array} $$

The Kumaraswamy normal (KN) distribution (Cordeiro and de Castro 2011) with density$$\begin{array}{@{}rcl@{}} f_{KN}(x)=\frac{\alpha\beta}{\sigma} \left\{\Phi\left[\left(\frac{x\mu}{\sigma}\right)\right]\right\}^{\alpha1} \left\{1\left[\Phi\left(\frac{x\mu}{\sigma}\right)\right]^{\alpha}\right\}^{\beta1}\phi\left(\frac{x\mu}{\sigma}\right). \end{array} $$

The odd loglogistic normal (OLLN) distribution (special case of OLLLN distribution when β→1) with density (Braga et al. 2016)$$\begin{array}{@{}rcl@{}} f_{OLLN}(x)=\frac{\alpha\, \phi\left(\frac{x\mu}{\sigma}\right)[\Phi\left(\frac{x\mu}{\sigma}\right) ]^{\alpha1} [1\Phi\left(\frac{x\mu}{\sigma}\right)]^{\alpha1}}{\sigma \{[1\Phi\left(\frac{x\mu}{\sigma}\right)]^{\alpha}+[\Phi\left(\frac{x\mu}{\sigma}\right)]^{\alpha}\}^{2}}, \end{array} $$
where \(x\in \mathbb {R}\), \(\mu \in \mathbb {R}\), α>0, β>0 and σ>0.
8.1 Application 1
First, we consider the data set representing the failure times of a particular windshield device. These data were also studied by Blischke and Murthy (2000) and Murthy et al. (2004). The data, referred as D1, are: 0.040, 1.866, 2.385, 3.443, 0.301, 1.876, 2.481, 3.467, 0.309, 1.899, 2.610, 3.478, 0.557, 1.911, 2.625, 3.578, 0.943, 1.912, 2.632, 3.595, 1.070, 1.914, 2.646, 3.699, 1.124, 1.981, 2.661, 3.779, 1.248, 2.010, 2.688, 3.924, 1.281, 2.038, 2.823, 4.035, 1.281, 2.085, 2.890, 4.121, 1.303, 2.089, 2.902, 4.167, 1.432, 2.097, 2.934, 4.240, 1.480, 2.135, 2.962, 4.255, 1.505, 2.154, 2.964, 4.278, 1.506, 2.190, 3.000, 4.305, 1.568, 2.194, 3.103, 4.376, 1.615, 2.223, 3.114, 4.449, 1.619, 2.224, 3.117, 4.485, 1.652, 2.229, 3.166, 4.570, 1.652, 2.300, 3.344, 4.602, 1.757, 2.324, 3.376, 4.663.
The MLEs of the parameters and SEs in parentheses and the goodnessoffit statistics for D1
Model  μ  σ  α  β  \(\hat \ell _{n}\)  W ^{∗}  A ^{∗}  KS  pvalue 

OLLLN  2.887  0.532  0.379  1.639  126.178  0.035  0.331  0.053  0.963 
(0.212)  (0.180)  (0.192)  (1.947)  
Normal  2.557  1.112  128.119  0.092  0.608  0.092  0.445  
(0.121)  (0.086)  
EN  1.832  1.337  1.939  128.064  0.075  0.522  0.085  0.557  
(2.366)  (0.708)  (3.878)  
BN  0.808  2.443  7.113  2.469  128.085  0.074  0.520  0.084  0.562 
(7.151)  (8.162)  (48.582)  (14.619)  
GN  2.805  0.541  0.290  0.197  127.757  0.058  0.438  0.075  0.7105 
(1.059)  (0.265)  (0.382)  (0.216)  
KN  1.654  0.748  0.920  0.320  127.848  0.063  0.469  0.079  0.642 
(1.067)  (0.539)  (1.017)  (0.524)  
LN  3.172  1.079  7.091  127.570  0.048  0.391  0.066  0.828  
(0.562)  (0.094)  (16.134)  
OLLN  2.626  0.603  0.452  127.062  0.076  0.523  0.095  0.408  
(0.126)  (0.218)  (0.232) 
8.2 Application 2
The second data set D2 consists of lifetimes of 43 blood cancer patients (in days) from one of the Health Hospitals in Saudi Arabia (Abouammoh and Abdulghani 1994). These data are: 115, 181, 255, 418, 441, 461, 516, 739, 743, 789, 807, 865, 924, 983, 1025, 1062, 1063, 1165, 1191, 1222, 1222, 1251, 1277, 1290, 1357, 1369, 1408, 1455, 1478, 1519, 1578, 1578, 1599, 1603, 1605, 1696, 1735, 1799, 1815,1852, 1899, 1925, 1965.
The MLEs of the parameters and SEs in parentheses and the goodnessoffit statistics for D2
Model  μ  σ  α  β  \(\hat \ell _{n}\)  W ^{∗}  A ^{∗}  KS  pvalue 

OLLLN  930.893  215.288  0.323  0.854  325.684  0.018  0.157  0.057  0.997 
(151.780)  (93.152)  (0.224)  (0.202)  
Normal  1191.628  500.709  328.303  0.068  0.489  0.083  0.901  
(76.357)  (53.993)  
EN  2083.910  52.5006  0.005  325.847  0.020  0.174  0.076  0.9485  
(81.898)  (14.640)  (0.003)  
BN  2153.003  173.89  0.0496  4.139  325.680  0.017  0.157  0.059  0.996 
(313.548)  (512.195)  (0.268)  (6.681)  
GN  2138.76  251.154  0.101  2.607  326.237  0.020  0.183  0.057  0.998 
(0.036)  (0.036)  (0.016)  (1.319)  
KN  1825.282  71.445  0.0097  0.529  325.4645  0.018  0.162  0.058  0.997 
(288.596)  (65.481)  (0.019)  (0.319)  
LN  748.777  472.416  0.974  326.628  0.026  0.221  0.064  0.991  
(275.414)  (71.311)  (0.070)  
OLLN  1131.115  223.172  0.338  327.292  0.058  0.415  0.125  0.473  
(75.566)  (99.640)  (0.231) 
8.3 Application 3
The MLEs of the parameters and SEs in parentheses and the goodnessoffit statistics for D3
Model  μ  σ  α  β  \(\hat \ell _{n}\)  W ^{∗}  A ^{∗}  KS  pvalue 

OLLLN  102.050  7.879  0.100  0.521  188.193  0.024  0.206  0.076  0.9705 
(5.611)  (5.275)  (0.119)  (0.381)  
Normal  110.214  37.873  192.021  0.109  0.671  0.144  0.378  
(6.144)  (4.344)  
EN  169.145  13.261  0.062  191.360  0.074  0.491  0.113  0.674  
(26.432)  (13.067)  (0.142)  
BN  172.325  18.326  0.109  1.785  191.273  0.068  0.465  0.105  0.759 
(22.216)  (14.485)  (0.160)  (2.635)  
GN  172.393  17.696  0.102  1.340  191.263  0.068  0.465  0.106  0.752 
(20.263)  (15.038)  (0.159)  (2.564)  
KN  129.188  13.163  0.082  0.361  191.071  0.065  0.450  0.111  0.695 
(30.610)  (5.980)  (0.104)  (0.297)  
LN  62.065  33.056  0.997  190.819  0.059  0.417  0.109  0.717  
(23.996)  (6.044)  (0.011)  
OLLN  105.546  7.830  0.098  188.655  0.034  0.271  0.110  0.7025  
(4.163)  (4.921)  (0.110) 
In summary, we conclude that the OLLLN distribution outperforms all the fitted competitive models under the selected criterion for D1, D2 and D3. For all three data sets, we verify that the fitted OLLLN distribution best captures the three histograms, especially for the third data set, which indicates the outstanding performance of this distribution.
8.4 Application 4: OLLLG longterm survival models
These data consist of n=493 lifetimes (t _{ i } in months) of patients diagnosed with breast cancer. The steps to construct these data can be found in Gendoo et al. (2015). In many applications there is qualitative information about the hazard shape, which can help for selecting a particular polyhazard model. In this context, a device called the total time on test (TTT) plot is useful. The TTT plot is obtained by plotting \(\mathsf {G}(r/n)=\left [\left (\sum _{i=1}^{r}T_{i:n}\right)+(nr)T_{r:n}\right ]/\left (\sum _{i=1}^{n}T_{i:n}\right)\), where r=1,…,n, and T _{ i:n } (for i=1,…,n) are the order statistics of the sample, against r/n. It is a straight diagonal for constant hazards leading to an exponential model. It is convex for decreasing hazards and concave for increasing hazards leading to a singleWeibull model. It is first convex and then concave if the hazard is bathtubshaped leading to a biWeibull model. It is first concave and then convex if the hazard is bimodalshaped leading to a loglogistic model. For multimodal hazards, the TTT plot contains several concave and convex regions. The TTT plot in Fig. 10 a indicates an increasingdecreasingincreasing hrf. So, the OLLLW distribution would be a good option to model these data.
MLEs, SEs in parentheses and goodnessoffit statistics for breast cancer data
Model  μ  σ  α  β  p  AIC  BIC 

OLLLWcr  5.933  0.006  0.317  9.131  0.575  1878.15  1899.15 
(0.079)  (0.001)  (0.084)  (4.319)  (0.061)  
OLLWcr  0.778  0.036  1.788    0.569  1886.37  1903.17 
(0.085)  (0.009)  (0.426)    (0.059)  
LWcr  1.581  0.007    1.334  0.597  1885.27  1902.07 
(0.188)  (0.001)    (1.712)  (0.044)  
Weibull cure rate  1.027  0.040      0.550  1900.03  1912.63 
(0.082)  (0.008)      (0.064) 
Conclusions
We study some mathematical properties of the odd loglogistic logarithmicG family of distributions with two extra shape parameters α>0 and β∈(0,1). We provide some special models, a very useful linear representation for the density function in terms of exponentiated densities, explicit expressions for the moments, generating function, entropies and order statistics. The model parameters are estimated by the method of maximum likelihood. We perform a simulation study to verify the adequacy of the estimators. We also introduce a longterm survival model based on the new family. The importance of the proposed models is illustrated by means of four real life data sets. The new models provide consistently better fits than other competitive models for the current data.
Appendix 1: Asymptotes and shapes
Corollary 1
Corollary 2
If x=x _{0} is a root of (18) then it corresponds to a local maximum (minimum) if λ(x _{0})<0 (λ(x _{0})>0). It refers to a point of inflexion if λ(x _{0})=0.
There may be more than one root to (19).
Appendix 2: Useful power series
The power series derived in this appendix are required for the proofs of the linear representations in Section 3. All power series given below are convergent for u≤1. In Sections 3 and 5.1, they can be applied for the support of X since the quantity \(\beta G(x,\boldsymbol {\xi })^{\alpha }/[G(x,\boldsymbol {\xi })^{\alpha }+\bar {G}(x,\boldsymbol {\xi })^{\alpha }]\) does belong to the interval (0,1) when β∈(0,1).
which holds for u≤1.
where \(s^{*}_{k}(\alpha,w,\gamma)=\sum \limits _{j=0}^{k}s_{1,j}^{*}(\alpha,w)\,s_{2,kj}^{*}(\alpha,w,\gamma)\). Equation (28) is the main result to obtain the Rényi entropy in Section 5.1.
Appendix 3: Properties for a special model
The OLLLE distribution is defined by inserting G(x)=1−e^{−λ x } and g(x)=λ e^{−λ x } in Eq. (3), where x>0 and λ>0. Let X be the random variable representing this distribution. We derive some statistical measures of X from the asymptotics in Appendix 1 and the general results in Sections 4 and 5.2
Corollary 3
Corollary 4
These equations can provide the effects of the parameters on the tails of the OLLLE distribution.
where d _{ k } is defined in Eq. (6).
where \(\gamma (a,z)=\int _{0}^{z}t^{a1}\,\mathrm {e}^{t}\,dt\) denotes the incomplete gamma function.
where s _{ k } is given by (11).
Declarations
Acknowledgements
The authors are very grateful to the editor Felix Famoye for helpful comments and suggestions.
Authors’ contributions
The five authors jointly participated of the study and elaborated the research. Gauss Cordeiro corrected the manuscript throughout. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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