A New Generalized Family of Odd Lindley-G Distributions With Application

A new family of distributions, namely the Kumaraswamy Odd Lindley-G distribution is developed. The new density function can be expressed as a linear combination of exponentiated-G densities. Statistical properties of the new family including hazard rate and quantile functions, moments and incomplete moments, Bonferroni and Lorenz curves, distribution of order statistics and Rényi entropy are derived. Some special cases are presented. We conduct some Monte Carlo simulations to examine the consistency of the maximum likelihood estimates. We provide an application of KOL-LLo distribution to a real data set.


Introduction
There are considerable amount of work in the literature on the modification of the beta distribution including work by (Eugene, Lee and Famoye 2002), (Nadarajah and Kotz, 2004), (Nadarajah and Kotz, 2006), (Cordeiro, Gomes, da Silva, and Ortega, 2013), (Oluyede and Yang, 2015), and (Makubate, Oluyede, Motobetso, Huang and Fagbamigbe, 2018), to mention a few. These extended models exhibited very interesting properties because of the two extra shape parameters that make it possible to explore skewness, kurtosis and tail properties inherent in some data. The core issue with beta generated distributions is lack of tractability and this is mainly caused by the involvement of the incomplete beta function in the cumulative distribution function (cdf). (Kumaraswamy, 1980) proposed a new distribution called the Kumaraswamy distribution. This new distribution has a wide application in hydrology. (Jones, 2009) studied the properties of the Kumaraswamy distribution and highlighted some of its similarities to the beta distribution and also its desirable tractability property over the beta distribution.
(Gomes-Silva et al., 2017) developed the odd Lindley-G (OL-G) distribution, whose cdf and probability density function

The Model
We develop the KOL-G distribution using the generalization proposed by (Cordeiro et al., 2011), and taking G(x) and g(x) to be respective cdf and pdf of the Odd Lindley-G distribution by (Gomes-Silva et al., 2017). The cdf and pdf of the new KOL-G distribution are given by ( 3) and f KOL−G (x) = ab λ 2 (1 + λ) respectively, for a, b, λ, ξ > 0 and x > 0. We consider the KOL-G distribution where the baseline cdf G(x; ξ) has at most two parameters. This allows us to avoid issues of overparametrization with respect to the number of parameters coming from the baseline cdf G(x; ξ).

Sub-Classes of KOL-G Class of Distributions
• We obtain odd Lindley-G distribution from KOL-G distribution by setting a = b = 1.
• We also obtain a power function distribution from KOL-G distribution by setting a = 1, with pdf given by • The exponentiated-odd Lindley-G (E-OLG) distribution is obtained by setting b = 1, with pdf given by

Expansion of the KOL-G Density
Consider , then using series expansion for |z| < 1 and b > 0, we can write the KOL-G pdf as follows: Now, using the following series expansion Applying the series expansion Applying the generalized binomial expansion we can write KOL-G pdf as Therefore, the KOL-G pdf can be written as where g p+q (x; ξ) = (p + q + 1)g(x; ξ)[G(x; ξ)] p+q is the exponentiated-G (Exp-G) density function with power parameter p + q and The KOL-G pdf is a mixture of the Exp-G densities, with parameter vector ξ and power parameter (p + q). Statistical properties of KOL-G distribution can be derived directly from the Exp-G family of distributions.

Hazard Rate and Quantile Functions
We get the hazard rate function of the KOL-G distribution by dividing KOL-G density function by its survival function.
where f KOL−G (x) is given by equation (4),
Consequently, the quantile function of the KOL-G family of distributions reduces to A table of quantiles for selected parameter values of the special case of KOL-log-logistic (KOL-LLo) distribution is given in section 5.

Moments, Incomplete Moments, Moment Generating and Characteristic Functions
We assume that W p+q ∼ Exp-G(p + q) and let X ∼ KOL-G(a, b, λ; ξ), then the s th moment can be obtained from equation (6) as follows: where E(W s p+q ) denotes the s th moment of W p+q which follows an Exp-G distribution with parameter (p + q) and v p,q is as defined in equation (7). Furthermore, the incomplete moments can be obtained as follows where I p+q (t) = t 0 x s g p+q (x; ξ)dx. The moment generating function (mgf) of X is given by where E(e tW p+q ) is the mgf of the Exp-G distribution and v p,q is as defined in equation (7). Furthermore, we can obtain the characteristic function and is given by where φ p+q (t) is the characteristic function of Exp-G distribution and v p,q is as defined in equation (7).
Note that the r th cumulant of the random variable X can be readily obtained from the recursive relationship: κ r = µ r − r−1 s=1 r−1 s−1 µ r−s κ s , where µ r = E(X − µ 1 ) r , so that the CS and CK are given by γ 1 = κ 3

Mean Deviations, Bonferroni and Lorenz Curves
Let X ∼ KOL-G(a, b, λ, ξ), the mean deviation about the mean and about the median are defined by respectively where µ = E(X) and M = Median(X). The deviations can also be expressed as and Bonferroni and Lorenz curves are given by and where I * p+q (t) = t 0 xg p+q (x; ξ)dx, is the first incomplete moment of the Exp-G distribution and v p,q is as given in equation (7).

Distribution of Order Statistics
Let X 1 , X 2 , ..., X n be independent and identically distributed (i.i.d) random variables distributed according to (4). The pdf of the i th order statistic X i:n , is given by Substituting the KOL-G cdf, we get , then using the generalized binomial series expansion , and applying the same generalized binomial series expansion to we have Again applying the generalized binomial series expansion to we can write the pdf of the i th order statistic as Using series expansion and generalized binomial expansion we can then write the pdf of i th order statistic as Substituting the expansion of f KOL−G (x) density, we get Consequently, the pdf of the i th order statistic of KOL-G distribution can be expressed as where g p+q+r+z (x; ξ) = (p+q+r +z+1)g(x; ξ)[G(x; ξ)] p+q+r+z is the Exp-G distribution with power parameter (p+q+r +z) and Therefore, the i th order statistic of KOL-G distribution is a mixture of Exp-G densities with power parameter (p+q+r +z). Results in equation (17) are important and can be used to find the moments of the i th order statistic of KOL-G distribution.

Rényi Entropy
Entropy measures variation of uncertainty and can be quantified by two popular measures, namely Shannon entropy by (Shannon, 1951) and Rényi entropy by (Rényi, 1961). Rényi entropy is defined by where ν > 0 and ν 1. We derive an expression of Rényi entropy of the KOL-G family of distributions. Substituting f (x; ξ) by KOL-G pdf and consider the following integral: Applying the generalized binomial series expansion to we get Using the series expansion binomial expansion followed by generalized binomial expansion which can be written as Consequently, Rényi entropy of KOL-G family of distributions can be expressed as where

Maximum Likelihood Estimation
Let Θ = (a, b, λ, ξ) T be a p × 1 parameter vector. The total log-likelihood function for Θ is given by The first derivative of the log-likelihood function with respect to the parameters Θ = (a, b, λ, ξ) T , are

Special Cases
Some cases which includes Kumaraswamy Odd Lindley-uniform (KOL-U), Kumaraswamy Odd Lindley-log-logistic (KOL-LLo), and Kumaraswamy Odd Lindley-Weibull (KOL-W) distributions are presented. Statistical properties, distribution of order statistics and Rényi entropy of these cases are also studied.

Kumaraswamy-Odd-Lindley-Uniform (KOL-U) Distribution
Suppose the baseline distribution is a uniform distribution on the interval (0, θ) with θ > 0. The baseline pdf and cdf are g(x; θ) = 1/θ and G(x; , θ) = x/θ, respectively. The KOL-U distribution have cdf and pdf given by and respectively. Graphs of the KOL-U pdf take various shapes including platykurtic, left skewed and unimodal. Some of the sub-models of the KOL-U distribution are presented below.
• We obtain Odd Lindley-Uniform (OL-U) distribution from KOL-U distribution by setting a = b = 1.
• We obtain a power function distribution denoted by KOL − U(1, b, λ, θ) from KOL-U distribution by setting a = 1, with pdf given by • The exponentiated-Odd Lindley-U (EOL-U) distribution denoted by KOL − U(a, 1, λ, θ) is obtained from the KOL-U distribution by setting b = 1, with pdf given by

Hazard Rate and Quantile Functions
Hazard rate of KOL-U distribution is computed by the following formula where f KOL−U (x) is given by equation (23),F KOL−U (x) = 1 − F KOL−U (x) and F KOL−U (x) is the KOL-U cdf as in equation (22). Hazard function for KOL-U distribution can take various shapes including bathtub and increasing for the selected parameter values.
We obtain the quantile function of KOL-U from equation (11) and is given by
KOL-LLo density function takes various shapes including right skewed, left skewed and symmetric.

Nested Models
• We obtain odd Lindley-LLo distribution from KOL-LLo distribution by setting a = b = 1.

Hazard Rate and Quantile Functions
The hazard function for KOL-LLo distribution is defined by where f KOL−LLo (x) is given by equation (26), is the KOL-LLo cdf as in equation (25).

Figure 4. Plots of KOL-LLo Hazard Rate Function
Hazard function for KOL-U distribution can take various shapes including bathtub, decreasing and increasing for the selected parameter values. We obtain the quantile function of KOL-LLo distribution from equation (11), and is given by

Moments
Let X ∼ KOL − LLo(a, b, λ; c), the s th moment of KOL-LLo distribution comes directly from equation (6), and is given by v p,q,m,n E(W s p+q ), where E(W s p+q ) denotes the s th moment of W p+q which follows an E-LLo distribution with power parameter (p + q) and v p,q,m,n is as defined in equation (7).
We present some results on the first six moments and the measures of dispersion for the KOL-LLo distribution for some parameter values. In the first set of results, we fixed the parameters a = 2.0 and b = 1.0 and fixing the parameters λ = 1.5 and c = 1.5 in the second set of results. Table 2. Moments of the KOLL-Lo distribution for some parameter values (fixing a = 2.0 and b = 1.0 and fixing λ = 1.5 and c = 1.5) λ=3.5, c=1.5 λ=3.5, c=0.8 λ=1.5, c=1.5 λ=0.8, c=3.5 a=1.0, b=1.5 a=2.5, b=2.5 a=1.0, b=1.0 a=1.5, b=1.

Kumaraswamy Odd Lindley-Weibull (KOL-W) Distribution
Suppose the baseline distribution is Weibull distribution with parameters c and k, with pdf and cdf given by g(x; c, k) = k c x c k−1 e −( x c ) k and G(x; c, k) = 1 − e −( x c ) k , respectively, then the cdf and pdf of KOL-W distribution are given by and respectively. Plots of the KOL-W density function takes various shapes including decreasing, unimodal, left and right skewed for the selected parameter values. • We also obtain a power function distribution from KOL-W distribution by setting a = 1, given by • The exponentiated-odd Lindley-Weibull (EOL-W) distribution is obtained by setting b = 1, with pdf given by • By setting k = 1 and k = 2, we obtain the Kumaraswamy odd Lindley-exponential (KOL-E) distribution and Kumaraswamy odd Lindley-Rayleigh (KOL-R) distribution with pdfs given by and respectively.

Hazard Rate and Quantile Functions
The hazard rate function of the KOL-W distribution was obtained by dividing KOL-W density function by its survival function where f KOL−W (x) is given by equation (28),F KOL−W (x) = 1 − F KOL−W (x) and F KOL−W (x) is the KOL-W cdf as in equation (27). Hazard function for KOL-W distribution can take various shapes including bathtub, decreasing and increasing for the selected parameter values. We obtain the quantile function of KOL-W distribution from equation (11), and is given by

Simulation Study
A simulation study for the KOL-LLo distribution was conducted. The simulation study was repeated for N=1000 times with sample size n= 25, 50, 100, 200, 400 and 800. The following parameter values were used in the simulation study: a = 1.0, b = 1.0, λ = 0.5, c = 0.5 and a = 1.5, b = 1.0, λ = 1.5, c = 0.5. The average bias (ABias) and Root Mean Square Errors (RMSEs) were computed and are given by: respectively. Tables 6.1 and 6.2 presents the mean MLEs of the model parameters along with the respective average bias and root mean square errors (RMSE) for the KOL-LLo distribution for selected parameter values. Note that from the tabulated results, we can verify that as the sample size increases, the mean estimates of the parameters tend to be closer to the true parameter values, since average bias and RMSEs decays toward zero for all the parameters.

Application
In this section, we present an application of the KOL-LLo distribution to a real data set to illustrate the flexibility of the new distribution. We estimated the model parameters using the bbmle package in R. The estimated values of the model parameters (standard error in parenthesis), −2 log-likelihood statistics (−2 ln(L)), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Consistent Akaike Information Criterion (AICC) The Cramer-von Mises and Anderson-Darling goodness-of-fit statistics W * and A * described by (Chen and Balakrishnan, 1995) are also presented. These statistics can be used to verify which distribution fits better to the data. In general, the smaller the values of W * and A * , the better the fit. The AdequacyModel package in R (R Development Core Team, 2011) was used to evaluate the fitted distributions.
We can use the likelihood ratio (LR) test to compare the fit of the KOL-LLo distribution with its sub-models for a given data set. For example, to test a = b = 1, the LR statistic is X 2 = 2[ln(L(â,b,λ,ĉ)) − ln(L(1, 1,λ,c))], whereâ,b,λ andĉ, are the unrestricted estimates, andλ, andc are the restricted estimates. The LR test rejects the null hypothesis if X 2 > χ 2 d , where χ 2 d denote the upper 100d% point of the χ 2 distribution with 2 degrees of freedom. Plots of fitted densities and the histogram of the data are given. Probability plots (Chambers, Cleveland, Kleiner and Tukey, 1983) are also presented. Furthermore, values of the sum of squares S S = n j=1 F KOL−LLo (x ( j) ) − j−0.375 n+0.25 2 is presented. This statistic is used to compare the distributions and are given in the probability plots.

Plasma Concentration Data
The KOL-LLo distribution was fitted to the plasma concentration data. The data set was taken from R base package in the Indometh object. The fitted KOL-LLo model for plasma concentration data was compared to the fits of its nested models and four non-nested models, namely exponentiated Dagum (ED), gamma log-logistic (GLLo) and gamma Burr III (GB) distributions (See (Oluyede, Huang and Pararai, 2014), for details) and the exponentiated power generalized Weibull distributions (EPGW) ( Fernando, Pe na-Ramírez, Renata, Cordeiro and Marinho, 2018) distributions. Gamma  (Oluyede et al., 2014) is given by When α = 1, β = 1 and λ = 1, GD distribution reduces to ED, GLLo ad GB distributions, respectively.
The pdf of EPGW is given by for λ, γ, α, β > 0, and x > 0. Likelihood ratio test results are shown in Table 6. From results of the likelihood ratio test, we conclude that there are significant differences between the KOL-LLo distribution and the nested models. Also based on the lowest values of the goodness-of-fit statistics A * , W * and K-S statistic and bigger p-value, the KOL-LLo model fits the data better than the ED, GLLo, GB and EPGW models.

Concluding Remarks
We have developed a new family of distributions, namely Kumaraswamy odd Lindley-G, which is a generalization of the odd Lindley-G distribution. We derived the statistical properties of the new family of distributions. The KOL-G new family can be expressed as a mixture of Ex-G densities. Special cases were also presented, namely, KOL-U, KOL-LLo and KOL-W distributions. Application of the KOL-LLo distribution to a real data set shows that it performed better than the nested models and non-nested ED, GLLo, GB and EPGW distributions for the plasma concentration data.