A New Lindley-Burr XII Distribution: Model, Properties and Applications

A new distribution called the Lindley-Burr XII (LBXII) distribution is proposed and studied. Some structural properties of the new distribution including moments, conditional moments, distribution of the order statistics and Rényi entropy are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study to examine the bias and mean square error of the maximum likelihood estimators is presented and applications to real data sets in order to illustrate the usefulness of the new distribution are given.


Introduction
There are several generalizations of the Lindley distribution that are considered to be useful life models, and are suitable for modeling data with different types of hazard rate functions including increasing, decreasing, bathtub and unimodal. (Lindley, 1958) used a mixture of exponential and length-biased exponential distributions to illustrate the difference between fiducial and posterior distributions. This mixture is called the Lindley (L) distribution. There are several generalizations of the Lindley distribution in the literature including the works by (Oluyede, Yang and Omolo, 2015) and (Nadarajah, Bakouch and Tahmasbi, 2011).  presented the beta generalized Lindley distribution. (Ghitany, Al-Mutairi, Balakrishnan and Al-Ezeni, 2013) presented a two-parameter power Lindley distribution. (Zakerzadeh and Dolati, 2009) studied an extension of the Lindley distribution. These models constitute flexible family of distributions in terms of the varieties of shapes and hazard functions. The cumulative distribution function (cdf) of the Lindley distribution is given by G L (x; λ) = 1 − λ + 1 + λx λ + 1 e −λx , for x > 0, and λ > 0.
We conisider the following generalization of the Lindley and Burr XII distributions. Let X 1 and X 2 be independent random variables with Lindley and Burr XII distributions, respectively. That is, F X 1 (x) = 1− λ+1+λx λ+1 e −λx , and F X 2 (x) = 1−(1+ x c ) −k , and for λ, c, k > 0. If a random variable X has the LBXII distribution, we write X ∼ LBXII (λ, c, k). The parameters c and k are shape parameters and λ is a scale parameter. Note that the LBXII pdf can be written as follows: g LBXII (x; λ, c, k) = g L (x; λ)G B (x; c, k) + G L (x; λ)g B (x; c, k), where G B (x; c, k) = 1 − G(x; c, k), and G L (x; λ) = 1 − G(x; λ) are the survival functions of Burr XII and Lindley distributions, respectively, and g L (x; λ), g B (x; c, k) are the pdf's of Lindley and Burr XII distributions.
Plots of the LBXII pdf shows different shapes including right skewed, almost symmetric and reverse J-shaped as indicated in Figure 1.

Sub-models of LBXII Distribution
In this subsection, we discuss some special models of the LBXII distribution.
• If c = k = 1, we have the one parameter distribution with cdf for x > 0, and λ > 0.

Hazard Rate and Quantile Functions
In this section, we present the hazard rate and quantile functions of the LBXII distribution. We note from equation (7) that the hazard rate function of the LBXII distribution is given by where h G L (x) and h G B (x) are the hazard rate functions of the Lindley and Burr XII distributions, respectively, that is, Plots of the LBXII hazard rate function shows different shapes including decreasing, increasing, as well as upside down bathtub shapes as shown in Figure 2. The quantile function of the LBXII distribution is obtained by solving the non-linear equation: Thus, random numbers can be readily generated from the LBXII distribution by numerically solving the non-linear equation (12). Quantiles of the LBXII distribution are given in Table 1.

Moments, Conditional Moments and Mean Deviations
In this section, we present the moments, conditional moments and mean deviations of the LBXII distribution.

Moments
The r th moment of the LBXII distribution is given by International Journal of Statistics and Probability Vol. 10, No. 4;2021 where we have applied the transformation t = (1 + x c ) −1 , and B(a, b) = ∫ 1 0 t a−1 (1 − t) b−1 dt is the complete beta function. The coefficients of variation (CV), Skewness (CS) and Kurtosis (CK) can be readily obtained. The variance (σ 2 ), Standard deviation (SD=σ), coefficient of variation (CV), coefficient of skewness (CS) and coefficient of kurtosis (CK) are given by respectively. Some moments for selected parameter values for the LBXII distribution are given in Table 2 and 3D plots for skewness and kurtosis are presented in Figure 3, Figure 4 and Figure 5, respectively. The 3D plots shows the dependence of skewness and kurtosis on the shape parameters.

Conditional Moments
In this subsection, the r th conditional moment is presented. The mean residual life function as well as income inequality measures such as Bonferroni and Lorenz curves as well as mean deviation about the mean and mean deviation about the median can be readily obtained from the conditional moments. The r th conditional moment is given by where

Mean Deviation, Lorenz and Bonferroni Curves
Mean deviation about the mean and mean deviation about the median as well as Lorenz and Bonferroni Curves for the LBXII distribution are presented in this subsection. The application of Lorenz and Bonferroni curves is not limited to economics for the study of income and poverty, but they can also be used to other field like reliability, demography, insurance and medicine.

Mean Deviations
The mean deviation about the mean and the mean deviation about the median are defined by respectively, where µ = E[X] and M = Median(X) denotes the mean and median, respectively. We note that δ 1 (x) and δ 2 (x) can be expressed as where

Bonferroni and Lorenz Curves
Bonferroni and Lorenz curves for the LBXII didtribution are given as

Order Statistics and Rényi Entropy
In this section, we present the pdf of the i th order statistic from the LBXII distribution as well as Rényi entropy.

Order Statistics
Suppose that X 1 , · · · , X n is a random sample of size n from a continuous pdf, g(x). Let X 1:n < X 2:n < · · · < X n:n denote the corresponding order statistics. If X 1 , · · · , X n is a random sample from LBXII distribution, it follows from the equations (5) and (6) that the pdf of the i th order statistic from the LBXII distribution is given by Therefore, the cdf of the i th order statistic is given by where Consequently, the pdf of the i th order statistic from the LBXII distribution is a linear combination of the exponentiated LBXII (ELBXII) pdf's with parameters: s + i > 0, λ > 0, c > 0, and k > 0.

Rényi Entropy
Rényi entropy (Rényi, 1960) is an extension of Shannon entropy. Rényi entropy is defined to be Rényi entropy tends to Shannon entropy as v → 1. Note that where B(a, b) = ∫ 1 0 t a−1 (1 − t) b−1 dt is the complete beta function. Consequently, Rényi entropy for the LBXII distribution is given by

Maximum Likelihood Estimates
The maximum likelihood estimates (MLEs) of the parameters of the LBXII distribution are presented in this section. Let X i ∼ LBXII(λ, c, k) and ∆ = (λ, c, k) T be the parameter vector. The log-likelihood ℓ = ℓ(∆) based on a random sample of size n is given by Elements of the score vector U = ( ∂ℓ ∂k , ∂ℓ ∂λ , ∂ℓ ∂c ) are given by: and respectively. The equations obtained by setting the above partial derivatives to zero are not in closed form and the values of the parameters k, λ, c must be found via iterative methods. The maximum likelihood estimates of the parameters, denoted by∆ is obtained by solving the non-linear equation ( ∂ℓ ∂k , ∂ℓ ∂λ , ∂ℓ ∂c ) T = 0, using a numerical method such as Newton-Raphson procedure. The Fisher information matrix is given by , i, j = 1, 2, 3, can be numerically obtained by MATLAB or NLMIXED in SAS or R software. The total Fisher information matrix nI(∆) can be approximated by For a given set of observations, the matrix given in equation (26) is obtained after the convergence of the Newton-Raphson procedure via NLMIXED in SAS or R software.

Monte Carlo Simulations
In this section, the performance of the maximum likelihood estimates is examined by conducting simulation studies for different sample sizes. We conducted various simulations for different sizes sample (n=25, 50, 100, 200, 400, 800) via the R package. We simulate N = 1000 samples for the true parameters values given in the Table 3 and Table 4. The Average Bias and Root Mean Square Error (RMSE) were computed. The average bias and RMSE for the estimated parameterθ, say, are given by: respectively. The tables list the mean MLEs of the parameters along with the respective root mean squared errors (RMSEs) and Average Bias.We note that as the sample size n increase the mean estimates of the parameters tend to be closer to the true parameter value and the RMSEs and Average Bias decay towards zero.
The maximum likelihood estimates (MLEs) of the LBXII parameters and its sub-models are computed by maximizing the objective function via the subroutine NLMIXED in SAS as well as the function nlm in R. The estimated values of the parameters (standard errors in parenthesis), -2log-likelihood statistic (−2 ln(L)), Akaike Information Criterion (AIC = 2p − 2 ln(L)) Bayesian Information Criterion (BIC = p ln(n) − 2 ln(L)), and Consistent Akaike Information Criterion (AICC = AIC + 2 p(p+1) n−p−1 ), where L = L(∆) is the value of the likelihood function evaluated at the parameter estimates, n is the number of observations, and p is the number of estimated parameters. Tables 6, 8 and 10 shows results for the data sets for LBXII distribution and several non-nested models.
Plots of the fitted densities, the histogram of the data and probability plots (Chambers, Cleveland, Kleiner and Turkey, 1983) are given in Figure 6, Figure 7, and Figure 8, 9, 10 and 11, respectively. For the probability plot, we plotted G LBXII (x ( j) ;λ,ĉ,k) against j − 0.375 n + 0.25 , j = 1, 2, · · · , n, where x ( j) are the ordered values of the observed data. The measures International Journal of Statistics and Probability Vol. 10, No. 4;2021 of closeness are given by the sum of squares The goodness-of-fit statistics W * and A * , described by (Chen and Balakrishnan, 1995) are presented in the tables. The Kolmogorov-Smirnov (K-S) statistic and its p-value 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 * , A * and K-S, the better the fit.
For the failure times of 50 components data, the values of the statistic AIC, AICC and BIC are the smallest for the L-BXII distribution. Also the goodness of fit statistics W * , A * and K − S , are the lowest. The P-value and the S S values from the probability plots indicate that indeed our model performs better than the non-nested models that were considered.

Epoxy Strands Failure at 90% Stress Level Data
The second data set represent the stress-rupture life of kevlar 49/epoxy strands which were subjected to constant sustained pressure at the 90% stress level until all had failed, so that we have complete data with exact times of failure, given by (Barlow, Toland and Freeman, 1984) and (Andrews and Herzberg, 1985).  The estimated variance-covariance matrix for the LBXII of the Epoxy strands failure at 90% stress level data is given by For epoxy strands failure at 90% stress level data, the values of the AIC, AICC and BIC are smaller for the LBXII distribution also the values of the statistics W * , A * , K − S and its P-value, and the S S value indicate that the LBXII distribution performs better than all the models as indicated in Table 8. Vol. 10, No. 4;2021 Figure 9. Probability plots for epoxy strands data

Wheaton River Data
The third data set are the exceedances of flood peaks (in m3/s) of the Wheaton River near Carcross in Yukon Territory, Canada. The data consist of 72 exceedances for the years 1958-1984. These data were also analyzed by (Akinsete, Famoye and Lee, 2008).   Figure 10. Fitted pdf's for Wheaton River data Similarly for Wheaton River data , the values of the statistics K − S , P-value, and S S from the probability plots shows that the LBXII distribution performs better for this data set. In overall the LBXII distribution performs better PGW, MW, EPL, LW, WE and the MOLLD distributions for the Wheaton River data.  Figure 11. Probability plots for Wheaton River data

Concluding Remarks
A new three parameter distribution called the Lindley-Burr XII (LBXII) distribution is presented, which extends the Burr XII (BXII) distribution. The LBXII distribution has two components namely the Lindley and the Burr XII distributions. The LBXII distribution possesses hazard function with flexible behaviour. We derive expressions for the moments, mean and median deviations, distribution of order statistics and entropy. Maximum likelihood estimation technique is used to estimate the model parameters. The performance of the LBXII distribution was examined by conducting various simulations for different sizes. Finally, the distribution was fitted to real data sets to illustrate empirically that the LBXII provides consistently better fits to two real data sets than other six non-nested distributions with same number of parameters.
The two examples proves that the LBXII is a desired alternative for modelling survival data. The LBXII include several other distributions, hence, it is our belief that the LBXII will attract wider applications in several different areas of research.