The Lindley Weibull Distribution : properties and applications

We introduce a new three-parameter lifetime model called the Lindley Weibull distribution, which accommodates unimodal and bathtub, and a broad variety of monotone failure rates. We provide a comprehensive account of some of its mathematical properties including ordinary and incomplete moments, quantile and generating functions and order statistics. The new density function can be expressed as a linear combination of exponentiated Weibull densities. The maximum likelihood method is used to estimate the model parameters. We present simulation results to assess the performance of the maximum likelihood estimation. We prove empirically the importance and flexibility of the new distribution in modeling two data sets.

We propose a new generalization of the Weibull (W) distribution named the Lindley Weibull (LiW) model.The W distribution has been widely used in reliability analysis and in applications of several different fields; see, for example Lai et al. (2003).Although its common use, a negative point of the distribution is the limited shape of its hazard rate function (hrf) that can only be monotonically increasing or decreasing or constant.
Generally, practical problems require a wider range of possibilities in the medium risk, for example, when the lifetime data present a bathtub shaped hazard function such as human mortality and machine life cycles.Researchers in the last years developed various extensions and modified forms of the W distribution to obtain more flexible distributions.A state-of-the-art survey on the class of such distributions can be found in Lai et al. (2001) and Nadarajah (2009).
The cdf and pdf of the W distribution are given by respectively, where α > 0 is a scale parameter and β > 0 is a shape parameter.
The main objectives of this paper is to obtain a more flexible model by inducting just one extra shape parameter to the W model and to improve goodness-of-fit to real data.The basic motivations for the LiW distribution in practice are: (i) to make the kurtosis more flexible as compared to the baseline model; (ii) to produce skewness for symmetrical distributions; (iii) to construct heavy-tailed distributions that are not longer-tailed for modeling real data; (iv) to generate distributions with symmetric, left-skewed, right-skewed and reversed-J shaped; (v) to provide consistently better fits than other generated models under the same underlying distribution.
In fact, we prove empirically that the proposed distribution provides better fits to two real data sets than other six extended W distributions with three and four parameters (see Section 7).These examples really show that the new distribution is a good alternative for modeling survival data.Further, the LiW density can be symmetric, left-skewed, right-skewed or reversed J-shape (see Figure 1), whereas the LiW hrf can be bathtub, unimodal, reversed J-shape, monotonically increasing and decreasing shapes (see Figure 2).The skewness of the LiW distribution can range in the interval (-0.9, 6.5), whereas the skewness of the W distribution varies only in the interval (-0.63, 3.12) when the scale parameter is one and the shape parameter takes values from 0.75 to 10. Further, the spread for the LiW kurtosis is much larger ranging from 2.7 to 82, whereas the spread for the W kurtosis only varies from 2.85 to 18.98 with the above parameter values.
Based on the Li-G family, we construct the LiW distribution and provide a comprehensive description of some of its mathematical properties.The paper is outlined as follows.In Section 2, we define the LiW distribution.In Section 3, we derive useful representations for the pdf and cdf of the new distribution.Some mathematical properties including the ordinary and incomplete moments and other types of moments, quantile function (qf), moment generating function (mgf), order statistics and quantile spread order are investigated in Section 4. In Section 5, we obtain the maximum likelihood estimates (MLEs) of the model parameters.In Section 6, we verify the consistency of the estimates by means of some simulations.In Section 7, we prove empirically that the LiW distribution provides better fits than other seven lifetime models, each one having the same number of parameters, by means of two applications to real data sets.Finally, in Section 8, we provide some concluding remarks.

-THE LIW DISTRIBUTION
In this section, we define the LiW model and provide some plots of its pdf and hrf.The LiW cdf is given by The pdf corresponding to ( 5) is given by The LiW model is very attractive to define special models with different types of hazard rates.Figure 1 displays some plots of the LiW density for different values of α, β and θ .These plots reveal that the LiW density can be symmetric, left-skewed, right-skewed or reversed J-shape.The hrf plots of the LiW distribution given in Figure 2 can be bathtub, unimodal, reversed J-shape, increasing and decreasing shapes.

-LINEAR REPRESENTATION
In this section, we obtain a very useful linear representation for the LiW density.An expansion for (6) can be derived using the very popular exponentiated Weibull (exp-W) distribution, whose applications have been widespread in several areas.A random variable Z has the exp-W density with the baseline W given in (3) and power parameter d > 0, say Z ∼exp-W(d), if its cdf and pdf (for z > 0) are given by Using the generalized binomial expansion, the Li-G cdf in (1) can be expressed as  Consider the logarithmic power series given by We can write and then equation ( 7) becomes Equivalently, we obtain where By differentiating the last equation, the LiW pdf reduces to An Acad Bras Cienc (2018) 90 (3)  0 where h d (x) denotes the exp-W density with power parameters d.Equation ( 8) reveals that the LiW density can be expressed as a linear combination of exp-W densities.Thus, several mathematical properties of the new model can be obtained from those properties of the exp-W distribution.This is the main result of this section.The hrf of the exp-W model allows for constant, monotonically increasing, monotonically decreasing, unimodal and bathtub shaped hazard rates.So, these forms also hold for the hrf of the new distribution (as shown in Figure 2).

-SOME PROPERTIES
The formulas derived in this section are simple and manageable, and with the use of modern computer resources and their numerical capabilities, the LiW model may prove to be a useful addition to those distributions applied for modeling data in economics, medicine, reliability, engineering, among other areas.

-ORDINARY AND INCOMPLETE MOMENTS
The several types of moments of a random variable are important especially in applied work.Some of the most important features and characteristics of a distribution can be studied through moments, e.g., tendency, dispersion, skewness and kurtosis, mean deviations, Bonferroni and Lorenz curves, etc.
First, we provide explicit formulas for the rth ordinary and incomplete moments of the exp-G random variable Z (defined in the last section) given by respectively, where b dx is the complete gamma function and γ(p, z) = z 0 x p−1 e −x dx is the lower incomplete gamma function.Second, the sth ordinary moment of X, say µ s , follows from (8) and the above results as The mean, variance, skewness and kurtosis of the LiW distribution are computed numerically for α = 1 and some selected values of β and θ using the R software.The numerical values displayed in Table I indicate that the skewness of the new distribution can range in the interval (−0.9, 6.5).The spread for its kurtosis is much larger ranging from 2.7 to 82.
Fourth, the nth moment of the residual life of X, say , n = 1, 2,… and z > 0, uniquely determines F(x), and it is given by Using equation ( 8), we can write where (ρ) s = Γ (ρ) /Γ (ρ − s) is the the falling factorial.Fifth, the nth moment of the reversed residual life of X, denoted by M n (z) = E [(z − X) n | X ≤ z] for z > 0 and n = 1, 2, . .., uniquely determines F(x), and it is defined by Then, M n (z) reduces to .
The mean residual life (MRL) and mean inactivity time (MIT) of X follow simply by setting n = 1 in m n (z) and M n (z), respectively.The MRL of X at age z represents the expected additional life length for a unit which is alive at age z, whereas the MIT of X at age z represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, z).

-QUANTILE AND GENERATING FUNCTIONS
The qf of X, say Q(u), is obtained by inverting the following equation Let z = 1 − G (x) and q = θ θ +1 .We have and then , where W (z) denotes the negative branch of the Lambert W-function (also known as product log function in the Wolfram Language) which is the inverse function of z = t e z .We can invert to obtain t = F(z) as We have checked the above power series expansion for F(z) = ProductLog[z] using the Mathematica software that gives F(z) as the principal solution for t in z = t e t .
The pdf of Z, h d (z), can be expanded using the power series Then, the pdf of Z can be expressed as where f α(i+1) 1/β (z) denotes a two-parameter W density with scale parameter α (i + 1) 1/β and shape parameter β .Equation ( 10) can be used to derive the mgf of the LiW distribution from that of a two-parameter W distribution.
Let p Ψ q (•) be the complex parameter Wright generalized hypergeometric function with p numerator and q denominator parameters (Kilbas et al. 2006) defined by the power series Then, following similar algebraic developments of Nadarajah et al. (2013), we can write the mgf of Z, say M Z (t; α, β , d), as where Hence, the mgf of the Li-W model follows from (8) as where Equation ( 12) can be easily evaluated by scripts of the Maple, Matlab and Mathematica plataforms.

-ORDER STATISTICS AND QUANTILE SPREAD ORDER
Order statistics make their appearance in many areas of statistical theory and practice.They enter in the problems of estimation and hypothesis tests in a variety of ways.We now discuss some properties of the order statistics for the LiW distribution.Let X i:n denote the ith order statistic from a random sample X 1 , . . ., X n from the LiW distribution.Then, the pdf f i:n (x) of the ith order statistic of X i:n is given by The quantile spread (QS) of the random variable X ∼LiW(θ , α, β ) having cdf ( 5) is given by where is the survival function.The QS of a distribution describes how the probability mass is placed symmetrically about its median and hence can be used to formalize concepts such as peakedness and tail weight traditionally associated with kurtosis.So, it allows us to separate concepts of kurtosis and peakedness for asymmetric models.Let X 1 and X 2 be two random variables following the LiW model with quantile spreads QS X 1 and QS X 2 , respectively.Then X 1 is called smaller than X 2 in quantile spread order, denoted as The following properties of the QS order can be obtained: • The order ≤ QS is location-free • The order ≤ QS is dilative • Let F X 1 and F X 2 be symmetric, then • The order ≤ QS implies ordering of the mean absolute deviation around the median, say MAD, where An Acad Bras Cienc (2018) 90 (3) THE LINDLEY WEIBULL DISTRIBUTION 2589

-MAXIMUM LIKELIHOOD ESTIMATION
The MLEs enjoy desirable properties and can be used for confidence intervals and test statistics.The normal approximation for these estimators in large sample theory is easily handled either analytically or numerically.Here, we determine the MLEs of the parameters of the LiW model from complete samples only.Further works could be addressed using different methods to estimate the LiW parameters such as moments, least squares, weighted least squares, bootstrap, Jackknife, Cramér-von-Mises, Anderson-Darling, Bayesian analysis, among others, and compare the estimators based on these methods.
Let x 1 , • • • , x n be a random sample from the LiW distribution with parameters θ , α and β .Let ϕ =(θ , α, β ) T be the parameter vector.Then, the log-likelihood function for ϕ, say = (ϕ), is given by Equation ( 13) can be maximized either directly by using the R (optim function), SAS (PROC NLMIXED) or Ox program (sub-routine MaxBFGS) or by solving the nonlinear likelihood equations obtained by differentiating (13).
The score vector components, say T , are available from the corresponding author.
Setting the nonlinear system of equations U θ = U α = U β = 0 and solving them simultaneously yields the MLE ϕ = ( θ , α, β ) T of ϕ = (θ , α, β ) T .These equations cannot be solved analytically but Newton-Raphson type algorithms can be used to solve them numerically.
For interval estimation of the model parameters, we require the observed information matrix J (ϕ) which comes as output using the above maximization procedures.Under standard regularity conditions when n → ∞, the distribution of ϕ can be approximated by a multivariate normal N 3 (0, J( ϕ) −1 ) distribution to construct approximate confidence intervals for the parameters.Here, J( ϕ) is the total observed information matrix evaluated at ϕ.The method of the re-sampling bootstrap can be adopted for correcting the biases of the MLEs of the model parameters.Interval estimates may also be obtained using the bootstrap percentile method.Likelihood ratio tests can be performed for the proposed family of distributions in the usual way.

-SIMULATION STUDY
We perform a Monte Carlo simulation study to verify the finite sample behavior of the MLEs of θ , α and β .All simulation results are obtained from 1, 000 Monte Carlo replications carried out using the statistical software R. In each replication, a random sample of size n is drawn from X ∼LiW(θ , α, β ) and the conjugate gradient method has been used for maximizing the total log-likelihood function.The LiW random number generation is performed using the inversion method via the qf Q(u) given in Section 4.2.Five different combinations of true parameter values in the first column of Tables II and III are adopted for the data generating process.Tables II and III list the mean values and mean square errors (MSEs) of the MLEs of the model parameters by taking sample sizes n = 30, 50, 200 and 1, 000.The figures in both tables indicate that the MSEs decrease when the sample size increases as expected under first-order asymptotic theory.
In order to compare the fits of the LiW model with other competing distributions, we consider the Anderson-Darling (A * ) and Cramé r-von Mises (W * ) statistics.The two statistics are widely used to compare non-nested models and to determine how closely a specific cdf fits the empirical distribution of a given data set.These statistics are given by and respectively, where z i = F (y j ) and the y j 's values are the ordered observations.
Tables IV and V list the values of the statistics A * and W * for eight fitted models to these two data sets.The MLEs and their corresponding standard errors (in parentheses) of the model parameters are also given in these tables.The figures in both tables reveal that the LiW distribution yields the lowest values of these statistics among the fitted models and then provides the best fit to both data sets.Hence, we prove empirically that the proposed distribution provides better fits in two applications than other six extended Weibull distributions with three and four parameters.There are too many models to fit and this fact really shows that the LiW distribution can be a good alternative for modeling survival data.
The figures in these tables are calculated using the MATHCAD program.In this program, we provide any initial values (in several cases from fits of special models) and then the program calculates the MLEs.After that, we update the initial estimates to obtain new values for the MLEs.This process continues up to obtain the final MLEs, which make the first derivatives of the log-likelihood function equal to zero.
More information is provided by a visual comparison of the histogram of the data with the fitted density functions.The plots of the fitted LiW, WLi, KwW, GW, BW, TCWG, WFr and W densities are displayed in Figures 3 and 4 for the two data sets, respectively.

Figure 1 -
Figure 1 -Plots of the LiW pdf for selected parameter values.

Figure 2 -
Figure 2 -Plots of the LiW hrf for selected parameter values.

TABLE I Mean, variance, skewness and kurtosis of the LiW distribution for
α = 1 and different values of β and θ .

TABLE IV MLEs (standard errors in parentheses) and the statistics
W * and A * for data set I.

TABLE V MLEs (standard errors in parentheses) and the statistics
W * and A * for data set II.