1INTRODUCTION
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 stateoftheart survey on the class of such distributions can be found in Lai et al. (^{2001}) and Nadarajah (^{2009}).
Some extensions of the W distribution are available in the literature such as the exponentiated W (expW) (Mudholkar et al. ^{1993}, ^{1995}, ^{1996}), additive W (^{Xie and Lai 1995}), Marshall–Olkin extended W (^{Ghitany et al. 2005}), modified W (Lai et al. 2003, ^{Sarhan and Zaindin 2009}), extended W (^{Xie et al. 2002}), betaW (^{Lee et al. 2007}), beta modified W (^{Silva et al. 2010}), Kumaraswamy W (^{Cordeiro et al. 2010}), transmuted W (^{Aryal and Tsokos 2011}), Kumaraswamy inverse W (^{Shahbaz et al. 2012}), exponentiated generalized W (^{Cordeiro et al. 2013}), McDonald modified W (^{Merovci and Elbatal 2013}), beta inverse W (^{Hanook et al. 2013}), transmuted additive W (^{Elbatal and Aryal 2013}), McDonald W (^{Cordeiro et al. 2014a}), Kumaraswamy modified W (^{Cordeiro et al. 2014b}), transmuted complementary W geometric (^{Afify et al. 2014}), transmuted exponentiated generalized W (^{Yousof et al. 2015}), Marshall–Olkin additive W (^{Afify et al. 2018}), Kumaraswamy transmuted exponentiated additive W (^{Nofal et al. 2016}), generalized transmuted W (^{Nofal et al. 2017}), ToppLeone generated W (^{Aryal et al. 2017}) and Kumaraswamy complementary W geometric (^{Afify et al. 2017}) distributions. Among these models, the expW is certainly the most popular one.
Recently, Cakmakyapan and Ozel (^{2017}) proposed a new class of distributions called the Lindley generator (LiG) with one extra parameter. For an arbitrary baseline cumulative distribution function (cdf)
and
respectively, where
The cdf and pdf of the W distribution are given by
and
respectively, where
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 goodnessoffit 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 heavytailed distributions that are not longertailed for modeling real data; (iv) to generate distributions with symmetric, leftskewed, rightskewed and reversedJ 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, leftskewed, rightskewed or reversed Jshape (see Figure 1), whereas the LiW hrf can be bathtub, unimodal, reversed Jshape, 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 LiG 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.
2THE 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
3LINEAR 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 (expW) distribution, whose applications have been widespread in several areas. A random variable
and
respectively.
Using the generalized binomial expansion, the LiG 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
where
4SOME 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.
4.1ORDINARY 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
and
respectively, where
Second, the
where
The mean, variance, skewness and kurtosis of the LiW distribution are computed numerically for


Mean  Variance  Skewness  Kurtosis 

0.75  0.75  3.0785  12.3301  2.4522  12.6252 
1.5  1.6000  9.7659  5.3718  56.6381  
3.5  0.2358  0.2465  5.9361  69.1826  
5  0.1067  0.0526  6.1391  74.213  
10  0.0236  0.0027  6.4156  81.6319  
1.5  0.75  1.5102  0.7978  0.7857  3.6215 
1.5  0.8726  0.3017  0.8783  3.7978  
3.5  0.4496  0.0883  0.9856  4.0844  
5  0.3430  0.0525  1.0168  4.1850  
10  0.2063  0.0194  1.0521  4.3112  
2.5  0.75  1.2230  0.2126  0.1393  2.7452 
1.5  0.8751  0.1230  0.2156  2.7272  
3.5  0.5853  0.0600  0.2990  2.7767  
5  0.4972  0.0441  0.3216  2.8015  
10  0.3661  0.0243  0.3459  2.8351  
5  0.75  1.0836  0.0486  0.4485  3.1601 
1.5  0.9144  0.0390  0.3759  2.9957  
3.5  0.7464  0.0281  0.3016  2.9034  
5  0.6876  0.0242  0.2829  2.8902  
10  0.5898  0.0181  0.2636  2.8817  
10  0.75  1.0350  0.0123  0.8301  4.1053 
1.5  0.9501  0.0117  0.7547  3.8264  
3.5  0.8579  0.0103  0.6815  3.6441  
5  0.8234  0.0096  0.6639  3.6105  
10  0.7626  0.0083  0.6461  3.5816 
Third, the
where
The first incomplete moment
The Bonferroni and Lorenz curves are defined (for a given probability
Fourth, the
Using equation (8), we can write
where
Fifth, the
Then,
The mean residual life (MRL) and mean inactivity time (MIT) of
4.2QUANTILE AND GENERATING FUNCTIONS
The qf of
Let
and then
where
We have checked the above power series expansion for
The pdf of
Then, the pdf of
where
Let
Then, following similar algebraic developments of Nadarajah et al. (^{2013}), we can write the mgf of
where
Hence, the mgf of the LiW model follows from (8) as
where
and
Equation (12) can be easily evaluated by scripts of the Maple, Matlab and Mathematica plataforms.
4.3ORDER 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
The quantile spread (QS) of the random variable
which implies
where
and
where
Finally
5MAXIMUM 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, CramérvonMises, AndersonDarling, Bayesian analysis, among others, and compare the estimators based on these methods.
Let
Equation (13) can be maximized either directly by using the R ( optim function), SAS (PROC NLMIXED) or Ox program (subroutine MaxBFGS) or by solving the nonlinear likelihood equations obtained by differentiating (13).
The score vector components, say
Setting the nonlinear system of equations
For interval estimation of the model parameters, we require the observed information matrix
6SIMULATION STUDY
We perform a Monte Carlo simulation study to verify the finite sample behavior of the MLEs of
7APPLICATIONS
We provide two applications of the LiW model to prove empirically its potentiality by comparing to the fits of other competitive models, namely: the Weibull Lindley (WLi) (^{Asgharzadeh et al. 2018}), Weibull Fréchet (WFr) (^{Afify et al. 2016b}), transmuted complementary Weibull geometric (TCWG) (Afify et al. 2014), Kumaraswamy Weibull (KwW) (Cordeiro et al. 2010), beta Weibull (BW) (Lee et al. 2007), gamma Weibull (GW) (^{Provost et al. 2011}) and W distributions, whose pdfs (for
In order to compare the fits of the LiW model with other competing distributions, we consider the AndersonDarling
and
respectively, where
Data Set I: Exceedances of Wheaton River Flood
The data represent the exceedances of flood peaks (in m
Data Set II: Failure Times of 84 Aircraft Windshield
The second data set consists of 84 failure times for a particular windshield device. These failures do not result in damage to the aircraft but do result in replacement of the windshield (^{Murthy et al. 2004}).
n  

30  50  
Parameters  Mean Values  MSEs  Mean Values  MSEs  















0.5  0.2  2.5  0.50991  0.19792  2.62564  0.00501  0.00030  0.17232  0.50649  0.19867  2.57862  0.00300  0.00020  0.10098 
0.5  0.2  4.0  0.51217  0.19906  4.18515  0.00493  0.00013  0.51064  0.50734  0.19860  4.14959  0.00286  0.00008  0.26106 
0.5  2.5  2.5  0.51175  2.47809  2.62782  0.00521  0.05289  0.19608  0.50506  2.48636  2.56986  0.00279  0.02888  0.09129 
0.5  2.5  4.0  0.50879  2.48038  4.19961  0.00467  0.01829  0.43254  0.50495  2.48680  4.12371  0.00303  0.01091  0.23466 
0.5  0.2  0.3  0.51061  0.24238  0.31474  0.00497  0.05728  0.00262  0.50846  0.21737  0.30899  0.00270  0.02213  0.00129 
1.6  0.2  0.3  1.64044  0.20130  0.31577  0.05970  0.00087  0.00258  1.62416  0.20055  0.30922  0.03668  0.00036  0.00127 
1.6  0.2  1.7  1.64080  0.19998  1.79482  0.06436  0.00002  0.08717  1.63232  0.20010  1.74988  0.03550  0.00001  0.04742 
1.6  0.2  5.0  1.62814  0.20005  5.22257  0.05475  0.00000  0.72139  1.61006  0.20004  5.14097  0.03047  0.00000  0.41215 
1.6  1.5  0.3  1.64172  1.51804  0.31409  0.05912  0.04400  0.00249  1.62863  1.50330  0.30960  0.03558  0.02118  0.00127 
1.6  1.5  5.0  1.62901  1.50058  5.22638  0.05627  0.00014  0.71332  1.60478  1.49987  5.15953  0.02978  0.00007  0.38982 
1.6  4.0  0.3  1.62918  4.06111  0.31276  0.05541  0.32855  0.00265  1.61808  3.99504  0.31052  0.03133  0.14399  0.00146 
1.6  4.0  1.7  1.64360  4.00722  1.77624  0.05722  0.00951  0.08558  1.61299  3.99800  1.75432  0.03555  0.00426  0.04493 
1.6  4.0  5.0  1.63797  4.00106  5.25027  0.06354  0.00108  0.71757  1.62001  4.00063  5.13799  0.03489  0.00061  0.41305 
3.0  0.2  0.3  3.08367  0.22157  0.31638  0.23409  0.00405  0.00245  3.04353  0.21084  0.30855  0.14662  0.00169  0.00130 
3.0  0.2  1.7  3.06657  0.20248  1.78674  0.23794  0.00009  0.08300  3.04115  0.20141  1.74705  0.14390  0.00004  0.04079 
3.0  0.2  5.0  3.05929  0.20089  5.27383  0.22984  0.00001  0.70851  3.04622  0.20041  5.12599  0.13519  0.00001  0.36311 
3.0  1.5  0.3  3.05429  1.64512  0.31480  0.23332  0.21787  0.00255  3.04415  1.57312  0.30713  0.13708  0.10655  0.00130 
3.0  1.5  1.7  3.06052  1.51710  1.78322  0.26068  0.00480  0.08165  3.03280  1.50993  1.74986  0.13679  0.00268  0.04491 
3.0  1.5  5.0  3.10828  1.50621  5.25946  0.23379  0.00060  0.74276  3.02710  1.50318  5.14911  0.13649  0.00031  0.39503 
3.0  4.0  0.3  3.07690  4.39372  0.31468  0.22985  1.66922  0.00262  3.05295  4.20288  0.30719  0.13699  0.84081  0.00134 
3.0  4.0  1.7  3.08967  4.05920  1.79784  0.24145  0.03831  0.08628  3.03756  4.02491  1.74494  0.13823  0.02043  0.04438 
3.0  4.0  5.0  3.09394  4.01259  5.21241  0.24109  0.00396  0.64305  3.03010  4.00985  5.15383  0.13879  0.00216  0.37027 
n  

200  100  
Parameters  Mean Values  MSE  Mean Values  MSE  















0.5  0.2  2.5  0.50246  0.19972  2.51801  0.00075  0.00005  0.02218  0.50004  0.19984  2.50568  0.00015  0.00001  0.00417 
0.5  0.2  4.0  0.50037  0.19989  4.02296  0.00061  0.00002  0.05245  0.50034  0.19990  4.00866  0.00013  0.00000  0.01026 
0.5  2.5  2.5  0.50087  2.49631  2.51720  0.00061  0.00707  0.02008  0.50069  2.50106  2.50053  0.00015  0.00145  0.00395 
0.5  2.5  4.0  0.50291  2.49697  4.03103  0.00072  0.00304  0.05666  0.49990  2.50012  4.00303  0.00013  0.00059  0.01041 
0.5  0.2  0.3  0.50202  0.20826  0.30125  0.00071  0.00416  0.00030  0.49999  0.20061  0.30055  0.00012  0.00069  0.00006 
1.6  0.2  0.3  1.60415  0.19958  0.30308  0.00786  0.00008  0.00030  1.60189  0.19977  0.30090  0.00173  0.00001  0.00006 
1.6  0.2  1.7  1.60751  0.20002  1.71108  0.00823  0.00000  0.01023  1.60040  0.20003  1.70097  0.00163  0.00000  0.00187 
1.6  0.2  5.0  1.60839  0.19998  5.04457  0.00786  0.00000  0.07630  1.59866  0.20000  5.00736  0.00171  0.00000  0.01641 
1.6  1.5  0.3  1.60704  1.49607  0.30331  0.00866  0.00439  0.00031  1.60120  1.50025  0.30041  0.00177  0.00080  0.00005 
1.6  1.5  5.0  1.60249  1.49988  5.04088  0.00808  0.00002  0.07825  1.60078  1.50001  5.00577  0.00165  0.00000  0.01627 
1.6  4.0  0.3  1.60858  4.00176  0.30213  0.00814  0.02899  0.00028  1.60095  3.99835  0.30055  0.00156  0.00570  0.00005 
1.6  4.0  1.7  1.60470  4.00205  1.70443  0.00760  0.00098  0.00932  1.60187  3.99998  1.70279  0.00161  0.00018  0.00183 
1.6  4.0  5.0  1.60656  3.99945  5.04988  0.00742  0.00011  0.08033  1.59865  3.99980  5.01257  0.00170  0.00002  0.01595 
3.0  0.2  0.3  3.01580  0.20290  0.30230  0.03511  0.00034  0.00028  3.00156  0.20041  0.30031  0.00620  0.00006  0.00005 
3.0  0.2  1.7  3.01093  0.20043  1.71386  0.03081  0.00001  0.00867  3.00384  0.20011  1.70353  0.00659  0.00000  0.00172 
3.0  0.2  5.0  3.00480  0.20014  5.04237  0.02864  0.00000  0.08716  3.00362  0.20001  5.00335  0.00615  0.00000  0.01527 
3.0  1.5  0.3  3.00938  1.51712  0.30173  0.03318  0.01950  0.00030  3.00637  1.50627  0.30073  0.00606  0.00380  0.00006 
3.0  1.5  1.7  3.01056  1.50259  1.71168  0.02985  0.00057  0.00916  3.00947  1.50054  1.70237  0.00652  0.00010  0.00168 
3.0  1.5  5.0  3.00359  1.50077  5.03344  0.03071  0.00006  0.07981  3.00540  1.50007  5.00299  0.00622  0.00001  0.01530 
3.0  4.0  0.3  3.01337  4.07077  0.30298  0.03184  0.12498  0.00026  3.00193  4.01575  0.30066  0.00617  0.02279  0.00005 
3.0  4.0  1.7  3.00841  4.00542  1.70934  0.03271  0.00448  0.01025  3.00348  4.00184  1.70300  0.00628  0.00081  0.00186 
3.0  4.0  5.0  3.01074  4.00164  5.02749  0.03215  0.00046  0.07799  3.00249  3.99993  4.99994  0.00587  0.00009  0.01526 
Tables IV and V list the values of the statistics
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 loglikelihood 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.
Distribution  Estimates 




LiW(

2.5365  0.0399  0.8705  0.12115  0.70761  
(4.2430)  (0.0880)  (0.1100)  
WLi(

0.0468  0.8027  0.0695  0.12278  0.71588  
(0.0400)  (0.1550)  (0.0270)  
W(

0.0860  0.9012  0.12568  0.72891  
(0.0120)  (0.0860)  
WFr(

0.1294  0.0681  2.2089  9.2457  0.12887  0.74176 
(0.8280)  (0.0110)  (10.115)  (4.7670)  
TCWG(

1.2612  0.6710  0.1458  0.9102  0.16233  0.91437 
(1.0410)  (0.0470)  (0.0800)  (0.3450)  
KwW(

8.5878  0.1118  19.1940  56.9153  0.16421  0.92299 
(31.558)  (0.016)  (10.1530)  (79.9300)  
BW(

1.8533  0.2772  9.3774  1.7812  0.2346  1.30499 
(1.7000)  (0.0650)  (2.2240)  (2.1480)  
GW(

2.5359  0.3397  1.4145  0.18747  1.0470  
(2.9470)  (0.1490)  (0.7730) 
Distribution  Estimates 




LiW(

0.3036  0.9209  1.8846  0.05022  0.49507  
(0.3138)  (0.6552)  (0.2465)  
WLi(

0.2311  3.0359  0.3104  0.07666  0.54281  
(0.0819)  (0.4246)  (0.0214)  
TCWG(

0.0188  0.9598  1.4081  0.6649  0.07758  0.57770 
(0.0610)  (0.7160)  (2.7490)  (0.294)  
W(

0.3493  2.3744  0.08296  0.74365  
(0.0170)  (0.2100)  
KwW(

14.4331  0.2041  34.6599  81.8459  0.18523  1.50591 
(27.095)  (0.0420)  (17.5270)  (52.0140)  
WFr(

630.9384  0.3024  416.0971  1.1664  0.25372  1.95739 
(697.942)  (0.032)  (232.359)  (0.357)  
BW(

1.3600  0.2981  34.1802  11.4956  0.46518  3.21973 
(1.0020)  (0.0600)  (14.8380)  (6.7300)  
GW(

2.3769  0.8481  3.5344  0.25533  1.94887  
(0.3780)  (0.0005)  (0.6650) 
CONCLUSIONS
In this paper, we propose a new threeparameter model, called the Lindley Weibull (LiW) distribution, which extends the Weibull (W) distribution. In fact, the LiW distribution is motivated by the wide use of the W distribution in many applied areas and also for the fact that the generalization provides more flexibility to analyze real data. The LiW density function can be expressed as a linear combination of the exponentiatedW (expW) densities. We derive explicit expressions for the ordinary and incomplete moments, moments of the (reversed) residual life, quantile and generating functions and order statistics. We discuss the maximum likelihood estimation of the model parameters. Two applications illustrate that the proposed model provides consistently better fit than other competitive models. We hope that the new distribution will attract wider applications in reliability, engineering and other areas of research. Estimation of the model parameters under the bayesian paradigm is currently underway and will be reported in a separate article elsewhere. However, we must make a note of the fact under the Bayesian setting, that a non informative prior approach is essentially maximum likelihood estimation under the classical approach. In the absence of an appropriate conjugate prior, the choice of prior will be a challenging in such a setting. As a future work we will consider bivariate and multivariate extension of the LiW distribution. In particular with the copula based construction method, trivariate reduction etc.