Odd Lindley-Lomax Model: Statistical Properties and Applications

In this work, we focus on some new theoretical and computational aspects of the Odd Lindley-Lomax model. The maximum likelihood estimation method is used to estimate the model parameters. We show empirically the importance and flexibility of the new model in modeling two types of aircraft windshield lifetime data. This model is much better than exponentiated Lomax, gamma Lomax, beta Lomax and Lomax models so the Odd Lindley-Lomax lifetime model is a good alternative to these models in modeling aircraft windshield data. A Monte Carlo simulation study is used to assess the performance of the maximum likelihood estimators.


Introduction
In the statistical literature, the Lomax (Lo) or Pareto II model was originally pioneered for modeling business failure data by Lomax (1954), and is related to the four-parameter type II generalized beta distribution and the three-parameter Singh-Maddala distribution, as well as the beta distribution of the second kind. The Lo distribution has found a wide application in many fields such as size of cities, income and wealth inequality, actuarial science, engineering, medical and biological sciences, reliability and lifetime modeling. It has been applied to model data obtained from income and wealth (see Harris 1968 and Atkinson and Harrison 1978), firm size (see Corbellini et al., 2007), reliability and life testing (see Hassan Al-Ghamdi 2009), Hirsch-related statistics (see Glanzel 2008), for modeling gauge lengths data (see Afify et al., 2015), for modeling bladder cancer patients data and remission times data (see Yousof  A random variable (rv) X has the Lo distribution with two parameters  and  if it has cumulative distribution function (cdf) (for >0 x ) given by ( ) ( ) where >0  and >0  are the shape and scale parameters, respectively. Then the corresponding pdf of (1) is where >0 a is the scale parameter, is the parameter vector of the baseline distribution and ( ) ( ) From Equation (4), the corresponding cdf of the pdf in (5) is given by is the Exp-Lo cdf. In this work, we focus on some new theoretical and computational aspects of the OLLo model. The rest of the paper is outlined as follows. In Section 2, we derive some of its statistical properties including moments, generating function, residual life and reversed residual life functions and order statistics and their moments are introduced at the end of the section. Maximum likelihood estimation of the model parameters is addressed in Section 3. In Section 4, simulation results to assess the performance of the proposed maximum likelihood estimations, we provide the applications to real data sets to illustrate the importance of the new family in Section 5. Finally, we offer some concluding remarks in Section 6.

Statistical properties
In this section, we provide some mathematical properties of the OLLo distribution. The formulas derived in this section are manageable and simple, and with the use of advanced computer resources and their numerical computing capabilities, the OLLo model may prove to be a useful addition to those distributions which are used for modeling data in reliability, economics, medicine engineering, among others.

Moments and cumulants
The various types of moments of a rv are important especially in applied areas. Many of the most important features and characteristics of a certain distribution can be studied through moments, e.g., tendency, dispersion, skewness and kurtosis etc. The th r ordinary moment of X is given by where ( ) is the complete beta function and The skewness and kurtosis measures also can be calculated from the ordinary moments using well-known relationships. For the skewness and kurtosis coefficients, we have   The mean, variance, skewness and kurtosis of the OLLo distribution are computed numerically for different values of parameters using the R software. The numerical values displayed in Table 1 indicate that the skewness of the OLLo distribution can range in the interval ( 0.023,4.23) − . The spread for its kurtosis is much larger ranging from 2.6 to 36 .

Generating function
The moment generating function (mgf) can be derived from equation

Incomplete moments and mean deviations
The main applications of the first incomplete moment refer to the mean deviations and the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, where ( , ) B  is the beta function. Substituting (5) and (6) in Equation (10)

Estimation
Several approaches for parameter estimation has been proposed in the literature but maximum likelihood method is the most commonly employed. The maximum likelihood estimators (MLEs) enjoy desirable properties and can be used for constructing confidence intervals and regions and also in test statistics. The normal approximation for these estimators in large samples can be easily handled either analytically or numerically. So, we consider the estimation of the unknown parameters of this family from complete samples only by maximum likelihood. Let 1 ,, n xx be a random sample from OLLo distribution with parameters ,, Here, () J h is the total observed information matrix evaluated at h . Further works could be addressed using different methods to estimate the OLLo parameters such as least squares, weighted least squares, moments, bootstrap, Jackknife, Anderson-Darling, Cramer-von-Mises, Bayesian analysis, among others, and compare the estimators based on these methods.

Simulation study
In this section, we conduct the simulation study to see the performance of the MLEs of the OLLo distribution with respect to sample size n. We generated = 1000 N samples of size = 20,30, ,500 n from OLLo distribution with = 12 a , = 0.5 .
The results are presented in Figure2. From Figure2, we can say that for three parameters the emprical means are very close to true parameter values and they are quite stable. Moreover, the bias, MSE and sd decrease as sample size increases. These above results are as expected.

Applications
In this section, we provide two applications to two real data sets to prove the importance and flexibility of the OLLo distribution. We compare the fit of the OLLo with competitve models namely: exponentiated Lomax (ELo) model (Gupta et al., 1998), gamma Lomax (KwLo) model , beta Lomax (BLo) model (Lemonte and Cordeiro, 2013) and Lo model. The cdfs of these distributions are, respectively, given by (for >0 x and , , , > 0 a    These data sets were recently studied by Tahir et al. (2015). The unit for measurement is 1000 h for both data sets.
In order to compare the distributions, the estimated log-likelihood values ˆ, Akaike Information Criteria (AIC), Cramer von Mises (W  ) and Anderson-Darling ( A  ) goodness of-fit statistics were calculated for all models. The statistics W  and A  are described in detail in Chen and Balakrishnan (1995). In general, it can be chosen as the best model which has the smaller values of the AIC, W  and A  statistics and the larger values of ˆ. The required computations are obtained by using the "maxLik" and "goftest" sub-routines in R-software. The analysis results of both these applications are listed in Tables 2 and 3. These results show that the OLLo distribution has the lowest AIC, W  and A  values and has the biggest estimated log-likelihood among all the fitted models. Hence, it could be chosen as the best model under these criteria.  The plots of the fitted densities, cdfs and and probability-probability (P-P) plot of OLLo model are displayed in Figures 3 and 4. We can see that the OLLo distribution provides a good fit and can be used as a competitive model to the other considered models from these Figures. Figure 3: The fitted pdfs (left), cdfs (middle) and P-P plot (right) for the first data set Figure 4: The fitted pdfs (left), cdfs (middle) and P-P plot (right) for the second data set

Conclusions
In this work, we focus on some new theoretical and computational aspects of the Odd Lindley-Lomax model. A Monte Carlo simulation study is used to assess the performance of the proposed maximum likelihood estimations. The maximum likelihood estimation method is used to estimate the model parameters. We show empirically the importance and flexibility of the new model in modeling two types of aircraft windshield lifetime data. We hope that the new model will attract a wider applications in engineering, reliability, and other areas of research. As a future work we will consider bivariate and multivariate extension of the Odd Lindley-Lomax distribution. In particular with the copula based construction method, trivariate reduction etc.