The Generalized Odd Weibull Generated Family of Distributions: Statistical Properties and Applications

In this work, we propose a new class of lifetime distributions calledthe generalized odd Weibull generatedfamily. It can provide better fits than some of the well known lifetime models and this fact represents a good characterization of this family. Some of its mathematical properties are derived. The maximum likelihood method is used for estimating the model parameters. We study the behaviour of the estimators by means of two Monte Carlo simulations. The importance of the family illustrated by means of two applications to real data sets.


Introduction
Recently, some attempts have been made to define new families of distributions that extend well-known distributions and at the same time provide great flexibility in modelling data in practice. So, several classes by adding one or more parameters to generate new distributions have been proposed in the statistical literature. Some wellknown generators are Gupta et al. (1998) who proposed the exponentiated-G class, which consists of raising the cumulative distribution function (cdf) to a positive power parameter. Many other classes cited by Marshall and Olkin (1997) The corresponding pdf is given by where ( ; ) is the baseline pdf. Hereafter, a random variable with density function (5) is denoted by ~GOWG-G( , , ). Further, we can omit sometimes the dependence on the vector of the parameters and write simply ( ) = ( ; ) and ( ) = ( ; ).
The hazard rate function (hrf) of becomes An interpretation of the GOW family (4) can be given as follows. Let be a random variable describing a stochastic system by the cdf ( ) (for > 0). Then, if the random variable represents the odds ratio, the risk that the system following the lifetime will be not working at time is given by ( ) /[1 − ( ) ]. Suppose that we are interested in modeling the randomness of the odds ratio using a Weibull model with cdf ( ) = 1 − exp(− ) (for > 0). Then, the cdf of can be written as which is exactly equal to the family (4).
If ~(0,1) then the solution of nonlinear equation By using taylor expansion and generalized binomial expansion we can demonstrate that the pdf (5) of has the expansion ( ; , , ) = ∑ ∞ , =0 , ℎ ( +1)+ ( ), and ℎ ( ) = −1 ( ) ( ) is the pdf of the Exp-G distribution with power parameter . The corresponding GOWG-G cdf is obtained by integrating (8) ( ; , , ) = ∑ ∞ , =0 , where ( ) = ( ) denotes the exponentiated-G ("Exp-G" for short) cumulative distribution. Equation (7) reveals that the GOWG-G density function is a linear combination of Exp-G density functions. Thus, some mathematical properties of the new model can be derived from those properties of the Exp-G distribution. For example, the ordinary and incomplete moments and moment generating function (mgf) of can be obtained from those quantities of the Exp-G distribution. Let = inf{ | ( ) > 0}, the asymptotics of equations (4), (5) and (6) as → are given by → . The asymptotics of equations (4), (5) and (6) as → ∞ are given by The rest of the paper is organized as follows. In Section 2, we derive some of the mathematical properties of the new family. Maximum likelihood estimation for the model parameters under uncensored data is addressed in Section 3. Two simulation studies are performed in Section 4 to assess the performance of the maximum likelihood estimations. In Section 5, potentiality of the proposed class is illustrated by means of two real data sets. Finally, Section 6 provides some concluding remarkes.

General properties
The ℎ moment of , say ′ , follows from (8) as Henceforth, denotes the Exp-G distribution with power parameter . For > 0, we have ( ) = ∫

Entropies
An entropy is a measure of variation or uncertainty of a random variable . Two popular entropy measures are the Rényi and Shannon entropies (Shannon, 1948;Renyi, 1961). The Rényi entropy of a random variable with pdf ( ) is defined as for > 0 and ≠ 1. The Shannon entropy of a random variable is defined by First we define and compute Using generalized binomial expansion and taylor expansion , we obtain

Proposition 1 Let X be a random variable with pdf (5). Then
The simplest formula for the entropy of is given by After some algebraic developments, we obtain an alternative expression for

Order statistics
Suppose 1 , 2 , … , is a random sample from the F-G distribution. Let : denote the th order statistic. The pdf of : can be expressed as Then, the density function of the : can be expressed as With using this expansion we can easily obtain moments,generating function and incomplete moment of order statistics from any . Equation (14) reveals that the pdf of the GOWG-G order statistic can be expressed as a linear combination of the Exp-G densities. Therefore, some statistical and mathematical properties of these order statistics can be obtained by using this result. Analogous to the ordinary moments we can get the L-moments but it can be estimated by the linear combinations of order statistics in (14). They exist as long the mean of the distribution exists, even if some higher moments may not exist, and are relatively robust to the effects of outliers. Based upon the moments in Equation (14), we can derive explicit expressions for the L-moments of as infinite weighted linear combinations of the means of suitable GOWG-G order statistics. They are linear functions of expected order statistics defined by = 1 ∑ −1 =0 (−1) ( − 1 ) ( − : ), ≥ 1.

Probability weighted moments
Generally, the probability weighted moments (PWMs) method can be used for estimating parameters of a distribution whose inverse form cannot be expressed explicitly. The PWMs are expectations of certain functions of a random variable and they can be defined for any random variable whose ordinary moments exist. They have low variance and no severe bias and can compare favorably with estimators obtained by the maximum likelihood method. The ( , ) ℎ PWM of following the GOW-G family of distribution, say , , is formally defined by (4) and (5) , we can write Finally, the ( , ) ℎ PWMs of can be obtained from an infinite linear combination of Exp-G moments given by , = ∑ ∞ , =0 Ω , ( ( +1)+ ).

Special GOWG models
In here, we obtain the new two extended models based on the new family. We also note that GOWG-G family reduces to odd Weibull-G (OW-G) family, introduced by Bourguignon et al. (2014), for = 1.

The GOWG-Weibull
Consider the cdf ( ) = 1 − exp[−( ) ] of the Weibull distribution with scale > 0 and shape > 0. The pdf of the GOWG-Weibull (GOWG-W) model (for > 0) follows from (5). Some plots of the GOWG-W pdf and hrf for selected parameter values are displayed in Figure 2. Figure 2 reveals that the GOWG-W density can be concave down, right skewed or bi-modal. The hrf of the XG-W model can be increasing, decreasing, bathtub or unimodal then bathtub.

Estimation
Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The MLEs enjoy desirable properties and can be used for constructing confidence intervals and also for test statistics. The normal approximation for these estimators in large samples can be easily handled either analytically or numerically. Here, we consider the estimation of the unknown parameters of the new family from complete samples only by maximum likelihood. Let 1 , … , be a random sample from the GOWG-G distribution with a ( + 2) × 1 parameter vector Θ =( , , ) ú , where is a × 1 baseline parameter vector. . Under standard regularity conditions when → ∞, the distribution of Θ can be approximated by a multivariate normal (0, (Θ) −1 ) distribution to construct approximate confidence intervals for the parameters. Here, (Θ) is the total observed information matrix evaluated at Θ. The method of the re-sampling bootstrap can be used for correcting the biases of the MLEs of the model parameters. Good interval estimates may also be obtained using the bootstrap percentile method.

Simulation studies
In this Section, we perform the two simulation studies by using the GOWG-N and GOWG-W distributions to see the performance of MLEs of these distribution. The random numbers generation is obtained by inverse of their cdfs. Inverse process and results of MLEs were obtained using optim-CG routine in the R programme.In the first simulation study, we obtain the graphical results. We generate = 1000 samples of size respectively. We give results of this simulation study in Figure 3.

Figure 3: Simulation results of the special GOWG-N distribution
In the second simulation study, we generate 1,000 samples of sizes 20,50 and 100 from selected GOWG-W distributions. For this simulation study, we obtain the empirical means and sd's of the parameters.The results of this simulation study are reported in Table 1.
From Figure 3, we observe that when the sample size increases, the empirical means approach to true parameter value. At the same time, the all biases and MSEs approach to 0. The standard deviations decrease in all the cases, while sample size increases. Table 1 shows that when the sample size increases, the empirical means approach to true parameter value and the sds decreases in all the cases as expected.

Real data applications
In this section, we illustrate the flexibility of the GOWG-N and GOWG-W models via two data sets. We compare these models with several extensions and generalizations of the normal and Weibull distributions in the literature. To determine the best model, we also computed the estimated log-likelihood values l, Kolmogorov-Smirnov (KS), Cramervon Mises ( * ) and Anderson-Darling ( * ) goodness of-fit statistics for distribution models. We note that the statistics * and * are described in detail in Chen and Balakrishnan (1995). In general, it can be chosen as the best model which has the smaller the values of the K-S, * and * statistics and the larger the values of l and pvalues. All computations are performed by the maxLik routine in the R programme. The details are the followings.

Windshield data set
As first example, we consider the data studied by Murthy et al.  Table 2 for this data. Table 2 shows that the GOWG-N model has the smallest values of the * and statistics, and has the biggest l value among the fitted models. Hence, it could be chosen as the best model. The the plots of the fitted pdfs and cdfs for models are shown in Figure 4. Also, Figure 5 displays the probability-probability (P-P) plots for the models. From these plots, we can conclude that the GOWG-N distribution is suitable to this data set. The GOWG-N model captures the data as bimodal.

Failure times data set
The following data set represents the failure times (in minutes) for a sample of 15 electronic components in an accelerated life test studied by Lawless (2003) Table 3 for this data. Table 3 shows that the GOWG-W model has the smallest values of the * , * and statistics, and has the biggest l value among the fitted models. Hence, it could be chosen as the best model. The the plots of the fitted pdfs and cdfs for models are shown in Figure 6. Also, Figure 7 displays the P-P plots for the models. From these plots, we can conclude that the GOWG-W distribution is suitable to this data set. The GOWG-W model also captures the data as bimodal. Figure 6: The fitted pdfs (left) and cdfs (right)for the second data set Figure 7: P-P plots for the second data set data

Conclusions
In this work, we propose a new class of lifetime distributions calledthe generalized odd Weibull generatedfamily. It can provide better fits than some of the well known lifetime models and this fact represents a good characterization of this family. Some of its mathematical properties are derived. The maximum likelihood method is used for estimating the model parameters. We study the behaviour of the estimators by means of two Monte Carlo simulations. The importance of the family illustrated by means of two applications to real data sets.