McDonald Generalized Linear Failure Rate Distribution

We introduce in this paper a new six-parameters generalized version of the generalized linear failure rate (GLFR) distribution which is called McDonald Generalized Linear failure rate (McGLFR) distribution. The new distribution is quite flexible and can be used effectively in modeling survival data and reliability problems. It can have a constant, decreasing, increasing, and upside down bathtub-and bathtub shaped failure rate function depending on its parameters. It includes some well-known lifetime distributions as special sub-models. Some structural properties of the new distribution are studied. Expressions for the density, moment generating function, conditional moments, mean deviation, Bonferroni and Lorentz curves also are obtained. Moreover we discuss maximum likelihood estimation of the unknown parameters of the new model.


Introduction
In analyzing lifetime data one often uses the exponential, Rayleigh, linear failure rate or generalized exponential distributions.It is well known that exponential can have only constant hazard function whereas Rayleigh, linear failure rate and generalized exponential distribution can have only monotone (increasing in case of Rayleigh or linear failure rate and increasing decreasing in case of generalized exponential distribution) hazard functions.Unfortunately, in practice often one needs to consider non-monotonic function such as bathtub shaped hazard function also, see, for example, Lai et al. (2001).The Generalized linear failure rate distribution generalizes both these distributions which may have non-increasing hazard function also.Also, the Weibull distribution, having the exponential, Rayleigh as special cases,is very popular distribution for modeling lifetime data and for modeling phenomenon with monotone failure rates, when modeling monotone hazard rates, the Weibull distribution may be an initial choice because of its negatively and positively skewed density shapes.However, the Weibull distribution does A random variable X is said to have the generalized linear failure rate distribution ) (GLFR with three parameters ) , , ( if its probability density function is given by while the cumulative distribution function is given by 0. > , , , where  , are scale parameters of the distribution whereas the parameter  denotes the shape parameters.
The aim of this paper is extend the ) (GLFR distribution by introducing three extra shape parameters to define a new distribution refereed to as the McDonald generalized linear failure rate ) (McGLFR distribution.The role of the three additional parameters is to introduce skewness and to vary tail weights and provide greater flexibility in the shape of the generalized distribution and consequently in modeling observed data.It may be mentioned that although several skewed distribution functions exist on the positive real axis, not many skewed distributions are available on the whole real line, which are easy to use for data analysis purpose.The main idea is to introduce three shape parameters, so that the ) (McGLFR distribution can be used to model skewed data, a feature which is very common in practice.

Mc-Donald Generalized Distribution
For an arbitrary parent cdf and Kumaraswamy (Kw) generalized distributions (Cordeiro & Castro, (2011)) for 1 = a .For random variable X with density function (3) denotes the incomplete beta function ratio (Gradshteyn & Ryzhik, (2000)).Equation ( 4) can also be rewritten as follows where is the well-known hypergeometric functions which are well established in the literature (see, Gradshteyn and Ryzhik (2000)).Some mathematical properties of the cdf in equation ( 5), could, in principle, follow from the properties of the hypergeometric function, which are well established in the literature (Gradshteyn and Ryzhik, 2000, Sec.9.1).One important benefit of this class is its ability to model skewed data that cannot properly be fitted by many other existing distributions.Mc-G family of densities allows for higher levels of flexibility of its tails and has a lot of applications in various fields including economics, finance, reliability, engineering, biology and medicine.
The hazard function (hf) and reverse hazard functions (rhf) of the Mc-G distribution are given by The rest of the article is organized as follows.In Section 2, we define the cumulative, density and hazard functions of the McGLFR distribution and some special cases.Section 3 includes the statistical properties such as th r moment , moment generating function .The distribution of the order statistics are proposed in Section 4. Least squares and weighted least squares estimators introduced in Section 5. Finally, maximum likelihood estimation of the parameters is determined in Section 6.
with color shapes purple, blue, orange, red, pink, green and black, respectively.

Submodels
The McDonald generalized linear failure rate ) (McGLFR distribution is very flexible model that approaches to different distributions when its parameters are changed.The McGLFR distribution contains as special-models the following well known distributions.If X is a random variable with pdf (7), using the notation then we have the following cases.

Statistical Properties
In this section we study the statistical properties of the ) (McGLFR distribution, specifically moments and moment generating function .Moments are necessary and important in any statistical analysis, especially in applications.It can be used to study the most important features and characteristics of a distribution (e.g., tendency, dispersion, skewness and kurtosis).

Theorem 3.1
The r th moment of . is given by Proof.
We start with the well known definition of the r th moment of the random variable X with probability density function substituting from ( 17) into ( 16), yields the integral in ( 18) can be computed as follows wich completes the proof.

Theorem 3.2.
The moment generating function of

Proof.
We start with the well known definition of the using the binomial series expansion given by ( 15) and ( 17) we get which completes the proof.

Conditional moments
For lifetime models , it is also of interest to find the conditional moments and the mean residual liftime function.The conditional moments for ) (McGLFR distribution is given by using ( 14), ( 15) and ( 17) , Equation (24) becomes The mean residual lifetime function is given by

Mean deviation
The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and median.These are known as the mean deviation about the mean and the mean deviation about the median are defined by and M=Median (X) denotes the median.The measures  can be calculated using the relationships we can calculate Equations ( 26) and (27) as follows, from equation (12) , when 1 = r we get 14), ( 15) and ( 17) , the formula above imply is the upper incomplete gamma function given by

Bonferroni and Lorenz Curves
In this section we proposed the Bonferroni and Lorenz Curves.The Bonferroni and Lorenz curves (Bonferroni 1930) and the Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine.The Bonferroni and Lorenz curves are defined by


, given by substituting from ( 7) and ( 8) into (32), we can express both X : can be calculated.

Estimation and Inference
In this section we determine the maximum likelihood estimates ) (MLEs of the parameters of the be the vector of the model parameters.The log likelihood function of ( 7) is defined as and c and setting the result equals to zero, we obtain maximum likelihood estimates.The partial derivatives of ) ( L with respect to each parameter or the score function is given by: There is no closed form solution to these equations, so numerical technique must be applied.These non-linear can be routinely solved using Newton's method or fixed point iteration techniques.The subroutines to solve non-linear optimization problem are available in R software namely optim(), nlm() and bbmle() etc.We used nlm() package.

Application
Now we use a real data set to show that the McGLFR distribution can be a better model than the GLFR distribution.
We work with nicotine measurements made from several brands of cigarettes in 1998.The data have been collected by the Federal Trade Commission which is an independent agency of the US government, whose main mission is the promotion of consumer protection.
The report entitled tar, nicotine, and carbon monoxide of the smoke of 1206 varieties of domestic cigarettes for the year of 1998 available at http://www.ftc.gov/reports/tobacco and consists of the data sets and some information about the source of the data, smoker's behaviour and beliefs about nicotine, tar and carbon monoxide contents in cigarettes.The free form data set can be found at http://pw1.netcom.com/rdavis2/smoke.html.
The site http://home.att.net/rdavis2/cigra.htmlcontains 384 = n observations.We analysed data on nicotine, measured in milligrams per cigarette, from several cigarette brands.Some summary statistics for the nicotine data are as follows: mean = 0.852, median = 0.9, minimum = 0.1 and maximum = 2.
, so we reject the null hypothesis.In order to compare the two distribution models, we consider criteria like KS (Kolmogorow Smirnow),  2  , AIC (Akaike information criterion), and AICC (corrected Akaike information criterion), for the data set.The better distribution corresponds to smaller KS,  2  , AIC and AICC values: where k is the number of parameters in the statistical model, n the sample size and  is the maximized value of the log-likelihood function under the considered model.Table 1 shows the MLEs under both distributions, Table 2 shows the values of KS,  2  , AIC, and AICC, values.The values in Table 2 indicate that the McGLFR distribution leads to a better fit than the GLFR distribution.

Conclusion
Here we propose a new model, the so-called the McGLFR distribution which extends the GLFR distribution in the analysis of data with real support.An obvious reason for generalizing a standard distribution is because the generalized form provides larger flexibility in modeling real data.We derive expansions for the moments and for the moment generating function.The estimation of parameters is approached by the method of maximum likelihood, also the information matrix is derived.We consider the likelihood ratio statistic to compare the model with its baseline model.An application of the McGLFR distribution to real data show that the new distribution can be used quite effectively to provide better fits than the GLFR distribution.

Appendix
The elements of Hessian matrix: class of distributions called the Mc-Donald generalized distributions (denoted with the prefix " Mc" for short) is defined by parameters .( See Corderio et al. (2012) for additional details).Note that ) (x g is the pdf of parent distribution , of distributions (3) includes as special sub-models the beta generalized (Eugene et al. (2002)) for 1 = c Cordeiro et al. (2012) presented results on the McDonald normal distribution, Cordeiro et al. (2012) proposed McDonald Weibull distribution, and Francisco et al. (2012) obtained the statistical properties of the  Mc  and applied the model to reliability data.Oluyede and Rajasooriya (2013) introduced the Mc-Dagum distribution and its Statistical Properties with Applications.
7) where  , are scale parameters the other positive parameters  , a , b and c are shape parameterscdf of the McGLFR distribution is given by

Figures 0 and 1
Figures 0 and 1 illustrates some of the possible shapes of the pdf and cdf of the McGLFR distribution for selected values of the parameters  , , , , c b a

Figure 1 :
Figure 1: The pdf's of various McGLFR distributions for values of parameters:

Figures 3
Figures 3 and 4 illustrates some of the possible shapes of the survival funvction and hazard function of the McGLFR distribution for selected values of the parameters  , , , , c b a

Figure 3 :
Figure 3: The survival function's of various McGLFR distributions for values of parameters: purple, blue, orange, red, pink, green and black, respectively.

Figure 4 :
Figure 4: The hazard function's of various McGLFR distributions for values of parameters: reduces to BLFR distribution which introduced by Jafari and Mahmoudi (2012). Kumaraswamy Generalized Linear Failure Rate distribution: For 1, = a the McGLFR distribution reduces to KGLFR distribution which introduced by Elbatal (2013).McGLFR distribution reduces to GE distribution which introduced by Gupta and Kundu (1999).distribution reduces to McR distribution.

4 .
Distribution of the order statisticsIn this section, we derive closed form expressions for the pdfs of the th r order statistic of the ) (McGLFR distribution, also, the measures of skewness and kurtosis of the distribution of the th r order statistic in a sample of size n for different choices of r n; are presented in this section.Let statistics obtained from this sample.We now give the probability density function of combination of the th k moments of the ) (McGLFR distribution with different shape parameters.Therefore, the measures of skewness and kurtosis of the distribution of n r