Transmuted Complementary Weibull Geometric Distribution

This paper provides a new generalization of the complementary Weibull geometric distribution introduced by Tojeiro et al. (2014), using the quadratic rank transmutation map studied by Shaw and Buckley (2007). The new distribution is referred to as transmuted complementary Weibull geometric distribution (TCWGD). The TCWG distribution includes as special cases 11 sub models such as the complementary Weibull geometric distribution (CWGD), complementary exponential geometric distribution (CEGD), Weibull distribution (WD), exponential distribution (ED) and three new submodels. Various structural properties of the new distribution including moments, quantiles, moment generating function and Rényi entropy of the subject distribution are derived. We proposed the method of maximum likelihood for estimating the model parameters. Two real data sets are used to compare the flexibility of the new model versus the complementary Weibull geometric distribution.


Introduction
The Weibull distribution is of utmost interest to theory-orientated statisticians because of its great number of special features and to practitioners because of its ability to fit to data from various fields, ranging from life data to observations made in economics and business administration, meteorology, hydrology, maintenance, replacement, quality control, acceptance sampling, statistical process control, inventory control, geology, geography, astronomy, medicine, psychology, pharmacy, material science, engineering, physics, chemistry, biology, warranty and weather data, see, e.g., Rinne (2009).For more than half a century the Weibull distribution has attracted the attention of statisticians working on theory and methods as well as in various fields of applied statistics.
However, the Weibull distribution does not provide a reasonable parametric fit for some practical applications where the underlying hazard rates may be bathtub or unimodal shapes.Recently, some generalizations of the Weibull distribution are considered.Aryal and Tsokos (2011) presented a new generalization of Weibull distribution called the transmuted Weibull distribution.Khan and King (2013) introduced the transmuted modified Weibull distribution.Ashour andEltehiwy (2013a, 2013b) studied the transmuted exponentiated modi.edWeibull and transmuted exponentiated Lomax distributions.Ebraheim (2014) introduced exponentiated transmuted Weibull distribution.
An interesting idea of generalizing a distribution, known in the literature by transmution, is derived by using the quadratic rank transmutation map studied by Shaw and Buckley (2007)

The cumulative distribution function (
) of the complementary Weibull geometric distribution (CWGD) is given by where is a scale parameter and and are shape parameters.The corresponding probability density function ( ) is given by ( , , , . ( The procedure of expanding a family of distributions for added flexibility or to construct covariate models is a well-known technique in the literature.In this article we present a new generalization of complementary Weibull geometric distribution called transmuted complementary Weibull geometric distribution.We derived the subject distribution using the quadratic rank transmutation map studied by Shaw and Buckley (2007).A random variable is said to have transmuted distribution if its cumulative distribution function ( ) is given by where ( ) is the ( ) of the base distribution, which on differentiation yields, where ( ) and ( ) are the corresponding associated with ( ) and ( ) respectively.For more information about the quadratic rank transmutation map is given in Shaw and Buckley (2007).Observe that at = 0, we have the base distribution.
In this paper we provide mathematical formulation of the transmuted complementary Weibull geometric distribution (TCWGD) and some of its properties.We will also provide possible area of applications.The rest of the paper is organized as follows.In Section 2 we demonstrate the subject distribution.In Section 3, we find the reliability function, hazard rate and cumulative hazard rate of the subject model.The statistical properties include quantile functions, random number generation, moments, moment generating functions and Rényi entropy are derived in Section 4. In section 5, the minimum, maximum and median order statistics models are discussed.We also demonstrate the joint density functions : ; , of the transmuted complementary Weibull geometric distribution.In Section 6, we demonstrate the maximum likelihood estimates (MLEs) and the asymptotic confidence intervals of the unknown parameters.In section 7, the TCWG distribution is applied to a real data set to illustrate its usefulness.Finally, in Section 8, we provide some concluding remarks.

Transmuted Complementary Weibull Geometric Distribution (TCWGD)
The transmuted complementary Weibull geometric distribution (TCWGD) and its submodels are presented in this section.A random variable is said to have transmuted complementary Weibull geometric distribution with parameters , , and if its cumulative distribution function ( ) is defined as Using the series expansion The of the transmuted complementary Weibull geometric in (3) can be expressed in the mixture form where ℓ = ( + − )/( + − 2 ), ℎ = + − 2 , is a scale parameter representing the characteristic life, and are the shape parameters representing the different patterns of the transmuted complementary Weibull geometric distribution and is the transmuted parameter.The corresponding probability density function ( ) of the transmuted complementary Weibull geometric distribution is given by Using the series expansion the of the transmuted complementary Weibull geometric distribution in ( 5) can be expressed in the mixture form where Figures 1 and 2 show some of the possible shapes of the and of TCWGD for different choices of the parameters , , and respectively.The transmuted complementary Weibull geometric distribution is very flexible model that approaches to different distributions when its parameters are changed.The subject distribution contains 11 sub models of well known and unknown probability distributions, as special cases, such as the transmuted complementary exponential geometric distribution (TCEGD), the transmuted complementary Rayleigh geometric distribution (TCRGD) and the complementary Rayleigh geometric distributio (CRGD).These three sub models are new distributions.The flexibility of the transmuted complementary Weibull geometric distribution is illustrated in the following .

Corollary 1
If is a random variable with in ( 5), then we have the following special cases.

Reliability Analysis
The characteristics in reliability analysis which are the reliability function (RF), the hazard rate function (HF), the cumulative hazard rate function (CHF) for the TCWGD( , , , , ) are introduced in this section.

Reliability Function
The reliability function ( ) also known as the survival function, which is the probability of an item not failing prior to some time , is defined by ( ) = 1 − ( ).
The reliability function of the transmuted complementary Weibull geometric distribution denoted by ( , , , , ), can be a useful characterization of life time data analysis.It can be defined as ( , , , , ) = 1 − ( , , , , . Using the series expansion the of the transmuted complementary Weibull geometric distribution in (7) can be expressed in the mixture form as follows where = ( − − 1)/ (1 − ), ≠ 1.
Using the series expansion, the of the transmuted complementary Weibull geometric distribution in (9) can be expressed in the mixture form as follows ℎ ( , , , , ) = ( ) where and are defined above.It is important to note that the units for ℎ ( ) is the probability of failure per unit of time, distance or cycles.These failure rates are defined with different choices of parameters.When = 0, the failure rate is same as the CWGD( , , , .

Cumulative Hazard Rate Function
The Cumulative hazard function ( ) of the transmuted complementary Weibull geometric distribution, denoted by ( , , , , ), is defined as ( , , , , ) = ℎ ( , , , , ) = −ln ( , , , , ), ( , , , , ) = ln We can express the of the TCWGD using the series expansion as follows where ℓ is defined above.It is important to note that the units for ( , , , , ) is the cumulative probability of failure or death per unit of time, distance or cycles.

Statistical properties
The statistical properties of the transmuted complementary Weibull geometric distribution including quantile and random number generation, moments, moment generating function and Rényi entropy are discussed in this section.
If we put = 0.5 in the above equation we can get the median of the TCWGD( , , , , ).The quantile of the transmuted complementary Weibull geometric distribution.The median life of the subject distribution is the 50-th percentile.In practice, this is the life by which 50 percent of the units will be expected to have failed and so it is the life at which 50 percent of the units would be expected to still survive.

Random Number Generation
The random number of the TCWGD( , , , , ) is defined by the following relation ( , , , , ) = ,where ~ (0,1), then Solving for ,we get Using a random number uniformly distributed from zero to one, we have solved the above equation to obtain a random number in .

Moments
The moment, denoted by ,of the TCWGD( , , , , ) is given by the following theorem.
Theorem 1.If is a continuous random variable has the TCWGD( , , , , ) with | | ≤ 1, then the non-central moment of is given as follows Proof: Γ 1 + .
Based on the above Theorem (1) the coefficient of variation, coefficient of skewness and coefficient of kurtosis of the TCWGD( , , , , ) distribution can be obtained according to the following relations .
For interval estimation of the model parameters, we require the 4 × 4 observed information matrix.Under standard regularity conditions, the multivariate normal N 0, V distribution can be used to construct approximate confidence intervals for the model parameters.Here, V is the total observed information matrix.Therefore, Approximate 100(1 − )% confidence intervals for , , and can be determined as: and ± , where is the upper th percentile of the standard normal distribution.

Application
Now, we present an application of the proposed TCWG distribution (and their sub models, TCWG, TCRG, TCEG, CWG, and W) in two real data sets to illustrate its potentiality.The first real data set given below represents survival in months of 20 acute myeloid leukemia patients.In the following, we shall compare the proposed TCWG distribution with their submodels, TCRG, TCEG, CWG and W distributions.We shall apply formal goodness-of-fit tests to verify which distribution fits better the real data sets.Here, we consider the Anderson-Darling (A * ) and Cramér-von Mises (W * ) statistics (full details can be found in Chen and Balakrishnan, 1995).In general, the distribution which has the smaller values of these statistics is the better the fit to the data.These results show that the TCRG, TCWG and TCEG distributions give better fit than the CWG and W distributions and the TCEG distribution has the lowest A * and W * values among all the fitted models, and so it could be chosen as the best model.Additionally, it is evident that the W distribution presents the worst fit according to the data Similarly, the results given in Table 2 illustrate that the TCRG, TCWG and TCEG distributions give better fit than the CWG and W distributions and the TCRG has the lowest A * and W * values among all the fitted models, so the TCRG is the best model to fit this real data set.

Conclusions
In this paper, a new lifetime (TCWG) distribution is provided and discussed.The TCWG distribution extends the CWG distribution proposed by Tojeiro et.al., (2014).We provide a mathematical treatment of the new distribution, including expansions for its density and cumulative distribution functions, We derive Various structural properties of the new distribution including explicit expressions for the density function, moments, generating and quantile functions.The estimation of parameters is approached by the method of maximum likelihood.Finally, we fit the TCWG model to two real data sets to show its flexibility and potentially as a lifetime distribution.The new model has 11 well known and unknown probability distributions as special cases, 3 of them are new models.We hope that this new distribution may attract wider applications in the lifetime literature.Finally according to the results in tables 1, and 2 it is obvisely that the TCEG and TCRG distributions (as two new sub models of our proposed model) have the lowest A * and W * values among all the fitted models, respectively.So they could be chosen as the best models.
In this paper we propose a new distribution family by extending the complementary Weibull geometric (CWG), introduced by Tojeiro et al. (2014) by using the quadratic rank transmutation map.Louzada et al. (2011) introduced the complementary exponential geometric distribution, which is complementary to the exponential geometric model proposed by Adamidis and Loukas (1998), based on a complementary risk problem (Basu and J., 1982) in presence of latent risks, in the sense that there is no information about which factor was responsible for the component failure but only the maximum lifetime value among all risks is observed.Louzada et al. (2013) introduced complementary exponentiated exponential geometric distribution which considered a generalization to the complementary exponential geometric distribution.Tojeiro et al. (2014) introduced the complementary Weibull geometric (CWG) as a complementary distribution to the Weibull geometric (WG) model proposed by Barreto-Souza et al. (2011).

Figure 3
Figure 3 illustrates the pattern of the transmuted complementary Weibull geometric distribution reliability function with different choices of parameters , , and .

Figure 3 :
Figure 3: Reliability Function of the TCWGD

Figure 4
Figure 4 illustrates some of the possible shapes of the hazard rate function of the transmuted complementary Weibull geometric distribution for different values of the parameters , , and .

Figure 4 :
Figure 4: Hazard Rate of the TEFD

Figure 5 :
Figure 5: Cumulative Hazard Rate of the TCWGD

Table 1
lists the MLEs of the model parameters for TCWG, TCRG, TCEG, CWG, and W models, the corresponding standard errors (given in parentheses) and the statistics A * and W * .

Table 2 : MLEs (standard errors in parentheses) for TCWG, TCRG, TCEG, CWG, and W models and the statistics
* and * (second data set)