The Complementary Exponentiated Inverted Weibull Power Series Family of Distributions and its Applications

DOI: 10.9734/BJMCS/2016/21903 Editor(s): (1) H. M. Srivastava, Department of Mathematics and Statistics, University of Victoria, Canada. Reviewers: (1) G. Y. Sheu, Chang-Jung Christian University, Tainan, Taiwan. (2) P. E. Oguntunde, Covenant University, Nigeria. (3) Thomas L. Toulias, Technological Educational Institute of Athens, Greece. Complete Peer review History: http://sciencedomain.org/review-history/12518


Introduction
In recent years, compound distributions arise and applied in several areas, such as public health, economics, engineering, and industrial reliability. New several compound models of distributions have been introduced follows the same way that was previously carried out in [1]. Many researchers have proposed a series of new compounding distributions by mixing the distribution when the lifetime can be expressed as the minimum or maximum of a sequence of independent and identically distributed, random variables which represent the failure time of a system.

Original Research Article
Several authors have proposed new compounding distributions by mixing the distribution of a maximum of a fixed number of independent for any continuous lifetime distributed random variables and discrete random variable. The exponential truncated Poisson with increasing failure rate has been considered in [2]. A new lifetime distribution, which is called the exponentiated Weibull-geometric distribution, was introduced in [3]. This new distribution was obtained by compounding the exponentiated Weibull together with geometric distributions. For more about some other compounding distributions; can be found in [4][5][6][7][8][9][10][11][12][13][14].
Furthermore, several authors derived new compounding distributions by mixing continuous distributions with power series distribution for both schemes (minimum or maximum). In [15] the Weibull power series family of distributions, which includes the sub-models of the exponential power series distributions, was defined. The generalized exponential power series distribution was introduced in [16]. A new class of extended Weibull power series distributions was introduced in [17]. Two compound families, namely the max-Erlang power series distribution and the min-Erlang power series distribution were proposed in [18]. A new family of Burr XII power series models was introduced in [19]. A new class of lifetime distributions called the Lindley power series was proposed in [20]. A new life time distribution with decreasing failure rate, called the inverse Weibull power series distribution was, introduced in [21].
In this article, a new lifetime family of distributions is introduced by compounding exponentiated inverted Weibull and power series distributions. The density, cumulative, survival and hazard rate functions of the new family are introduced in Section 2. In Section 3, some mathematical properties of the new family are obtained such as, quantiles and moments. In the same section, some distributions of order statistics are derived. In Section 4, some special sub-models and mathematical properties of the complementary exponentiated inverted Weibull Poisson and the complementary exponentiated inverted Weibull logarithmic distributions are given. In Section 5, maximum likelihood estimator of the unknown parameters for the family is obtained. In Section 6, three illustrative examples based on real data set are provided. Finally, the concluding remarks are addressed in Section 7.

The New Family of Distributions
The inverse Weibull distribution is one of the most popular probability distributions to model life time data with some monotone failure rates. The usefulness and applications of the inverse Weibull distribution in various areas including reliability and branching processes, can be seen in [22] and in references therein. Exponentiated (generalized) inverted Weibull distribution is a generalization to the inverted Weibull distribution through adding a new shape parameter. The probability density function (pdf) of exponentiated inverted Weibull (EIW) with two shape parameters; takes the following form The corresponding cumulative distribution function (cdf) is given by For = 1, it represents the standard inverted Weibull distribution, for = 1, it represents the exponentiated standard inverted exponential distribution as mentioned in [23].

Let
= max{ } , be independent and identically distributed random variables following the exponentiated inverted Weibull distribution whose density function is given by (1) with shape parameters > 0 and > 0, while is a discrete random variable following the power series with probability mass function (truncated at zero).
In [6], a compound class of distributions was introduced. These distributions were obtained by mixing the maximum of a sequence of any identically independent lifetime distributed random variables together with a power series random variable. The cumulative distribution function of this class is given by where ( ) is the cdf of any lifetime distribution. Therefore, the distribution function of the complementary exponentiated inverted Weibull power series (CEIWPS) family of distributions is obtained by substituting cdf (2) in cdf (4) as follows The pdf corresponding to (5) is given by, A random variable distributed as in (5) shall be denoted by ∼ CEIWPS , where ≡ ( , , ) is an unknown vector of parameters. Furthermore, the reliability and hazard rate functions take the following forms and ℎ ; =

Properties of the Family
In this section, some statistical properties of the new family, including quantiles, moments and moment generating function are obtained. In addition, some distributions of order statistics are studied.

A Useful Expansion
Some properties of the cdf (5) and pdf (6) will be studied through the following two Propositions.

Proposition 1
The EIW distribution is a limiting special case of EIWPS family of distributions when → 0.  Proof (6), then we get

Quantiles and Moments
Quantile functions are used in theoretical aspects of probability theory, statistical applications and simulations. Simulation methods utilize quantile function to produce simulated random variables for classical and new continuous distributions. The quantile function, say ( ) of is given by after some simplifications, it reduces to the following form Where, is considered as a uniform random variable on the unit interval (0,1) and (. ) is the inverse function of (. ). The r-th moment of about the origin follows from proportion 2 as follows According to [23], the r-th moment for random variable ′ , which follows an EIW distribution with parameters and is given by Γ 1 − . It is easy to show that, by using this property as mentioned in [23], the r-th moment of following the CEIWPS distribution with parameters ( , ) is given by Also, it is easy to show that, where, ̀ is the r-th moment, while ( ) denotes the moment generating function (mgf) of . Then by using (10), the mgf of can be written as

Some Distributions of Order Statistics
Let : < : < … < : be the order statistics of a random sample of size following the exponentiated inverted Weibull power series family of distributions, with parameters , , and , then as mentioned in [24], the pdf of the i-th order statistic, can be written as follows where B(. , . ) denotes the beta function. The expansion for is given by for positive integer as mentioned in [25]), then By using the expansion for ′ ( ) ≔ ∑ , > 0, ∞ substituting (12) and (13) where,

The Complementary Exponentiated Inverted Weibull Poisson Distribution
In this subsection, the pdf, reliability, hazard function, quantile and moments for the complementary exponentiated inverted Weibull Poisson distribution are studied in more details. The pdf of the CEIWP corresponding to (15) is as follows Figs. 1 and 2 represent the pdf and the cdf of the CEIWP distribution for some selected values of parameters , and .

Fig. 1. Plots of the CEIWP densities for some parameter values
The reliability and hazard rate functions of the CEIWP are obtained respectively, as follows where is a uniform random variable on the unit interval (0,1). In particular, the median of the CEIWP distribution, say , is obtained by setting : = into (20), i.e.
∞ In particular, setting = 1 in (22), the mean of the CEIWP distribution is given by The variance of the CEIWP distribution takes the following form

The Complementary Exponentiated Inverted Weibull logarithmic Distribution
In this subsection, the pdf, reliability, hazard function, quantile, and moments for the complementary exponentiated inverted Weibull logarithmic distribution are discussed.
The probability density function of the CEIWL distribution corresponding to (16) is given by Figs. 5 and 6 represent the pdf and the cdf of the CEIWL distribution for selected values of parameters , and .
The reliability and hazard rate functions of the CEIWL distribution are obtained respectively, as follows where is a uniform random variable on the unit interval (0,1). In particular, the median of the CEIWL distribution, say , is obtained by setting, : = into (27), i.e.

Fig. 8. Plots of the CEIWL hazard rates functions for some parameter values
In particular, setting = 1 in (28), the mean of is given by The variance of the CEIWL distribution takes the following form

Parameter Estimation of the Family
In this section, estimation of the model parameters of CEIWPS family of distributions is obtained by using the maximum likelihood method.
Let , , … , be a random sample from the CEIWPS family of distributions with parameters , and .
The likelihood function based on the observed random sample of size is given by The natural logarithm of the likelihood function, ln ≡ ; , is given by The maximum likelihood estimators of , , , say , , , are obtained by setting the first partial derivatives of ln L to be zero with respect to , and as follows, The maximum likelihood estimator of , and , say , and is obtained by solving numerically the nonlinear system of equations ≔ 0, ≔ 0 and : = 0.

Applications
In this section, the flexibility and potentiality of some special models of EIWPS family are examined using three real data sets. Applications of the CEIWPS distributions based on their sub-models; namely, CEIWP, CEIWL and EIW are considered.

The First Real Data Set
The first data set is taken from [26], where the vinyl chloride data is obtained from clean upgradient ground -water monitoring wells in mg/L; the data set is as follows:  Table 2 provides the maximum-likelihood estimates (MLEs), the AIC, BIC, and the CAIC values as well as the Kolmogorov-Smirnov statistics for the first data set. Based on the values of AIC, CAIC, BIC and K-S in Table 2, the CEIWP model provide better fit than the CEIWL and EIW models.
Figs. 9 and 10 provide the plots of estimated cumulative and estimated densities of the fitted CEIWP, CEIWL and EIW models for the first data set.
It is clear from Fig. 10 that the fitted density for CEIWP model is closer to the empirical histogram than CEIWL and EIW models.

The Second Real Data Set
The second data set is obtained from [27].  The following Table 3 provides the MLEs, the AIC, BIC and CAIC values as well as the K-S statistics for the second data set. In order to assess whether the CEIWPS models are appropriate, Fig. 11 provides the histograms of the data set as well as the plots of the fitted CEIWP, CEIWL and EIW density functions. From Fig. 11, it is concluded that the CEIWP distribution is quite suitable for this data set.
Figs. 11 and 12 provide some plots of the estimated cdf as well as the estimated probability densities of the fitted CEIWP, CEIWL and EIW models for the second data set.

The Third Real Data Set
The third data set is taken from [28]. The corresponding data are referring to the time between failures for a repairable item. The data set is as follows: The following Table 4 provides the MLEs, the AIC, BIC and CAIC values as well as K-S statistics for the third data set. The values in Table 4, indicate that the complementary exponentiated inverted Weibull Poisson model is a strong competitor among other distributions that was used here to fit this data set. Figs. 13 and 14 provide some plots of the estimated cdf and pdf of the fitted CEIWP, CEIWL and EIW models for the third data set.
In Fig. 14 the fitted densities of the CEIWP, CEIWL and EIW models are compared to the empirical histogram of the observed third data set. Based on the plots of Fig. 14, one can notice that the fitted density of the CEIWP model is closer to the empirical histogram than the corresponding densities of the CEIWL and EIW models. In this paper, a new family of lifetime distributions, called the complementary exponentiated inverted Weibull power series was introduced. This family was obtained by compounding the exponentiated inverted Weibull and power series distributions. The properties of the proposed family were discussed, including the quantile function, the moments and the moment generating function. Some distributions of order statistics were also obtained. The estimation of the model parameters was performed by the maximum likelihood method. Two special sub-models of the new family were investigated, namely, the complementary exponentiated inverted Weibull Poisson distribution and the complementary exponentiated inverted Weibull logarithmic distribution. Some mathematical properties of the new distributions were also discussed. Finally, the complementary exponentiated inverted Weibull Power series models were fitted to real data sets revealing the flexibility and potentiality of the introduced CEIWPS family of distributions.