Odd Generalized Exponential Flexible Weibull Extension Distribution
- Abdelfattah Mustafaabdelfatah_mustafa@yahoo.comMathematics Department, Faculty of Science, Mansoura University, Mansoura, EgyptBeih S. El-Desoukyb_desouky@yahoo.comMathematics Department, Faculty of Science, Mansoura University, Mansoura, EgyptShamsan AL-Garashshamsan_algarash@hotmail.comMathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
- https://doi.org/10.2991/jsta.2018.17.1.6How to use a DOI?
- Odd Generalized Exponential Family, Flexible Weibull Extension distribution, Generalized Weibull, Odd Generalized Exponential Flexible Weibull distribution, Maximum likelihood estimation
In this article we introduce a new four - parameters model called the odd generalized exponential flexible Weibull extension (OGE-FWE) distribution which exhibits bathtub-shaped hazard rate. Some of it’s statistical properties are obtained including ordinary and incomplete moments, quantile and mode, the moment generating functions, reliability and order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed Fisher’s information matrix is given. Moreover, we give the advantage of the OGE-FWE distribution by an application using real data.
- Copyright © 2018, the Authors. Published by Atlantis Press.
- Open Access
- This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).
The Weibull distribution is a highly known distribution due to its utility in modelling lifetime data where the hazard rate function is monotone, . Recently new classes of distributions were proposed based on modifications of the Weibull distribution to provide a good fit to data set with bathtub hazard failure rate, see . Among of these, Exponentiated Weibull family, , Modified Weibull distribution, [11, 19], Beta-Weibull distribution, , A flexible Weibull extension, , Extended flexible Weibull, , Generalized modified Weibull distribution, , Kumaraswamy Weibull distribution, , Beta modified Weibull distribution, [16,22], Beta generalized Weibull distribution, , A new modified weibull distribution,  and Exponentiated modified Weibull extension distribution, , among others. A good review of these models is presented in [3,15,17].
The flexible Weibull extension (FWE) distribution,  has many applications in life testing experiments, reliability analysis, applied statistics and clinical studies. For more details on this distribution, see .
A random variable X is said to have the Flexible Weibull Extension (FWE) distribution with parameters α, β > 0 if it’s probability density function (pdf) is given bywhile the cumulative distribution function (cdf) is given by The survival function is given by the equation and the hazard rate function is Gupta and Kundu  proposed a generalization of the exponential distribution named as Generalized Exponential (GE) distribution. The GE distribution with parameters ϑ, γ > 0, has the following distribution function Recently, a new class of univariate continuous distributions named as the odd generalized exponential (OGE) class introduced in [7, 24]. This class is flexible because of the hazard rate shapes could be increasing, decreasing, bathtub and upside down bathtub. The odd generalized exponential (OGE) class is defined as follows.
If G(x), x > 0 is cumulative distribution function (cdf) of a random variable X, then the corresponding survival function is and the probability density function is g(x), then we define the cdf of the OGE class by replacing x in the distribution function of generalized exponential (GE) distribution given in equation (1.5) by leading toThe probability density function corresponding to (1.6) is given by where x > 0, ϑ > 0, γ > 0. In this article we present a new distribution depending on flexible Weibull extension distribution referred to as the odd generalized exponential flexible Weibull extension (OGE-FWE) distribution by using the class of univariate distributions defined above.
This paper can be organized as follows, we define the cumulative, density and hazard functions of the odd generalized exponential flexible Weibull extension (OGE-FWE) distribution in Section 2. In Sections 3, we present some statistical properties including, quantile function and median, the mode, rth moment, skewness and kurtosis. In Sections 4, we introduce the moment generating function. The distribution of the order statistics is expressed in Section 5. The maximum likelihood estimation of the parameters is determined in Section 6. We use real data sets and analyzed it by an application in Section 7 and the results are compared with existing distributions. Finally, we present a conclusion in Section 8.
2. The Odd Generalized Exponential Flexible Weibull Extension Distribution
In this section we studied the four parameters odd generalized exponential flexible Weibull extension OGE-FWE (ϑ, γ, α, β) distribution. Using G(x) from Eq. (1.2) and g(x) from Eq. (1.1) to obtain the cdf and pdf of Eqs. (1.6) and (1.7), respectively. The cumulative distribution function cdf of the odd generalized exponential flexible Weibull extension distribution (OGE-FWE) is given byThe pdf corresponding to Eq. (2.1) is given by where x > 0 and, α, β> 0 are two additional shape parameters.
The survival function S(x), hazard rate function h(x) and reversed hazard rate function r(x) of X ~ OGE-FWE (ϑ, γ, α, β) are given byrespectively, x > 0 and ϑ, γ, α, β > 0.
3. Statistical Properties
In this section, we will study some statistical properties for the OGE-FWE distribution, specially quantile function and simulation median, the mode, moments, skewness and kurtosis.
3.1. Quantile and median
We determine the explicit formulas of the quantile and simulation median of the OGE-FWE distribution. The quantile xq of the OGE-FWE(ϑ, γ, α, β) distribution is given byFrom Eq. (2.1), we have we obtain xq by solving the following equation. where So, the simulation of the OGE-FWE random variable is straightforward. If U is a uniform random variable on unit interval (0, 1). Then, by means of the inverse transformation method, we can obtain the random variable X as follows Since the median of OGE-FWE distribution can be obtain by setting q = 0.5 in Eq. (3.3).
3.2. The mode
In this subsection, we can obtain the mode of the OGE-FWE distribution by differentiating its probability density function pdf with respect to x and equaling it to zero. The mode is the solution the following equationSince Then from Eq. (3.5), we have where h(x; ϑ, γ, α, β) is hazard rate function of OGE-FWE distribution Eq. (2.4), and S(x; ϑ, γ, α, β) is survival function of OGE-FWE Eq. (2.3).
It is difficult to get an analytic solution in x to Eq. (3.6) in the general case. So, it has to be obtained by numerically methods.
3.3. Skewness and Kurtosis
In this subsection, we obtain the skewness and kurtosis based on quantile measures. The Moors Kurtosis is given by, and the Bowely’s skewness based on quartiles is given by,  where q(.) is quantile function.
3.4. The Moments
We derive the rth moment for the OGE-FWE distribution in Theorem (3.1)
If X has OGE-FWE (ϑ, γ, α, β) distribution, then The rth moments of random variable X, is given by
We start with the well known distribution of the rth moment of the random variable X with probability density function f(x) given bySubstituting from Eq. (2.2) into Eq. (3.10) we get since for x > 0, the binomial series expansion of yields then we get using series expansion we obtain Using series expansion hence using series expansion we obtain Using series expansion gives by using the definition of gamma function in the form, , Finally, we obtain the rth moment of OGE-FWE distribution in the form This completes the proof.
4. The Moment Generating Function
The moment generating function (mgf) of the OGE-FWE distribution is given by Theorem (4.1).
The moment generating function (mgf) of OGE-FWE distribution is given by
5. Order Statistics
In this section, we derive closed form expressions for the PDFs of the rth order statistic of the OGE-FWE distribution. Let X1:n, X2:n,··· ,Xn:n denote the order statistics obtained from a random sample X1, X2, ··· , Xn which taken from a continuous population with cumulative distribution function cdf F(x; φ) and probability density function pdf f(x; φ), then the pdf of Xr:n is given as followswhere f(x; φ) and F(x; φ) are the pdf and cdf of OGE-FWE(φ) distribution given by Eq. (2.2) and Eq. (1.7) respectively, φ = (ϑ, γ, α, β) and B(.,.) is the Beta function, also we define first order statistics X1:n = min(X1, X2,··· ,Xn) and the last order statistics as Xn:n = max(X1, X2,··· ,Xn). Since 0 < F(x; φ) < 1 for x > 0, we can use the binomial expansion of [1 − F(x; φ)]n−r as follows Substituting from Eq. (5.2) into Eq. (5.1), we obtain Substituting from Eq. (2.1) and Eq. (2.2) into Eq. (5.3), we obtain the probability density function for rth order statistics.
Relation (5.3) show that fr:n(x; φ) is the weighted average of the OGE-FWE distribution with different shape parameters.
6. Parameters Estimation
In this section, point and interval estimation of the unknown parameters of the OGE-FWE distribution are derived by using the maximum likelihood method based on a complete sample.
6.1. Maximum Likelihood Estimation:
Let x1, x2,··· ,xn denote a random sample of complete data from the OGE-FWE distribution. The Likelihood function is given assubstituting from (2.2) into (6.1), we have The log-likelihood function is The maximum likelihood estimation of the parameters (ϑ, γ, α, β) are obtained by differentiated the log-likelihood function ℒ with respect to the parameters ϑ, γ, α and β and setting the result to zero, we have the following normal equations. where . The MLEs can be obtained by solving the equations, (6.3) – (6.6), numerically for ϑ, γ, α and β.
6.2. Asymptotic confidence bounds
In this section, we derive the asymptotic confidence intervals of these parameters when ϑ, γ, α > 0 and β > 0 as the MLEs of the unknown parameters ϑ, γ, α > 0 and β > 0 can not be obtained in closed forms, by using variance covariance matrix I −1 see , where I −1 is the inverse of the observed information matrix which defined as followswhere where We can obtain the (1 − δ)100% confidence intervals of the parameters ϑ, γ, α and β by using variance matrix as in the following forms where is the upper -th percentile of the standard normal distribution.
In this section, we will analysis of a real data set using the OGE-FWE (ϑ, γ, α, β) model and compare it with the other fitted models like a flexible Weibull extension distributions using Kolmogorov Smirnov (K-S) statistic, as well as Akaike information criterion(AIC), [?], Akaike Information Citerion with correction (AICC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC) and Schwarz information criterion (SIC) values, . The data have been obtained from , it is for the time between failures (thousands of hours) of secondary reactor pumps, Table 1.
Time between failures of secondary reactor pumps.
|Reduced Additive Weibull||0.0070||1.7292||0.0452||–||–||0.1619|
MLEs and K–S of parameters for secondary reactor pumps.
|Reduced additive Weibull||−86.0728||178.1456||179.4088||181.5521||12.6168||181.5521|
Log-likelihood, AIC, AICC, BIC, HQIC and SIC values of models fitted.
Substituting the MLEs of the unknown parameters ϑ, γ, α, β into (6.7), we obtain estimation of the variance covariance matrix as the followingThe approximate 95% two sided confidence intervals of the unknown parameters ϑ, γ, α and β are [0, 0.230], [0.072, 0.154], [0.125, 0.351] and [0.31, 3.83], respectively.
The nonparametric estimate of the survival function S(x) using the Kaplan-Meier method and its fitted parametric estimations when the distribution is assumed to be OGE-FWE, FW, W, MW, RAW and EW are computed and plotted in Figure 4.
Figure 5 gives the form of the hazard rate h(x) and cumulative density function cdf for the OGE-FWE, FW, W, MW, RAW and EW which are used to fit the data after the unknown parameters included in each distribution are replaced by their MLEs.
We find that the OGE-FWE distribution with the four - parameters provides a better fit than the previous new modified a flexible Weibull extension distribution(FWE) which was the best in . It has the largest likelihood, and the smallest AIC, AICC, BIC, HQIC and SIC values among those considered in this paper.
We proposed a new distribution, based on odd generalized exponential family distributions, this distribution is named the odd generalized exponential flexible Weibull extension OGE-FWE distribution. Some its statistical properties are studied. The maximum likelihood method is used for estimating the parameters model. Finally, we introduce an application using real data. We show that the OGE-FWE distribution fits certain well known data sets better than existing modifications of the Weibull and flexible Weibull extension distributions.
Cite this article
TY - JOUR AU - Abdelfattah Mustafa AU - Beih S. El-Desouky AU - Shamsan AL-Garash PY - 2018 DA - 2018/03 TI - Odd Generalized Exponential Flexible Weibull Extension Distribution JO - Journal of Statistical Theory and Applications SP - 77 EP - 90 VL - 17 IS - 1 SN - 2214-1766 UR - https://doi.org/10.2991/jsta.2018.17.1.6 DO - https://doi.org/10.2991/jsta.2018.17.1.6 ID - Mustafa2018 ER -