# Odd Generalized Exponential Flexible Weibull Extension Distribution

- DOI
- https://doi.org/10.2991/jsta.2018.17.1.6How to use a DOI?
- Keywords
- Odd Generalized Exponential Family, Flexible Weibull Extension distribution, Generalized Weibull, Odd Generalized Exponential Flexible Weibull distribution, Maximum likelihood estimation
- Abstract
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
- 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/).

## 1. Introduction

The Weibull distribution is a highly known distribution due to its utility in modelling lifetime data where the hazard rate function is monotone, [27]. 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 [25]. Among of these, Exponentiated Weibull family, [14], Modified Weibull distribution, [11, 19], Beta-Weibull distribution, [8], A flexible Weibull extension, [4], Extended flexible Weibull, [4], Generalized modified Weibull distribution, [5], Kumaraswamy Weibull distribution, [6], Beta modified Weibull distribution, [16,22], Beta generalized Weibull distribution, [23], A new modified weibull distribution, [2] and Exponentiated modified Weibull extension distribution, [20], among others. A good review of these models is presented in [3,15,17].

The flexible Weibull extension (FWE) distribution, [4] has many applications in life testing experiments, reliability analysis, applied statistics and clinical studies. For more details on this distribution, see [4].

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 by

*ϑ*,

*γ*> 0, has the following distribution function

If *G*(*x*), *x* > 0 is cumulative distribution function (cdf) of a random variable *X*, then the corresponding survival 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

*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 by

*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 by

*x*> 0 and

*ϑ*,

*γ*,

*α*,

*β*> 0.

Figures (1–3) display the cdf, pdf, survival, hazard rate and reversed hazard rate function of the OGE-FWE (*ϑ*, *γ*, *α*,*β*) distribution for some parameter values.

## 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 *x _{q}* of the OGE-FWE(

*ϑ*,

*γ*,

*α*,

*β*) distribution is given by

*x*by solving the following equation.

_{q}*X*as follows

*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 equation

*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, [13]

*q*(.) is quantile function.

## 3.4. The Moments

We derive the *r*th moment for the OGE-FWE distribution in Theorem (3.1)

### Theorem 3.1.

*If X has OGE-FWE* (*ϑ*, *γ*, *α*, *β*) *distribution*, *then The rth moments of random variable X*, *is given by*

### Proof.

We start with the well known distribution of the *r*th moment of the random variable *X* with probability density function *f*(*x*) given by

*x*> 0, the binomial series expansion of

*r*th moment of OGE-FWE distribution in the form

## 4. The Moment Generating Function

The moment generating function (mgf) of the OGE-FWE distribution is given by Theorem (4.1).

### Theorem 4.1.

*The moment generating function (mgf) of OGE-FWE distribution is given by*

### Proof.

The moment generating function *M _{X}*(

*t*) of the random variable

*X*with probability density function

*f*(

*x*) is given by

*e*, we obtain

^{tx}## 5. Order Statistics

In this section, we derive closed form expressions for the PDFs of the *r*th order statistic of the OGE-FWE distribution. Let *X*_{1:n}, *X*_{2:n},··· ,*X*_{n:n} denote the order statistics obtained from a random sample *X*_{1}, *X*_{2}, ··· , *X _{n}* 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

*X*

_{r:n}is given as follows

*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

*X*

_{1:n}= min(

*X*

_{1},

*X*

_{2},··· ,

*X*) and the last order statistics as

_{n}*X*

_{n:n}= max(

*X*

_{1},

*X*

_{2},··· ,

*X*). Since 0 <

_{n}*F*(

*x*;

*φ*) < 1 for

*x*> 0, we can use the binomial expansion of [1

*− F*(

*x*;

*φ*)]

*as follows*

^{n−r}*r*th order statistics.

Relation (5.3) show that *f*_{r: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 *x*_{1}, *x*_{2},··· ,*x _{n}* denote a random sample of complete data from the OGE-FWE distribution. The Likelihood function is given as

*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.

*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 [12], where *I
*^{−}^{1}
is the inverse of the observed information matrix which defined as follows

*− δ*)100% confidence intervals of the parameters

*ϑ*,

*γ*,

*α*and

*β*by using variance matrix as in the following forms

## 7. Application

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, [21]. The data have been obtained from [18], it is for the time between failures (thousands of hours) of secondary reactor pumps, Table 1.

2.160 | 0.746 | 0.402 | 0.954 | 0.491 | 6.560 | 4.992 | 0.347 |

0.150 | 0.358 | 0.101 | 1.359 | 3.465 | 1.060 | 0.614 | 1.921 |

4.082 | 0.199 | 0.605 | 0.273 | 0.070 | 0.062 | 5.320 |

Time between failures of secondary reactor pumps.

Table 2 gives MLEs of parameters of the OGE-FWE and K-S Statistics. The values of the log-likelihood functions, AIC, AICC, BIC, HQIC, and SIC are in Table 3.

Model | K-S | |||||
---|---|---|---|---|---|---|

OGE-FWE | 0.2380 | 2.0700 | – | 0.069 | 0.113 | 0.0760 |

Flexible Weibull | 0.0207 | 2.5875 | – | – | – | 0.1342 |

Weibull | 0.8077 | 13.9148 | – | – | – | 0.1173 |

Modified Weibull | 0.1213 | 0.7924 | 0.0009 | – | – | 0.1188 |

Reduced Additive Weibull | 0.0070 | 1.7292 | 0.0452 | – | – | 0.1619 |

Extended Weibull | 0.4189 | 1.0212 | 10.2778 | – | – | 0.1057 |

MLEs and K–S of parameters for secondary reactor pumps.

Model | L | AIC | AICC | BIC | HQIC | SIC |
---|---|---|---|---|---|---|

OGE-FEW | −29.2980 | 66.5960 | 68.8182 | 71.1380 | 10.5590 | 71.1380 |

Flexible Weibull | −83.3424 | 170.6848 | 171.2848 | 172.9558 | 12.5416 | 172.9558 |

Weibull | −85.4734 | 174.9468 | 175.5468 | 177.2178 | 12.5915 | 177.2178 |

Modified Weibull | −85.4677 | 176.9354 | 178.1986 | 180.3419 | 12.6029 | 180.3419 |

Reduced additive Weibull | −86.0728 | 178.1456 | 179.4088 | 181.5521 | 12.6168 | 181.5521 |

Extended Weibull | −86.6343 | 179.2686 | 180.5318 | 182.6751 | 12.6296 | 182.6751 |

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 following

*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 [4]. It has the largest likelihood, and the smallest AIC, AICC, BIC, HQIC and SIC values among those considered in this paper.

## 8. Conclusions

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.

## References

*arXiv preprint arXiv:1507.06400*

### 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 -