The Odd Generalized Exponential Linear Failure Rate Distribution
M.A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky, M.E. Mustafa
TThe Odd Generalized Exponential Linear Failure RateDistribution
M. A. El-Damcese , Abdelfattah Mustafa , ∗ ,B. S. El-Desouky and M. E. Mustafa Tanta University, Faculty of Science, Mathematics Department, Egypt. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.
Abstract
In this paper we propose a new lifetime model, called the odd generalized exponentiallinear failure rate distribution. Some statistical properties of the proposed distribution suchas the moments, the quantiles, the median, and the mode are investigated. The method ofmaximum likelihood is used for estimating the model parameters. An applications to realdata is carried out to illustrate that the new distribution is more flexible and effective thanother popular distributions in modeling lifetime data.
Keywords:
Hazard function; Moments; Maximum likelihood estimation; Linear failure rate distribution.
Some distributions such as the exponential (E), Rayleigh (R), generalized exponential (GE), and linearexponential (LE) are used for modelling the lifetime data in reliability. These distributions have severaldesirable properties and satisfactory interpretations which enable them to be used frequently. It is well-known that the exponential distribution can have only constant hazard rate function whereas, Rayleigh,linear failure rate, and generalized exponential distributions can have only monotone (increasing in caseof linear failure rate distribution and increasing/decreasing in case of generalized exponential distribu-tion) failure rate functions. However, the above distributions sometimes have some respective drawbacksin analyzing lifetime data. Gupta and Kundu [5] proposed a generalization of the exponential distri-bution named as Generalized Exponential (GE) distribution. The two-parameter GE distribution withparameters α > and β > , has the following distribution function F ( x ) = (cid:2) − e − αx (cid:3) β , x > , α > , β > . (1.1)The linear exponential (LE) distribution is also known as the Linear Failure Rate (LFR) distribution, hav-ing exponential and Rayleigh distributions as special cases, Bain [2]. The two-parameter LE distributionwith parameters a > and b > , has the following distribution function F ( x ) = 1 − e − ax − b x , x > , a > , b > . (1.2) ∗ Corresponding author: abdelfatah [email protected] a r X i v : . [ m a t h . S T ] O c t .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa Sarhan and Kundu [10] presented a three-parameter generalized linear failure rate (GLFR) distribution byexponentiating the LFR distribution as was done for the exponentiated Weibull distribution by Mudholkaret al. [8]. The exponentiation introduces an extra shape parameter in the model, which may yield moreflexibility in the shape of the probability density function (pdf) and hazard function. The distributionfunction of the generalized linear failure rate (GLFR) distribution is given as F ( x ) = (cid:104) − e − ax − b x (cid:105) β , x > , a > , b > , β > . (1.3)It is observed that the GLFR distribution has decreasing or unimodal pdf and it can have increasing,decreasing, and bathtub-shaped hazard functions. Another important characteristic of GLFR distribu-tion is that it contains, as special sub-models, the generalized exponential (GE), generalized Rayleigh(GR), Linear failure rate (LFR), exponential (E), and Rayleigh (R) distributions, [4, 10]. Jamkhaneh[6] introduced four-parameter distribution called the modified generalized linear failure rate (MGLFR)distribution. Mahmoud and Alam [9] proposed a generalization of linear exponential distribution calledthe generalized linear exponential (GLE) distribution. Anew four-parameter generalization of the linearfailure rate (LFR) distribution which is called Beta-linear failure rate (BLFR) distribution is introducedby Jafari and Mahmoudi [7]. The BLFR distribution is quite flexible and can be used effectively in mod-eling survival data and reliability problems. It can have a constant, decreasing, increasing, upside-downbathtub (unimodal) and bathtub-shaped failure rate function depending on its parameters, and it alsoincludes some well-known lifetime distributions as special sub-models. Another generalized version oflinear exponential distribution introduced by Yuzhu tiana et al. [11] called the new generalized linearexponential (NGLE) distribution and discuss some of its properties, it also includes some well-knownlifetime distributions as special sub-models. Yuzhu tiana et al. [12] also presented another generalizationof linear exponential distribution called the transmuted linear exponential (TLE) distribution. Recently,a new class of univariate continuous distributions called the odd generalized exponential (OGE) classintroduced by [3, 13]. This class is flexible because of the hazard rate shapes could be increasing, de-creasing, bathtub and upside down bathtub. The odd generalized exponential (OGE) class is definedas follows. If G ( x ) , x > is cumulative distribution function (cdf) of a random variable X, then thecorresponding survival function is G ( x ) = 1 − G ( x ) and the probability density function is g ( x ) , thenwe define the cdf of the OGE class by replacing x in the distribution function of generalized exponential(GE) distribution given in equation (1.1) by G ( x ) G ( x ) leading to F ( x ) = (cid:20) − e − α G ( x ) G ( x ) (cid:21) β , x > , α > , β > . (1.4)The probability density function corresponding to (1.4) is given by f ( x ) = αβg ( x ) G ( x ) e − α G ( x ) G ( x ) (cid:20) − e − α G ( x ) G ( x ) (cid:21) β − , x > , α > , β > . (1.5)In this article we present a new distribution depending on Linear Failure Rate distribution called the OddGeneralized Exponential-Linear Failure Rate (OGE-LFR) distribution by using the class of univariatedistributions defined above.This paper is organized as follows. In Section 2 we define the cumulative distribution function, densityfunction, reliability function, hazard function and the reversed hazard function of the odd generalizedexponential-linear failure rate (OGE-LFR) distribution. In Section 3 we study some different propertiesof (OGE-LFR) distribution include, the quantile function, median, mode, and the moments. Section .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa In this subsection we present a new four parameters distribution called Odd Generalized Exponential-Linear Failure Rate (OGE-LFR) distribution with parameters α, a, b, and β written as OGE-LFR( Ψ) ,where the vector Ψ is defined in the form Ψ = ( α, a, b, β ) .A random variable X is said to have OGE-LFR with parameters α, a, b, and β if its cumulative distribu-tion function (cdf) given as F ( x ) = (cid:34) − e − α (cid:18) e ax + b x − (cid:19) (cid:35) β , x > , α, a, b, β > . (2.1)The corresponding pdf has the form f ( x ) = αβ ( a + bx ) e ax + b x e − α (cid:18) e ax + b x − (cid:19) (cid:34) − e − α (cid:18) e ax + b x − (cid:19) (cid:35) β − , (2.2)where x > , α, a, b, β > . If a random variable X has cdf in (2.1), then the corresponding survival function is given by S ( x ) = 1 − (cid:34) − e − α (cid:18) e ax + b x − (cid:19) (cid:35) β . (2.3)The hazard function of OGE-LFR( Ψ ) is defined as follow h ( x ) = αβ ( a + bx ) e ax + b x e − α (cid:18) e ax + b x − (cid:19) (cid:34) − e − α (cid:18) e ax + b x − (cid:19) (cid:35) β − − (cid:34) − e − α (cid:18) e ax + b x − (cid:19) (cid:35) β . (2.4)Also the reversed hazard function of OGE-LFR( Ψ ) is given as follow r ( x ) = αβ ( a + bx ) e ax + b x e α (cid:18) e ax + b x − (cid:19) − . (2.5) .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa This section is devoted for studying some statistical properties for the odd generalized exponential-linearfailure rate (OGE-LFR), specifically quantile function, median and the moments.
The quantile of the OGE-LFR( Ψ ) distribution is simply the solution of the following equation, withrespect to x q , < q < q = F ( x q ) = (cid:34) − e − α (cid:18) e axq + b x q − (cid:19) (cid:35) β . (3.1)By solving equation (3.1), we obtain x q as follow x q = − a ± (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) a + 2 b ln ln (cid:18) − q b (cid:19) α b . Since the quantile x q is positive, then we obtain the quantile as follow x q = − a + (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) a + 2 b ln ln (cid:18) − q b (cid:19) α b . (3.2) .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa The median can be derived from (3.2) be setting q = . That is, the median is given by the followingrelation M ed = − a + (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) a + 2 b ln ln (cid:18) − ( ) b (cid:19) α b . (3.3)Moreover, the mode of the OGE-LFR( Ψ ) distribution can be obtained by deriving its pdf with respect to x and equal it to zero. Thus the mode of the OGE-LFR( Ψ ) distribution can be obtained as a nonnegativesolution of the following nonlinear equation b ( a + bx ) − − αe ax + b x − β − e α (cid:18) e ax + b x − (cid:19) − = 0 . (3.4)It is not possible to get an explicit solution of (3.4) in the general case. Numerical methods should beused such as fixed-point or bisection method to solve it. In this subsection, we will derive the rth moments of the OGE-LFR( Ψ ) distribution as infinite seriesexpansion. Theorem 3.1.
The rth moment of a random variable X ∼ OGE-LFR( Ψ ), where Ψ = ( α, a, b, β ) is givenby µ ´ r = ∞ (cid:88) i =0 ∞ (cid:88) j =0 j (cid:88) k =0 ∞ (cid:88) L =0 (cid:0) β − i (cid:1)(cid:0) jk (cid:1) ( − i + j + k βα j +1 b L ( i + 1) j j ! L !2 L × (cid:20) ( r + 2 L )! a r +2 L ( j − k + 1) r +2 L +1 + b ( r + 2 L + 1)! a r +2 L +2 ( j − k + 1) r +2 L +2 (cid:21) . Proof.
The rth moment of a random variable X with pdf f ( x ) is defined by µ ´ r = (cid:90) ∞ x r f ( x ) dx. (3.5)Substituting from (2.2) into (3.5), we obtain µ ´ r = (cid:90) ∞ x r αβ ( a + bx ) e ax + b x e − α (cid:18) e ax + b x − (cid:19) (cid:34) − e − α (cid:18) e ax + b x − (cid:19) (cid:35) β − dx. (3.6)Sinec < (cid:34) − e − α (cid:18) e ax + b x − (cid:19) (cid:35) < for x > , we obtain (cid:34) − e − α (cid:18) e ax + b x − (cid:19) (cid:35) β − = ∞ (cid:88) i =0 (cid:0) β − i (cid:1) ( − i e − αi (cid:18) e ax + b x − (cid:19) . (3.7) .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa Substituting from (3.7) into (3.6), we get µ ´ r = ∞ (cid:88) i =0 (cid:0) β − i (cid:1) ( − i αβ (cid:90) ∞ x r ( a + bx ) e ax + b x e − α ( i +1) (cid:18) e ax + b x − (cid:19) dx. Using series expansion of e − α ( i +1) (cid:18) e ax + b x − (cid:19) , we obtain µ ´ r = ∞ (cid:88) i =0 ∞ (cid:88) j =0 (cid:0) β − i (cid:1) ( − i + j βα j +1 ( i + 1) j j ! (cid:90) ∞ x r ( a + bx ) e ax + b x (cid:104) e ax + b x − (cid:105) j dx. Using binomial expansion of (cid:104) e ax + b x − (cid:105) j , we obtain µ ´ r = ∞ (cid:88) i =0 ∞ (cid:88) j =0 j (cid:88) k =0 (cid:0) β − i (cid:1)(cid:0) jk (cid:1) ( − i + j + k βα j +1 ( i + 1) j j ! × (cid:90) ∞ x r ( a + bx ) e a ( j − k +1) x e b ( j − k +1) x dx. Using series expansion of e b ( j − k +1) x , we obtain µ ´ r = ∞ (cid:88) i =0 ∞ (cid:88) j =0 j (cid:88) k =0 ∞ (cid:88) L =0 (cid:0) β − i (cid:1)(cid:0) jk (cid:1) ( − i + j + k βα j +1 b L ( i + 1) j ( j − k + 1) L j ! L !2 L × (cid:20) a (cid:90) ∞ x r +2 L e a ( i − k +1) x dx + b (cid:90) ∞ x r +2 L +1 e a ( i − k +1) x dx (cid:21) . By using the definition of gamma function in the form, Zwillinger [14], Γ( z ) = x z (cid:90) ∞ e tx t z − dt, z, x > . Finally, we obtain the rth moment of OGE-LFR in the form µ ´ r = ∞ (cid:88) i =0 ∞ (cid:88) j =0 j (cid:88) k =0 ∞ (cid:88) L =0 (cid:0) β − i (cid:1)(cid:0) jk (cid:1) ( − i + j + k βα j +1 b L ( i + 1) j ( j − k + 1) L j ! L !2 L × (cid:20) ( r + 2 L )! a r +2 L ( j − k + 1) r +2 L +1 + b ( r + 2 L + 1)! a r +2 L +2 ( j − k + 1) r +2 L +2 (cid:21) . This completes the proof.
Let X n , X n , · · · , X n : n denote the order statistics obtained from a random sample X , X , · · · , X n which taken from a continuous population with cumulative distribution function (cdf) F ( x, Ψ) and prob-ability density function (pdf) f ( x, Ψ) , then the probability density function of X r : n is given by f r : n ( x, Ψ) = 1 B ( r, n − r + 1) [ F ( x, Ψ)] r − [1 − F ( x, Ψ)] n − r f ( x, Ψ) , (4.1) .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa where f ( x, Ψ) , F ( x, Ψ) are the pdf and cdf of OGE-LFR( Ψ ) distribution given by ( ?? ) and ( ?? ) respec-tively and B ( ., . ) is the beta function, also we define first order statistics X n = min( X , X , · · · , X n ) ,and the last order statistics as X n : n = max( X , X , · · · , X n ) . Since < F ( x, Ψ) < for x > , wecan use the binomial expansion of [1 − F ( x, Ψ)] n − r given as follows [1 − F ( x, Ψ)] n − r = n − r (cid:88) i =0 (cid:18) n − ri (cid:19) ( − i [ F ( x, Ψ)] i . (4.2)Substituting from (4.2) into (4.1), we obtain f r : n ( x, Ψ) = 1 B ( r, n − r + 1) f ( x ; Ψ) n − r (cid:88) i =0 (cid:18) n − ri (cid:19) ( − i [ F ( x, Ψ)] i + r − . (4.3)Substituting from (2.1) and (2.2) into (4.3), we obtain f r : n ( x ; α, a, b, β ) = n − r (cid:88) i =0 ( − i n ! i !( r − n − r − i )!( r + i ) f ( α, a, b, ( r + i ) β ) . (4.4)Relation (4.4) shows that f r : n ( x, Ψ) is the weighted average of the odd generalized exponential-linearfailure rate with different shape parameters. Now, we discuss the estimation of the OGE-LFR( α, a, b, β ) parameters by using the method of maximumlikelihood based on a complete sample.
Let X , X , · · · , X n be a random sample of size n from X ∼ OGE-LFR( α, a, b, β ) with observed values x , x , · · · , x n , then the log-likelihood function can be written as L = n (cid:89) i =1 f ( x i ; α, a, b, β ) . (5.1)Substituting from (2.2) into (5.1), we get L = n (cid:89) i =1 αβ [ a + bx i ] e ax i + b x i e − α (cid:20) e axi + b x i − (cid:21) (cid:34) − e − α (cid:20) e axi + b x i − (cid:21) (cid:35) β − . The log-likelihood function can be written as L = n ln( α ) + n ln( β ) + n (cid:88) i =1 ln [ a + bx i ] + n (cid:88) i =1 (cid:20) ax i + b x i (cid:21) − α n (cid:88) i =1 (cid:104) e ax i + b x i − (cid:105) + ( β − n (cid:88) i =1 ln (cid:34) − e − α (cid:18) e axi + b x i − (cid:19) (cid:35) . (5.2) .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa The maximum likelihood estimates of the parameters are obtained by Differentiating the log-likelihoodfunction L with respect to the parameters α, a, b and β and setting the result to zero ∂L∂β = nβ + n (cid:88) i =1 ln (cid:34) − e − α (cid:18) e axi + b x i − (cid:19) (cid:35) = 0 , (5.3) ∂L∂α = nα − n (cid:88) i =1 [ ϕ ( x i , a, b ) −
1] + ( β − n (cid:88) i =1 [ ϕ ( x i , a, b ) − ψ ( x i , α, a, b ) = 0 , (5.4) ∂L∂a = n (cid:88) i =1 a + bx i + n (cid:88) i =1 x i − α n (cid:88) i =1 ϕ ( x i , a, b ) x i + ( β − α n (cid:88) i =1 ϕ ( x i , a, b ) x i ψ ( x i , α, a, b ) = 0 , (5.5) ∂L∂b = n (cid:88) i =1 xa + bx i + 12 n (cid:88) i =1 x i − α n (cid:88) i =1 ϕ ( x i , a, b ) x i + ( β − α n (cid:88) i =1 ϕ ( x i , a, b ) x i ψ ( x i , α, a, b ) = 0 , (5.6)Where the nonlinear functions ψ ( x i , α, a, b ) and ϕ ( x i , a, b ) are given by ϕ ( x i , a, b ) = e ax i + b x i ,ψ ( x i , α, a, b ) = e α (cid:20) e axi + b x i − (cid:21) − . From equation (5.3), we obtain the maximum likelihood estimate of β in a closed form as follow ˆ β = − n (cid:80) ni =1 ln (cid:34) − e − α (cid:20) e axi + b x i − (cid:21) (cid:35) . (5.7)Substituting from (5.7) into (5.4), (5.5) and (5.6), we get the MLEs of α, a, b by solving the followingsystem of non-linear equations n ˆ α − n (cid:88) i =1 (cid:104) ϕ ( x i , ˆ a, ˆ b ) − (cid:105) + ( ˆ β − n (cid:88) i =1 (cid:104) ϕ ( x i , ˆ a, ˆ b ) − (cid:105) ψ ( x i , ˆ α, ˆ a, ˆ b ) = 0 , n (cid:88) i =1 a + ˆ bx i + n (cid:88) i =1 x i − ˆ α n (cid:88) i =1 ϕ ( x i , ˆ a, ˆ b ) x i + ( ˆ β −
1) ˆ α n (cid:88) i =1 ϕ ( x i , ˆ a, ˆ b ) x i ψ ( x i , ˆ α, ˆ a, ˆ b ) = 0 , n (cid:88) i =1 x ˆ a + ˆ bx i + 12 n (cid:88) i =1 x i − ˆ α n (cid:88) i =1 ϕ ( x i , ˆ a, ˆ b ) x i + ( ˆ β −
1) ˆ α n (cid:88) i =1 ϕ ( x i , ˆ a, ˆ b ) x i ψ ( x i , ˆ α, ˆ a, ˆ b ) = 0 . There is no closed form solution to these equations, so statistical software or numerical technique mustbe applied.
In this subsection, we derive the asymptotic confidence intervals of the unknown parameters α, a, b and β. As the sample size n −→ ∞ , then ( ˆ α − α, ˆ a − a, ˆ b − b, ˆ β − β ) approaches a multivariate normal vector .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa with zero means and the variance I − ( ˆ α, ˆ a, ˆ b, ˆ β ) , where I − ( ˆ α, ˆ a, ˆ b, ˆ β ) is the inverse of the observedinformation matrix which defined as follows I − = − ∂ ∂α ∂ ∂α∂a ∂ ∂α∂b ∂ ∂α∂β∂ ∂a∂α ∂ ∂a ∂ ∂a∂b ∂ ∂a∂β∂ ∂b∂α ∂ ∂b∂a ∂ ∂b ∂ ∂b∂β∂ ∂β∂α ∂ ∂β∂a ∂ ∂β∂b ∂ ∂β − = V ar ( ˆ α ) cov ( ˆ α, ˆ a ) cov ( ˆ α, ˆ b ) cov ( ˆ α, ˆ β ) cov (ˆ a, ˆ α ) V ar (ˆ a ) cov (ˆ a, ˆ b ) cov (ˆ a, ˆ β ) cov (ˆ b, ˆ α ) cov (ˆ b, ˆ a ) V ar (ˆ b ) cov (ˆ b, ˆ β ) cov ( ˆ β, ˆ α ) cov ( ˆ β, ˆ a ) cov ( ˆ β, ˆ b ) V ar ( ˆ β ) . (5.8)The second partial derivatives included in I − are given as follows ∂ L∂β = − nβ , ∂ L∂β∂α = n (cid:88) i =1 ϕ ( x i , a, b ) − ψ ( x i , α, a, b ) ,∂ L∂β∂a = α n (cid:88) i =1 x i ϕ ( x i , a, b ) ψ ( x i , α, a, b ) , ∂ L∂β∂b = α n (cid:88) i =1 x i ϕ ( x i , a, b ) ψ ( x i , α, a, b ) ,∂ L∂α = − nα − ( β − n (cid:88) i =1 [ ϕ ( x i , a, b ) − [ ψ ( x i , α, a, b ) + 1][ ψ ( x i , α, a, b )] ,∂ L∂α∂a = − n (cid:88) i =1 x i ϕ ( x i , a, b ) + ( β − n (cid:88) i =1 x i ϕ ( x i , a, b ) h ( x i , α, a, b )[ ψ ( x i , α, a, b )] ,∂ L∂α∂b = − n (cid:88) i =1 x i ϕ ( x i , a, b ) + ( β − n (cid:88) i =1 x i ϕ ( x i , a, b ) h ( x i , α, a, b )[ ψ ( x i , α, a, b )] ,∂ L∂a = − n (cid:88) i =1 a + bx i ) − α n (cid:88) i =1 x i ϕ ( x i , a, b ) + ( β − α × n (cid:88) i =1 x i ϕ ( x i , a, b ) τ ( x i , α, a, b )[ ψ ( x i , α, a, b )] ,∂ L∂a∂b = − n (cid:88) i =1 x i ( a + bx i ) − α n (cid:88) i =1 x i ϕ ( x i , a, b ) + ( β − α × n (cid:88) i =1 x i ϕ ( x i , a, b ) τ ( x i , α, a, b )[ ψ ( x i , α, a, b )] ,∂ L∂b = − n (cid:88) i =1 x i ( a + bx i ) − α n (cid:88) i =1 x i ϕ ( x i , a, b ) + ( β − α n (cid:88) i =1 x i ϕ ( x i , a, b ) τ ( x i , α, a, b )[ ψ ( x i , α, a, b )] , where the nonlinear functions ψ ( x i , α, a, b ) , ϕ ( x i , a, b ) , h ( x i , α, a, b ) and τ ( x i , α, a, b ) are given by ϕ ( x i , a, b ) = e ax i + b x i , ψ ( x i , α, a, b ) = e α (cid:18) e axi + b x i − (cid:19) − , .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa h ( x i , α, a, b ) = e α (cid:18) e axi + b x i − (cid:19) (cid:104) − α (cid:16) e ax i + b x i − (cid:17)(cid:105) − ,τ ( x i , α, a, b ) = e α (cid:18) e axi + b x i − (cid:19) (cid:104) − αe ax i + b x i (cid:105) − . The above approach is used to derive the (1 − δ )100% confidence intervals for the parameters α, a, b and β as in the following forms ˆ α ± Z δ (cid:112) V ar ( ˆ α ) , ˆ a ± Z δ (cid:112) V ar (ˆ a ) , ˆ b ± Z δ (cid:113) V ar (ˆ b ) , ˆ β ± Z δ (cid:113) V ar ( ˆ β ) , where Z δ is the upper ( δ )th percentile of the standard normal distribution. Now we use a real data set to show that the OGE-LFR distribution can be a better model, compar-ing with many known distributions such as the Exponential(E), Generalized Exponential(GE), LinearFailure Rate(LFR), New Generalized Linear Exponential (NGLE) and Transmuted Linear Exponential(TLE). Consider the data have been obtained from Aarset [1], and widely reported in many literatures. Itrepresents the lifetimes of 50 devices, and also, possess a bathtub-shaped failure rate property, Table 1.Table 1: The data from Aarset [1].0.1 0.2 1 1 1 1 1 2 3 6 7 11 12 1818 18 18 18 21 32 36 40 45 46 47 50 55 6063 63 67 67 67 67 72 75 79 82 82 83 84 8484 85 85 85 85 85 86 86The MLEs of the unknown parameters and the corresponding Kolmogorov–Smirnov(K–S) test statisticfor the six models are given in Table 2.Table 2: The MLES of the parameters, the K–S values and p-values.The model MLE of the parameters K–S P-value(K-S)E ˆ α = 0.0219 0.1911 0.0519GE ˆ α = 0.0212, ˆ β = 0.9012 0.1940 0.0514LFR ˆ a = 0.014, ˆ b = 2.4 × − ˆ a = 0.0012, ˆ b = 0.0127, ˆ c =1.0682, ˆ β = 0.7231 0.2030 0.0276TLE ˆ a = 0.0145, ˆ b = 2.4186 × − , ˆ λ = -0.0948 0.1740 0.0855OGE-LFR ˆ α = 472.404, ˆ a = 8.218 × − , ˆ b = 6.427 × − , ˆ β = 0.529 0.1627 0.12830The values of the log-likelihood functions (-L), AIC (Akaike Information Criterion), the statistics AICC(Akaike Information Citerion with correction), BIC (Bayesian Information Criterion) and HQIC (Hannan-Quinn information criterion) are calculated in Table 3 for the six distributions in order to verify whichdistribution fits better to these data. .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa Table 3: The –L, AIC, AICC, BIC and HQIC for devices data.The model –L AIC AICC BIC HQAICE 241.090 484.1792 484.2625 486.0912 484.908GE 240.3855 484.7710 485.0264 488.5951 486.227LFR 238.064 480.128 480.383 483.952 481.584NGLE 239.49 486.98 487.869 494.6281 489.892TLE 238.01 482.02 482.54 487.756 484.204OGE-LFR 232.865 473.730 474.618 481.378 476.642Based on Tables 2 and 3, it is shown that OGE-LFR( α, a, b, β ) model provide better fit to the data ratherthan other distributions which we compared with because it has the smallest value of (K-S), AIC, AICC,BIC and HQIC test.To show that the likelihood equation have unique solution, we plot the profiles of the log-likelihoodfunction of α, a, b and β in Figures 5-6. Figure 5: The profile of the log-likelihood function of α, a .Figure 6: The profile of the log-likelihood function of b, β . .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa The nonparametric estimate of the survival function using the Kaplan-Meier method and its fitted para-metric estimations when the distributions is assumed to be E, GE, LFR, NGLE, TLE and OGE-LFR arecomputed and plotted in Figure 7.
Figure 7: The Kaplan-Meier estimate of survival function and fitted survival functions.
Figures 8 and 9, give the form of the probability density functions and the hazard functions for the ED,GED, LFRD, NGLED, TLED, OGE-LFRD distributions which are used to fit the data after replacing theunknown parameters included in each distribution by their MLE.
Figure 8: The Fitted hazard functions for the data. .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa
In this article, we have introduced a new four-parameter model called odd generalized exponential linearfailure rate distribution. We refer to the new model as the OGE-LFR distribution and study some of itsmathematical and statistical properties. We provide the pdf, the cdf, the hazard rate function and thereversed hazard function for the new model also we provide an explicit expression for the moments. Themodel parameters are estimated by maximum likelihood method. We use application on set of real data tocompare the OGE-LFR with other known distributions such as Exponential (E), Generalized Exponential(GE), Linear Failure Rate (LFR), New Generalized Linear Exponential (NGLE) and Transmuted LinearExponential (TLE). Applications on set of real data showed that the OGE-LFR is the best distributionfor fitting these data sets compared with ED, GED, LFRD, NGLED and TLED distributions.
References [1] Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability 36(1),106–108.[2] Bain, L. J. (1974). Analysis for the linear failure-rate life-testing distribution. Technometrics, 16(4),551–559.[3] El-Damcese, M. A., Mustafa, A., El-Desouky, B. S., & Mustafa, M. E. (2015). The odd generalizedexponential gompertz. arXiv preprint arXiv:1507.06400.[4] Gupta, R. D., & Kundu, D. (2007). Generalized exponential distribution: Existing results and somerecent developments. Journal of Statistical Planning and Inference, 137(11), 3537–3547.[5] Gupta, R. D., & Kundu, D. (1999). Theory & methods: Generalized exponential distributions.Australian & New Zealand Journal of Statistics, 41(2), 173–188.[6] Jamkhaneh, E. (2014). Modified generalized linear failure rate distribution: Properties and reliabil-ity analysis. International Journal of Industrial Engineering Computations, 5(3), 375–386.[7] Jafari, A. A., & Mahmoudi, E. (2012). Beta-linear failure rate distribution and its applications.arXiv preprint arXiv:1212.5615. .A. El-Damcese, Abdelfattah Mustafa, B.S. El-Desouky and M.E. Mustafa15