The Adjusted Log-logistic Generalized Exponential Distribution with Application to Lifetime Data

I. E. Okorie1, A. C. Akpanta2, J. Ohakwe3, D. C. Chikezie2, C. U. Onyemachi2, E. O. Obi4 1 School of Mathematics, University of Manchester, Manchester, M13 9PL, UK 2 Department of Statistics, Abia State University, Uturu, Abia State, Nigeria 3 Department of Mathematics & Statistics, Faculty of Sciences, Federal University Otuoke, Bayelsa State, P. M. 126 Yenagoa, Bayelsa, Nigeria 4 Department of Strategic Knowledge Management, NACA, Abuja, Nigeria


Introduction
Due to the lack of fits that characterize the standard probability distributions in modelling various complex real data sets a lot of effort have been expended by researchers in developing new distributions as a way of circumventing the problem of inadequate fits of the already existing distributions.The new distributions often referred to as as the generalized class of distributions have consistently been shown to provide better fits than the existing (standard) ones.Almost all the available methods of generating new distributions in statistical literature depends on the cumulative distribution function of the standard distributions; for a holistic up-to-date review of these methods see; Nadarajah and Rocha (2016).
The main motivation of this paper stem from the trending literature of probability distribution construction and generalization; which in principle, entails the injection of one distribution into another distribution, in order to extend the injected distribution to a wider family of distribution with added flexibility.In the literature, the generalized distributions are often referred to as the G-distributions.For example, 1. Eugene et al. (2002) defined a new class of distribution called the Beta-G family of distributions as the logit of the beta random variable with cumulative density function (cdf) as and probability density function (pdf) as 3. Zografos and Balakrishnan (2009) proposed the Gamma-G family of distributions based on baseline continuous distribution with reliability function Ḡ(x) = 1 − G(x).The cdf of the Gamma-G distributions is defined as ∫ − log Ḡ(x) 0 y δ−1 e −y dy; x ∈ R, δ > 0, and pdf as [− log Ḡ(x)] δ−1 g(x); x ∈ R, δ > 0.
4. An alternative version of Zografos and Balakrishnan (2009) was proposed by Ristić and Balakrishnan (2012).The cdf of the gamma generator due to Ristić and Balakrishnan (2012) is defined as while its pdf is given by; 5. Alzaatreh et al. (2013) introduced the Weibull-G distributions whose cdf is defined as and pdf as In all cases, G(x) is the cdf of the injected distribution (or baseline distribution) with g(x) as the corresponding pdf, F(x) is the cdf of the new generalized version of G(x) with f (x) as the corresponding pdf, and f (x) have the same support as g(x).
and so on.
The aim of this paper is two-fold; first to introduce a new generator of distributions-the adjusted log-logistic generalized (ALLoG) distribution which as far as we know have not appeared in the literature before now and secondly, to introduce and give explicit statistical properties of the adjusted log-logistic generalized exponential (ALLoGExp) distribution as a sub-model of the ALLoG distribution which generalizes the standard one parameter exponential distribution.The cdf G(x) of the log-logistic distribution is given by; with the corresponding pdf g(x) defined as where δ is the scale parameter and θ is the shape parameter.
with the corresponding pdf f(x) as where δ is the scale parameter, θ is the shape parameters and η is the rate parameter.

Quantile Function and Random Number Generation
By using (6), we obtain the quantile function of the ALLoGExp distribution as Random samples from the ALLoGExp distribution can be obtained through the inverse transformation method of random number generation by simply substituting p in (8) with a Uniform (0, 1) variate.Also, it is easy to obtain the median of the ALLoGExp distribution by simply substituting p = 0.5 in (8), which is given by;

Moments
In statistics and applications the moments of a random variable say X are important, they are used to characterize the underlying distribution.For example, to measure the center, spread/variation of the distribution, and to ascertain the degree of deviation from normality (skewness and kurtosis) of the distribution, etc.;  Theorem 2.1.If X follows the ALLoGExp distribution with pdf defined in (7) then its kth crude moment is given by; where B(•) is the beta function.
Proof.By using (7) we have Where ( 10) is computed as follows substituting y = e −ηx in (11) we have by expanding (12) we have Corollary 2.1.1.Evaluating (13) at k=1, 2, 3, and 4 we have the first four crude moments of the ALLoGExp distribution as follows: and where B(•) is the beta function and Ψ(•) is the psi or digamma function.
Theorem 2.2.If X follows the ALLoGExp distribution with pdf defined in ( 7) then its moment generating function (mgf) is given by; Proof.Using the definition of the mgf of the continuous random variable say X which is defined as and by substituting ( 13) into ( 19) it is clear that the mgf of the ALLoGExp distribution is as presented in (18).

Entropy Measure
In this section we present the Rényi entropy measure of the ALLoGExp distribution.The Rényi entropy measure is used to quantify the uncertainty of variation in a random variable say X and the Rényi entropy measure of a continuous random variable is generally given by; Theorem 2.3.If X follows the ALLoGExp distribution with pdf defined in (7) then its Rényi entropy measure is given by; where Γ(•) is the gamma function and B(•) is the beta function.
Proof.Substituting ( 7) into (20), setting ∫ R f φ (x)dx to A φ and evaluating the integral on the support [0, ∞) gives By expanding (21) we have and by substituting y = e −ηx into (22) we have which further simplifies to and finally Thus, substituting ( 23) into (20) completes the proof.

Order Statistics
Order statistics is an essential tool in reliability and life testing analysis.For instance, suppose the following n-sized random sample X 1 , X 2 , . . ., X n are drawn from the ALLoGExp distribution with cd f and pd f corresponding to ( 6) and ( 7).Let X 1,n ≤ X 2,n ≤ . . .≤ X n,n represent the ith order statistics denoted by X i,n then, X i,n could be interpreted as the lifetime of the (n − i + 1)th item of the total nth independent and identical components.The density of X i,n could be expressed as The cdf of the ALLoGExp distribution in (6) to the (i + ℓ − 1)th power is given by; and if θ ∈ N, where N\{0} is a natural number; then we have the series representation of (34) as Also, the series representation of ( 7) is given by; Therefore substituting ( 26) and ( 27) into (24) gives the density of the ith order statistics of the ALLoGExp distribution as The density of the smallest order statistics of the ALLoGExp distribution is given by; while the density of largest order statistics of the ALLoGExp distribution is given by;

Moment of the Order Statistics
Theorem 2.4.If X follows the ALLoGExp distribution with pdf of the ith order statistics f X (i) (x) then, its pth crude moment is given by; where Γ(•) is the gamma function and x (i) is defined below.
Proof.By the definition of moment of a continuous random variable we have, by substituting y = η(h + j + k + n + 1)x in (28) we have Where, ) .

Estimation
Here, we estimate the parameters of the ALLoGExp distribution by the method of maximum likelihood estimation.Suppose the random sample x 1 , x 2 , x 3 , . . ., x n of size n is drawn from the ALLoGExp distribution with pdf f(x) in ( 7) then the maximum likelihood estimation (mle) procedure for estimating its parameters is as follows: The likelihood (L) equation is given by; and the log-likelihood function is given by; Taking the partial derivatives of ( 30) with respect to δ, θ and η gives and Furthermore, setting ( 31)-( 33) to zero results to a system of three equations in three unknowns which has no analytical solutions.
However, the estimates δ, θ and η can only be obtained by solving ( 31)-( 33) by some non linear numerical optimization methods eg.; the Newton-Raphson or quasi-Newton-Raphson's technique.

Reliability
The reliability function R( x) is an important tool in reliability analysis for characterizing life phenomena.The reliability function is mathematically expressed as 1 − F(x).Under certain predefined conditions R(x) generally gives the estimate of the probability that, a system will not fail given that it has operated without failure up to time x.The reliability function of the ALLoGExp distribution is given by; Another important reliability characteristics is the failure rate function h(x).The failure rate function gives the probability of failure, for a system that has not failed up-to time x.The failure rate function is mathematically expressed as f (x)/R(x).

Shapes and Asymptotics
(a) The pdf of the ALLoGExp distribution could either be a unimodal or monotonic decreasing function of x depending on the value of δ and θ, while F(x) is an increasing function of x for all possible values of δ and θ parameters (see; Figures 1 and 2), and the asymptotic behaviour of the pdf is and the asymptotic behaviour of the cdf is lim x→∞ F(x) = 1 while lim x→0 F(x) = 0.
(b) The reliability function R(x) of the ALLoGExp distribution is generally a monotonic decreasing function of x for all possible values of δ and θ parameters (see; Figure 3), and lim x→∞ R(x) = 0 while lim x→0 R(x) = 1.
(c) The failure rate function (frf) h(x) of the ALLoGExp distribution could be a decreasing, increasing or upside-down bathtub shaped function of x depending on the value of the δ and θ parameters (see; Figure 4), and lim x→∞ h(x) = 0, while  The loss of memory property of the exponential distribution and the shape limitation of its failure rate are well known.
The major advantage of the new distribution over the baseline model is its added tail and skewness flexibility due to the presence of δ and θ.The ALLoGExp is suitable for modelling lifetime data sets with increasing, decreasing, unimodal and upside-down bathtub failure rate characteristics.

Monte-Carlo Simulation
In this section, we investigate the consistency of the mle estimates of the ALLoGExp distribution with different sample size (n), through a Monte-Carlo study.The simulation procedure as outlined below was implemented in R (Statistical software): 1. simulate a random sample of size n from the ALLoGExp distribution with parameters δ = 0.5, θ = 4.0 and η = 6.0 using the inversion of the cdf method with Equation (8).
2. compute the mle of the parameters of the ALLoGExp distribution.
4. compute the mean, standard deviation (standard error), bias and mean square error (mse) of the 5000 estimates of each of the parameters (δ, θ and η).
Results from the Monte-Carlo simulation study are tabulated in Tables 1 and 2. The simulation results in Table 1 indicates that the mle estimates of the ALLoGExp distribution is generally consistent for n; while the standard error, bias and mse approaches zero as n becomes large.

Application
This section illustrates the applicability and flexibility of the ALLoGExp distribution with a real data set.The goodness of fit of the new lifetime distribution would be assessed by a comparison of its performance in modelling real data with the following five distributions: (i) The exponentiated exponential (EE) distribution due to Gupta and Kundu (1999), (ii) The Log-logistic (LLo) distribution, ) θ ] 2 ; x, θ, δ > 0.

Concluding Remarks
In this paper we have introduced a new generator of distributions called the adjusted log-logistic generalized (ALLoG) distribution and a new lifetime distribution-the adjusted log-logistic generalized exponential (ALLoGExp) distribution is also introduced as a sub-model of the ALLoG distribution.The new lifetime distribution generalizes the exponential (Exp) distribution.We have given explicit expressions of some of its basic statistical properties such as the probability density function, cumulative density function, kth raw moment, mean, variance, coefficient of variation, skewness, kurtosis, moment generating function, pth quantile function, the ith order statistics, and the Rényi's entropy measure.Also, some of its reliability characteristics like the reliability function and the failure rate function were provided; the failure rate could be monotonically decreasing, increasing or upside-down bathtub shaped depending on the value of the scale parameter δ and shape parameter θ.Estimation of the model parameters was approached through the method of maximum likelihood estimation mle and the stability of the mle estimates was verified through a Monte-Carlo simulation study.The applicability and goodness of fit of the ALLoGExp distribution was illustrated with the active repair times data and the results based on the AIC, AICc, HQC K-S and L-S statistics shows that the ALLoGExp distribution provides a better fit than the Exp, EE, LE, LLo, and NH distribution, also, the density plot of the ALLoGExp distribution comparatively provides the best fit to the histogram of the empirical data.We strongly recommend the ALLoGExp distribution for effective modelling of life time data because of its flexible failure rate characteristics.

Figure 1 .
Figure 1.Possible shapes of the probability density function pd f f (x) for some parameter values.

Figure 2 .
Figure 2. Possible shapes of the cumulative density function cd f F(x) for some parameter values.

Figure 3 .
Figure 3. Possible shapes of the reliability function R(x) for some parameter values.

Figure 4 .
Figure 4. Possible shapes of the hazard rate function h(x) for some parameter values.

Figure 5 .
Figure 5.The estimated pdf F(x) (left panel) and cdf F(x) (right panel) plots of the fitted distributions superimposed on the empirical pdf F(x) and cdf F(x) of the active repair time data.

Table 1 .
Simulation results of the estimates and standard errors of the ALLoGExp distribution parameters for different sample sizes

Table 2 .
Simulation results of the bias and mse of the ALLoGExp distribution parameters for different sample sizes