The Odd Log-Logistic Dagum Distribution: Properties and Applications

This paper introduces a new four-parameter lifetime model called the odd log-logistic Dagum distribution. The new model has the advantage of being capable of modeling various shapes of aging and failure criteria. We derive some structural properties of the model odd log-logistic Dagum such as order statistics and incomplete moments. The maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimation are discussed. We prove empirically the importance and exibility of the new model in modeling real data.


Introduction
In 1977, the Professor Camilo Dagum has derived the distribution function, which is called Dagum distribution, from a set of assumptions characterizing the observed regularities in the income distributions from both developed and developing countries (Dagum 1977). Then the Dagum distribution has been extensively and successfully used in researches on income and wealth. Based on this distribution, many authors have extensively studied the characteristics and properties of the Dagum model. In addition to, many works published by the same Dagum, we point out the papers of Kleiber (1996Kleiber ( , 1999, Quintano & D'Agostino (2006), Domma (2007), Pérez & Alaiz (2011), just to name a few. Kleiber & Kotz (2003) and Kleiber (2008) provided an accurate and in-depth reviews of the genesis of Dagum distribution and its empirical applications. Domma (2002) showed that the hazard rate of the Dagum distribution, according to the values of parameters, can be monotonically decreasing, upside-down bathtub, and, nally, bathtub then upside-down bathtub. This particular exibility of the hazard rate has led to several authors apply the Dagum distribution in dierent elds, such as survival analysis and reliability theory (see, for example, Domma, Giordano & Zenga, 2011. In order to increase the exibility of the Dagum distribution, some authors have proposed dierent transformations of such distribution. Domma & Condino (2013) introduced a ve parameter Beta-Dagum distribution, the special case of four-parameter of Beta Dagum has been used by Domma & Condino (2016) for modeling hydrologic data. Oluyede & Rajasooriya (2013) proposed the six paprameter Mc-Dagum Distribution. Among the various transformations found in the literature, we remember the Exponentiated Kumaraswamy Dagum distribution (Huang & Oluyede 2014), the Weighted Dagum (Oluyede & Ye 2014), the transmuted Dagum distribution (Elbatal & Aryal 2015), the Weibull Dagum distribution (Tahir, Cordeiro, Mansoor, Zubair & Alizadeh 2016), the Dagum Poisson Distribution (Oluyede, Motsewabagale, Huang, Warahena-Liyanage & Pararai 2016).
Recently, Gleaton & Lynch (2006) dened a new transformation of distribution function called the odd log-logistic-G (OLL-G) family with one additional shape parameter α > 0 by the cumulative distribution function (cdf) where G(x; ϕ) = 1 − G(x; ϕ). The corresponding probability density function (pdf) of (1) is given by In general a random variable X with pdf (2) is denoted by X ∼ OLL-G(α, ϕ).
An interpretation of the OLL family (1) can be given as follows. Let T be a random variable describing a stochastic system by the cdf G(x). If the random variable X represents the odds, the risk that the system following the lifetime T will be not working at time x is given by G(x)/[1 − G(x)]. If we are interested in modeling the randomness of the odds by the log-logistic pdf r(t) = α t α−1 /(1+t α ) 2 (for t > 0), the cdf of X is given by where Π(t) = t α 1+t α is the cdf of log-logistic distribution. For generating data from this distribution, if u ∼ u(0, 1) then The aim of this paper is to dene and study a new lifetime model called the odd log-logistic Dagum (OLLDa) distribution. Its main feature is that one additional shape parameter is inserted in (3) to provide greater exibility for the generated distribution. Based on the odd log-logistic-G (OLL-G) family of distributions, we construct the four-parameter OLLDa model and give a comprehensive description of some of its mathematical properties in order that it will attract wider applications in reliability, engineering and other areas of research. In fact, the OLLDa model can provide better ts than other competitive models. This paper is organized as follows. In Section 2, we dene the OLLDa distribution. We derive useful mixture representations for the pdf and cdf and give some plots for its pdf and hazard rate function (hrf) in Section 3. We provide in Section 4 some mathematical properties of the OLLDa distribution including quantile function, ordinary and incomplete moments, moment generating function (mgf), mean deviations, probability weighted moments (PWMs), moments of the residual life, reversed residual life and order statistics and their moments are determined. The maximum likelihood estimates (MLEs) of the unknown parameters are obtained in Section 5. In Section 6, the OLLDa distribution is applied to two real data sets to illustrate its potentiality. Finally, in Section 7, we provide a simulation study.

The OLLDa Model
The cdf and pdf of the Dagum distribution are given by (for x > 0) and respectively, where b and c are positive shape parameters and a is a scale parameter.
Using (3) and (1), the cdf and pdf of the OLLDa distribution are given (for x > 0) by and respectively, where b, c and α are positive shape parameters and a > 0 is a scale parameter. If X is a random variable with pdf (6), we denote X ∼OLLDa(α, a, b, c).
The survival function, hrf and cumulative hazard rate function of X are, respectively, given by Quantile function is considered as an important quantity in each distribution and is used widely for simulation of that distribution and nding key percentiles. One can obtain the quantile function of OLLDa distribution by inverting (5) as follows An application of the quantile function is to generate numbers of the distribution. We can easily simulate the OLLDa random variable based on the quantile function. If U is a random variable from Uniform distribution on the unit interval, i.e. U ∼ u(0, 1), then the random variable X = Q X (U ) follows (6), that is X ∼OLLDa(α, a, b, c). In addition, one is interested to analyze the variability of the kurtosis and skewness on the basis of the shape parameters of X. This purpose can be obtained using quantile function and based on the Moors kurtosis (Moors 1988) and Bowley skewness (Kenney & Keeping 1962 The OLL-Da distribution has interesting asymptotic properties based on the cdf, pdf and hrf. The properties are obtained when either x → 0 or x → ∞. In what follows, we present two propositions for asymptotic properties of the OLL-Da distribution.
Proposition 1. he symptotis of yvvEh distriution for dfD pdf nd hrf s x → 0 re given y Proposition 2. he symptotis of yvvEh distriution for dfD pdf nd hrf s x → ∞ re given y hese equtions show the e'et of prmeters on tils of yvvEh distriutionF
By dierentiating of (10) we have denotes the pdf of the Dagum distribution with parameters a, b and ck. The mixture representation (11) is useful to calculate the moments and incomplete moments in an immediate way from those properties of the Dagum distribution.
Let Z be a random variable having the Dagum distribution (3) with parameters a, b and c. For r ≤ b, the rth ordinary and incomplete moments of Z are given by Tahir et al. (2016) are the complete and incomplete beta functions, respectively. So, several structural properties of the OLLDa model can be obtained from (11) and those are properties of the Dagum distribution.

The OLLDa Properties
We investigate mathematical properties of the OLLDa distribution including ordinary and incomplete moments, PWMs and order statistics. It is better to obtain some structural properties of the OLLDa distribution by establishing algebraic expansions than computing those directly by numerical integration of its density function. To this end, we apply the obtained results in the last section.

Ordinary and Incomplete Moments
The nth ordinary moment of X is given by Setting n = 1 in (12), we have the mean of X.
The sth central moment (M s ) and cumulants (κ s ) of X, are given by respectively, where κ 1 = µ 1 . The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships.
Using denition of the moment generating function we have The nth incomplete moment, say ϕ n (t), of the OLLDa distribution is given by ϕ n (t) = t 0 x n f (x)dx. Therefore, we can write from equation (11) ϕ n (t) = and then using the lower incomplete gamma function, we obtain (for n ≤ b) The rst incomplete moment of X, denoted by ϕ 1 (t) , is simply determined from the equation (13) by setting s = 1. It should be noted that the nth incomplete moment of X can be determined on the basis of the quantile function but using (7) and properties of power series (Gradshteyn & Ryzhik 2007, p.17), we can write The rst incomplete moment has important applications related to the Bonferroni and Lorenz curves and the mean residual life and the mean waiting time. Furthermore, the amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and median. The mean deviations, about the mean and about the median of X, depend on ϕ 1 (t).

Residual and Reversed Residual Lifes
The nth moment of the residual life, denoted by m n (t) = E[(X − t) n | X > t] for t > 0 and n = 1, 2, . . . , uniquely determines F (x) (see Navarro, Franco & Ruiz 1998). The nth moment of the residual life of X is given by Based on (15) we can write where R(t) = 1 − F (t) and (n) i = Γ(n + 1)/Γ(n − i + 1) is the falling factorial.
Another interesting function is the mean residual life function or the life expectation at age x dened by m 1 (t) = E [(X − t) | X > t], which represents the expected additional life length for a unit which is alive at age t. The mean residual life of the OLLDa distribution can be obtained by setting n = 1 in the last equation. Navarro et al. (1998) proved that the nth moment (for n = 1, 2, . . .) of the reversed residual life, say M n (t) = E [(t − X) n | X ≤ t] for t > 0, uniquely determines F (x). By denition, the nth moment of the reversed residual life is given by So, it is clear that the nth moment of the reversed residual life of X is as follows The mean inactivity time or mean waiting time, also called the mean reversed residual life function, say M 1 (t) = E[(t − X) | X ≤ t], represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, t). The mean inactivity time of X can be obtained by setting n = 1 in the above equation.

Probability Weighted Moments
The PWMs are used to derive estimators of the parameters and quantiles of generalized distributions. These moments have low variance and no severe bias, and they compare favorably with estimators obtained by the maximum likelihood method. The (s, r)th PWM of X (for r ≥ 1, s ≥ 0) is formally dened by Using equation (10), we can write where q k,l = (−1) k s + k + 1 k Γ(α (s + k + 1) + l) l!Γ(α (s + k + 1) + 1) .
The last equation can be rewritten as follows Then, we have The above equation reveals that the (s, r)th PWM of X is a linear combination of the Da densities. So, we can write ρ r,s = a r ∞ k,l=0 q k,l c [α (s + k + 1) + l] B(1 − r/b, c [α (s + k + 1) + l] + r/b).

Order Statistics
In this part, we obtain the pdf of the ith order statistic from a random sample with size n. We denote the ith order statistic and corresponding pdf from OLLD distribution by X i:n and f i:n (x), respectively. By suppressing the parameters, we have (for i = 1, 2, . . . , n) but based on the equations (5) and (6) we have , and then where Using (17), it can be easily shown that It is clear that the density function of the ith order statistic from the OLLD distribution is a linear combination of the Dagum densities. So, we can easily obtain the mathematical properties for X i:n . For example, the rth moment of X i:n follows from

Estimation
Several methods to estimate parameters were proposed in the literature, but the most popular method is maximum likelihood method. Based on this method, one obtains the maximum likelihood estimations (MLEs) for the parameters. After that, using asymptotic property of MLE, it is easy to derive a 100(1 − γ)% approximate condence interval for each parameter. Here, for simplify, we use g(x) and G(x) as the pdf and cdf of Dagum distribution and obtain the MLEs of parameters for OLLDa(α, a, b, c) distribution. To this purpose, let X 1 , X 2 , . . . , X n be a random sample of size n from the OLLDa distribution given by (6). The log-likelihood function of the vector parameters θ = (α, a, b, c) is as follows It can be shown thaṫ ∂j for any j = α, a, b, c. For more details, see Appendix.
The MLEs can be obtained using the solutions of above equations, but these equations are nonlinear and one can be solved them iteratively.
On the other hand, to obtain the 100(1 − γ)% approximate condence interval for each parameter, we need to the Fisher information matrix and for this purpose, we should obtain the following expectation

Application
In this section, we show that the OLLDa distribution can be a better model than the Dagum, Beta Burr XII (BBXII) ( Histogram of x  Tables 1 and 2 for the rst and second data sets, respectively. Figure 5(a) and (b) displays the tted densities for the rst and second data sets obtained using kernel density estimation based on Gaussian kernel function, respectively. Suppose that X 1 , . . . , X n is a random sample of independent and identically distributed random variables with an unknown pdf f . Then the kernel density estimator of f is derived bŷ where k(·) is the kernel function and h is a smoothing parameter or bandwidth. Silverman (1986) is an appropriate reference for more details of kernel estimation and its properties. In Figure 9, we have considered the standard normal as kernel function, k(·), with bandwidth h = (4σ/3n) 1/5 , whereσ is the standard deviation of the interested sample.

Simulation Study
In order to assess the performance of the MLEs, a small simulation study is performed using the statistical software R through the package (stats4), command MLE. The number of Monte Carlo replications was 20,000. For maximizing the loglikelihood function, we use the MaxBFGS subroutine with analytical derivatives. The evaluation of the estimates was performed based on the following quantities for each sample size: the empirical mean squared errors (MSEs) are calculated using the R package from the Monte Carlo replications. The MLEs are determined for each simulated data, say, (α i ,β i ,â i ,b i ) for i = 1, 2, . . . , 10, 000 and the biases and MSEs are computed by bias h (n) = 1 10000 for h = a, b, c, α. We consider the sample sizes at n = 100, 150 and 300 and consider dierent values for the parameters. The empirical results are given in Table 3. The gures in Table 3 indicate that the estimates are quite stable and, more importantly, are close to the true values for the these sample sizes. Furthermore, as the sample size increases, the MSEs decreases as expected.