A New Generalized Log-logistic and Modified Weibull Distribution with Applications

A new generalized distribution called the log-logistic modified Weibull (LLoGMW) distribution is presented. This distribution includes many submodels such as the log-logistic modified Rayleigh, log-logistic modified exponential, log-logistic Weibull, log-logistic Rayleigh, log-logistic exponential, log-logistic, Weibull, Rayleigh and exponential distributions as special cases. Structural properties of the distribution including the hazard function, reverse hazard function, quantile function, probability weighted moments, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics, L-moments and Rényi entropy are derived. Model parameters are estimated based on the method of maximum likelihood. Finally, real data examples are presented to illustrate the usefulness and applicability of the model.


Introduction
Distribution theory because of its widespread applications in many fields such as finance, economics, physics, just to cite a few, has spawned the statistical literature. From univariate to multivariate distributions, a lot of important new contributions have been presented. Such contributions include those of Eugene, Famoye & Lee (2002) dealing with the beta-normal distribution,  on the beta generalized Lindley distribution, Oluyede, Huang & Pararai (2014) on the gamma-Dagum distribution, as well general family of univariate distributions generated from Weibull distribution introduced by Gurvich, DiBenedetto & Ranade (1997). The cumulative distribution function (cdf) given by Gurvich et al. (1997) is where C is a subset of R and H(x; θ) is a non-negative monotonically increasing function that depends on the vector of parameters θ. The corresponding probability density function (pdf) is given by where h(x; θ) is the derivative of H(x; θ). Different choices of the function H(x; θ) lead to different models. Such choices are for example, H(x; θ) = x, for the exponential distribution, H(x; θ) = x 2 , for the Rayleigh distribution, H(x; θ) = log(x/k), for the Pareto distribution, and H(x; θ) = ( exp(βx) − 1 ) /β, for the Gompertz distribution.
Several useful ways for generating new probability distributions from classic ones to relative new distributions have been proposed in the literature. Nelson (1982) stated that distributions with bathtub-shaped failure rate are often complex and difficult to model. One example of such a distribution was proposed by Hjorth (1980). To lessen the complexity of these distribution, while Rajarshi and Rajarshi (1988) presented a revised version of these distributions, Haupt and Schabe (1992) on the other hand put forward a new lifetime model with bathtub-shaped failure rates. These models are unfortunately not sufficient enough to address various practical situations. Then, new classes of distributions based on modified versions of the Weibull distribution were presented to satisfy non-monotonic failure rate. For such distributions, the reader can refer to Mudholkar and Srivastava (1993), and Pham and Lai (2007) for more details. These authors in their papers, summarized some generalizations of Weibull distribution. Additional generalized distributions include the exponentiated Weibull (EW) (Gupta and Kundu (2001)), the modified Weibull (MW) (Lai, Xie & Murthy (2003)) and the beta exponential (BE) (Nadarajah & Kotz (2005)). Recent extensions are the generalized modified Weibull (GMW) (Carrasco, Ortega & Cordeiro (2008)), the beta modified Weibull (BMW) (Nadarajah Cordeiro & Ortega (2011)), (Siva, Ortega & Cordeiro (2010)), the Weibull-G family (Bourguignon, Silva & Cordeiro (2014)), the gamma-exponentiated Weibull (GEW) (Pinho, Cordeiro & Nobre (2012)), gamma Weibull-G family (Oluyede, Pu, Makubate & Qui (2018)) and the gamma generalized modified Weibull (GGMW) (Oluyede, Huang & Yang (2015)). A new statistical distribution for characterizing the random strength of brittle materials was developed in Gurvich et al. (1997).
In this paper, we propose and study a new generalized distribution called the log-logistic modified Weibull distribution. The primary motivations for considering this new distribution are the advantages it offers with respect to having hazard functions that exhibits increasing, decreasing and bathtub shapes, as well as the versatility and flexibility in modeling lifetime data. This new distribution inherits these desirable properties and also has quite a variety of shapes.
There is another added advantage to this model, in that it also has added dispersion parameter, depending on the overall form that accounts for the scale of the underlying random variable. This distribution also has exponential dumping in the upper tail making the distribution suitable for modeling samples that display power behavior for intermediate observations and decrease in tail probability for large observations or beyond a certain threshold or specified value. Our proposed new distribution also generalizes the log-logistic and modified Weibull distributions. Some structural properties of this distribution are obtained and based on the method of maximum likelihood, estimates of the model parameters are presented.
To this end, the remainder of the paper is organized as follows. Section 2 is concerned with the presentation of the generalized distribution, log-logistic modified Weibull distribution as well as it's hazard and reverse hazard functions, quantile function, and various sub-models. In section 3, probability weighted moments, moments, conditional moments and moment generating function are presented. Mean deviations, Bonferroni and Lorenz curves are derived in section 4. Section 5 is concerned with Rényi entropy, distribution of order statistics and L-moments. Estimates of model parameters and comparisons with other models are presented in section 6. A Monte Carlo simulation study is conducted to examine the bias and mean square error of the maximum likelihood estimates in section 7. Estimation with censored samples are given in section 8. Applications of the proposed model to a real data are given in section 9. The paper end with concluding remarks in section 10.

Generalized Extended Weibull Distribution
The generalized class of distributions referred to as the generalized extended Weibull (GEW) distribution is given by where B(x; ξ) > 0 is a continuous function, H(x; θ) is a non-negative monotonically increasing function that depends on the vector of parameters θ, and ξ is a vector of parameters corresponding to the function B(x; ξ). Putting H(x; θ) = x β e λx , the generalized extended modified Weibull distribution is obtained. For B(x; ξ) = (1 + ( x s ) c ) −1 and H(x; θ) = x β e λx , we get the log logistic modified Weibull distribution. The choice of the functions B(x; ξ) and H(x; θ) including in particular, H(x; θ) = x β e λx , defines a large class of distributions with applications in several areas including reliability, biometry, economics, finance, medicine and survival analysis to mention just a few areas. To avoid potential problems or issues with over parametrization, the parameter vectors ξ and θ can be restricted to two component vectors, respectively.
We can also have the following motivation for the model under study. Consider a series system and assume that the lifetime of the components follow the log-logistic and modified Weibull distributions with reliability functions R 1 (t) = (1 + ( t s ) c ) −1 and R 2 (t) = e −αt β e λt , respectively. The reliability R(t) = P(T > t) of the system is given by . This leads to the log-logistic modified Weibull reliability function. Note that in some context, a series model is referred to as a competing risk model.

The Log-Logistic Modified Weibull Distribution
Some statistical properties of the new log-logistic modified Weibull (LLoGMW) distribution, including pdf, cdf, hazard and reverse hazard functions are presented. Plots of the hazard rate function for selected values of the model parameters are also given. First of all, we present the log-logistic and modified Weibull distributions. The log-logistic distribution attract special attention because it has varying degrees of skewness and kurtosis. Furthermore, this distribution have applications in a wide variety of areas in statistics and applied mathematics including modeling events associated with fracture roughness, life testing, operational risk, option market price distributions, forestry, meteorology, modeling crop prices, software reliability growth, and reliability analysis. The cdf of the log-logistic distribution is given by International Journal of Statistics and Probability Vol. 7, No. 3; The parameters c and s are shape and scale parameters, respectively. The log-logistic distribution is a special case of the Burr XII distribution. See Burr (1942Burr ( ), (1973 for additional details. The modified Weibull (MW) (Lai et al. (2003)) distribution is given by The corresponding pdf is given by for x ≥ 0, α > 0, β > 0, and λ ≥ 0. It is noted that α controls the scale of the distribution, β controls the shape, whereas λ can be considered to be an accelerating factor in the imperfection time and a factor of fragility in the survival of the individual as time increases. The Weibull distribution is obtained by setting λ = 0. This distribution is well known and has been extensively used for modeling data in several areas including reliability. Now, consider the log-logistic modified Weibull (LLoGMW) distribution obtained by setting H(x; θ) = x β e λx and B(x; ξ) = (1 + ( x s ) c ) −1 , (or via the competing risk model), to obtain for s, c, α, β > 0, λ ≥ 0 and x ≥ 0. The corresponding LLoGMW pdf is given by s, c, α, β > 0, λ ≥ 0 and x ≥ 0. For simplicity, we suppress the index-name on G LLoGMW and g LLoGMW so that G and g now stand for the cdf and pdf of the LLoGMW, respectively.
As shown in the plot below, depending on the values of the parameters, the density and hazard functions can exhibit different behavior. Figure 1 displays graphs of the pdf for selected values of the model parameters. The plot suggests that the LLoGMW pdf can be right skewed or decreasing for the selected values of the parameters.

Some New and Known Sub-models
Under the considered choice of the functions B(x; ξ) and H(x; θ), several new and well known distributions can be obtained. For the LLoGMW distribution, the sub-models include the following distributions: • If β = 1, we obtain the log-logistic modified exponential (LLoGME) distribution.
• If c = β = 1 and λ = 0, then the LLoGMW cdf reduces to the two parameter distribution given by for s, α > 0, and x ≥ 0.

Hazard and Reverse Hazard Functions
We present the hazard and reverse hazard functions of the LLoGMW distribution. The hazard and reverse hazard functions of the LLoGMW distribution are respectively given by for x ≥ 0, s, c, α, β > 0, λ ≥ 0, where h F MW (x) and h F B (x) are the hazard functions of the modified Weibull and Burr XII distributions. The limiting behavior of the hazard function of the LLoGMW distribution can be readily established and is as follows: • Note that, • For β > 0, and for each c > 0, lim x→∞ h G (x) = ∞.

Quantile Function
The quantile function of the LLoGMW distribution is obtained by inverting G(x) = u, where 0 ≤ u ≤ 1, and This is equivalent to solving the non-linear equation which can be done using numerical methods. Consequently, random number can be generated based on equation (14). Some quantile results for selected parameters are given in Table 1.

Probability Weighted Moments, Moments, Conditional Moments and Moment Generating Function
We now derive probability weighted moments (PWMs) (Greenwood, Landwehr, Matalas, and Willis (1997)), moments, moment generating function and conditional moments for the LLoGMW distribution. We also present the PWMs for the special case when λ = 0, that is, the PWMs for the log-logistic Weibull distribution. The PWMs of the submodels given in section 2 can be readily obtained from those of the generalized distribution. Moments are necessary and crucial in any statistical analysis, especially in applications. Moments can be used to study the most important features and characteristics of a distribution (e.g., tendency, dispersion, skewness and kurtosis). The probability weighted moments (PWMs) of the LLoGMW distribution is given by http://ijsp.ccsenet.org International Journal of Statistics and Probability Vol. 7, No. 3; Now, by setting y Special Cases: For λ = 0.
• When l = m = 0, we obtain the r th non-central moments given by ) .

Conditional Moments
When it comes to lifetime models, it is often of interest to find the conditional moments and the mean residual lifetime function. The r th conditional moments for LLoGMW distribution is given by Consequently, the r th conditional moment of the LLoGMW distribution is given by

Mean Deviations
The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and median. These are known as the mean deviation about the mean and the mean deviation about the median are defined by δ 1 ( and M =Median (X). The measures δ 1 (x) and δ 2 (x) can be calculated using the relationships δ 1 ( Consequently, the mean deviation about the mean and the mean deviation about the median are δ 1 (x) = 2µG(µ) − 2µ + 2T (µ) and δ 2 (x) = −µ + 2T (M), respectively.

Bonferroni and Lorenz Curves
Bonferroni and Lorenz curves have applications not only in economics for the study income and poverty, but also in other fields such as reliability, demography, insurance and medicine. Bonferroni and Lorenz curves for the LLoGMW distribution are given by

Order Statistics, Moments of Order Statistics, L-Moments and Rényi Entropy
The concept of entropy plays a vital role in information theory. The entropy of a random variable is defined in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty. In this section, we present the distribution of the order statistic, L-moments and Rényi entropy for the LLoGMW distribution.

Order Statistics
Order statistics play an important role in probability and statistics. In this section, we present the distribution of the i th order statistic from the LLoGMW distribution. The pdf of the i th order statistic from the LLoGMW pdf is given by where g m+i (x) is the pdf of the exponentiated LLoGMW distribution with parameters s, c, α, β, λ and (m + i), B(., .) is the beta function and the weights w i,m are given by .

Moments of Order Statistics
The t th moment of the i th order statistics from the LLoGMW distribution can be derived via a result of Barakat and Abdelkader (2004) as follows: Note that .

Rényi Entropy
Recall that an entropy is a measure of uncertainty or variation of a random variable. Rényi entropy (1960) is an extension of Shannon entropy. It is defined by and tends to Shannon entropy as v → 1. Putting [g(x; s, c, α, β, λ)] v = g v (x), we can be write Consequently, Rényi entropy is given by for v 1, and v > 0.

Maximum Likelihood Estimation
Let X ∼ LLoGMW(α, β, λ, s, c) and θ = (α, β, λ, s, c) T be the parameter vector. The log-likelihood ℓ = ℓ(θ) for a single observation x of X is given by The first derivative of the log-likelihood function with respect to θ are given by Given a random sample of size n drawn from the LLoGMW distribution, the total log-likelihood function is given by ℓ * = 1, 2, ....., n is given by equation (21). The equations obtained by setting the above partial derivatives to zero are not in closed form and the values of the parameters s, c, α, β, λ must be found by using iterative methods. The maximum likelihood estimates of the parameters, denoted by θ is obtained by solving the nonlinear equation ) T = 0, using a numerical method such as Newton-Raphson procedure.
International Journal of Statistics and Probability Vol. 7, No. 3;
(a) Mean estimate of the MLEθ of the parameter ϑ = s, c, α, β, λ : (b) Average bias of the MLEθ of the parameter ϑ = s, c, α, β, λ : (c) Root mean squared error (RMSE) of the MLEθ of the parameter ϑ = s, c, α, β, λ :  Table 3, we can verify that as the sample size n increases, the RMSEs decay toward zero. We also observe that for all the parameter values, the biases decrease as the sample size n increases. Consequently, the MLE's and their asymptotic results can be used for estimating and constructing confidence intervals even for reasonably small sample sizes.

Estimation with Censored Samples
In this section, we present maximum likelihood estimates of the parameters of the LLoGMW distribution under Type I right censoring scheme. An application is presented in section 9.3.

MLE for Type I Right Censored Data
Type I right censored data arises when the study is conducted over a specified time period that can terminate before all the units have failed. Each individual has a fixed censoring time C i , so that the complete failure time of an individual will be known only if it is less than or equal to the censoring time T i ≤ C i ; otherwise, only a lower bound of the individual lifetime is available T i > C i . The data for this design are conveniently indicated by pairs of random variables (T i , ϵ i ), i = 1, . . . , n.

Applications
To illustrate the flexibility of the LLoGMW distribution and its sub-models for data modeling, real data examples are presented.
Estimates of the parameters of LLoGMW distribution with standard errors (in parentheses), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Cramer von Mises (W * ), Anderson-Darling (A * ), and sum of squares (SS) from the probability plots are displayed for each data set. The LLoGMW distribution is also compared with the gamma-Dagum (GD) (Oluyede et al. (2014)) and beta modified Weibull (BMW) (Nadarajah et al. (2011)) distributions. The GD and BMW pdfs are given by The likelihood ratio (LR) test can be used to compare the fit of the LLoGMW distribution with its sub-models for a given data set.
Plots of the histogram of the data together with the fitted densities, as well as probability plots (Chambers, Cleveland, Kleiner and Tukey (1983)) are presented. Probability graphs are obtained by plotting G LLoGMW (x ( j) ;ŝ,ĉ,α,β,λ) against ( j − 0.375)/(n + 0.25), j = 1, 2, · · · , n, where x ( j) are the ordered values of the observed data. The measures of closeness are given by the sum of squares (SS), where As described by Chen and Balakrishnan (1995), W * and A * represent the goodness-of-fit statistics. These statistics can be used to verify which distribution fits better to the data. In general, the smaller the values of W * and A * , the better the fit.
International Journal of Statistics and Probability Vol. 7, No. 3; 9.1 Failure Times The data represent the failure times of 50 components (per 1000h), (Murthy et al. (2004)). When optimizing the maximum likelihood function, initial values for the LLoGMW model in the R code are set at s = 0.1, c = 0.1, α = 0.1, β = 0.2, λ = 0.2. Estimates of the parameters of LLoGMW distribution and its related sub-models, AIC, BIC, W * , A * and SS are give in Table 4. Plots of the fitted densities and the histogram, observed probability vs predicted probability are given in Figures 3 and 4. The estimated variancecovariance matrix for the LLoGMW distribution is given by:   however the goodness-of-fit statistics W * and A * are the smallest and definitely points to the LLoGMW distribution as the "best"fit for the failure time data when compared to the values for the sub-models. The goodness-of-fit statistics W * and A * are also better for the LLoGMW distribution when compared to the values for the non-nested GD and BMW distributions. Thus, there is evidence that the LLoGMW distribution is the "best"fit for the failure times data.

Windshield Data
This example consist of 84 Aircraft Windshield data, (Murthy et al. (2004)). When optimizing the maximum likelihood function, initial values for the LLoGMW model in the R code are set at s = 1, c = 1, α = 1, β = 1, λ = 1. The parameters estimates of the LLoGMW distribution and its related sub-models as well as the AIC, BIC, W * , A * and SS are reported in Table 5. While Figure 5 contains the histogram and graphs of the fitted densities, Figure 6 displays observed probability against predicted probability. The estimated variance-covariance matrix for the LLoGMW distribution is given by: , and the 95% confidence intervals for the model parameters are given by s ∈ (3.0369 ± 1.96 × 0.4872), c ∈ (2.9168 ± 1.96 × 0.6356), α ∈ (0.0154 ± 1.96 × 0.0180), β ∈ (0.3996 ± 1.96 × 0.3884) and λ ∈ (0.9283 ± 1.96 × 0.3319), respectively. Figure 5. Fitted Densities, where Model 1 is LLoGMW(s, 1, α, β, λ); Model 2 is LLoGMW(s, 1, α, 1, 0); and Model 3 is LLoGMW(s, 1, α, 2, 0). Figure 6. Probability Plots for Aircraft Windshield data, where 'four'is the four parameter model: LLoGMW(s, 1, α, β, λ); two1 is the model: LLoGMW(s, 1, α, 1, 0); and two2 is the model: LLoGMW(s, 1, α, 2, 0). The LR test statistic for testing H 0 : LLoG against H a : LLoGMW and H 0 : LLoGE against H a : LLoGMW are 28.1716 (p-value < 0.0001) and 7.9451 (p-value = 0.0048). We can conclude that there are significant difference between LLoGMW and LLoG distributions as well between LLoGMW and LLoGE distributions. There is also a significant difference between Weibull and LLoGMW distributions based on the LR test statistic value of 9.12 (p-value=0.02774 < 0.05) at the 5% level. The values of AIC, AICC and BIC as well as the goodness-of-fit statistics W * and A * shows that the LLOGMW distribution is significantly better than the non-nested GD and comparable to the BMW distributions. The submodel LLoGR distribution seem to the better fit for the data based on the goodness-of-fit statistics W * and A * .

Type I Right Censored Data: Remission Times of Cancer Patients
This data set is remission times (in months) for 137 cancer patients, (Lee and Wang (2003)). The data is given in Table 6. Estimates of the parameters under Type I right censoring, AIC, AICC, BIC and SS are given in Table 7. The LLoGMW distribution is compared to the non-nested gamma log-logistic Weibull (GLLoGW) distribution (Foya, Oluyede, Fagbamigbe & Makubate (2017)), the beta Weibull Poisson (BWP) distribution (Percontini, Blas & Cordeiro (2013)) and the gamma Dagum (GD) distribution (Oluyede et al.   The LR test statistic of the hypothesis H 0 : LLoGMW against H a : LLoGMR is 7.54 (p-value = 0.006). Also, LLoGMW distribution gives the smallest, AIC, BIC and SS values. We can conclude that LLoGMW is the best fit for remission times data under Type I right censoring scheme.

Concluding Remarks
This paper presents a new generalized distribution called the LLoGMW distribution that is shown to be suitable for applications in various areas including reliability, survival analysis, just to mention a few areas. Some of its structural properties including hazard and reverse hazard functions, quantile function, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, Rényi entropy, distribution of order statistics, maximum likelihood estimates, asymptotic confidence intervals are presented. Applications of the model to real data sets are given in order to illustrate the applicability and usefulness of the proposed distribution.