Applications of Delay Differential Equations in Biological Systems 2021View this Special Issue
Marshall–Olkin Extended Gumbel Type-II Distribution: Properties and Applications
Due to the advance computer technology, the use of probability distributions has been raised up to solve the real life problems. These applications are found in reliability engineering, computer sciences, economics, psychology, survival analysis, and some others. This study offers a new probability model called Marshall–Olkin Extended Gumbel Type-II (MOEGT-II) which can model various shapes of the failure rate function. The proposed distribution is capable to model increasing, decreasing, reverse J-shaped, and upside down bathtub shapes of the failure rate function. Various statistical properties of the proposed distribution are derived such as alternate expressions for the density and distribution function, special cases of MOEGT-II distribution, quantile function, Lorenz curve, and Bonferroni curve. Estimation of the unknown parameters is carried out by the method of maximum likelihood. A simulation study is conducted using three different iterative methods with different samples of sizes n. The usefulness and potentiality of the MOEGT-II distribution have been shown using three real life data sets. The MOEGT-II distribution has been demonstrated as better fit than Exponentiated Gumbel Type-II (EGT-II), Marshall–Olkin Gumbel Type-II (MOGT-II), Gumbel Type-II (GT-II), Marshall–Olkin–Frechet (MOF), Frechet (F), Burr III, Log Logistic (LL), Beta Inverse Weibull (BIW), and Kumaraswamy Inverse Weibull (KIW) distributions.
Probability distributions are mostly used in survival analysis for modeling data because it provides insight interest in the nature of various parameters and functions mainly in the failure rate function. The lifetime distributions are used in reliability engineering and life data analysis. For example, Aksoy  used the Gamma distribution to model the rainfall data, the Weibull distribution has been used by Keshavan et al.  to model fracture strength of glass data, Al-Hasan and Nigmatullin  worked out the analysis of wind speed data, and environmental radioactivity data were analyzed by Dahm et al.  since the real data set values have different structures and we cannot model these complex data sets with the existing distributions. The existing verities of models can only be applied to the data that have monotonic increasing or decreasing failure function. Surprisingly, this gap has been covered by many researchers working in distribution theory. In this way, the exponential distribution has been modified by many researchers; for example, Gupta and Kundu [5, 6] defined new families of exponential distribution; exponentiated generalized class of distributions was introduced by Cordeiro et al. ; and Exponentiated Generalized Gumbel distribution was proposed by Andrade et al. .
A family of distribution was introduced by Marshall and Olkin  by developing a new method for adding a parameter to the existing probability distributions. Special cases of this family were discussed by using Exponential, Weibull, Gamma, and Log Normal distribution. The survival function of a two parametric exponential distribution was obtained by using the survival function of one-parametric exponential distribution. As a result, it was found that the new extended family might seem as much better competitor than the gamma and Weibull families; likewise, two-parametric Weibull distribution was used to derive the survival function of three-parametric Weibull distribution.
The cumulative distribution function (cdf) of Marshall–Olkin family is given bywhere F(x) is the cdf of a continuous type distribution and “ ” is shape parameter. The corresponding probability density function (pdf) of Marshall–Olkin family is
In recent years, many researchers have used Marshall–Olkin technique to modify different existing models. For example, Marshall–Olkin Bivariate semipareto (MO-BSP) distribution and Marshall–Olkin Bivariate Pareto (MO-BP) distribution were proposed by Alice and Jose . A three-parameter lifetime distribution named as Marshall–Olkin extended Weibull distribution was introduced by Ghitany et al. . Ghitany et al.  derived a new probability distribution to model life time data by extending the Lomax distribution, and this new distribution is called as the Marshall–Olkin Extended Lomax distribution. A four-parameter continuous distribution named as Marshall–Olkin q Weibull distribution and max-min processes was proposed by Jose et al. . Jose  presented a new Marshall–Olkin Extended Uniform distribution by using the Marshall–Olkin model on Extended Uniform distribution. The Marshall–Olkin Extended Lindley distribution was introduced by Ghitany et al. ; The Marshall–Olkin–Frechet Distribution was proposed by Krishna et al. . Al-Saiari et al.  introduced a new three-parameter distribution named as Marshall–Olkin Extended Burr Type XII Distribution.A four-parameter distribution named as a Marshall–Olkin Exponential Weibull (MOEW) distribution for skewed positive data was proposed by Pogány et al. . Saboor and Pogány  suggested a new distribution named as the Marshall–Olkin Gamma Weibull (MOGW) distribution. Marshall–Olkin Extended Weibull Distribution was proposed by Ahmad et al. . Marshall–Olkin additive Weibull distribution was proposed by Afify et al. . Exponentiated Gumbel type-II distributed was discovered by Biswas and Gupta . Gillariose et al.  introduced a new Marshall–Olkin Modified Lindley Distribution; Marshall Olkin Power Lomax distribution was proposed by Haq et al. ; Ogunde et al.  proposed the Extended Gumbel Type-II distribution; and Almetwally et al.  derived various properties of Marshall–Olkin Alpha Power Weibull distribution.
The generalized form of the standard Gumbel type-II distribution was derived by Okorie et al.  called Exponentiated Gumbel type-II (EGT-II). The cdf, survival function, and pdf arewhere “” is the scale parameter and “” and “” are shape parameters.
The Exponentiated Gumbel Type-II Distribution is used for modeling complex data sets
However, a Gumbel Type-II distribution cannot handle the analysis of complex data sets. To deal with this problem, some shape parameters have been added to Gumbel Type-II to improve the model flexibility.
The current research study is motivated by producing new probability models by adding a shape and scale parameter. More surprisingly, there was a gap to modify the Gumbel type-II distribution by using the Marshall–Olkin family of distributions. In this study, we derive a new probability distribution by employing the cdf and pdf of Gumbel type-II in the Marshall–Olkin family of distributions called Marshall–Olkin Extended Gumbel Type-II (MOEGT-II) distribution.
This article is organized as follows. In section 2, we present Marshall–Olkin Extended Gumbel Type-II (MOEGT-II) distribution. In section 3, reliability analysis of MOEGT-II distribution is derived. Several mathematical properties of MOEGT-II distribution are derived in section 4. Measure of uncertainty and inequality of our proposed distribution are derived in section 5. Order statistics of the proposed distribution is derived in section 6. Estimation of the parameters by maximum likelihood is given in section 7, and in section 8, a simulation study is discussed. In section 9, we analyze three real data sets and compared our proposed distribution with different competent distributions. Finally, we conclude the paper in section 10.
2. Marshall–Olkin Extended Gumbel Type-II Distribution
Figures 1 and 2 define the plot of pdf with different values of parameters. We observed that the shape of density function is positively skewed, modified bathtub shape, and reverse J-shaped curve. From Figure 2, it can be observed that the peak of the curves shows a gradual rising behavior; however, a reverse bathtub curve can be also examined.
3. Reliability Analysis of Marshall–Olkin Extended Gumbel Type-II Distribution
In this section, the survival function (SF), hazard rate function (HRF), reverse hazard rate function (RHRF), and cumulative hazard rate function (CHRF) of MOEGT-II distribution are derived.
The survival function of MOEGT-II distribution is
The graph of survival function of MOEGT-II distribution is shown in Figure 3. It is demonstrated from Figures 3(a)–3(c) that curves are in a downward direction; however, varying the parameters produces a decreasing trend also.
The hazard rate function of MOEGT-II distribution is
The graph of hazard rate function of MOEGT-II distribution is presented in Figure 4. From Figure 4, it can be observed that for different values of parameters, the failure rate of MOEGT-II distribution is increasing, decreasing, reverse J-shaped, and upside down bathtub shape.
The reverse hazard rate function of MOEGT-II distribution is
The cumulative hazard rate function of MOEGT-II distribution is as follows:
4. Properties of MOEGT-II Distribution
4.1. Alternate Expression for Distribution Function
For any positive real number “a” and for , we have the generalized binomial expansion:
Consider the power series expansion:which is valid for and any real noninteger .
4.2. Alternate Expression for Density Function
4.3. Special Cases
For different values of parameters, the following distributions are obtained as special cases of MOEGT-II distribution.
Case 1. If substitute in (6), we get the Exponentiated Gumbel Type-II (EGT-II) distribution.
Case 2. If substitute in (6), we get the Marshall–Olkin Gumbel Type-II (MOGT-II) distribution.
Case 3. If substitute in (6), we get the Gumbel Type-II (GT-II) distribution.
Case 4. If substitute and in (6), we get the Exponentiated Frechet (EF) distribution.
Case 5. If substitute and , we get the Frechet (F) distribution.
Case 6. If substitute and , we get the Marshall–Olkin–Frechet (MOF) distribution.
4.4. Quantile Function
Quantile function gives the value of the random variable at a particular probability level in the given probability distribution. Quantile function is also known as inverse cumulative distribution.
The median for MOEGT-II distribution can be obtained by substituting in the above expression:
4.5. Characteristic Function
The characteristic function of MOEGT-II distribution is obtained by substituting (16) in characteristic function formula:
Making the substitution and after simplification, we get
Using the above expression, we get
4.6. Moment Generating Function
Moment generating function can be defined as follows:where
Making the substitution and using expansion (29) in the above equation, we getunder condition ().
4.7. Moments about Mean
Moments about means are under the condition () for = 1, 2. under the condition () for = 1, 2, 3. under the condition () for = 1, 2, 3, 4.
4.8. Mean, Variance, Skewness, and Kurtosis
4.9. Mean Deviation
The mean deviation can be defined from mean or a median.
4.9.1. Mean Deviation about Mean
The mean deviation about mean is defined by Nadarajah and Kotz  with the following form:
Substitute (16) in the above expression,making the substitution , and after simplification, we get
From (14), substituting , we get
4.9.2. Mean Deviation about Median
The mean deviation about median is defined by Nadarajah and Kotz  with the following form:
Substituting in (44), we get
4.10. Geometric Mean
The geometric mean of a random variable is defined by Abd El-Monsef and Ghoneim  with the following form:where
Substituting (16) in the above equation, making the substitution , and after simplification, we getas
Substituting the result in the above equation, we get the geometric mean of MOEGT-II distribution:
4.11. Harmonic Mean
The Harmonic mean is defined as
Substituting (16) in the above expression, making the substitution , and after simplification, we get
The Gamma function is
So, using Gamma function (57) in the above equation, we get the Harmonic mean of MOEGT-II distribution:
5. Measure of Uncertainty and Inequality
In this section, the measure of uncertainty of MOEGT-II distribution has been derived including Renyi entropy and entropy. Lorenz and Bonferroni curves are also derived as a measure of inequalities of MOEGT-II distribution.
5.1. Renyi Entropy
Renyi entropy is defined aswhere
Making the substitution and after simplification, we get
As we know,
Using expansion (63) in the above expression,
Applying log on the above equation and then substituting the resulting equation in (59), we get the Renyi entropy of MOEGT-II distribution: under the condition ().
It can be observed from Table 1 that as values of parameters decrease, the value of Renyi entropy increases.
q-entropy can be defined as
Substituting = q in equation (64) and then substituting the resulting equation in the above expression, we get the entropy of MOEGT-II distribution:where
It can be observed from Table 2 that the value of q-Renyi entropy decreases as the values of parameters decrease.
5.3. Lorenz Curve
L (x) denotes the Lorenz curve and is defined aswhere
Making the substitution and after using upper incomplete gamma function, we get
5.4. Bonferroni Curve
B(x) denotes the Bonferroni Curve and is defined as
Figure 7 defines the graph of Bonferroni and Lorenz curve.
6. Order Statistics
In this section, the density of order statistics for MOEGT-II distribution has been derived. The moments for the density of order statistics are obtained. The distribution of the median is also discussed.
6.1. Density of Order Statistics
Let be a random sample of size n from MOEGT-II density function, (x). Let denote the order statistics. Then, the density function of order statistics of the random variable can be obtained using
Substituting (6) and (7) in the above equation, after applying the generalized binomial expansion (12) and power series expansion (13), we get the final expression of density of order statistics of MOEGT-II distribution:whereand
denotes the Gumbel type-II density function with parameter and . So, the density function of the order statistics is simply an infinite linear combination of Gumbel type-II density.
6.2. Moments of Order Statistics
The moment for the density of order statistics can be obtained as follows:
For different values of “,” i.e., = 1, 2, 3, 4, in (81), we can obtain first four arbitrary moments of the order statistic.
6.3. Distribution of Median
In many situations, when data are skewed or when the outliers are present in the data, usually the median is found to be the best measure amongst all. The distribution of median can be obtained using the density of order statistics for odd and even sample sizes.
When n is odd, say n = 2l + 1, the median is the order statistic. By substituting k = l + 1 in the density function of order statistic, the distribution of median can be obtained.
From (77), the density of order statistic and the density of median (m) are obtained as follows:
When n is even, say , the median is then obtained as and density can be obtained from the joint density function of and order statistics.
The joint density function of and can be found using
Now, using the expansion,
Substituting (88) and (89) in the above equation, then applying the binomial expansion (89) twice, then solving integration term, and after simplification, we get the distribution of median for MOEGT-II distribution: for .
7. Maximum Likelihood Estimation
Let be a random sample of size “n” from the MOEGT-II distribution defined in (7). Then, the likelihood function can be expressed as follows:
Taking on both sides, we get
Partially differentiating (92) w.r.t “,” “,” “,” and “,” we get
The maximum likelihood estimates of parameters, , , and can be obtained from the solution of equations given in (93). Since these are nonlinear equations and their solution in closed form cannot be obtained, therefore any numerical method can be used to obtain MLE.
8. Simulation Study
To check the flexibility of the MOEGT-II, we discuss the simulation study of the proposed distribution using Monte Carlo simulations. All results are obtained for 5000 Monte Carlo runs, and the simulations are performed using R. In each replication, random samples of size (n = 50, 100, 300, 500, and 1000) are generated for different combinations of parameters from MOEGT-II (, , , ) distribution, and three different iterative procedures named as Nelder–Mead, Quasi-Newton, and Conjugate-Gradients (CG) method (also known as Fletcher–Reeves method) have been used. The means and standard deviations of the MLEs are listed in Tables 3 and 4.
In this section, the MOEGT-II distribution is applied to three real data sets to illustrate the usefulness and applicability of the proposed model. The MLEs of the proposed model parameters are computed, and some goodness-of-fit statistics for the fitted models are computed in R by using the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method. The estimated cdfs of the data sets and the pdfs of the fitted distributions are plotted. In addition, the probability plot (P-P) is also discussed. We compare the MOEGT-II distribution with nine different models that are the Exponentiated Gumbel Type-II (EGT-II) distribution, Marshall–Olkin Gumbel Type-II (MOGT-II) distribution, Gumbel Type-II (GT-II) distribution, Marshall Olkin Frechet (MOF) distribution, Frechet (F) distribution, Burr III distribution, Log Logistic (LL) distribution, Beta Inverse Weibull (BIW) distribution, and Kumaraswamy Inverse Weibull (KIW) distribution.
The first example is a data set from Nichols and Padgett  consisting of 100 observations on breaking stress of carbon fibres (in Gba). Recently, Afify et al.  and Saboor et al.  used these data to analyze the distributions. The data values are 3.7, 2.74, 2.73, 2.5, 3.6, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.9, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.2, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.7, 2.03, 1.8, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65, 2.341, 4.628, 1.244, 2.435, 4.806, 1.249, 2.464, 4.881, 1.262, 2.543, 5.140.
The second example is a data set corresponding to service times for a particular windshield model including 63 observations that are classified as service times of windshields from Murthy et al. . Recently, Tahir et al.  applied these data to the Weibull–Lomax distribution. The data values are 0.046, 1.436, 2.592, 0.140, 1.492, 2.600, 0.150, 1.580, 2.670, 0.248, 1.719, 2.717, 0.280, 1.794, 2.819, 0.313, 1.915, 2.820, 0.389, 1.920, 2.878, 0.487, 1.963, 2.950, 0.622, 1.978, 3.003, 0.900, 2.053, 3.102, 0.952, 2.065, 3.304, 0.996, 2.117, 3.483, 1.003, 2.137, 3.500, 1.010, 2.141, 3.622, 1.085, 2.163, 3.665, 1.092, 2.183, 3.695, 1.152, 2.240, 4.015, 1.183, 2.341, 4.628, 1.244, 2.435, 4.806, 1.249, 2.464, 4.881, 1.262, 2.543, 5.140.
The third data set represents the fracture toughness of alumina (Al2O3) (in the units of MPa m1 = 2) by Nadarajah . Recently, Arifa et al.  and Ghitany et al.  used these data to analyze the distributions. The data values are 5.5, 5, 4.9, 6.4, 5.1, 5.2, 5.2, 5, 4.7, 4, 4.5, 4.2, 4.1, 4.56, 5.01, 4.7, 3.13, 3.12, 2.68, 2.77, 2.7, 2.36, 4.38, 5.73, 4.35, 6.81, 1.91, 2.66, 2.61, 1.68, 2.04, 2.08, 2.13, 3.8, 3.73, 3.71, 3.28, 3.9, 4, 3.8, 4.1, 3.9, 4.05, 4, 3.95, 4, 4.5, 4.5, 4.2, 4.55, 4.65, 4.1, 4.25, 4.3, 4.5, 4.7, 5.15, 4.3, 4.5, 4.9, 5, 5.35, 5.15, 5.25, 5.8, 5.85, 5.9, 5.75, 6.25, 6.05, 5.9, 3.6, 4.1, 4.5, 5.3, 4.85, 5.3, 5.45, 5.1, 5.3, 5.2, 5.3, 5.25, 4.75, 4.5, 4.2, 4, 4.15, 4.25, 4.3, 3.75, 3.95, 3.51, 4.13, 5.4, 5, 2.1, 4.6, 3.2, 2.5, 4.1, 3.5, 3.2, 3.3, 4.6, 4.3, 4.3, 4.5, 5.5, 4.6, 4.9, 4.3, 3, 3.4, 3.7, 4.4, 4.9, 4.9, 5.
Tables 5–7 provide the MLEs and their corresponding standard errors (SEs) (in parentheses); and the result of AIC, CAIC, BIC, HQIC, log-likelihood, Cramer-von Mises , Anderson–Darling , and Kolmogorov–Smirnov (KS) statistics and its value for each distribution are presented in Tables 8–10. The MOEGT-II distribution has the smallest values for the AIC, CAIC, BIC, HQIC, log-likelihood, Cramer-von Mises , Anderson–Darling , and Kolmogorov-Smirnov (KS) statistics as compared with other fitted distributions, suggesting that the MOEGT-II model provides the good fit. The estimated cdfs, histogram of three data sets, pdfs, and P-P are presented in Figures 8–13, respectively. It is clear from Tables 8–10 and Figures 8–13 that the MOEGT-II distribution provides better fit than EGT-II, MOGT-II, GT-II, MOF, Frechet (F), Burr III, LL, BIW, and KIW models for given three data sets.
A new probability model called MOEGT-II is derived by employing the cdf of EGT-II in Marshall–Olkin family of distributions. The proposed model is important due to its modeling of various shapes of the hazard rate function such as increasing, decreasing, reverse J-shaped, and upside down bathtub. The special cases of MOEGT-II are called EGT-II, MOGT-II, GT-II, EF, F, and MOF distribution. Mathematical properties of the proposed distribution were studied including survival function, hazard rate function, reverse hazard rate function, Renyi entropy, entropy, Lorenz curve, and Bonferroni curve. Expressions for the order statistics and their moments are also derived. Performance of the maximum likelihood estimates is also investigated through Monte Carlo simulation. We observed that with the increase in the sample size, the better results for maximum likelihood estimates are obtained. Also, the Conjugate-Gradients (CG) method (Fletcher–Reeves method) provides better estimates as compared with the Nelder–Mead method and Quasi-Newton method with increasing sample size. At the end, we have considered three data sets, and it has been shown that MOEGT-II distribution provides better performance than EGT-II, MOGT-II, GT-II, MOF, F, Burr III, LL, BIW, and KIW distributions.
The data sets have been taken from the literature and the references are given at the end.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The following are the contributions of each author to the paper. Farwa Willayat was responsible for main analysis wrote original article, provided software, proposed methodology, and visualized the study. Naz Saud wrote original draft, supervised the study, reviewed the article, provided suggestions, developed methodology, and visualized the study. Muhammad Ijaz conceptualized the study, visualized the study, reviewed and edited the article, developed methodology, and provided suggestions. Anita Silvianita and M. Mahmoud El-Morshedy reviewed and edited the manuscript, provide suggestions, and validated the study.
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