https://orcid.org/0000-0002-5887-8858
Correction
12 Oct 2023: Klakattawi H, Alsulami D, Elaal MA, Dey S, Baharith L (2023) Correction: A new generalized family of distributions based on combining Marshal-Olkin transformation with T-X family. PLOS ONE 18(10): e0293100. https://doi.org/10.1371/journal.pone.0293100 View correction
Figures
Abstract
Data analysis in real life often relies mainly on statistical probability distributions. However, data arising from different fields such as environmental, financial, biomedical sciences and other areas may not fit the classical distributions. Therefore, the need arises for developing new distributions that would capture high degree of skewness and kurtosis and enhance the goodness-of-fit in empirical distribution. In this paper, we introduce a novel family of distributions which can extend some popular classes of distributions to include different new versions of the baseline distributions. The proposed family of distributions is referred as the Marshall-Olkin Weibull generated family. The proposed family of distributions is a combination of Marshall-Olkin transformation and the Weibull generated family. Two special members of the proposed family are investigated. A variety of shapes for the densities and hazard rate are presented of the considered sub-models. Some of the main mathematical properties of this family are derived. The estimation for the parameters is obtained via the maximum likelihood method. Moreover, the performance of the estimators for the considered members is examined through simulation studies in terms of bias and root mean square error. Besides, based on the new generated family, the log Marshall-Olkin Weibull-Weibull regression model for censored data is proposed. Finally, COVID-19 data and three lifetime data sets are used to demonstrate the importance of the newly proposed family. Through such an applications, it is shown that this family of distributions provides a better fit when compared with other competitive distributions.
Citation: Klakattawi H, Alsulami D, Elaal MA, Dey S, Baharith L (2022) A new generalized family of distributions based on combining Marshal-Olkin transformation with T-X family. PLoS ONE 17(2): e0263673. https://doi.org/10.1371/journal.pone.0263673
Editor: Qichun Zhang, University of Bradford, UNITED KINGDOM
Received: October 26, 2021; Accepted: January 24, 2022; Published: February 9, 2022
Copyright: © 2022 Klakattawi et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: This study is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (72317). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Selecting a suitable statistical distribution for modeling and analyzing data is of great importance in order to achieve more accurate decisions. Over the years, many statistical distributions have been proposed to fit different shapes of the data. Applying classical distributions to fit these data sets may lead to inaccurate results. Hence, the need for modifying the standard distributions is clearly evident.
Recently, many methods have been proposed to obtain more flexible distributions that can reflect the data behavior in many situations more accurately. Adding a new parameter to existing distributions is one popular method that has attracted more attention in literature. [1] proposed the exponentiated method by raising a new parameter to the cumulative distribution function (cdf) of any distribution.
Additionally, [2], presented a new modified method for obtaining more adaptable distributions called Marshall-Olkin (MO) family of distributions. The method relies on extending a given distribution by including an additional shape parameter called a tilt parameter, [3]. To illustrate, consider a random variable X from a baseline distribution and such variable has a cdf G(x; τ). Then, for the MO family, its cdf and its probability density function (pdf) are, respectively, given by
(1)
and
(2)
where α > 0 is the shape (tilt) parameter,
and τ is a vector of parameters for the baseline model. The MO transformation has some attractable characteristics which can add more flexibility to any baseline distribution. This encouraged many researchers to apply the MO transformation to common classical distributions such as the Pareto distribution by [4], Weibull distribution by [5, 6], the three parameter Weibull by [7], Lomax distribution by [8], gamma distribution by [9], Uniform distribution by [10], log-logistic distribution by [11], inverted Kumaraswamy distribution by [12], inverse Lomax distribution by [13], kappa distribution by [14], generalized Pareto distribution by [15], Gumbel-Lomax distribution by [16], power Lomax distribution by [17] and inverse log-logistic distribution by [18], among others.
The MO transformation can be applied on family of distributions. Specifically, a compounding technique can be used to combine the MO family with other generated classes to obtain some new families that provide distribution with greater flexibility in modeling. This method can be defined by taking any family of distributions as the baseline cdf in Eq (1). Several families have been generalized using this method, for instance, [19] introduced the MO extended Weibull family, [20] suggested the MO Kumarswamy-G family, [21] proposed the MO odd Lindley-G family, [22] proposed the MO alpha power family of distribution, [23] considered the MO Topp Leone-G family and [24] suggested the MO odd Burr III-G family and among many others.
Another way to develop a new flexible distribution is by the transformed-transformer (T-X) family proposed by [25]. The T-X method is widely applicable because any continuous distribution can be used as a generator. In other words, for any continuous random variable X, the cdf and pdf for the T-X family with a vector of parameters ζ = (θ, τ) are, respectively, defined by
where a is a real number, r(t; θ) is the generator pdf of a random variable T and W(G(x; τ)) is function of the cdf of the random variable X. [25, 26] presented the Weibull-G family as an example of the T-X family based on the Weibull distribution with parameters c and β as a generator. That is, for any distribution with cdf G(x; τ) and vector of parameters; τ, the cdf of the Weibull-G family can be expressed as follows
(3)
and the corresponding pdf is of the form
(4)
where c > 0 is the shape parameter and β > 0 is a scale parameter.
Many studies have been considered of some members of the Weibull-G family, such as the Weibull-Pareto by [27], the Weibull-Rayleigh by [28], the Weibull-Lomax by [29] and more lately the Weibull-gamma by [30].
Recently, attempts have been made to generalize the Weibull-G family. For example, [31] introduced a Beta Weibull-G class of distributions, [32] generalized the Weibull-G family by the combination with Kumaraswamy-G family to obtain the Kumaraswamy Weibull-G family and also the Weibull-G family was combined by [33] with the gamma-generator for defining the gamma Weibull-G, among others.
The first objective of this paper is to introduce a novel generalization for the Weibull-G family, called MO Weibull-G (MOW-G) family and derive some of its properties. The construction of this family is the combination of the MO family and the Weibull-G. Marshall-Olkin Weibull-Exponential (MOW-E) and Marshall-Olkin Weibull-Weibull (MOW-W) distributions are special cases of the MOW-G distribution. The proposed family of distributions provide better fits than some well known lifetime distributions. The importance of this new family of distributions is the ability of describing decreasing, increasing, bath-tub and upside down bath-tub shaped hazard rate functions which is extensively used in many real life data. Besides, MOW-G family is a suitable model for fitting positively skewed data which may not be adequately modelled by many other distributions. Thus, it can be used to fit data related to public health, biomedical studies, industrial reliability, survival analysis and several other areas. Second objective is to estimate the unknown model parameters using maximum likelihood method for different sample sizes and different parameter values. To evaluate the performance of the estimators, a simulation study is carried out. In addition, four real life data sets have been analyzed for illustrative purposes. Third objective is to obtain the maximum likelihood estimators (MLEs) of the log Marshall-Olkin Weibull-Weibull (LMOW-W) regression model for censored data to show the flexibility of the log Marshall-Olkin Weibull-Weibull regression model. Thus far we have not come across any report on estimation of parameters for the considered distribution along with regression model for censored data.
We have organized the remainder of this paper in the following way: Section 2 discusses the new MOW-G family. Some sub-models of the family are presented in Section 3 while several statistical properties of this family are given in section 4. Section 5 presents the maximum likelihood estimation for the new family’s parameters. A Monte Carlo simulation study is provided in section 6 to evaluate the performance of the estimators for the MOW-G family. Section 7 illustrates the flexibility of the new family compared to other families using four real data sets. In Section 8, log Marshall-Olkin Weibull-Weibull regression model for censored data is presented. Finally, we conclude the paper in Section 9.
2 The MOW-G family
Replacing G and g in Eqs (1) and (2) by cdf and pdf of the Weibull-G in Eqs (3) and (4), then MOW-G family is obtained. That is, for any baseline distribution with cdf G(x; τ), the respective MOW-G family associated cdf with parameters ζ = (α, c, β, τ) is defined as
(5)
Consequently, its pdf can be obtained as
(6)
where α, c, β and τ are defined in Eqs (2) and (4).
3 Sub-models of the MOW-G family
Two special distributions are considered as members of the MOW-G family: the MOW-exponential (MOW-E) distribution and the MOW-Weibull (MOW-W) distribution. That is, for the MOW-E distribution, its corresponding cdf and pdf can be correspondingly obtained as
(7)
where λ > 0 is the scale parameter, α and c, β are defined in (2) and (4), respectively.
Similarly, for the MOW-W distribution, its respective cdf and pdf can be found as
(8)
where γ > 0 is the shape parameter, δ > 0 is the scale parameter, α and c, β are defined in Eqs (2) and (4), respectively.
Possible shapes for the density and hazard functions for MOW-E and MOW-W distributions are, respectively shown in Figs 1 and 2. It can be seen from Fig 1 that, the pdf for MOW-E and MOW-W exhibit symmetrical, right-skewed, left-skewed, J shaped and reversed-J shaped densities. Moreover, from Fig 2 we can see that, the hazard for these distributions exhibits increasing, decreasing, bathtub, upside-down bathtub, S shaped, J shaped, and reversed-J shapes. S1 Appendix provides a list of other sub-models associated with the MOW-G family.
Plots for the pdf of two members from the MOW-G, with different values of the parameters: (a)MOW-E distribution and (b)MOW-W distribution.
Plots for the hazard function of two members from the MOW-G, with different values of the parameters: (a)MOW-E distribution and (b)MOW-W distribution.
4 Statistical properties of the MOW-G family
4.1 Expansion of the density function
For α ∈ (0,1), the binomial expansion states that
(9)
where Γ(.) is the gamma function.
Then, it follows from Eq (9) that
Therefore, the MOW-G family’s pdf can be obtained as
Considering the expansion of the exponential function as
(10)
we can have,
(11)
For a > 0, we have the following formula
(12)
where the constant Pt,m is defined as
(13)
Also, from Eq (9), we have
(14)
Therefore, from Eqs (12) and (14) and for 0 < α < 1, the pdf of MOW-G is given by
For α > 1, the pdf of the MOW-G family can be expressed as follows
which can be written as
From Eq (9), we can have
(15)
and by using the binomial theorem
(16)
Eq (15) can be written as
Thus, by replacing by
, the MOW-G family’s pdf is expressed as
Applying Eq (10), the pdf can be given by
Now by using Eq (9) for [1 − G(x; τ)]−1 and Eq (12) for (−log(1 − G(x; τ)))c(i+1)−1, the pdf for MOW-G family with α > 1 is obtained by
Therefore, the pdf of MOW-G family is written as
(17)
where
(18)
and
(19)
for Pt,m is defined by (13).
At this point, the cdf and pdf for the exponentiated-G (Exp-G) distribution, denoted by fExpG(x), with a parameter c can be respectively defined for an arbitrary G(x) as
Then, a linear combination of the Exp-G density functions serves as a way of expressing the pdf of the MOW-G family. Hence, starting from the properties of the Exp-G class of distributions, some features of the MOW-G family of distributions can be obtained. Such features have been investigated in several studies such as, [34–36], among others.
4.2 Moments
Let Yj is a random variable from Exp-G distribution with parameter (c(i + 1) + m + r), then the rth moment of the MOW-G family is given by
where ωj is defined in Eq (18).
4.3 Moment generating function
The moment generating function of the MOW-G family can be derived in term of the Exp-G distribution as follows
where Yj is a random variable from Exp-G distribution with parameter (c(i + 1) + m + r) and ωj is defined in Eq (18).
4.4 Incomplete moments
The sth incomplete moment for MOW-G family is obtained as follows
where
is the incomplete sth moment for the Exp-G distribution with parameter (c(i + 1) + m + r) and ωj is defined in Eq (18).
4.6 Rényi entropy
The Rényi entropy for the MOW-G family is given by
For 0 < α < 1, (fMOWG(x))R can be obtained using Eq (9) as
Now, using Eq (10), we have
Applying Eqs (9) and (12), respectively, for (1 − G(x; τ))−R and (−log(1 − G(x; τ)))R(c−1)+ci, the Rényi entropy for the MOW-G family for α ∈ (0, 1), is expressed as
(21)
(22)
where Γ(.) is the gamma function, Pt,m is defined in Eq (13) and
is the Rényi entropy for the Exp-G distribution with parameter (
).
Now for α > 1 and using Eq (9), we have
Applying Eq (16), and replacing by
, the entropy is written as
where A is defined in Eq (20).
Thus, for α > 1, the Rényi entropy for the MOW-G family is written as
where Γ(.) is the gamma function, Pt,m is defined in Eq (13) and
is the the Rényi entropy for the Exp-G distribution with parameter (
).
4.7 Distribution of order statistics
Consider the random sample x1, x2, …, xn from MOW-G distribution with order statistics x1:n < x2:n < … < xn:n. Then the distribution of the qth order statistics of MOW-G distribution is given by
where from Eq (16),
Then, for for α ∈ (0, 1) and after some algebra, the distribution of the qth order statistics of the MOW-G is expressed as follows
(24)
Similarly, for α > 1, we can obtain
(25)
where Γ(.) is the gamma function and z is defined in Eq (19).
From Eqs (24) and (25), we can see that the pdf for the order statistic of the MOW-G family is an infinite linear combination of the pdf of the Exp-G with parameter (c(i + 1) + m + r). Thus, many proprieties of these order statistics can be achieved from the proprieties of the Exp-G.
5 MOW-G family’s parameters estimation
To obtain the estimators for the parameters of any member from the MOW-G family, the maximum likelihood method can be applied. That is, if we have x1, x2, …, xn follows a distribution associated to the MOW-G class, with a parameters vector φ = (α, β, c, τ), where τ represents the vector of parameters of the baseline distribution G, then the log-likelihood function (ℓ) is defined by
(26)
The complexity of the above log-likelihood makes it difficult to solve analytically. Therefore, we take the derivative of Eq (26) with respect to the parameters as follows
and
Thus, the MLEs of the parameters vector φ can be acquired by obtaining the solutions in iterative way of the above nonlinear equations using the well-known Newton-Raphson method or any other numerical method. Instead, employing any standard non-linear optimization technique, the log-likelihood in Eq (26) can be directly maximized.
6 Simulation study
For evaluating the performance of the MLEs, some simulation studies is conducted for the two particular members of the MOW-G family namely; MOW-E distribution with rate parameter λ and MOW-W distribution with shape parameter γ and scale parameter δ. For each distribution, the simulation is performed over the number of iterations, nsimu = 1000. Four distinct sample sizes n = 25, 50, 100, 200, 500 are considered with the following cases for the true parameters, θtru.
We compute the mean square errors (MSEs) and biases of the MLEs of the parameters based on 1000 iterations, where
The R programming language [37] has been used to conduct the Monte Carlo simulation studies. Tables 1 and 2 report the results for the MLEs for parameters of the MOW-E and MOW-W along with their corresponding absolute average bias and RMSE, respectively.
The “optim” function in R software is used to achieve all estimation results. The results of the simulation study are reported in Tables 1 and 2. Tables 1 and 2 show that the RMSEs decrease as the sample size increases for all parameters. The bias tend to zero for large n, that is, the estimates are asymptotically unbiased. The RMSEs tend to zero as n increases, which indicates that the estimates are consistent. This provides evidence that the maximum likelihood method has a good performance when estimating the parameters of the proposed family.
7 Applications
Four real-world data sets with the exponential and Weibull distributions as the G, are considered to investigate the flexibility of the proposed MOW-G family. The first data set is fitted to MOW-E distribution and compared with Marshall Olkin-G defined by [2, 38] and Weibull-G defined by [39] considering G as exponential and Weibull. Second and third data sets are also fitted to MO extended Weibull (MOEW) defined by [6]. While the fourth data set is fitted to MOW-W distribution and compared with MO-exponential (MO-E), Weibull-G and Exp-G defined by [1] considering G as exponential and Weibull.
The proposed model is compared with other models according to some criteria, include the value of the log likelihood function (ℓ), Akaike information criterion (AIC), Bayesian information criterion (BIC), consistent Akaike information criterion (CAIC), Cramér-von Mises (W*), Anderson-Darling (A*), Kolmogorov-Smirnov (KS) and P-value statistics. These statistics or criteria are widely used to assess the performance of a distribution in modeling a data set. Smaller values of these statistics, indicate a better fit.
Data set I: The data represents COVID-19 drought mortality rate of Canada for 36 days, in the period 10 April to 15 May 2020, at https://covid19.who.int. The data is obtained from [40] and listed in Table 3.
Data set II: Represents tde number of successive failures for tde air conditioning system of each member in a fleet of 13 Boeing 720 jet airplanes [41]. Tde data is listed in Table 4.
Data set III: Represents strengtd data measured in GPA for single carbon fibers which were tested under tension at gauge lengtds of 20 mm and impregnated 1000-carbon fiber tows. [42] originally reported tdis data witd 63 observations in Table 5.
Data set IV: Represents experimental data of tde strengtd of glass fibers witd lengtds 1.5 cm. Tde source of such data is tde National Physical Laboratory in England and it is composed of 63 observations and listed in Table 6, [43].
Tables 7–14 report the MLE, standard error (SE), summary of the model selection criteria and showing the −ℓ, AIC, CAIC, BIC, W*, AD*, KS and P-value statistics in each model, for the four data sets respectively. In addition, the plots of the estimated pdf and cdf with empirical data in Figs 3–6 show closeness, confirming that MOW-E and MOW-W have a better fit for all data sets compared to other competing distributions.
The results in Tables 8–14 along with the plots in Figs 3–6 indicate the potentiality of the proposed distributions when comparing with competitive distributions using the selected data.
8 Regression modeling
Practically speaking, lifetimes are usually affected by some explanatory variables. Parametric models can be used to estimate univariate survival functions as well as can be solved regression problems. In this section, we present a new location regression model based on the MOW-W distribution.
8.1 The log MOW-W distribution
The regression model can be obtained by considering the log MOW-W (LMOW-W) distribution in which the location parameter depends on some explanatory variables v. Re-defining some parameters of the random variable X which have the MOW-W in Eq (8), as and δ = eμ, then, the random variable Y = log(X) will have the LMOW-W distribution with the following pdf and survival function respectively
(27)
and
(28)
where α, c, β are defined in Eq (8), σ > 0 is the scale parameter and
is a location parameter.
For the standardized random variable , its pdf can be derived as
(29)
8.2 The LMOW-W regression model for censored data
Assuming the survival time Xi of the ith individual in the sample, for i = 1, …, n, and considering a set of p covariates such that vi = (1, vi1, vi2, …, vip)T, where the 1 is for the intercept term, the location-scale regression model relating to the response variable yi = log(xi) with the explanatory variable vector v can be mathematically described from the LMOW-W as follows
(30)
where the random error zi has density function in Eq (29) with unknown parameters σ, α, c, β > 0 and τ = (τ0, τ1, τ2, …, τp)T are the unknown regression coefficients of the p explanatory variables. To illustrate, yi has the location parameter
, in which the vector μ = (μ1, μ2, …, μn)T of the location parameter is described by the linear model μ = Vτ, such that V = (v1, v2, …, vn)T is a known model matrix.
For an independent sample of n observations (y1 v1), …, (yn, vn), assume that F and C be the sets of individuals for which the response variable yi is the log-lifetime and log-censoring, respectively. Considering the non-informative censoring such that the observed lifetimes and censoring times are independent, the random response is determined by yi = {log(xi), log(ci)}. Then, the log-likelihood function for the vector of parameters ζ = (α, c, β, σ, τT)T from model (30) can be obtained from
where f(yi) is the density function in Eq (27) and S(yi) is the survival function in Eq (28) of Yi. Thus, the log-likelihood function for ζ defined as
(31)
where r is the number of uncensored observations (failures) and
, for
.
The log-likelihood function in Eq (31) is then, maximized to find the MLEs of ζ. For this case, numerical non-linear optimization procedures are commonly applied. The optim function in R can be used in order to find the MLEs.
8.3 Application for log Marshall-Olkin Weibull-Weibull regression model for censored data
The LMOW-W regression model is applied to fit data from an earlier study by [44], which compared treatment with radiotherapy only (Arm A) and radiotherapy plus chemotherapy (Arm B), for patients with head and neck cancer. The study included the survival time of 51 patients in arm A and 45 patients in arm B. Particularly, 9 patients in arm A and 14 patients in arm B failed to follow up and were considered censored. Only one predictor is considered in this study; v1 which represents the two-arms (Arm A = 0, Arm B = 1). Therefore, the LMOW-W regression model is considered as
where z1, …, z96 are independent random variables with pdf in Eq 29.
The results from LMOW-W model are compared with some alternative regression models, namely; the log-Weibull (LW) in [45], the log-gamma-Weibull (LGW) in [46], the Kumaraswamy-log-logistic (KumL) [47], the log-modified Weibull (LMW) in [48], the log-generalized modified Weibull (LGMW) in [49] and the log-Pareto Weibull generalized lambda (LPWGL) in [50].
Results in Table 15 suggest that the LMOW-W model has the best fit to the data set as it has the lowest AIC and BIC values.
9 Conclusion
Weibull-G family introduced in [25] has advantage of adding more flexibility in shapes of a base line distribution and can be generalized in some other forms. This paper presented and studied the MOW-G family which combines both MO transformation with the Weibull-G to produce compound distributions with better performance. Specifically, two sub-models namely, MOW-E distribution and MOW-W distribution are presented. Some of the statistical properties of the proposed family are studied. Some of the properties related to moments, moment generating function, incomplete moments, Quantile, median, Renyi entropy, and distribution of order statistics are presented. To estimate the parameters of the model, the method of maximum likelihood is applied. Additionally, the performance of the MLEs of the two selected members is assessed via Monte Carlo simulation studies based on two criteria; bias and RMSE. The study exhibits a good performance when estimating the parameters of the proposed family using the maximum likelihood method. Four real data sets are fitted by the proposed distributions as well as compared with some selected competitive distributions. The results have been compared based on some efficiency measurements. These results demonstrate that both MOW-E and MOW-W outperform other distributions in terms of goodness-of-fit. Furthermore, we suggested the LMOW-W regression model, which provides more versatility, as demonstrated by a real-world application. Hence, it can be concluded that the newly proposed family of distributions have a wider range of applications in several disciplines.
Acknowledgments
It is with great gratitude that the authors acknowledge the editor in chief and reviewers for their valuable suggestions for enhancing the quality of this manuscript.
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