A novel technique for generating families of continuous distributions

In this paper, we present the generalized flexible-G family for creating several continuous distributions. Our new technique features are that it adds only two extra shape parameters to any chosen continuous distribution and is not derived from any parent distribution that currently exists. Several special cases of this family are provided. The generalized flexible-G family offers significant improvements in flexibility, fit, and applicability across a wide range of fields. The family’s model parameters are estimated using the maximum likelihood estimation method. A simulation study is conducted to assess the consistency of the maximum likelihood estimates. The generalized flexible log-logistic, a specific case of our novel family, is applied to both patient’s analgesia and reliability data in order to illustrate the significance of the family. The generalized flexible log-logistic outperforms several competitive models provided in this paper. Furthermore, the generalized flexible log-logistic performs better than traditional distributions such as the BurrXII, Gumbel, and Weibull models.


Introduction
In our real-life applications, statistics is essential through the use of statistical methods that heavily rely on the standard probability distributions.Nonetheless, a number of statistical issues defy the assumptions of these traditional models.Many real-world phenomena exhibit data patterns that are not adequately captured by these models.Different fields have unique requirements.For instance, hydrologists might need distributions that accurately model extreme rainfall, while reliability engineers might require models for lifetimes of complex systems.Also, existing distributions might fail to capture certain empirical regularities observed in specific domains, such as the heavy tails in financial returns or the asymmetry in biological measurements, flexibility and a closer fit to empirical data, capturing features such as skewness, kurtosis, and multimodality.As a result of this, there is a growing need to create flexible distributions in order to draw trustworthy conclusions and make informed judgments [1,2,6,11,13,14,15,22].By expanding the toolbox of available distributions, statisticians and researchers can achieve more accurate and insightful analyses.More work on the development of new models was done (see [8,9,10,12,19,24]) Recently, [25] introduced a new technique of creating families of distributions called the new flexible generalized family (NFGF).This family was primarily designed to take account of the non-symmetrical behavior of the parent distribution.For any arbitrary baseline cdf distribution G(t), the cdf and pdf of the NFGF are given by where ϕ and ψ are positive non zero shape parameters.Thus, this family of distributions has its pdf as f (t; ϕ, ψ, φ) = ψϕg(t; φ) Ḡ(t; φ) ϕG(t;φ) G(t; φ) Ḡ(t; φ) − log( Ḡ(t; φ)) 1 − Ḡ(t; φ) ϕG(t;φ) The GFG has its survival function and hazard rate function (hrf) as and hrf (t) = ψϕg(t; φ) Ḡ(t; φ) ϕG(t;φ) G(t;φ) Ḡ(t;φ) − log( Ḡ(t; φ)) 1 − Ḡ(t; φ) ϕG(t;φ) ψ−1 respectively.For u ∈ (0, 1), the GFG has its quantile function as ϕG(t; φ) log( Ḡ(t; φ) = log(1 − u 1/ψ ).(7) Equation (7) can be solved numerically via some softwares such as R, MATHEMATICA, MAPLE and Ox.The strength of this research is solely based on the fact that the family of distributions defined in (3) is not developed from any well-known parent model similar to the T-X family [1], cubic rank transmuted-G [2], generalized exponentiated-G family [6], exponentiated-G family [11], alpha power transformation family [13], Marshall Olkin family [14], alpha log power transformed-G family [15], transmuted-G family [22] and the new flexible-G family [25].This research was primarily driven by a combination of practical needs, theoretical advancements, the desire to improve statistical modeling and the need to develop a novel technique for generating families of continuous distributions.This novel technique can provide better fits to empirical data, unify existing models, offer more flexible parameterizations, and address specific challenges in various applications.

Special Cases
Here we consider examples of the family in (3) for different standard distributions, namely for uniform (U), exponential (E), log-logistic (LLoG), Topp-Leone (TL), Pareto (P) and half-logistic (HL) distributions.Figures 1, 2, 3, 4, 5, and 6 demonstrate how the hrf of the GFG distribution can take on a variety of very flexible shapes, including bathtub, bathtub followed by upside-down bathtub, upside-down bathtub, constant, increasing and decreasing.We also provide moments, standard deviation (SD), variance (V ar(T )), skewness (S), kurtosis (K) and quantiles for selected values of Ω = (ϕ, ψ, δ).If the parent distribution is uniform, such that G(t; δ) = t/δ and g(t; δ) = 1/δ where 0 < t < δ, then the cdf and pdf of the generalized flexible uniform (GFU) model are respectively given by and The GFU has its i th moment and quantile function as and respectively.Moments associated with equation (10) for selected values of Ω are given in Table 1, whereas Table 2 shows some quantiles for selected values of Ω.

Generalized flexible exponential distribution
Let the parent distribution be exponential with cdf and pdf given by G(t; δ) = 1 − e −δt and g(t; δ) = δe −δt for non negative δ, then the cdf and pdf of the generalized flexible exponential (GFE) model are respectively given by and

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A NOVEL TECHNIQUE FOR GENERATING FAMILIES OF CONTINUOUS DISTRIBUTIONS  The GFE has its i th moment and quantile function as and Stat., Optim.Inf.Comput.Vol. 12, September 2024 B. MAKUBATE AND R. R. MUSEKWA 1235 respectively.For some selected values of Ω, moments associated with equation ( 14) and quantiles associated with Equation ( 15) are given in Table 3 and Table 4, respectively.

Generalized flexible log-logistic distribution
For a log-logistic parent distribution with cdf and pdf given by G(t; δ) = 1 − (1 + t δ ) −1 and g(t; δ) = δt δ−1 (1 + t δ ) −2 for non negative δ, then the cdf and pdf of the generalized flexible log-logistic (GFLLoG) model are respectively given by and Consequently, the i th moment and the quantile funtion of the GFLLoG distribution are given by Stat., Optim.Inf.Comput.Vol. 12, September 2024 respectively.For some selected values of Ω, moments associated with equation ( 18) and quantiles associated with (19) are given in Tables 5 and 6, respectively.
where δ is non negative, then the cdf and pdf of the generalized flexible Topp-Leone (GFTL) model are respectively given by and Figure 4. GFTL distribution pdf and hrf plots The GFTL distribution has its i th moment and quantile function as and respectively.Tables 7 and 8 show, respectively, moments associated with equation ( 22) and quantiles associated with Equation (23) for some selected values of Ω.

Generalized flexible Pareto distribution
Let the parent distribution be Pareto with cdf and pdf given by G(t; δ) = 1 − 1/t δ and g(t; δ) = δ/x δ+1 for non negative δ, then the cdf and pdf of the generalized flexible Pareto (GFP) model are respectively given by A NOVEL TECHNIQUE FOR GENERATING FAMILIES OF CONTINUOUS DISTRIBUTIONS  The i th moment and the quantile function of the GFP distribution are given by and respectively.Moments associated with Equation ( 26) and quantiles associated with Equation ( 27) for some selected values of Ω are given in Tables 9 and 10, respectively.

Generalized flexible half-logistic distribution
For a half-logistic parent distribution with cdf and pdf given by G(t; δ) = 1−e −δt 1+e −δt and g(t; δ) = 2δe −δt (1+e −δt ) 2 for non negative δ, then the cdf and pdf of the generalized flexible half-logistic (GFHL) model are respectively given by The GFHL distribution has its i th moment and quantile function as and respectively.For some selected values of Ω, moments and quantiles of the GFHL model are given in Tables 11 and  12, respectively.

Estimation
Let t 1 , t 2 , t 3 , ..., t i be values of a random sample of size i from the GFG family.The log-likelihood function log(L(Ω)) = ℓ(Ω), for Ω = (ϕ, ψ, φ) of the GFG family is given by  The log-likelihood in (32) has its score functions as and The score functions are not linear in the parameters.Hence iterative methods are required to solve them [16,18].

Simulation Study
In this section, we evaluate the effectiveness of the maximum likelihood estimators (MLEs) in the proposed model using the GFLLoG distribution.The algorithm of the simulation process is as follows: i) Select initial parameter values for the GFLLoG distribution; ii) Generate n random values from a uniform distribution with pdf f (x) = 1; 0 < x < 1; iii) Use Equation 19to compute n values of the GFLLoG distribution defined in step i); iv) Repeat the above steps 2000 times; The performance of the estimators is evaluated using the Average Bias (ABIAS) and Root Mean Square Error (RMSE).These metrics are calculated based on 2000 samples of each selected sample size.The simulation results, presented in Tables 13, 14 and 15, showcase the performance under selected initial parameter values.It is observed that both ABIAS and RMSE decrease as the sample size n increases, indicating the consistency of the MLEs.These findings demonstrate that the MLEs provide reliable results when estimating model parameters in the GFG distribution.

Patients receiving an analgesic
The data collection contains information on patients' lifetime alleviation times (measured in minutes) after taking an analgesic as reported [3].
Stat., Optim.Inf.Comput.Vol. 12, September 2024 Tables 16 and 17 show maximum likelihood estimates (standard errors in parenthesis) and GoF statistics of the fitted models.In table 17, it is clear that the GFLLoG consistently has the lowest GoF statistics values and a corresponding high value of the K-S statistic as compared to comparative models of different number of parameters.It is evident that the GFLLoG distribution is the best fit for data set on patients receiving an analgesic.

Reliability data
The reliability data considered here consisting of 20 mechanical components failure times as reported [4].Table 18 shows some parameter estimates and standard errors in parenthesis of the fitted models.In table 19, it is evident that the GFLLoG distribution is the best fit for reliability data since the GFLLoG has the lowest GoF statistics values and the highest K-S statistic value as compared to comparative models presented in the table.

Concluding Remarks
We introduce an original flexible generalized family for univariate distributions called the generalized flexible-G family, which has the flexible-G family as its sub-model and was not created using any well-known parent model.The hazard rate function of the generalized flexible-G family can take on a variety of very flexible shapes, such as bathtub, bathtub followed by upside-down bathtub, upside-down bathtub, constant, increasing  and decreasing.Because of its appealing flexibility, the generalized flexible-G family's hazard rate function can be used to non-monotonic empirical hazard behaviors, which are more likely to occur in or be seen in real-world scenarios.We used the log-logistic as our baseline model for the presented simulation study and data analysis.Our generalized flexible log-logistic fit the two real-life datasets better than compared models presented in this article.This technique was limited to the univariate case, in the same spirit, more work can be done including the multivariate extension, truncation, censoring schemes and regression.

Figure 2 .
Figure 2. GFE distribution pdf and hrf plots

Figure 5 .
Figure 5. GFP distribution pdf and hrf plots

Figure 7 .Figure 9 .
Figure 7. Fitted density, probability plot and KM survival plot of the GFLLoG distribution for patients receiving an analgesic data

Figure 10 .
Figure 10.ECDF, estimated hrf plot and TTT of the GFLLoG distribution for reliability data

Table 1 .
GFU distribution table of moments

Table 2 .
GFU distribution table of quantiles

Table 3 .
GFE table of moments

Table 7 .
GFTL distribution table of moments

Table 8 .
GFTL distribution table of quantiles

Table 9 .
GFP distribution table of moments

Table 10 .
GFP distribution table of quantiles

Table 11 .
GFHL distribution table of moments

Table 12 .
GFHL distribution table of quantiles

Table 13 .
Parameter estimation from the GFLLoG distribution Results 1

Table 14 .
Parameter estimation from the GFLLoG distribution Results 2

Table 15 .
Parameter estimation from the GFLLoG distribution Results 3