A new probabilistic model: Theory, simulation and applications to sports and failure times data

In applied sectors, data modeling/analysis is very important for decision-making and future predictions. Data analysis in applied sectors mainly relies on probability distributions. Data arising from numerous sectors such as engineering-related fields have complex structures. For such kinds of data having complex structures, the implementation of classical distributions is not a suitable choice. Therefore, researchers often need to look for more flexible models that might have the capability of capturing a high degree of kurtosis and increasing the fitting power of the classical models. Taking motivation from the above theory, to achieve these goals, we study a new probabilistic model, which we named a new beta power flexible Weibull (NBPF-Weibull) distribution. We derive some of the main distributional properties of the NBPF-Weibull model. The estimators for the parameters of the NBPF-Weibull distribution are derived. The performances of these estimators are judged by incorporating a simulation study for different selected values of the parameters. Three data sets are used to demonstrate the applicability of the NBPF-Weibull model. The first data set is observed from sports. It represents the re-injury rate of various football players. While the other two data sets are observed from the reliability zone. By adopting certain diagnostic criteria, it is proven that the NBPF-Weibull model repeatedly surpasses well-known classical and modified models.


Introduction
Assessing the multiform of data is very important to achieve more accurate and precise decisions.Selecting an appropriate model for capturing the complex form of data is very crucial.Over recent years, numerous probability models have been suggested to fit data sets with different behaviors and shapes.However, adopting a classical distribution to fit complex data sets can lead to X. Tang, J.-T.Seong, R. Alharbi et al. unreliable results.Hence, there is clearly evidence for researchers to modify the classical models to enhance their fitting capability [1][2][3][4][5][6][7][8][9][10][11].
In the group of existing probabilistic models (for instance, introduced, studied, and implemented models), the Weibull distribution has gained considerable attention.The Weibull distribution has certain advantages over the other classical distributions because of its different monotonic hazard shapes and closed-form distribution function.For reliability and healthcare data analyses, most often the utilization/employment of the Weibull model is considered [12][13][14][15][16][17][18][19].
From Fig. 1, it is obvious that when data sets have complex hazard shapes (i.e., non-monotonic), the Weibull model may not be an optimal model for these types of data sets.To overcome this weakness of the Weibull model, its several flexible extensions have been proposed.One of the most interesting modifications of Eq. ( 1) is called flexible Weibull (F-Weibull) distribution.Its CDF is  (; , ) = 1 −  − where  > 0 and  > 0. With linked to Eq. ( 2), the associated PDF (probability density function) is ( The F-Weibull model is furthermore extended and studied by (ii) El-Gohary et al. [20] introducing the inverse form of the F-Weibull model, (iii) El-Damcese et al. [21] introducing the Kumaraswamy version of the F-Weibull model, and (iv) El-Morshedy et al. [22] proposing the exponentiated inverse form of the F-Weibull model.
This study also provides a improvised version of Eq. ( 2) by implementing the beta power transformed method [23].For  ∈ ℝ the CDF of the new beta power transformed method is with PDF where Ξ Ξ Ξ is a parameter vector linked with  (;Ξ Ξ Ξ).It is important to keep in view that when  = 1, then Eq. ( 4) reduces  (;Ξ Ξ Ξ).
In the very next section, we use Eq. ( 2) along with Eq. ( 4) to study another improvised extension of the F-Weibull distribution.The improvised probability model is called a NBPF-Weibull distribution.Some visual displays of its PDF and hazard function (HF)

Estimation and simulation
This section is reserved for carrying out two objectives.The first objective is to obtain the maximum likelihood estimators (MLEs) ( α , δ , β ) for (, , ) of the NBPF-Weibull model.Second, it provides a simulation study to see the behaviors of α , δ , and β . and where   ().

Simulation
Here, we cover the second aim of this section by carrying out a simulation study.For this purpose, we generate random numbers from the NBPF-Weibull distribution.Random numbers from the NBPF-Weibull distribution of sizes, say,  = 100, 200, 300, ..., 1000 are generated.The simulation study of the NBPF-Weibull distribution is conducted for three sets of , , and .
We choose two evaluation criteria and implement them for checking the performances of α , δ , and β .These criteria are and where    = (, , ).
It is a known fact that simulation studies are based on default values (or predefined values) of parameters.As we have mentioned the range of , and  (  ∈ ℝ + ) .So, within their given ranges, we can choose any value of , , and  to carry out the simulation study.

Applications
Here, we fit the NBPF-Weibull model to three practical data sets, aiming to show the usefulness of the NBPF-Weibull distribution in applied sectors.These datasets are taken from sports and different fields of engineering.

Data sets
This subsection considers the numerical and visual evaluation of the sports and engineering data sets.

Sports data
The first data set (we set out to represent it with Data 1) is selected from sports sciences, specifically, it is observed from various football matches.It represents the rate of the re-injury of various players in different football matches [24].

Reliability data
The second data set (we set out to represent it with Data 2) shows the failure time of the electronic reactor pumps.More detail about this data can be found in [25].The waiting time is measured in hours.The key measures of the waiting time data are: the smallest value = 0.062, maximum value = 6.560,  1 = 0.310, median = 0.614,  3 = 2.041, mean = 1.578, skewness = X.Tang, J.-T.Seong, R. Alharbi et al.

Competing distributions
We accomplish the superiority of the NBPF-Weibull distribution over the Weibull distribution and F-Weibull distribution.We also compare its fitting results with other modifications of the F-Weibull and Weibull distributions, namely, exponentiated F-Weibull    (EF-Weibull) and exponentiated Weibull (E-Weibull) distributions, respectively.The SFs of the selected models (mentioned above) are given by • Weibull distribution [27]  () =  −  , ,>0.

Information criteria
The comparison between NBPF-Weibull and the above-selected models is made using well-known statistical approaches (i.e., certain information criteria).The information criteria (IC) chosen as comparative tools are obtained as follows • Akaike IC (AIC) 2 − 2.
In these criteria,  represents the parameters of the probability distribution applied to the data,  represents the number of observations of the underlined data, and  represents the LLF.These criteria are widely implemented to discriminate the most appropriate and optimal model among the rival models.The values of the IC are computed using the - with  method.Furthermore, the empirical plots of the NBPF-Weibull distribution are also obtained using the -.

Analysis of Data 1
This subsection considers the performances of the applied distributions using the re-injury rate (or Data 1).We apply the NBPF-Weibull and all the above-competing distributions to Data 1.The values of α , δ , β , and γ are presented in Table 4. Furthermore, Fig. 10(a-c) illustrates the plots for the profiles of α , δ , and β of the NBPF-Weibull distribution.
Using Data 1, Table 5 provides the IC values of the NBPF-Weibull distribution and its related competing distributions.For the NBPF-Weibull model, we have AIC = 223.057,CAIC = 223.915,BIC = 227.455,and HQIC = 224.515;see Table 5.Thus, Table 5 categorically confirms that the NBPF-Weibull distribution has the lowest IC values for Data 1.Therefore, it is obvious like a crystal that the NBPF-Weibull distribution can be ranked as the 1  as it provides a closer fit to Data 1.
For Data 1, we also show the performance of the NBPF-Weibull distribution graphically.For visually illustrating the fitted distributions, we select the QQ (quantile-quantile), empirical PDF, SF, PP (probability probability), and estimated CDF; see Fig. 11(a-d).
The graphical checking tools also support the close fitting ability of the NBPF-Weibull distribution.To support the given illustration in Table 7, we also visually compare and classify the fitted models.For visually illustrating the fitted distributions, we select the QQ, fitted PDF, SF, PP, and empirical CDF; see Fig. 13(a-d).The plots in Fig. 13(a-d) show that the graphical checking tools also support the close fitting ability of the NBPF-Weibull distribution.

Analysis of Data 3
Here, we show the performances of the applied distributions using data on the failure times of electronic machines (i.e., Data 3).
We fit the NBPF-Weibull along with competing distributions to Data 3.The values of α , δ , β , and γ are given in Table 8.Furthermore, Fig. 14(a-c) illustrates the plots for the profiles of α , δ , and β of the NBPF-Weibull distribution.
Furthermore, the values of the IC for the fitted models are obtained in Table 9.The given results/facts in Table 7 again categorically confirm that the NBPF-Weibull model has the lowest values for Data 3. Using Data 3, for the NBPF-Weibull distribution, we AIC = 163.882,CAIC = 164.404,BIC = 169.619,and HQIC = 166.067.These facts confirm that the NBPF-Weibull distribution is the most optimal and appropriate model for Data 3.
Besides the numerical comparison of the fitted models, we again show the performances of the NBPF-Weibull model visually.The visual results in Fig. 15(a-d) also show that Data 3 is closely fitted by the NBPF-Weibull distribution.

Final remarks
This study explored a new extension of the F-Weibull distribution.The new extension of the F-Weibull model was named a NBPF-Weibull distribution.Among certain desirable properties, the NBPF-Weibull distribution produces different patterns of HF including increasing, uni-model, and modified uni-model.Whereas, the PDF of the NBPF-Weibull distribution was capable of capturing right-X.Tang, J.-T.Seong, R. Alharbi et al.

Table 9
The criterion values of the fitted models using Data 3.  Furthermore, we also proved its applicability through two practical examples.Despite these virtues, the NBPF-Weibull distribution also has some deficiencies/limitations.For instance, the NBPF-Weibull distribution may not provide the best fit for bimodal data sets, because its PDF does not have a bimodal shape.Moreover, it is also not able to apply the discrete data sets as its continuous model.In the future, one can work further carry this research work to addresses the said demerits of the NBPF-Weibull distribution However, our intentions regarding future research studies or covering the existing research gaps, including the • development of the bivariate modification of the NBPF-Weibull distribution that is given by Eq. ( 5),

Table 7
The criterion values of the fitted models using Data 2.