# A New Alpha Power Weibull Model for Analyzing Time-to-Event Data: A Case Study from Football

Authors: Gao Shengjie; Alisa Craig; Getachew Tekle Mekiso
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

Statistical methodologies have wider applications in exercise science, sports medicine, sports management, sports marketing, sports science, and other related sciences. These methods can be used to predict the winning probability of a team or individual in a match, the number of minutes that an individual player will spend on the ground, the number of goals to be scored by an individual player, the number of red/yellow cards that will be issued to an individual player or a team, etc. Keeping in view the importance and applicability of the statistical methodologies in sport sciences, healthcare, and other related sectors, this paper introduces a novel family of statistical models called new alpha power family of distributions. It is shown that numerous properties of the suggested method are similar to those of the new Weibull-X and exponential type distributions. Based on the novel method, a special model, namely, a new alpha power Weibull distribution, is studied. The new model is very flexible because the shape of its probability density function can either be right-skewed, decreasing, left-skewed, or increasing. Furthermore, the new distribution is also able to model real phenomena with bathtub-shaped failure rates. Finally, the usefulness/applicability of the proposed distribution is shown by analyzing the time-to-event datasets selected from different football matches during 1964–2018.


---

## Body

## 1. Introduction

In the practice of healthcare [1], reliability [2], education [3], hydrology [4], management [5], metrology [6], and sports sciences [7], statistical modeling and predicting real-life events are very crucial. Numerous statistical models (traditional/classical and modified distributions) have been suggested to model data in these sectors.In the recent era, the development of generating new families of statistical models has received considerable attention [8]. Recently, Alzaatreh et al. [9] introduced an interesting method, namely, the T-X approach, to update the distributional flexibility of the existing models. The DF (distribution function) Ky;Φ,Ξ of the T-X method is given by(1)Ky;Φ,Ξ=∫aFGy;Φvt;Ξdt,where the function FGy;Φ fulfills some certain conditions (for details, see [9]). In equation (1), Gy;Φ is a baseline DF with parameter vector Φ and vt;Ξ is the probability density function (PDF) of the parent model with parameter vector Φ. Corresponding to Ky;Φ,Ξ, the PDF ky;Φ,Ξ is given by(2)ky;Φ,Ξ=ddyFGy;ΦrFGy;Φ.Most of the exponential type distributions belong to the class defined in equation (1). Some implemented functions of FGy;Φ are provided in Table 1.Table 1
Some used functions ofFGy;Φ via the T-X method.
S. no.FGy;ΦRange ofTMembers ofT-X family1Gy;Φ[0, 1][10]2−logGy;Φ0,∞[11]3−log1−Gy;Φ0,∞[12]4−log1−Gαy;Φ0,∞[13]5Gy;Φ/1−Gy;Φ0,∞[14]6−log1−Gy;Φ/eGy;Φ0,∞[15]7−log1−Gy;Φ/βGy;Φ0,∞[16]8−log1−Gαy;Φ/eGαy;Φ0,∞[17]9−logσ1−Gαy;Φ/σ−Gαy;Φ0,∞[18]10−log1−Gy;Φ/1−Gy;Φ0,∞[19] (implemented)11−logGy;Φ/1−Gy;Φ−∞,∞[20]12log−log1−Gy;Φ−∞,∞[21]Recently, Ahmad et al. [19] implemented the T-X approach and introduced an interesting method, namely, a new Weibull-X (NWei-X) family, to update the distributional flexibility of the classical or modified distributions. They introduced the NWei-X family by using FGy;Φ=−log1−Gy;Φ/1−Gy;Φ in equation (1) with vt;Ξ=αtα−1e−tα, where vt;Ξ is the PDF of the one-parameter Weibull model with parameter vector Ξ=α. The DF of the NWei-X family is given by(3)Ky;Δ=1−exp−−log1−Gy;Φ1−Gy;Φα,α>0,y∈ℝ,where Δ is a parameter vector. The expression Ky;Δ can also be written as(4)Ky;Δ=1−exp−Hy;Φ1−Gy;Φα,α>0,y∈ℝ,where Hy;Φ is the CHF (cumulative hazard function) associated with DF Gy;Φ.In terms of CHFMy;Δ, the DF Wy;Δ of the exponential type distributions is defined as(5)Wy;Δ=1−e−My;Δ,y∈ℝ,where My;Δ must fulfil certain conditions as given below:(i)
My;Δ is a nonnegative, differentiable, and increasing function of y.(ii)
limy⟶−∞My;Δ⟶0, and limy⟶∞My;Δ⟶∞.The traditional exponential, Rayleigh, Weibull, and other extended lifetime distributions belong to the class defined in equation (5) (see [22]). In link to equation (5), the PDF wy;Δ is given by(6)wy;Δ=my;Δe−My;Δ,y∈ℝ,where d/dyMy;Δ=my;Δ.Here, we introduce an additional parameter in equation (5), by replacing the exponent term with α to propose a very flexible family by the DF(7)Wy;Δ=1−α−My;Δ,α>1,α≠e,y∈ℝ.Forα=e, the DF in equation (7) becomes similar to equation (5). The function My;Δ fulfills the conditions given in (i) and (ii). As it is quite clear that(8)0≤Wy;Δ≤1,for clarification, let(9)limy⟶−∞Wy;Δ=limy⟶−∞1−α−My;Δ=0,(10)limy⟶∞Wy;Δ=limy⟶∞1−α−My;Δ=1.Hence, using the results obtained in equations (9) and (10), we observed that the function Wy;Δ defined in equation (7) is a proper DF. The expression in equation (7) is very interesting and can be useful to generate new statistical models belonging to the T-X family of distributions.Now, we introduce the proposed family called new alpha power (NAPow) family by usingWy;Δ=−log1−Gy;Φ/1−Gy;Φ in equation (7). The DF of the NAPow family is given by(11)Wy;Δ=1−α−−log1−Gy;Φ/1−Gy;Φ,α>1,α≠e,y∈ℝ.The DF in equation (11) can also be written as(12)Wy;Δ=1−α−Hy;Φ/1−Gy;Φ,α>1,α≠e,y∈ℝ.Corresponding toWy;Δ, the PDF wy;Δ, SF (survival function) Sy;Δ=1−Wy;Δ, and HF (hazard function) hy;Δ=wy;Δ/1−Wy;Δ are given by(13)wy;Δ=logαgy;Φ1+Hy;Φ1−Gy;Φ2α−Hy;Φ/1−Gy;Φ,α>1,α≠e,y∈ℝ,Sy;Δ=α−Hy;Φ/1−Gy;Φ,α>1,α≠e,y∈ℝ,hy;Δ=logαgy;Φ1+Hy;Φ1−Gy;Φ2,α>1,α≠e,y∈ℝ,respectively.Due to an extra parameter in the place of the exponent term in equation (7), the NAPow family delivers greater flexibility.


## 2. A New Alpha Power Weibull Model

Here, we discuss a special member of the NAPow family called new alpha power Weibull (NAPow-Weibull) model. The NAPow-Weibull model is introduced by using the DF of the Weibull (two-parameter) distribution as a parent model. A random variableY has the Weibull model, if its DF Gy;Φ is given by(14)Gy;Φ=1−e−φyϕ,y≥0,ϕ>0,φ>0,where Φ=ϕ,φ. Corresponding to Gy;Φ, the PDF gy;Φ is given by(15)gy;Φ=ϕφyϕ−1e−φyϕ,y>0,ϕ>0,φ>0.Corresponding toGy;Φ and gy;Φ, the HF hy;Φ and CHF Hy;Φ are given by(16)hy;Φ=ϕφyϕ−1,y>0,ϕ>0,φ>0,(17)Hy;Φ=φyϕ,y>0,ϕ>0,φ>0,respectively.UsingHy;Φ=φyϕ and Gy;Φ=1−e−φyϕ in equation (12), we get the DF Wy;Δ of the NAPow-Weibull model, given by(18)Wy;Δ=1−α−φyϕeφyϕ,y≥0,ϕ>0,φ>0,α>1,α≠e,where Δ=α,ϕ,φ.Corresponding toWy;Δ, the SF Sy;Δ is given by(19)Sy;Δ=α−φyϕeφyϕ,y>0,ϕ>0,φ>0,α>1,α≠e.A visual display ofWy;Δ and Sy;Δ for α=1.1,φ=1.6, and ϕ=1.9 is presented in Figure 1.Figure 1
A visual display ofWy;Δ and Sy;Δ.In link toWy;Δ, the PDF wy;Δ is given by(20)wy;Δ=logαϕφyϕ−11+φyϕα−φyϕeφyϕ,y>0,ϕ>0,φ>0,α>1,α≠e.In Figure2, a visual illustration of wy;Δ is presented for (i) α=1.2,φ=0.9,ϕ=1.5 (red line), (ii) α=1.9,φ=0.9,ϕ=0.7 (green line), (iii) α=1.1,φ=0.9,ϕ=2.1 (blue line), and (iv) α=1.1,φ=0.1,ϕ=4.1 (gold line).Figure 2
A visual illustration ofwy;Δ for different values of α, φ, and ϕ.Furthermore, in link toWy;Δ, the HF hy;Δ is given by(21)hy;Δ=logαϕφyϕ−11+φyϕ,y>0,ϕ>0,φ>0,α>1,α≠e.Some graphical sketches ofhy;Δ are provided in Figure 3. The plots of hy;Δ in Figure 3 are obtained for (i) α=1.5,φ=0.1,ϕ=3.8 (blue line), (ii) α=2.5,φ=0.7,ϕ=0.08 (green line), and (iii) α=9,φ=0.4,ϕ=0.8 (red line).Figure 3
A visual illustration ofhy;Δ for different values of α, φ, and ϕ.


## 3. Estimation

Here, we derive the estimatorsα^,φ^,ϕ^ of the parameter α,φ,ϕ of the NAPow-Weibull model with PDF wy;Δ. Consider a RS (random sample) say Y1,Y2,…,Yp from PDF wy;Δ. Then, corresponding to wy;Δ, the log likelihood function λΔ is(22)λΔ=ploglogα+plogϕ+plogφ+ϕ−1∑v=1plogyv+φ∑v=1pyvϕ+∑v=1plog1+φyvϕ−φ∑v=1pyvϕeφyvϕlogα.The partial derivatives ofλΔ are given by(23)∂∂αλΔ=plogαα−φα∑v=1pyvϕeφyvϕ,∂∂φλΔ=pφ+∑v=1pyvϕ+∑v=1pyvϕ1+φyvϕ−logα∑v=1pyvϕeφyvϕ1+φyvϕ,(24)∂∂ϕλΔ=pϕ+∑v=1plogyv+φ∑v=1plogyvyvϕ+φ∑v=1plogyvyvϕ1+φyvϕ−logαφ∑v=1plogyvyvϕeφyvϕ1+φyvϕ,respectively.The estimators ofα,φ, and ϕ can be obtained by solving ∂/∂αλΔ=0, ∂/∂φλΔ=0, and ∂/∂ϕλΔ, respectively.


## 4. Data Analyses

Here, we show the applicability of the NAPow-Weibull model by analyzing two datasets. These datasets are taken from the sports sciences.The first dataset consists of seventy-eight (78) observations and represents the waiting time duration till the first goal was scored in different football matches during 1964–2018. The first sports dataset is given by 14.00, 14.00, 13.00, 13.00, 12.00, 12.00, 12.00, 12.00, 12.00, 12.00, 12.00, 11.00, 11.00, 11.00, 10.80, 10.69, 10.12, 10.00, 10.00, 10.00, 10.00, 9.90, 9.60, 9.55, 9.00, 9.00, 9.00, 9.00, 8.70, 8.30, 8.10, 8.10, 8.00, 8.00, 8.00, 8.00, 8.00, 8.00, 8.00, 7.80, 7.70, 7.69, 7.66, 7.42, 7.30, 7.27, 7.22, 7.00, 7.00, 7.00, 7.00, 7.00, 7.00, 6.00, 6.00, 6.00, 6.00, 6.00, 6.00, 6.00, 5.00, 5.00, 4.00, 4.00, 4.00, 3.90, 3.69, 3.60, 3.57, 3.55, 3.17, 3.00, 3.00, 2.80, 2.80, 2.56, 2.20, 2.10.The second dataset consists of seventy-six (78) observations and is also taken from the sports sciences. The second dataset also represents the waiting time duration till the first goal was scored in different football matches. The second dataset is given by 2.3520, 2.4640, 2.8672, 3.1360, 3.1360, 3.3600, 3.3600, 3.5504, 3.9760, 3.9984, 4.0320, 4.1328, 4.3680, 4.4800, 4.4800, 4.4800, 5.6000, 5.6000, 6.7200, 6.7200, 6.7200, 6.7200, 6.7200, 6.7200, 6.7200, 7.8400, 7.8400, 7.8400, 7.8400, 7.8400, 7.8400, 8.0864, 8.1424, 8.1760, 8.3104, 8.5792, 8.6128, 8.6240, 8.7360, 8.9600, 8.9600, 8.9600, 8.9600, 8.9600, 8.9600, 8.9600, 9.0720, 9.0720, 9.2960, 9.7440, 10.0800, 10.0800, 10.0800, 10.0800, 10.6960, 10.7520, 11.0880, 11.2000, 11.2000, 11.2000, 11.2000, 11.3344, 11.9728, 12.0960, 12.3200, 12.3200, 12.3200, 13.4400, 13.4400, 13.4400, 13.4400, 13.4400, 13.4400, 13.4400, 14.5600, 14.5600.The complete information about these datasets can retrieved athttps://en.wikipedia.org/wiki/Fastest_goals_in_association_football.We fit the NAPow-Weibull and three other models (competitive models) to these datasets. The SFs of the competitive models (i) Weibull (Wei), (ii) exponentiated Weibull (Exp-Wei), and (iii) Kumaraswamy Weibull (Ku-Wei) are given by(25)Fy;φ,ϕ=1−eφyϕ,y≥0,φ>0,ϕ>0,Fy;φ,ϕ,λ1=1−1−e−φyϕλ1,y≥0,φ>0,ϕ>0,λ1>0, (26)Fy;φ,ϕ,λ1,λ2=1−1−e−φyϕλ1λ2,y≥0,φ>0,ϕ>0,λ1>0,λ2>0,respectively.To figure out the flexibility/applicability of the fitted models (NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei) using these two datasets, certain IC (information criteria) such as (i) AIC [23], (ii) BIC [24], (iii) CAIC [25], and (iv) HQIC [26] are considered. In addition to the IC, three g-o-f measures (goodness-of-fit measures) such as (i) Cramer–von Mises (CM) test statistic, (ii) Anderson–Darling (AD) test statistic, and (iii) Kolmogorov–Smirnov (KS) test statistic are also considered. Besides these seven statistical quantities, the p value of the fitted models is also derived.


### 4.1. Data Analysis Using the First Dataset

In this section, we apply the NAPow-Weibull in comparison with the Wei, Exp-Wei, and Ku-Wei models to the first dataset taken from the sports sciences. The summary measures of the first sports dataset are given by minimum = 2.100, 1st quartile = 6.000, median = 7.900, mean = 7.729, 3rd quartile = 10.000, maximum = 14.000, variance = 9.428, standard deviation = 3.070, and range = 11.900. Corresponding to the first sports dataset, the plots of hist (histogram), PB (boxplot), and TTT (total time test) are provided in Figure4.Figure 4
The graphs of BP, hist, and TTT corresponding to the first sports dataset.In link to the first sports dataset, (i) the values ofα^MLE,φ^MLE,ϕ^MLE,λ1^MLE, and λ2^MLE of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models are obtained in Table 2, (ii) the values of the IC measures of the fitted models are reported in Table 3, and (iii) the values of the g-o-f measures are provided in Table 4.Table 2
The values ofα^MLE,ϕ^MLE,φ^MLE,λ^1MLE, and λ^2MLE using the first sports dataset.
Modelα^MLEϕ^MLEφ^MLEλ^1MLEλ^2MLENAPow-Weibull5.59586221.95701760.0053182——Wei—2.81225480.0023228——Ex-Wei—2.40558800.00643251.1416816—Ku-Wei—1.91645660.00757471.67561714.1447374Table 3
The IC measures of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models using the first sports dataset.
ModelAICCAICBICHQICNAPow-Weibull397.9690398.2933405.0392400.7993Wei398.9563399.1163406.6697301.8431Exp-Wei400.1195400.4439407.1897402.9498Ku-Wei437.1103437.6582446.5371440.8840Table 4
The g-o-f measures of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models using the first sports dataset.
ModelCMADKSp valueNAPow-Weibull0.0819150.5531620.0876550.5867Wei0.1038790.6947360.1042100.3653Ex-Wei0.1227460.8021270.1331500.1259Ku-Wei0.1307640.8496050.1173100.2333Furthermore, for the first sports dataset, the (i) estimated DF, (ii) estimated SF, (iii) PP (probability-probability), and (iv) QQ (quantile-quantile) plots of the NAPow-Weibull model are sketched in Figure5. Corresponding to the first sports dataset, the plots of the fitted DF and SF of the NAPow-Weibull model are obtained using the functions Wy;Φ^ and Sy;Φ^, respectively. The mathematical expressions of Wy;Φ^ and Sy;Φ^ are given by(27)Wy;Φ^=1−5.5958622−0.0053182y1.9570176e0.0053182y1.9570176,y≥0,(28)Sy;Φ^=5.5958622−0.0053182y1.9570176e0.0053182y1.9570176,y>0,respectively.Figure 5
The estimated DF, SF, PP, and QQ plots of the NAPow-Weibull model corresponding to the first sports dataset.


### 4.2. Data Analysis Using the Second Dataset

In this section, we again implement the NAPow-Weibull distribution in comparison with the Wei, Exp-Wei, and Ku-Wei models to the second sports dataset. The summary measures of the second sports dataset are given by minimum = 2.352, 1st quartile = 6.720, median = 8.680, mean = 8.472, 3rd quartile = 11.116, maximum = 14.560, variance = 10.792, standard deviation = 3.285, and range = 12.208. Corresponding to the first sports data, the plots of hist, PB, and TTT are presented in Figure6.Figure 6
The plots of BP, hist, and TTT corresponding to the second sports dataset.Corresponding to the second sports dataset, (i) the values ofα^MLE,φ^MLE,ϕ^MLE,λ1^MLE, and λ2^MLE of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei distributions are provided in Table 5, (ii) the values of the IC measures of the fitted models are presented in Table 6, and (iii) the values of the g-o-f measures are reported in Table 7.Table 5
The values ofα^MLE,ϕ^MLE,φ^MLE,λ^1MLE, and λ^2MLE using the second sports dataset.
Modelα^MLEϕ^MLEφ^MLEλ^1MLEλ^2MLENAPow-Weibull5.36743042.08736010.0033880——Wei—2.70019480.0023373——Ex-Wei—2.16878000.01067781.5569487—Ku-Wei—1.18277050.08983113.82954333.1540636Table 6
The IC measures of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models using the second sports dataset.
ModelAICCAICBICHQICNAPow-Weibull396.5795396.9128403.5717399.3739Wei397.8422398.0066402.5037399.7052Exp-Wei401.9609402.2943408.9531404.7554Ku-Wei407.2465407.8099416.5695410.9724Table 7
The g-o-f measures of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models using the second sports dataset.
ModelCMADKSp valueNAPow-Weibull0.0854180.6183610.0868390.6153Wei0.1298710.8724880.1263400.1766Ex-Wei0.1701031.0927990.1284600.1627Ku-Wei0.2237171.3836920.1419800.0933In link to the second sports dataset, the (i) estimated DF, (ii) estimated SF, (iii) PP, and (iv) QQ plots of the NAPow-Weibull model are presented in Figure7. Corresponding to the second sports dataset, the plots of the fitted DF and SF of the NAPow-Weibull model are obtained via the expressions Wy;Φ^ and Sy;Φ^, respectively. For the second sports dataset, the mathematical expressions of Wy;Φ^ and Sy;Φ^ are, respectively, given by(29)Wy;Φ^=1−5.3674304−0.0033880y2.0873601e0.0033880y2.0873601,y≥0,(30)Sy;Φ^=5.3674304−0.0033880y2.0873601e0.0033880y2.0873601,y>0.Figure 7
The estimated DF, SF, PP, and QQ plots of the NAPow-Weibull model corresponding to the second sports dataset.


### 4.3. Discussion

In this section, the NAPow-Weibull model was applied to two datasets taken from the sports sciences. The first dataset consists of seventy-eight observations and was taken from the sports sciences. The NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models were applied to these data. Based on the IC measures, it is observed that the NAPow-Weibull was the best model. Furthermore, it is also observed that the Wei model was the second-best competitor in terms of the IC measures, whereas the Exp-Weibull model was observed to be the second-best model based on thep value.The second dataset consists of seventy-six observations and was also taken from the sports sciences. Again, the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models were applied to the second sports dataset. Based on the IC and g-o-f measures, it is again observed that the NAPow-Weibull was the best competitor than the Wei, Exp-Wei, and Ku-Wei distributions.


## 4.1. Data Analysis Using the First Dataset

In this section, we apply the NAPow-Weibull in comparison with the Wei, Exp-Wei, and Ku-Wei models to the first dataset taken from the sports sciences. The summary measures of the first sports dataset are given by minimum = 2.100, 1st quartile = 6.000, median = 7.900, mean = 7.729, 3rd quartile = 10.000, maximum = 14.000, variance = 9.428, standard deviation = 3.070, and range = 11.900. Corresponding to the first sports dataset, the plots of hist (histogram), PB (boxplot), and TTT (total time test) are provided in Figure4.Figure 4
The graphs of BP, hist, and TTT corresponding to the first sports dataset.In link to the first sports dataset, (i) the values ofα^MLE,φ^MLE,ϕ^MLE,λ1^MLE, and λ2^MLE of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models are obtained in Table 2, (ii) the values of the IC measures of the fitted models are reported in Table 3, and (iii) the values of the g-o-f measures are provided in Table 4.Table 2
The values ofα^MLE,ϕ^MLE,φ^MLE,λ^1MLE, and λ^2MLE using the first sports dataset.
Modelα^MLEϕ^MLEφ^MLEλ^1MLEλ^2MLENAPow-Weibull5.59586221.95701760.0053182——Wei—2.81225480.0023228——Ex-Wei—2.40558800.00643251.1416816—Ku-Wei—1.91645660.00757471.67561714.1447374Table 3
The IC measures of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models using the first sports dataset.
ModelAICCAICBICHQICNAPow-Weibull397.9690398.2933405.0392400.7993Wei398.9563399.1163406.6697301.8431Exp-Wei400.1195400.4439407.1897402.9498Ku-Wei437.1103437.6582446.5371440.8840Table 4
The g-o-f measures of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models using the first sports dataset.
ModelCMADKSp valueNAPow-Weibull0.0819150.5531620.0876550.5867Wei0.1038790.6947360.1042100.3653Ex-Wei0.1227460.8021270.1331500.1259Ku-Wei0.1307640.8496050.1173100.2333Furthermore, for the first sports dataset, the (i) estimated DF, (ii) estimated SF, (iii) PP (probability-probability), and (iv) QQ (quantile-quantile) plots of the NAPow-Weibull model are sketched in Figure5. Corresponding to the first sports dataset, the plots of the fitted DF and SF of the NAPow-Weibull model are obtained using the functions Wy;Φ^ and Sy;Φ^, respectively. The mathematical expressions of Wy;Φ^ and Sy;Φ^ are given by(27)Wy;Φ^=1−5.5958622−0.0053182y1.9570176e0.0053182y1.9570176,y≥0,(28)Sy;Φ^=5.5958622−0.0053182y1.9570176e0.0053182y1.9570176,y>0,respectively.Figure 5
The estimated DF, SF, PP, and QQ plots of the NAPow-Weibull model corresponding to the first sports dataset.


## 4.2. Data Analysis Using the Second Dataset

In this section, we again implement the NAPow-Weibull distribution in comparison with the Wei, Exp-Wei, and Ku-Wei models to the second sports dataset. The summary measures of the second sports dataset are given by minimum = 2.352, 1st quartile = 6.720, median = 8.680, mean = 8.472, 3rd quartile = 11.116, maximum = 14.560, variance = 10.792, standard deviation = 3.285, and range = 12.208. Corresponding to the first sports data, the plots of hist, PB, and TTT are presented in Figure6.Figure 6
The plots of BP, hist, and TTT corresponding to the second sports dataset.Corresponding to the second sports dataset, (i) the values ofα^MLE,φ^MLE,ϕ^MLE,λ1^MLE, and λ2^MLE of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei distributions are provided in Table 5, (ii) the values of the IC measures of the fitted models are presented in Table 6, and (iii) the values of the g-o-f measures are reported in Table 7.Table 5
The values ofα^MLE,ϕ^MLE,φ^MLE,λ^1MLE, and λ^2MLE using the second sports dataset.
Modelα^MLEϕ^MLEφ^MLEλ^1MLEλ^2MLENAPow-Weibull5.36743042.08736010.0033880——Wei—2.70019480.0023373——Ex-Wei—2.16878000.01067781.5569487—Ku-Wei—1.18277050.08983113.82954333.1540636Table 6
The IC measures of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models using the second sports dataset.
ModelAICCAICBICHQICNAPow-Weibull396.5795396.9128403.5717399.3739Wei397.8422398.0066402.5037399.7052Exp-Wei401.9609402.2943408.9531404.7554Ku-Wei407.2465407.8099416.5695410.9724Table 7
The g-o-f measures of the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models using the second sports dataset.
ModelCMADKSp valueNAPow-Weibull0.0854180.6183610.0868390.6153Wei0.1298710.8724880.1263400.1766Ex-Wei0.1701031.0927990.1284600.1627Ku-Wei0.2237171.3836920.1419800.0933In link to the second sports dataset, the (i) estimated DF, (ii) estimated SF, (iii) PP, and (iv) QQ plots of the NAPow-Weibull model are presented in Figure7. Corresponding to the second sports dataset, the plots of the fitted DF and SF of the NAPow-Weibull model are obtained via the expressions Wy;Φ^ and Sy;Φ^, respectively. For the second sports dataset, the mathematical expressions of Wy;Φ^ and Sy;Φ^ are, respectively, given by(29)Wy;Φ^=1−5.3674304−0.0033880y2.0873601e0.0033880y2.0873601,y≥0,(30)Sy;Φ^=5.3674304−0.0033880y2.0873601e0.0033880y2.0873601,y>0.Figure 7
The estimated DF, SF, PP, and QQ plots of the NAPow-Weibull model corresponding to the second sports dataset.


## 4.3. Discussion

In this section, the NAPow-Weibull model was applied to two datasets taken from the sports sciences. The first dataset consists of seventy-eight observations and was taken from the sports sciences. The NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models were applied to these data. Based on the IC measures, it is observed that the NAPow-Weibull was the best model. Furthermore, it is also observed that the Wei model was the second-best competitor in terms of the IC measures, whereas the Exp-Weibull model was observed to be the second-best model based on thep value.The second dataset consists of seventy-six observations and was also taken from the sports sciences. Again, the NAPow-Weibull, Wei, Exp-Wei, and Ku-Wei models were applied to the second sports dataset. Based on the IC and g-o-f measures, it is again observed that the NAPow-Weibull was the best competitor than the Wei, Exp-Wei, and Ku-Wei distributions.


## 5. Final Remarks

In this article, a new family, that can be used as an alternative to a NWei-X family, is introduced. Numerous distributional properties of the new family are obtained. There are several advantages of using the proposed family. First, a very simple approach was proposed to modify the existing distributions by adding an extra parameter. Second, the proposed method can be useful to introduce new distributions belonging to T-X family. Third, the existing distributions having a closed-form of DF were extended. Furthermore, a special model (NAPow-Weibull) of the new alpha power family was studied. Some graphical behaviors of the PDF and HF of the NAPow-Weibull model were obtained. Finally, two applications were discussed to illustrate the NAPow-Weibull distribution. Based on the findings using these two datasets, it is shown that the NAPow-Weibull model was an appropriate model for dealing with the data in sports and other related sciences.



---

*Source: 7257264-2022-02-09.xml*