# Exponentiated Gull Alpha Exponential Distribution with Application to COVID-19 Data

Authors: Hazar A. Khogeer; Amani Alrumayh; M. M. Abd El-Raouf; Mutua Kilai; Ramy Aldallal
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, the main aim is to define a statistical distribution that can be used to model COVID-19 data in Mexico and Canada. Using the method of exponentiation on the gull alpha exponential distribution introduces a new distribution with three parameters called the exponentiated gull alpha power exponential (EGAPE) distribution. The distribution has the benefit of being able to represent monotonic and nonmonotonic failure rates, both of which are often seen in dependability issues. It is possible to determine the quantile function as well as the skewness, kurtosis, and order statistics of the suggested distribution. The approach of maximum likelihood is used in order to calculate the parameters of the model, and the RMSE and average bias are utilised in order to evaluate how successful the strategy is. In conclusion, the flexibility of the new distribution is demonstrated by modeling COVID-19 data. From the practical application, we can conclude that the proposed model outperformed the competing models and therefore can be used as a better option for modeling COVID-19 and other related datasets.


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## Body

## 1. Introduction

Over the last decade, modeling using probability distributions has attracted the attention of most researchers. There are several ways to modify a distribution, and the most versatile approach is the introduction of an extra shape parameter so as to increase the flexibility of the distribution. Creating a new family of distributions and basing a new distribution off of an already existing baseline distribution is still another technique that may be used. Many families have been developed in the literature, for instance, Mudholkar and Srivastava1 developed the exponentiated Weibull family of distributions, Nadarajah and Kotz 2 developed the exponentiated types of distributions, the logistic-X family was proposed by Tahir et al. [3], the exponentiated TX family was proposed by Alzaghal et al.4, and Kumaraswamy–Marshall–Olkin family was proposed by Alizadeh et al. 5 (for further reading on other families of distributions, the reader is referred to [6–12]).The exponential distribution has been widely used to model survival analysis data. However, the shortcoming of the exponential distribution is that it only models data with constant hazard function. As a consequence of this, numerous adjustments to the exponential distributions have been produced in the scientific literature. For instance, Bhati et al.13 were the ones who first recommended using the Lindley exponential distribution. and extended new generalized exponential distribution was proposed by Eghwerido et al. 14. For further reading on the extensions of the exponential distribution, the reader is referred to [15–21].Exponentiated gull alpha power exponential distribution is a highly innovative and simple model introduced in this study. In Section2, we give the details of the brand-new design. Section 3 derives a few additional distributional features. The maximum likelihood approach of parameter estimation is described. It is used to examine the performance of the estimators in Section 4. Section 5 focuses on applying the theory to real-world data. Section 6 focuses on the findings.


## 2. The New EGAPE Distribution

The gull alpha power exponential distribution has been widely used in survival analysis. The CDF and the PDF of the GAPE distribution can be written and formulated as shown below:(1)Fx=α1−e−λxα1−e−λx,fx=λe−λxαe−λx1−1−e−λxlogα.X is assumed to be distributed with the EGAPE distribution with three parameters α,λ,a if the CDF and PDF, respectively, are given as(2)Fx=α1−e−λxα1−e−λxa.The following is the formula for the equivalent PDF to equation (2):(3)fx=aλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−1.The following formula may be used to describe the survival function of the EGAPE distribution:(4)Sx=1−α1−e−λxα1−e−λxa.The hazard function is given as(5)hx=aλe−λxαe−λx1−1−e−λxlogαα1−e−λx/α1−e−λxa−11−α1−e−λx/α1−e−λxa.Equations (6) and (7) give the reversed and cumulative hazard functions, respectively.(6)τx=aλe−λxαe−λx1−1−e−λxlogαα1−e−λx/α1−e−λxa−1α1−e−λx/α1−e−λxa,(7)Hx=−ln1−α1−e−λxα1−e−λxa.Figures1 and 2 depict the different contours of the probability density function (PDF) as well as the hazard rate curves, respectively. As observed, the PDF can exhibit shapes as unimodal, decreasing, or right skewed.Figure 1
EGAPE distribution PDF plots. (a)α = 1.5, λ = 3.2, a = 1.2. (b) α = 0.8, λ = 0.5, a = 2.3. (c) α = 0.3, λ = 2.7, a = 0.5. (d) α = 0.2, λ = 0.3, a = 1.5.
(a)(b)(c)(d)Figure 2
EGAPE distribution hrf plots. (a)α = 1.5, λ = 3.2, a = 1.2. (b) α = 0.8, λ = 0.05, a = 2.3. (c) α = 0.03, λ = 2.7, a = 0.05. (d) α = 0.2, λ = 0.3, a = 1.5.
(a)(b)(c)(d)


### 2.1. EGAPE Submodels

The EGAPE distribution includes some well-known submodels which include(a)
Ifa=1, then we obtain gull alpha power exponential distribution.(b)
Ifa=α=1, we have exponential distribution.(c)
Ifα=1, then we obtain exponentiated exponential distribution.


## 2.1. EGAPE Submodels

The EGAPE distribution includes some well-known submodels which include(a)
Ifa=1, then we obtain gull alpha power exponential distribution.(b)
Ifa=α=1, we have exponential distribution.(c)
Ifα=1, then we obtain exponentiated exponential distribution.


## 3. Important Mathematical Properties

In this section, all the formulae derived here including moments, incomplete moments, entropies, and order statistics can be handled by many computational software programs.


### 3.1. Quantile Function

Random samples from the EGAPE can be generated by inverting equation (2).(8)xq=−1λloglogα+W−1−logα×elogu/a/αlogα.To obtain the median of EGAPE, putu=0.5, and we have(9)x0.5=−1λloglogα+W−1−logα×elog0.5/a/αlogα.Some quantile values for the EGAPE are displayed in Table1.Table 1
Quantile values for EGAPE distribution.
Quantile(0.4, 0.3, 0.8)(0.5, 0.8, 0.3)(1.2, 0.7, 2.3)(0.9, 0.7, 0.5)0.10.0040.1430.0450.0590.20.0380.3520.1220.1630.30.1450.6070.2230.2980.40.3590.9130.3480.4640.50.7061.2810.5010.6680.61.2071.7340.6900.9210.71.9082.3200.9361.2480.82.9293.1441.2811.7080.94.6874.5461.8672.489


### 3.2. Moments

The EGAPE distribution’srth moment is defined as(10)Exr=∫0∞xrfxdx,Exr=∫0∞xraλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−1dx.The above equation does not have an explicit expression, and therefore we can find the moments numerically by using R version 4.1.2. Table2 gives the moments for the EGAPE for selected parameter values given as I:α=1.9,λ=1.0,a=0.9, II:α=1.4,λ=1.8,a=0.2, III:α=2.4,λ=1.5,β=1.3,a=0.9, and IV:α=1.4,λ=0.5,a=0.9.Table 2
Moments for EGAPE distribution.
IIIIIIIVμ11.8900.2870.5980.674μ27.0090.2550.5141.121μ340.5160.3880.7793.038μ4318.2160.8761.66311.513SD1.8520.4160.3950.816CV0.9791.4580.6611.209CS2.2472.9964.5862.543CK10.51317.83921.10012.977


### 3.3. Order Statistics

For an ordered random sampleX1,X2,……,Xn, from EGAPE distribution, the PDF of the ith minimum and maximum order statistics are provided by the following equations:(11)fX1x=nfx1−Fxn−1,fXnx=nfxFxn−1.Thus,(12)fX1x=naλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−11−α1−e−λxα1−e−λxan−1,fXnx=naλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−1α1−e−λxα1−e−λxan−1.


### 3.4. Skewness and Kurtosis

In the context of the EGAPE distribution, the definitions of Galton skewness and Moors kurtosis are stated as follows:(13)ZK=Q1/8+Q3/4−Q1/4−Q2/4Q3/4−Q1/4,ZMx=Q1/8+Q3/8−Q5/8−Q1/8Q3/4−Q1/4,where Q describe different quartile values. Clearly, the extra shape parameters α and a have an effect on the skewness and kurtosis values Figures 3 and 4.Figure 3
Kurtosis values for EGAPE distribution.Figure 4
Skewness values for EGAPE distribution.


### 3.5. Entropy

The Renyi entropy of EGAPE distribution:(14)RHx=11−plog∫0∞fpxdx.From equation (11), the Renyi entropy RHx becomes(15)RHx=11−plog∫0∞aλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−1pdx.


## 3.1. Quantile Function

Random samples from the EGAPE can be generated by inverting equation (2).(8)xq=−1λloglogα+W−1−logα×elogu/a/αlogα.To obtain the median of EGAPE, putu=0.5, and we have(9)x0.5=−1λloglogα+W−1−logα×elog0.5/a/αlogα.Some quantile values for the EGAPE are displayed in Table1.Table 1
Quantile values for EGAPE distribution.
Quantile(0.4, 0.3, 0.8)(0.5, 0.8, 0.3)(1.2, 0.7, 2.3)(0.9, 0.7, 0.5)0.10.0040.1430.0450.0590.20.0380.3520.1220.1630.30.1450.6070.2230.2980.40.3590.9130.3480.4640.50.7061.2810.5010.6680.61.2071.7340.6900.9210.71.9082.3200.9361.2480.82.9293.1441.2811.7080.94.6874.5461.8672.489


## 3.2. Moments

The EGAPE distribution’srth moment is defined as(10)Exr=∫0∞xrfxdx,Exr=∫0∞xraλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−1dx.The above equation does not have an explicit expression, and therefore we can find the moments numerically by using R version 4.1.2. Table2 gives the moments for the EGAPE for selected parameter values given as I:α=1.9,λ=1.0,a=0.9, II:α=1.4,λ=1.8,a=0.2, III:α=2.4,λ=1.5,β=1.3,a=0.9, and IV:α=1.4,λ=0.5,a=0.9.Table 2
Moments for EGAPE distribution.
IIIIIIIVμ11.8900.2870.5980.674μ27.0090.2550.5141.121μ340.5160.3880.7793.038μ4318.2160.8761.66311.513SD1.8520.4160.3950.816CV0.9791.4580.6611.209CS2.2472.9964.5862.543CK10.51317.83921.10012.977


## 3.3. Order Statistics

For an ordered random sampleX1,X2,……,Xn, from EGAPE distribution, the PDF of the ith minimum and maximum order statistics are provided by the following equations:(11)fX1x=nfx1−Fxn−1,fXnx=nfxFxn−1.Thus,(12)fX1x=naλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−11−α1−e−λxα1−e−λxan−1,fXnx=naλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−1α1−e−λxα1−e−λxan−1.


## 3.4. Skewness and Kurtosis

In the context of the EGAPE distribution, the definitions of Galton skewness and Moors kurtosis are stated as follows:(13)ZK=Q1/8+Q3/4−Q1/4−Q2/4Q3/4−Q1/4,ZMx=Q1/8+Q3/8−Q5/8−Q1/8Q3/4−Q1/4,where Q describe different quartile values. Clearly, the extra shape parameters α and a have an effect on the skewness and kurtosis values Figures 3 and 4.Figure 3
Kurtosis values for EGAPE distribution.Figure 4
Skewness values for EGAPE distribution.


## 3.5. Entropy

The Renyi entropy of EGAPE distribution:(14)RHx=11−plog∫0∞fpxdx.From equation (11), the Renyi entropy RHx becomes(15)RHx=11−plog∫0∞aλe−λxαe−λx1−1−e−λxlogαα1−e−λxα1−e−λxa−1pdx.


## 4. Parameter Estimation

To determine the MLEs of the given parameter estimation, for the model parametersλ,α,a, we use the log likelihood function, which may be expressed as(16)lnL=2nlnαλ+∑i=1n2e−λxilnα−2∑i=1nλxi+lnlnα−∑i=1nln1−e−λxi+2a−1∑i=1nlnα1−e−λxiα1−e−λxi.The equations of the EGAPE that give the maximum likelihood are provided by(17)∂l∂a=2∑i=1nlnα1−e−λxiα1−e−λxi,(18)∂l∂a=2nα+αlnα+∑i=1n2a−11−e−λxi/α1−e−λxi−1−e−λxi2/α1−e−λxiα1−e−λxiα1−e−λxi,(19)∂l∂λ=2nλ−∑i=1n2xi−∑i=1nxie−λxi1−e−λxi+∑i=1n2a−1αxie−λxi/α1−e−λxi−λ1−e−λxixie−λxilnα/α1−e−λxiα1−e−λxiα1−e−λxi.Equating equations (17)–(19) to zero and solving simultaneously, the maximum likelihood estimators of the parameters are obtained.


### 4.1. Monte Carlo Simulation

The effectiveness of the maximum likelihood method of estimate is examined by means of a simulation study, with attention given to both the root mean squared errors and the average bias. This evaluation is carried out with respect to the maximum likelihood technique. The simulation was carried out using version 4.1.2 of the R program, and the following is the technique that was followed for it:(a)
2000 random samples from different sample sizesn=100,200,300,…,2000 are generated using the quantile function.(b)
Two different sets of parameter values are considered (Set I:λ=0.2,α=0.3,a=0.5 and Set II λ=0.3,α=0.9,a=0.4).(c)
The calculated root mean squared errors as well as the average bias are then presented when the result has been obtained.


#### 4.1.1. Remarks on Simulation

From Tables2 and 3, we have recognized that by increasing the size of the samplethe average bias of the parameter estimates gets smaller. The same observation is made on the root mean squared error. This shows clearly that the maximum likelihood method performs well as a method of estimating the parameters from Tables 3 and 4.Table 3
RMSE and AB for Set I:λ=0.2,α=0.3,a=0.5.
RMSEAverage biasnλαaλαa1000.040060.3120.19650.004210.0864−0.18972000.02740.20060.19680.001230.0481−0.19353000.02230.15670.19770.001560.0246−0.19544000.01890.12840.19690.00005710.02462−0.19475000.01780.11470.19720.0005270.0180−0.19526000.01560.10530.19730.0003380.0175−0.19587000.01420.10050.19610.00014860.0207−0.19428000.01360.09030.19610.0003760.01686−0.19419000.01250.08530.19650.0002850.01359−0.194410000.01210.08170.19650.0002780.01333−0.192711000.01180.07620.19610.00007420.0123−0.193712000.01090.07370.1961−0.00009820.01197−0.193313000.01080.06920.1965−0.000045080.01061−0.193014000.01000.06780.19650.000005050.00975−0.194115000.00920.06250.19600.0002100.00785−0.193216000.009500.06190.1955−0.00004250.00938−0.192217000.009290.06040.19520.0002590.00756−0.191418000.00900.05790.19450.0002390.00793−0.190319000.00860.056580.19580.0001010.0068−0.192620000.00840.05620.19420.00010230.00719−0.1896Table 4
RMSE and AB for Set I:λ=0.3,α=0.4,a=0.6.
RMSEBiasnλαaλαa1000.060530.37410.20460.003560.0861−0.19092000.04210.25570.20310.001850.0402−0.19643000.03310.19620.20100.0003640.0335−0.19634000.02830.16120.20020.0001790.0236−0.19695000.02550.14370.20010.0005870.0193−0.19726000.02350.13040.20090.001350.01124−0.19867000.02160.11960.20020.001070.01107−0.19838000.01990.11080.20000.00002440.01322−0.19839000.01890.10480.20000.00046890.01069−0.198610000.01780.09890.19990.0005330.00965−0.198511000.01740.09270.20000.0005910.00608−0.198912000.01640.09060.19960.00003850.008727−0.198813000.01580.08730.1999−0.0002530.008357−0.198914000.01490.08440.1999−0.00003640.00895−0.198615000.01440.08090.19950.0003090.00701−0.198516000.01410.07860.19980.0002700.00606−0.198917000.01350.07370.20000.0005180.00355−0.199318000.01330.07140.19970.0004340.00407−0.199019000.01260.07020.19970.00009240.003956−0.198920000.01250.06870.19960.0004380.004439−0.1989


## 4.1. Monte Carlo Simulation

The effectiveness of the maximum likelihood method of estimate is examined by means of a simulation study, with attention given to both the root mean squared errors and the average bias. This evaluation is carried out with respect to the maximum likelihood technique. The simulation was carried out using version 4.1.2 of the R program, and the following is the technique that was followed for it:(a)
2000 random samples from different sample sizesn=100,200,300,…,2000 are generated using the quantile function.(b)
Two different sets of parameter values are considered (Set I:λ=0.2,α=0.3,a=0.5 and Set II λ=0.3,α=0.9,a=0.4).(c)
The calculated root mean squared errors as well as the average bias are then presented when the result has been obtained.


### 4.1.1. Remarks on Simulation

From Tables2 and 3, we have recognized that by increasing the size of the samplethe average bias of the parameter estimates gets smaller. The same observation is made on the root mean squared error. This shows clearly that the maximum likelihood method performs well as a method of estimating the parameters from Tables 3 and 4.Table 3
RMSE and AB for Set I:λ=0.2,α=0.3,a=0.5.
RMSEAverage biasnλαaλαa1000.040060.3120.19650.004210.0864−0.18972000.02740.20060.19680.001230.0481−0.19353000.02230.15670.19770.001560.0246−0.19544000.01890.12840.19690.00005710.02462−0.19475000.01780.11470.19720.0005270.0180−0.19526000.01560.10530.19730.0003380.0175−0.19587000.01420.10050.19610.00014860.0207−0.19428000.01360.09030.19610.0003760.01686−0.19419000.01250.08530.19650.0002850.01359−0.194410000.01210.08170.19650.0002780.01333−0.192711000.01180.07620.19610.00007420.0123−0.193712000.01090.07370.1961−0.00009820.01197−0.193313000.01080.06920.1965−0.000045080.01061−0.193014000.01000.06780.19650.000005050.00975−0.194115000.00920.06250.19600.0002100.00785−0.193216000.009500.06190.1955−0.00004250.00938−0.192217000.009290.06040.19520.0002590.00756−0.191418000.00900.05790.19450.0002390.00793−0.190319000.00860.056580.19580.0001010.0068−0.192620000.00840.05620.19420.00010230.00719−0.1896Table 4
RMSE and AB for Set I:λ=0.3,α=0.4,a=0.6.
RMSEBiasnλαaλαa1000.060530.37410.20460.003560.0861−0.19092000.04210.25570.20310.001850.0402−0.19643000.03310.19620.20100.0003640.0335−0.19634000.02830.16120.20020.0001790.0236−0.19695000.02550.14370.20010.0005870.0193−0.19726000.02350.13040.20090.001350.01124−0.19867000.02160.11960.20020.001070.01107−0.19838000.01990.11080.20000.00002440.01322−0.19839000.01890.10480.20000.00046890.01069−0.198610000.01780.09890.19990.0005330.00965−0.198511000.01740.09270.20000.0005910.00608−0.198912000.01640.09060.19960.00003850.008727−0.198813000.01580.08730.1999−0.0002530.008357−0.198914000.01490.08440.1999−0.00003640.00895−0.198615000.01440.08090.19950.0003090.00701−0.198516000.01410.07860.19980.0002700.00606−0.198917000.01350.07370.20000.0005180.00355−0.199318000.01330.07140.19970.0004340.00407−0.199019000.01260.07020.19970.00009240.003956−0.198920000.01250.06870.19960.0004380.004439−0.1989


## 4.1.1. Remarks on Simulation

From Tables2 and 3, we have recognized that by increasing the size of the samplethe average bias of the parameter estimates gets smaller. The same observation is made on the root mean squared error. This shows clearly that the maximum likelihood method performs well as a method of estimating the parameters from Tables 3 and 4.Table 3
RMSE and AB for Set I:λ=0.2,α=0.3,a=0.5.
RMSEAverage biasnλαaλαa1000.040060.3120.19650.004210.0864−0.18972000.02740.20060.19680.001230.0481−0.19353000.02230.15670.19770.001560.0246−0.19544000.01890.12840.19690.00005710.02462−0.19475000.01780.11470.19720.0005270.0180−0.19526000.01560.10530.19730.0003380.0175−0.19587000.01420.10050.19610.00014860.0207−0.19428000.01360.09030.19610.0003760.01686−0.19419000.01250.08530.19650.0002850.01359−0.194410000.01210.08170.19650.0002780.01333−0.192711000.01180.07620.19610.00007420.0123−0.193712000.01090.07370.1961−0.00009820.01197−0.193313000.01080.06920.1965−0.000045080.01061−0.193014000.01000.06780.19650.000005050.00975−0.194115000.00920.06250.19600.0002100.00785−0.193216000.009500.06190.1955−0.00004250.00938−0.192217000.009290.06040.19520.0002590.00756−0.191418000.00900.05790.19450.0002390.00793−0.190319000.00860.056580.19580.0001010.0068−0.192620000.00840.05620.19420.00010230.00719−0.1896Table 4
RMSE and AB for Set I:λ=0.3,α=0.4,a=0.6.
RMSEBiasnλαaλαa1000.060530.37410.20460.003560.0861−0.19092000.04210.25570.20310.001850.0402−0.19643000.03310.19620.20100.0003640.0335−0.19634000.02830.16120.20020.0001790.0236−0.19695000.02550.14370.20010.0005870.0193−0.19726000.02350.13040.20090.001350.01124−0.19867000.02160.11960.20020.001070.01107−0.19838000.01990.11080.20000.00002440.01322−0.19839000.01890.10480.20000.00046890.01069−0.198610000.01780.09890.19990.0005330.00965−0.198511000.01740.09270.20000.0005910.00608−0.198912000.01640.09060.19960.00003850.008727−0.198813000.01580.08730.1999−0.0002530.008357−0.198914000.01490.08440.1999−0.00003640.00895−0.198615000.01440.08090.19950.0003090.00701−0.198516000.01410.07860.19980.0002700.00606−0.198917000.01350.07370.20000.0005180.00355−0.199318000.01330.07140.19970.0004340.00407−0.199019000.01260.07020.19970.00009240.003956−0.198920000.01250.06870.19960.0004380.004439−0.1989


## 5. Applications

This section presents the effectiveness and the flexibility of the EGAPE distribution by two real datasets. The datasets are on the COVID-19 mortality rates for different countries. In addition, we provide the evaluation of the goodness of fit of the distribution and comparison to other competing models. The measures include the BIC, AIC, HQIC, and CAIC. The smaller the values of these statistics, the better the model. The proposed distribution is compared to the following:(i)
Exponential distribution (E).The CDF is given byFx=1−e−λx.(ii)
Gull alpha power exponential (GAPE).The CDF is given byFx=α1−e−λx/α1−e−λx.(iii)
The exponentiated generalized exponential distribution (EGE).The CDF is given asFx=1−e−λxab.(iv)
The Marshall–Olkin generalized exponential distribution (MOGE).The CDF is given by(20)Fx=1−e−λxaθ+1−θ1−e−λxα.(v)
Marshall–Olkin alpha power inverted exponential (MOAPIE) distribution.The CDF is given as(21)Fx=αe−λx−1−1α−1θ−θ−1αe−λx−1−1.(vi)
Alpha power inverted exponential (APIE) distribution.The CDF is given as(22)Fx=αe−λx−1−1α−1.(vii)
Weibull moment exponential distribution (WME) distribution.The CDF is given as(23)Fx=1−exp−a1−1+x/βe−x/β1+x/βe−x/βb.


### 5.1. Dataset I: Mexico Mortality Data

The first dataset depicts the Mexico mortality rate obtained from the following link:https://covid19.who.int/. Figure 5 shows the TTT plot for the Mexico mortality rate data. The data have an increasing hazard rate. Results for the Mexico application data are displayed in Tables 5 and 6.Figure 5
(a) TTT plot, (b) boxplot, and (c) histogram for Mexico data.
(a)(b)(c)Table 5
Estimates and SE (in parentheses).
DistributionEstimates (SE in parenthesis)EGAPEλ,α,a5.284 (2.476)1.456 (0.768)0.339 (0.059)GAPEλ,α0.400 (0.035)0.013 (0.011)Eλ0.174 (0.016)EEλ,α0.360 (0.034)3.982 (0.672)MOAPIEλ,θ,a0.003 (0.002)1.254 (0.381)8.887 (0.917)MOGEλ,θ,a0.360 (0.072)4.109 (0.784)0.956 (0.677)APIEλ,α9.247 (1.018)0.007 (0.008)Table 6
Goodness of fit measures for Mexico mortality rate data.
DistributionAICBICCAICHQICKSWAEGAPE537.983546.034538.216541.2470.0690.0560.308GAPE539.358544.720539.350541.5330.0790.0830.491EGE538.356546.403538.580541.6190.0720.0590.329E596.131598.81596.160597.2180.2410.0620.363MOAPIE551.267559.314551.267554.5300.1180.1771.069MOGE538.355546.140538.586541.6180.0740.0590.328APIE550.634555.998550.748552.8090.0890.1951.212


### 5.2. Dataset II: Canada Data

The second dataset gives the COVID-19 mortality rate data for Canada for a length of 36 days obtained from the following link:https://covid19.who.int/. As depicted in Figure 6, the data are characterized by an increasing hazard rate. The results for dataset I are presented in Tables 7 and 8.Figure 6
(a) TTT plot, (b) boxplot, and (c) histogram for Canada data.
(a)(b)(c)Table 7
Estimates and SE (in parentheses).
DistributionEstimates (SE in parenthesis)EGAPEλ,α,a5.420 (5.267)0.007 (0.041)1.219 (0.152)Eλ1.058 (0.0507)EEλ,α1.058 (0.122)17.977 (5.931)MOAPIEλ,θ,a8.396 (7.967)0.005 (0.003)11.984 (1.856)EGEλ,θ,a0.654 (0.400)1.522 (0.950)16.799 (0.023)APIEλ,α2.719 (0.672)0.211 (0.185)Table 8
Goodness of fit measures for Canada mortality rate data.
DistributionAICBICCAICHQICKSWAEGAPE102.746107.49103.496104.4040.1210.1060.636EGE104.634109.386105.385106.2930.1310.1040.614E159.559161.112159.559160.1120.4090.0990.574MOAPIE112.433117.183113.183114.0910.4090.0991.022EE102.341105.508102.705103.4470.1510.1040.622APIE160.153163.320160.517161.2580.3900.1390.843


### 5.3. Concluding Remarks on the Two Applications

Based on the application from the two datasets, the following conclusions are drawn:(1)
Referring to dataset I, it can be observed that EGAPE provides the lowest values for the Kolmogorov–Smirnov and the W∗ and A∗ distances.(2)
With regard to dataset II, we can deduce that EGAPE provides the lowest values for the Kolmogorov–Smirnov and the W∗ and A∗ distances.(3)
Since the datasets are on COVID-19 mortality rates, the EGAPE model is the best model with comparison to the competing models to handle this set of data.(4)
In future, the study may be extended to consider other estimation methods of parameters like maximum product spacing and weighted least squares, among others.(5)
Since the study considered complete samples, future research may incorporate censored data.(6)
For future research, the regression framework may be considered to incorporate covariates.


## 5.1. Dataset I: Mexico Mortality Data

The first dataset depicts the Mexico mortality rate obtained from the following link:https://covid19.who.int/. Figure 5 shows the TTT plot for the Mexico mortality rate data. The data have an increasing hazard rate. Results for the Mexico application data are displayed in Tables 5 and 6.Figure 5
(a) TTT plot, (b) boxplot, and (c) histogram for Mexico data.
(a)(b)(c)Table 5
Estimates and SE (in parentheses).
DistributionEstimates (SE in parenthesis)EGAPEλ,α,a5.284 (2.476)1.456 (0.768)0.339 (0.059)GAPEλ,α0.400 (0.035)0.013 (0.011)Eλ0.174 (0.016)EEλ,α0.360 (0.034)3.982 (0.672)MOAPIEλ,θ,a0.003 (0.002)1.254 (0.381)8.887 (0.917)MOGEλ,θ,a0.360 (0.072)4.109 (0.784)0.956 (0.677)APIEλ,α9.247 (1.018)0.007 (0.008)Table 6
Goodness of fit measures for Mexico mortality rate data.
DistributionAICBICCAICHQICKSWAEGAPE537.983546.034538.216541.2470.0690.0560.308GAPE539.358544.720539.350541.5330.0790.0830.491EGE538.356546.403538.580541.6190.0720.0590.329E596.131598.81596.160597.2180.2410.0620.363MOAPIE551.267559.314551.267554.5300.1180.1771.069MOGE538.355546.140538.586541.6180.0740.0590.328APIE550.634555.998550.748552.8090.0890.1951.212


## 5.2. Dataset II: Canada Data

The second dataset gives the COVID-19 mortality rate data for Canada for a length of 36 days obtained from the following link:https://covid19.who.int/. As depicted in Figure 6, the data are characterized by an increasing hazard rate. The results for dataset I are presented in Tables 7 and 8.Figure 6
(a) TTT plot, (b) boxplot, and (c) histogram for Canada data.
(a)(b)(c)Table 7
Estimates and SE (in parentheses).
DistributionEstimates (SE in parenthesis)EGAPEλ,α,a5.420 (5.267)0.007 (0.041)1.219 (0.152)Eλ1.058 (0.0507)EEλ,α1.058 (0.122)17.977 (5.931)MOAPIEλ,θ,a8.396 (7.967)0.005 (0.003)11.984 (1.856)EGEλ,θ,a0.654 (0.400)1.522 (0.950)16.799 (0.023)APIEλ,α2.719 (0.672)0.211 (0.185)Table 8
Goodness of fit measures for Canada mortality rate data.
DistributionAICBICCAICHQICKSWAEGAPE102.746107.49103.496104.4040.1210.1060.636EGE104.634109.386105.385106.2930.1310.1040.614E159.559161.112159.559160.1120.4090.0990.574MOAPIE112.433117.183113.183114.0910.4090.0991.022EE102.341105.508102.705103.4470.1510.1040.622APIE160.153163.320160.517161.2580.3900.1390.843


## 5.3. Concluding Remarks on the Two Applications

Based on the application from the two datasets, the following conclusions are drawn:(1)
Referring to dataset I, it can be observed that EGAPE provides the lowest values for the Kolmogorov–Smirnov and the W∗ and A∗ distances.(2)
With regard to dataset II, we can deduce that EGAPE provides the lowest values for the Kolmogorov–Smirnov and the W∗ and A∗ distances.(3)
Since the datasets are on COVID-19 mortality rates, the EGAPE model is the best model with comparison to the competing models to handle this set of data.(4)
In future, the study may be extended to consider other estimation methods of parameters like maximum product spacing and weighted least squares, among others.(5)
Since the study considered complete samples, future research may incorporate censored data.(6)
For future research, the regression framework may be considered to incorporate covariates.


## 6. Conclusion

In this paper, the main purpose was to develop a new three-parameter distribution called the EGAPE distribution which can be useful in modeling datasets that exhibit both monotone and nonmonotone hazard shapes. Several mathematical properties of the EGAPE distribution were derived. The estimation of the parameters of the distribution was estimated through the maximum likelihood method. Simulation study was performed to investigate the effectiveness of the method of estimation, and it was found that the root mean squared error and the average bias decrease with increase in sample size. Two real datasets were used to demonstrate the flexibility of the EGAPE distribution against its competing distribution. By use of goodness of fit tests, we demonstrated that the EGAPE distribution provides a better fit compared to that of competing distributions considered. With regard to application of the proposed model, other domains apart from COVID-19-related data can be considered like reliability engineering and financial sciences. Also, we will extend our work in the future by making regression analyses of the COVID-19 infections in future infections. This may help researchers and scientists to prepare suitable amounts of vaccines and enough space in hospitals. Also in our future work, we will study the vaccination rate and its effect on the mortality rate. Another approach that could be done is to use machine learning to predict future infections. Also, we will work on censored samples to avoid time-consuming and reduce the costs of the experiments. We will use different censoring schemes such as type-I and type-II censored schemes.



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*Source: 4255079-2022-06-08.xml*