International Journal of Analysis and Applications 

Volume 19, Number 5 (2021), 695-708 

URL: https://doi.org/10.28924/2291-8639 

DOI: 10.28924/2291-8639-19-2021-695 

 

 

Received June 2nd, 2021; accepted July 8th, 2021; published August 2nd, 2021. 

2010 Mathematics Subject Classification. 62E10. 

Key words and phrases. discretization; inverted Topp-Leone distribution; moments; count data; data analysis. 

©2021 Authors retain the copyrights 

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 

695 

 

A DISCRETE ANALOG OF INVERTED TOPP-LEONE DISTRIBUTION: PROPERTIES, 

ESTIMATION AND APPLICATIONS 

 

AHMED SEDKY ELDEEB1,2,*, MUHAMMAD AHSAN-UL-HAQ3,4, AYESHA BABAR5 

 

1Department of Business Administration, College of Business, King Khaled University, Saudi Arabia 

2 Department of Statistics, Mathematics and Insurance, Alexandria University, Egypt 

3College of Statistical & Actuarial Sciences, University of the Punjab, Pakistan 

4Quality Enhancement Cell, National College of Arts, Lahore Pakistan 

5School of Statistics, Minhaj University Lahore Pakistan 

*Corresponding author: asedky@kku.edu.sa 

 

ABSTRACT. In this study, a discrete inverted Topp-Leone (DITL) distribution is proposed by utilizing the survival 

discretization approach. The proposed distribution's mathematical features were derived. The maximum likelihood 

(ML), method of least squares (LS), weighted least squares (WLS), and Cramer Von-Mises (CVM) estimation techniques 

were used to estimate the parameter. The theoretical results of the ML, LS, WLS, and CVM estimators were 

demonstrated via a comprehensive simulation study. The proposed DITL distribution has been applied to analyze two 

count data sets number of deaths due to Covid-19 in Pakistan and India and the findings show the relevance of the 

proposed distribution. 

 

 

1. INTRODUCTION 

In December 2019, the first incidence of COVID-19 was reported in the Chinese city of 

Wuhan. COVID-19 is an extremely contagious disease. In Pakistan, the first case was reported on 

February 26, 2020 [1]. The first death was reported in Pakistan on March 20, 2020.  

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-695


Int. J. Anal. Appl. 19 (5) (2021) 696 

 

Many researchers make efforts to study the patterns of pandemic Covid-19 and provide 

models which better fit the data and can be used to have an idea about the expected number of 

cases to help the government to take decisions regarding precautionary measures. These efforts 

include the derivation of different probabilistic models and time series modeling of the data such 

as a new discrete Lindley [2], discrete Marshall-Olkin generalized exponential distribution to 

model the daily new cases in Egypt [3], a new discrete generalized distribution to analyze the 

count of daily cases in Hong Kong and Iran [4], a mathematical model known as SIR is used to 

predict the daily new cases in China [5,6], the logistic growth model is used to estimate the final 

size and its peak time of coronavirus epidemic [7], autoregressive time series model is used to 

forecast the recovered and confirmed cases [8],  and discrete Marshall-Olkin Lomax distribution 

is used to estimate the daily new cases in Australia [9]. 

Practically, lifetime data sets are often recorded as whole numbers of counts. To model 

the count data, there are few classical distributions as geometric, Poisson, negative binomial, etc. 

these models sometimes do not provide a better fit due to the complex behavior of data. From the 

last few decades, researchers have paid attention to introduced discrete type distributions which 

meet the required need to model the complex behavior of data sets. Several discretization 

approaches are available in the literature. A detailed systematic review was conducted on 

discretization approaches [10]. Among all approaches, one of the most important is the survival 

discretizing approach due to its important feature of keeping the original form of the survival 

function. 

“Let X a continuous random variable. If X has a survival function 𝑆𝑋(𝑥), then the discrete random 

Variable 𝑌 = [𝑋] , where [𝑋]  indicates the smallest integer part or equal to 𝑋,  have probability mass 

function (PMF) written as”:  

𝑃(𝑌 =  𝑘) =  ∑(−1)𝑖
1

𝑖=0

𝑆𝑋 (𝑘 + 𝑖). 𝑘 = 0,1,2,3, …                                    (1) 

Using survival discretization approach authors have derived discrete distributions. Some 

of these include discrete Weibull [11], discrete Rayleigh [12], discrete Burr and Pareto [13], 

discrete inverse Weibull [14], discrete Lindley [15], discrete generalized Rayleigh [16], discrete 

Bilal [17], discrete Nadarajah-Haghighi [18], discrete Burr-Hutke [19] and many others. 



Int. J. Anal. Appl. 19 (5) (2021) 697 

 

The inverted Topp-Leone distribution [20] was derived for the analysis of reliability 

observations. A comprehensive discussion about its mathematical properties, reliability 

characteristics, stochastic ordering, and parameter estimation via complete and censored samples, 

among others is also presented in the mentioned paper. The corresponding survival function is 

given by 

𝑆(𝑥) =  
(1 + 2𝑥)𝛿

(1 + 𝑥)2𝛿
, 𝑥 > 0, 𝛿 > 0.                                                                     (2) 

where 𝛿 is the shape parameter.  

The goal of this study is to introduce a new one-parameter discrete model called the 

discrete inverted Topp-Leone (DITL) distribution, which is based on the survival function 

approach of discretization. The DITL distribution can be used to model the over-dispersed data 

sets. We derive some of its properties such as, survival and hazard function, quantile function, 

moments, and generating function. The maximum likelihood, Cramer-von Mises, least-square, 

and weighted least square estimation methods are used to estimate the model's parameter. A 

simulation study is conducted to elaborate on the performance of these estimation methods. In 

the end, we will use data sets about the number of deaths due to coronavirus in Pakistan and 

India to illustrate the importance of the proposed distribution. 

The organization of the article is as follows. Section 2 contains the derivation of the 

proposed distribution and its features. Section 3 addressed maximum likelihood estimation, as 

well as the least-squares, weighted least squares, and Cramer von Mises approach. In Section 4, 

a complete simulation study is used to assess the behavior of these estimators. The proposed 

distribution's application is discussed in Section 5. The conclusion has been presented in the final 

section. 

2. THE DITL DISTRIBUTION AND PROPERTIES 

Using a discretization approach based on the survival function, the discrete inverted Topp-Leone 

distribution has been developed. The probability mass function (pmf) of DITL distribution can 

be represented as  

𝑃(𝑥) =
(1 + 2𝑥)𝛿

(1 + 𝑥)2𝛿
−

(3 + 2𝑥)𝛿

(2 + 𝑥)2𝛿
,   𝛿 > 0, 𝑥 = 0,1,2, …                                               (3) 



Int. J. Anal. Appl. 19 (5) (2021) 698 

 

The pmf plots of the DITL distribution with some selected values of parameter δ are presented in 

Figure 1.   

 

 

Figure 1: Behavior of pmf of DITLD for different parameter values 

 

The cumulative distribution and survival functions of DITL are given, respectively as  

𝐹(𝑥) = 1 −
(3 + 2𝑥)𝛿

(2 + 𝑥)2𝛿
                                                                           (4) 

and,  

𝑆(𝑥) =
(3 + 2𝑥)𝛿

(2 + 𝑥)2𝛿
                                                                                   (5) 

The hazard function of the DITL distribution is obtained using Eq. (3) and Eq. (5). The behavior 

of the hazard function is illustrated in Figure 2.  

ℎ(𝑥) =

(1+2𝑥)𝛿

(1+𝑥)2𝛿
−

(3+2𝑥)𝛿

(2+𝑥)2𝛿

(3+2𝑥)𝛿

(2+𝑥)2𝛿

 



Int. J. Anal. Appl. 19 (5) (2021) 699 

 

or 

ℎ(𝑥) =  
(1 + 2𝑥)𝛿 (2 + 𝑥)2𝛿

(1 + 𝑥)2𝛿 (3 + 2𝑥)𝛿
− 1                                                                             (6) 

where 𝛿 > 0 &  𝑥 = 0,1,2, …. Note that, for 𝑥 → 0 the HRF turn into  

ℎ(0) = [ (
4

3
)

𝛿

− 1]. 

 

 

Figure 2: Behavior of HRF of DITLD for different parameter values 

 

The reverse hazard function of DITL is given as 

𝑟∗(𝑥) =
𝑃(𝑥)

𝐹(𝑥)
=

(1 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (1 + 𝑥)2𝛿

(1 + 𝑥)2𝛿 [(2 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 ]
                       (7) 

The second rate of failure of DITL is  

r∗∗(x) =  log [
𝑆(𝑥)

𝑆(𝑥 + 1)
] = log [

(3 + 2𝑥)𝛿 (3 + 𝑥)2𝛿

(2 + 𝑥)2𝛿 (5 + 2𝑥)𝛿
]                                 (8) 

The recurrence relation for generating the probabilities of discrete DITL distribution is given by 



Int. J. Anal. Appl. 19 (5) (2021) 700 

 

𝑃(𝑥 + 1)

𝑃(𝑥)
=  

(1 + 𝑥)2𝛿 (2 + 𝑥)2𝛿 [(3 + 2𝑥)𝛿 (3 + 𝑥)2𝛿 − (5 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 ]

(2 + 𝑥)2𝛿 (3 + 𝑥)2𝛿 [(1 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 ]
 

or  

𝑃(𝑥 + 1) =  
(1 + 𝑥)2𝛿 [(3 + 2𝑥)𝛿 (3 + 𝑥)2𝛿 − (5 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 ]

(3 + 𝑥)2𝛿 [(1 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 ]
  𝑃(𝑥). 

2.1. Quantile Function 

The pth quantile function of DITL distribution is given by  

𝑥𝑝 =
√4((1 − 𝑝)(1 𝛿)⁄ − 1)2 − 4(1 − 𝑝)(1 𝛿) ⁄ ((1 − 𝑝)1 𝛿⁄ − 1) − 2((1 − 𝑝)1 𝛿⁄ − 1)

2(1 − 𝑝)(1 𝛿)⁄
. 

2.2. The moments of DITL distribution  

The non-central moments of DITL distribution can be obtained using Eq. (3) as follows:  

𝜇𝑟
ʹ =  ∑ 𝑥𝑟

∞

𝑥=0

𝑃(𝑥) 

𝜇𝑟
ʹ =  ∑ 𝑥𝑟

∞

𝑥=0

[
(1 + 2𝑥)𝛿

(1 + 𝑥)2𝛿
−

(3 + 2𝑥)𝛿

(2 + 𝑥)2𝛿
]                                                                                     (9) 

In particular, the mean of DITL distribution is  

𝜇1
ʹ =  ∑ 𝑥

∞

𝑥=0

[
(1 + 2𝑥)𝛿

(1 + 𝑥)2𝛿
−

(3 + 2𝑥)𝛿

(2 + 𝑥)2𝛿
] 

The central moments of DITL distribution can be obtained using the following relation  

𝜇𝑟 =  ∑ (
𝑟
𝑗 )

𝑟

𝑗=0

(−1)𝑗 𝜇 𝑗 𝜇𝑟−𝑗
′  

The variance of DITL distribution is given as  

𝑉𝑎𝑟(𝑋) = ∑ 𝑥2
∞

𝑥=0

[
(1 + 2𝑥)𝛿

(1 + 𝑥)2𝛿
−

(3 + 2𝑥)𝛿

(2 + 𝑥)2𝛿
] − {∑ 𝑥

∞

𝑥=0

[
(1 + 2𝑥)𝛿

(1 + 𝑥)2𝛿
−

(3 + 2𝑥)𝛿

(2 + 𝑥)2𝛿
]}

2

 

The dispersion index can be calculated using the expression  

𝐷𝐼 =
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝐷𝐼𝑇𝐷

𝑀𝑒𝑎𝑛 𝑜𝑓 𝐷𝐼𝑇𝐿𝐷
 

Since the above equation cannot be solved, so the descriptive measures, i.e., mean and variance 

are computed numerically. These measures are presented in Figure 3.  



Int. J. Anal. Appl. 19 (5) (2021) 701 

 

 

Figure 3: Plots of Mean, Variance and Dispersion Index for DITL distribution 

 

From Figure 3, it is apparent that the mean, variance, and DI of the DITL distribution have 

decreasing behavior with an increase in parameter δ. 

 

3. PARAMETER ESTIMATION OF DITL DISTRIBUTION  

This section describes the parameter estimation of DITL distribution using four different 

estimation methods. These methods are Maximum Likelihood Estimator (MLE), Cramer Von-

Mises Estimator (CVM), Least Square Estimator (LS), and Weighted Least Square Estimator 

(WLSE).  

3.1. Method of Maximum Likelihood 

Let 𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑛 be a random sample of size ‘n’ from DITL distribution with parameter 𝛿. The 

likelihood function is given by 

𝐿 =  ∑ log[(1 + 2𝑥𝑖 )
𝛿 (2 + 𝑥𝑖 )

2𝛿 − (3 + 2𝑥𝑖 )
𝛿 (1 + 𝑥𝑖 )

2𝛿 ] 

𝑛

𝑖=1

− 2𝛿 ∑ log(1 + 𝑥𝑖 )

𝑛

𝑖=1

− 2𝛿 ∑ log(2 + 𝑥𝑖 ) 

𝑛

𝑖=1

                                                                                                               (10) 

Now partially differentiate with respect to parameter δ.  



Int. J. Anal. Appl. 19 (5) (2021) 702 

 

𝜕𝐿

𝜕𝛿
=  ∑

{
2(1 + 2𝑥𝑖 )

𝛿 (2 + 𝑥𝑖 )
2𝛿 log(2 + 𝑥𝑖 ) + (2 + 𝑥𝑖 )

2𝛿 (1 + 2𝑥𝑖 )
𝛿 log(1 + 2𝑥𝑖 )

−2(3 + 2𝑥𝑖 )
𝛿 (1 + 𝑥𝑖 )

2𝛿 log(1 + 𝑥𝑖 ) − (3 + 2𝑥𝑖 )
𝛿 log(3 + 2𝑥𝑖 ) (1 + 𝑥𝑖 )

2𝛿
}

[(1 + 2𝑥𝑖 )
𝛿 (2 + 𝑥𝑖 )

2𝛿 − (3 + 2𝑥𝑖 )
𝛿 (1 + 𝑥𝑖 )

2𝛿 ]
 

𝑛

𝑖=1

− 2 ∑ log(1 + 𝑥𝑖 )

𝑛

𝑖=1

− 2 ∑ log(2 + 𝑥𝑖 ) 

𝑛

𝑖=1

                                                                             (11) 

The MLE of DITL distribution parameter δ can be obtained from the above Equation (11). The 

exact solution is not possible, so we can be obtained numerically. 

3.2. Method of Cramer von-Mises  

The Cramèr-von-Mises estimators (CVME), can be determined depending on the difference 

between both the estimated and exact distributions. The CVME estimator (𝛿𝐶𝑉𝑀 ) can be obtained 

by minimizing 

𝐶𝑉𝑀(𝛿) =
1

12𝑛
+ ∑ [𝐹(𝑥𝑖 ) −

2𝑖 − 1

2𝑛
]

2𝑛

𝑖=1

                                                                 (12) 

3.3. Method of Least Squares and Weighted Least Squares  

The least-square estimator can be obtained by minimizing the sum of the square of residuals.  

𝐿𝑆(𝛿) = ∑ [𝐹(𝑥𝑖 ) −
𝑖

𝑛 + 1
]

2𝑛

𝑖=1

                                                                 (13) 

The weighted least squares estimators of the parameter of DITL distribution are obtained by 

minimizing  

𝑊𝐿𝑆(𝛿) = ∑
(𝑛 + 1)2(𝑛 + 2)

𝑛 − 𝑖 + 1
[𝐹(𝑥𝑖 ) −

𝑖

𝑛 + 1
]

2𝑛

𝑖=1

                                                                 (14) 

with respect to parameter δ. 

4. SIMULATION  

In this section, a simulation analysis evaluates the output of four different estimators of the DITL 

for different values of parameter δ. We consider the different sample sizes n = 10, 20, 50, and 100 

for the different values of parameter = (0.8, 1, 1.5, 2, 3, 5, 10). From DITL distribution, we generate 

10000 iterations of random samples. For each computation, we get the average estimations (AEs) 

and mean square error (MSE). The output of considered estimators is compared in terms of MSE, 

with the lowest MSE values indicating the best successful technique of estimation. The R program 

is used to obtain simulation results. The AE and MSE values for the MLE, CVM, LS, and WLS 



Int. J. Anal. Appl. 19 (5) (2021) 703 

 

approaches are shown in Table 1. Methods tend towards the true parameter values, suggesting 

that all estimators are asymptotically unbiased.  

Table 1:  The simulation results for the parameter δ 

Para. 
n 

MLE CVME LSE WSE 

𝛿 AEs MSEs AEs MSEs AEs MSEs AEs MSEs 

0.8 

10 0.9138 0.1310 1.0117 0.2847 0.9922 0.3008 0.9590 0.2406 

20 0.8460 0.0376 0.9669 0.1519 0.9203 0.1012 0.9341 0.1144 

50 0.8137 0.0142 0.9142 0.0476 0.9046 0.0414 0.8965 0.0380 

100 0.8042 0.0065 0.8961 0.0224 0.8972 0.0249 0.8923 0.0210 

1.0 

10 1.1199 0.1786 1.3289 0.5450 1.2527 0.4231 1.2878 0.5169 

20 1.0729 0.0675 1.2515 0.2398 1.2063 0.2054 1.2225 0.2150 

50 1.0200 0.0212 1.1863 0.0908 1.1841 0.0879 1.1623 0.0745 

100 1.0113 0.0101 1.1653 0.0539 1.1742 0.0590 1.1414 0.0433 

1.5 

10 1.6754 0.4174 2.0778 1.2522 1.9456 0.9305 1.9506 0.9876 

20 1.5762 0.1524 1.9852 0.6515 1.9267 0.5357 1.8867 0.4822 

50 1.5332 0.0543 1.9298 0.3412 1.9285 0.3425 1.8565 0.2574 

100 1.5079 0.0213 1.9242 0.2546 1.8986 0.2330 1.8440 0.1781 

2.0 

10 2.2598 0.7833 2.8469 2.1167 2.6438 1.5017 2.6569 1.6724 

20 2.0934 0.2675 2.7375 1.2770 2.6655 1.0143 2.6394 1.0230 

50 2.0488 0.1046 2.6844 0.7219 2.6551 0.6799 2.5891 0.5918 

100 2.0202 0.0468 2.6759 0.5741 2.6522 0.5508 2.5782 0.4564 

3.0 

10 3.6942 29.170 4.2539 3.9863 3.9756 2.8443 3.9646 3.1796 

20 3.2014 0.7266 4.2718 2.8786 4.0647 2.2260 4.1605 2.7242 

50 3.0637 0.2182 4.1455 1.7956 4.1121 1.7172 4.1168 1.7456 

100 3.0366 0.1003 4.1571 1.5673 4.1206 1.4607 4.1275 1.5432 

5.0 

10 11.356 497.89 6.6005 7.1286 5.7374 2.9089 6.0616 3.5941 

20 5.7253 29.774 6.8668 6.2545 6.4529 4.2814 6.6718 4.9023 

50 5.1582 0.7127 6.8237 4.3197 6.6485 3.6260 7.0444 5.4704 

100 5.0605 0.3035 6.7486 3.5393 6.7070 3.3384 7.1014 5.0482 

10.0 

10 51.192 3263.8 9.6885 0.9487 9.5835 1.5339 9.0152 2.6714 

20 31.884 1671.7 9.8330 0.5029 9.6529 1.0747 9.7720 1.1258 

50 15.195 355.64 9.9858 0.0241 9.9235 0.2293 11.7927 4.8997 

100 10.609 24.082 9.9994 0.0081 9.9972 0.0099 12.7972 9.2633 

 



Int. J. Anal. Appl. 19 (5) (2021) 704 

 

If 𝛿=0.8, 1, 1.5, and 2, the MLE is the best estimation method in all sample sizes. If 𝛿 =3 

with sample size n=10, the LSE method is the best estimation method while in other sample sizes 

the MLE is the best method of estimation. If 𝛿=5 with sample sizes n=10 and 20 the LS is the best 

estimation method while the MLE is the best method in sample sizes n= 50 and 100. If 𝛿= 10 the 

CVME is the best estimation method in all sample sizes. 

 

5. APPLICATION 

In this section, we illustrate the importance of the proposed distribution using two data 

sets. Both data sets are counts. Five one-parameter competitive distributions of the DITL 

distribution are Poison distribution, discrete Pareto distribution [13], discrete Rayleigh 

distribution [12], discrete inverse Rayleigh distribution [21], and discrete Burr-Hutke distribution 

[19].  

The first data represents the number of deaths due to coronavirus in Pakistan. A sample of 

44 deaths is considered from March 18, 2020, to April 30, 2020.  

The second data set represents the number of deaths due to coronavirus in India. A sample 

of 51 deaths is considered from March 11, 2020, to April 30, 2020. Both data sets are plotted in 

Figure 4.  

 

Figure 4. Plots for COVID-19 daily deaths in Pakistan and India 

The results presented in Tables 2-3 and Figures 5-6 demonstrate the sufficiency and superiority 

of the proposed distribution in modeling the data sets when compared to the competitive 

distributions.  



Int. J. Anal. Appl. 19 (5) (2021) 705 

 

Table 2: The MLEs, standard errors of the competing models for number of deaths in Pakistan 

Model 
MLE Goodness-of-fit Measures 

𝐸(𝛿) SE ℓ AIC BIC KS 

DITLD 0.70958 0.10705 -153.04 308.09 309.87 0.1619 

DPoi 9.47694 0.46409 -283.94 569.89 571.67 0.3910 

DPr 0.50220 0.07576 -162.19 326.38 328.17 0.4010 

DR 9.98734 0.75335 -168.85 339.70 341.49 0.3390 

DIR 7.43010 1.26260 -166.31 334.61 336.40 0.3820 

DBHD 0.99483 0.01148 -175.37 352.74 354.52 0.6470 

 

Table 3: The MLEs, standard errors of the competing models for number of deaths in India 

Model 
MLE Goodness-of-fit Measures 

𝐸(𝛿) SE ℓ AIC BIC KS 

DITLD 0.52980 0.07421 -218.93 439.86 441.79 0.2190 

DPoi 22.6661 0.66665 -740.90 1483.8 1485.7 0.4930 

DPr 0.40864 0.05726 -221.91 445.82 447.75 0.2470 

DR 22.8734 1.60190 -259.23 520.46 522.39 0.3780 

DIR 3.74690 0.59100 -306.05 614.10 616.03 0.5470 

DBHD 0.99905 0.00443 -249.99 501.97 503.91 0.5100 

 

 

Figure 5: Density plot for the deaths due to COVID-19 in Pakistan 



Int. J. Anal. Appl. 19 (5) (2021) 706 

 

 

Figure 6: Density plot for the deaths due to COVID-19 in India 

 

6. CONCLUSION  

In this study, a new one-parameter discrete inverted Topp-Leone distribution has been proposed 

using a survival discretizing approach. Some structural properties of the proposed distribution 

are discussed. Different estimation methods including maximum likelihood estimation, least-

square and weighted least squares estimation, and Cramer Von-Mises estimation have been 

presented. We carried out a simulation study to illustrate the performance of the parameter by 

different methods. Among all methods, the maximum likelihood estimator performs better for a 

large sample size. Application of the proposed DITL distribution in the analysis of two discrete 

data sets about Covid-19 has been presented. The DITL distribution, which has flexible features, 

is expected to make a significant contribution to the field of count data modeling. 

 

Acknowledgement: Authors would like to express their gratitude to King Khalid University, 

Saudi Arabia for providing administrative and technical support. 

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the 

publication of this paper. 

 

  



Int. J. Anal. Appl. 19 (5) (2021) 707 

 

REFERENCES 

[1] M. Ahsan-ul-Haq, M. Ahmed, J. Zafar, and P. L. Ramos, Modeling of COVID‑19 Cases in Pakistan 

Using Lifetime, Ann. Data Sci. (2021), https://doi.org/10.1007/s40745-021-00338-9.  

[2] A. A. Al-Babtain, Abdul Hadi N. Ahmed, and A. Z. Afify, A New Discrete Analog of the Continuous 

Lindley Distribution, with Reliability Applications, Entropy, 22(6) (2020), 603. 

[3] E. M. Almetwally, H. M. Almongy and H. A. Saleh, Managing risk of spreading “COVID-19” in Egypt: 

Modelling using a discrete Marshall-Olkin generalized exponential distribution, Int. J. Probab. Stat. 9(2) 

(2020), 33-41.  

[4] E. M. Almetwally, and G. M. Ibrahim, Discrete Alpha Power Inverse Lomax Distribution with 

Application of COVID-19 Data, Int. J. Appl. Math. 9(6) (2020), 11-22.  

[5] B. Milan, Estimation of the Final Size of the Second Phase of the Coronavirus COVID 19 Epidemic by 

the Logistic Model. MedRxiv (Preprint), (2020b), https://doi.org/10.1101/2020.03.11.20024901. 

[6] I. Nesteruk, Statistics Based Predictions of Coronavirus 2019-NCoV Spreading in Mainland China, 

MedRxiv, (2020), https://doi.org/10.1101/2020.02.12.20021931.  

[7] B. Milan, Estimation of the Final Size of the COVID-19 Epidemic. MedRxiv, (2020), 

https://doi.org/10.1101/2020.02.16.20023606. 

[8] M. Maleki, M. R. Mahmoudi, D. Wraith, and K. Pho, Time series modelling to forecast the confirmed 

and recovered cases of COVID-19, Travel Med. Infect. Dis.  37 (2020), 101742.  

[9] G. M. Ibrahim, and E. M. Almetwally, Discrete Marshall-Olkin Lomax Distribution Application of 

COVID-19, Biomed. J. Sci. Techn. Res. 32(5) (2021), 25381-25390.  

[10] S. Chakraborty, Generating Discrete Analogues of Continuous Probability Distributions-A Survey of 

Methods and Constructions, J. Stat. Distrib. Appl. 2 (2015), 6. 

[11] T. Nakagawa, and S. Osaki, The discrete Weibull distribution. IEEE Trans. Reliab. 24(5) (1975), 300-301. 

[12] Roy, Dilip. Discrete Rayleigh distribution, IEEE Trans. Reliab. 53(2) (2004), 255-260.  

[13] H. Krishna, and P. S. Pundir, Discrete Burr and Discrete Pareto Distributions, Stat. Methodol. 6(2) (2009), 

177–88.  

[14] M. A. Jazi, D. L. Chin, and H. A. Mohammad, A Discrete Inverse Weibull Distribution and Estimation 

of Its Parameters. Stat. Methodol. 7(2) (2010), 121–32.  

[15] E. Gómez-Déniz, and C. Enrique, The Discrete Lindley Distribution: Properties and Applications, J. 

Stat. Comput. Simul. 81(11) (2010), 1405–16.  

[16] M. H. Alamatsaz, S. Dey, T. Dey and S. S. Harandi, Discrete Generalized Rayleigh Distribution, Pak. J. 

Stat. 32(1) (2016), 1-20. 



Int. J. Anal. Appl. 19 (5) (2021) 708 

 

[17] E. Altun, M. El-Morshedy, and M. S. Eliwa, A Study on Discrete Bilal Distribution With Properties and 

Applications on integer-valued Autoregressive Process, Revstat. Stat. J. 18 (2020), 70-99. 

[18] M. Shafqat, S. Ali, I. Shah and S. Dey, Univariate Discrete Nadarajah and Haghighi Distribution: 

Properties and Different Methods of Estimation, Statistica, 80(3) (2020), 301-330.  

[19] M. El-Morshedy, M. S. Eliwa, and E. Altun, Discrete Burr-Hatke Distribution with Properties, 

Estimation Methods and Regression Model, IEEE Access, 8 (2020), 74359–70. 

[20] A. S. Hassan, M. Elgarhy and R. Ragab, Statistical properties and estimation of inverted Topp-Leone 

distribution, J. Stat. Appl. Probab. 9(2) (2020), 319-331.  

[21] T. Hussain, and M. Ahmad, Discrete Inverse Rayleigh Distribution, Pak. J. Stat. 30(2) (2014), 203-222.