The new discrete distribution with application to COVID-19 Data.

This research aims to model the COVID-19 in different countries, including Italy, Puerto Rico, and Singapore. Due to the great applicability of the discrete distributions in analyzing count data, we model a new novel discrete distribution by using the survival discretization method. Because of importance Marshall-Olkin family and the inverse Toppe-Leone distribution, both of them were used to introduce a new discrete distribution called Marshall-Olkin inverse Toppe-Leone distribution, this new distribution namely the new discrete distribution called discrete Marshall-Olkin Inverse Toppe-Leone (DMOITL). This new model possesses only two parameters, also many properties have been obtained such as reliability measures and moment functions. The classical method as likelihood method and Bayesian estimation methods are applied to estimate the unknown parameters of DMOITL distributions. The Monte-Carlo simulation procedure is carried out to compare the maximum likelihood and Bayesian estimation methods. The highest posterior density (HPD) confidence intervals are used to discuss credible confidence intervals of parameters of new discrete distribution for the results of the Markov Chain Monte Carlo technique (MCMC).

Introduction

Corona viruses are a huge family of viruses that can cause a variety of diseases varying from the common cold to much more serious conditions such as Middle East Respiratory Syndrome (MERS) and Severe Acute Respiratory Syndrome (SARS). In Wuhan, China, a new Coronavirus (COVID-19) was discovered in 2019. This is an extremely new coronavirus that has not been found in people before. The coronavirus disease 2019 (COVID-19) has been declared a pandemic by the World Health Organization (WHO). To stop the virus from spreading further, a concerted global effort is required. A pandemic affected a wide geographic area and affecting an exceptionally high proportion of the population”. The H1N1 flu pandemic in 2009 is the last pandemic reported in the world.

There are numerous scientists that examined the pandemic Covid-19 and created models to match the data and offer predictions about the projected number of cases to aid the nations to make choices about prevention strategies. For example, see El-Morshedy et al. [1] he presented a new discrete distribution, a discrete generalized Lindley, for analyzing everyday coronavirus infections in Hong Kong and daily new fatalities in Iran. Maleki et al. [2] he predicted recovered and verified COVID19 cases using an autoregressive time series model based on the two-piece scale mixture normal distribution. Nesteruk [3] and Batista [4] they studied the daily new COVID-19 cases in China were anticipated using a mathematical model dubbed susceptible, infected, and recovered (SIR). Almongy et al. [5] introduced a new modeling of the COVID-19 mortality rates in Italy, Mexico, and the Netherlands. Liu et al. [6] discussed new modeling of the survival times for the COVID-19 patients in China.

By using the inverse transformation to random variables, we proposed the inverse distributions. These distributions display different features in the behavior of the density and hazard rate shapes. Many authors discussed the inverted distributions and their applications. Some of the well-known inverted models are inverse Weibull distribution (Calabria and Pulcini [7], Muhammed and Almetwally [8], [9]), inverted Topp–Leone (ITL) (Hassan et al. [10], Almetwally et al. [11], Hassan et al. [12] and Almetwally [13]) among others. Hassan et al. [10] proposed the ITL with CDF given by where is the shape parameter. The probability mass function (PMF) related to Eq. (2) is given by

We utilize discrete distributions in countable data analysis since most existing continuous distributions do not produce appropriate results for modeling COVID-19 cases, and counts of deaths or daily new cases exhibit significant dispersion. The survival discretization method is the most often used method for generating discrete distributions, and it necessitates the presence of a cumulative distribution function (CDF). Time is divided into unit intervals, and the survival function should be continuous and non-negative. Roy [14] defines the discrete distribution PMF as follows: Where , where is a continuous distribution CDF and is a parameter vector. If the CDF of the random variable has , it is considered to have a discrete distribution. The hazard rate is given by . The discrete distribution’s reversed failure rate is given as .

Discrete Burr type XII and discrete Lomax distributions were proposed by Para, and Jan [15]. Discrete data with heavy tails can be modeled using Discrete Lomax(DL) distribution. Nakagawa and Osaki [16] proposed the discrete Weibull (DW) model, Krishna, and Pundir [17] introduced the discrete Buur (DB) model, Gómez-Déniz and Calderín-Ojeda [18] introduced discrete Lindley (DL), Nekoukhou et al. [19] suggested discrete generalized exponential (DGEx), Al-Babtain et al. [20] introduced the natural discrete Lindley (NDL), and Eliwa et al. [21] introduced the discrete Gompertz Exponential (DGzEx). Gillariose et al. [22] introduced a discrete Weibull Marshall–Olkin exponential distribution. Almetwally et al. [23] introduced Discrete Marshall–Olkin generalized exponential distribution.

Marshal and Olkin [24] introduced a novel technique for adding a new parameter to an existing distribution, resulting in a new distribution known as the Marshall–Olkin(MO) extended distribution. This new distribution includes the original distribution as a unique feature and gives the model more flexibility. Sankaran and Jayakumar [25] have presented a detailed analysis on the physical interpretation of the MO family. Let denote the survivor function of a continuous random variable X. The MO extended distribution has a survival function if is the density function connected to the cumulative distribution function (CDF) F(x). where is survival function, , indicate the survival function (S) of a baseline model.

Because , , is a special case of . The probability mass function (PMF) for Eq. (4) has the following shape:

Our aim is to introduce discrete Marshal Olkin inverted Topp–Leone (DMOITL) and use this distribution to model the Covid-19 data from different countries. We made point estimation of the unknown parameters by using the maximum likelihood estimation method and Bayesian estimation. The HPD Intervals are used to discuss credible confidence intervals of parameters of new discrete distribution for the results of the MCMC. We computed the confidence intervals (CI) for the DMOITL distribution’s unknown parameters using asymptotic confidence intervals (ACI) as well.

The rest of this study is organized as follows. In Section ‘DMOITL distribution’. We define DMOITL distribution. In Section ‘Statistical Properties’, we introduce the statistical properties of DMOITL distribution. The Two parameters of the distribution were estimated by two classical and Bayesian point estimation methods in Section ‘Parameter estimation’. While Section ‘Confidence intervals’ is concerned with the interval estimation methods. In Section ‘Simulation analysis’ we made a simulation study to compare the performance of the estimating approaches. Three real data sets from COVID-19 in different countries, including Italy, Puerto Rico, and Singapore, are used in Section ‘Data analysis’ to prove the efficiency of the DMOITL distribution with respect to other distributions. Finally, conclusions and major findings are given in Section ‘Conclusion’.

DMOITL distribution

In this part, we introduced the Marshall–Olkin inverted Topp–Leone (MOITL) distribution and converted this new continuous distribution to discrete distribution as discrete MOITL (DMOITL) distribution.

By using Eqs. (4), and Eq. (1), the survival function of MOITL distribution can obtained and written as follows: where is defined as a vector parameters of MOITL distribution , and . The DMOITL distribution is obtained based on survival discretization method. Eq. (6) is used as the survival function of a baseline MOITL model using the parameter vector . As a result, the CDF of the DMOITL distribution is: The corresponding PMF of Eq. (7) is defined by where is positive vector parameters.  DMOITL() indicates the random variable with PMF (8).

Fig. 1 is a graphical representation for various shapes of the PMF of the DMOITL distribution. These figures show that the PMF of the DMOITL distribution can be right-skewed, symmetric, or decreasing curves. The DMOITL distribution, as seen in the application section, has a lot of versatility and can be used to simulate skewed data. Therefore it is extensively utilized in fields like biomedical studies, biology, dependability, physical engineering, and survival analysis.

Fig. 1

PMF of DMOITL distribution.

Sub-models of the DMOITL model for selected values of the parameters are presented as: If , the DITL distribution with the PMF, and the CDF of the DITL distribution is given by:

Statistical properties

The DMOITL distribution’s reliability measures, moments, and moment generating function (MGF) are shown here.

Reliability measures

The hazard rate function (HR) of the DMOITL distribution are given by The survival functions of DMOITL is given as

There are some important shapes of the HR of the DMOITL distribution in Figs. 2. The HR of the DMOITL distribution has some important shapes, containing decreasing, and upside down curve, which are appealing features for various count models.

Fig. 2

HRF of DMOITL distribution.

The reverse hazard function of DMOITL is given as The second rate of failure (srf) of DMOITL is

th-moment function

The non-central th-moment of DMOITL distribution can be derived using Eq. (8) as follows: In particular, the mean of DMOITL distribution is The variance of DMOITL distribution is given as The dispersion index (DI) may be determined with the help of the following expression:

The skewness value (SKV) for DMOITL distribution, can be positive, zero, negative, or undefined. It can be expressed in terms of the third raw moment: The kurtosis value (KTV) for DMOITL distribution can be expressed in terms of the four raw moment:

From Table 1, it is apparent that the mean, , variance, DI, , , SKV, and KTV of the DMOITL distribution with different parameters and .

Table 1

Different measures by moment function of DMOITL distribution.

α	ϑ	Mean	μ2′	Var	DI	μ3′	μ4′	SKV	KTV
0.5	0.6	39.900	3.94E＋04	3.77E＋04	965.629	5.26E＋07	7.22E＋10	6.544	44.993
	0.9	7.120	555.720	505.026	72.378	77002.960	1.16E＋07	5.803	37.695
	1.5	2.140	21.020	16.440	7.839	408.820	9692.300	4.402	24.816
	3	0.840	2.000	1.294	1.572	7.320	34.640	2.353	10.157
	5	0.460	0.660	0.448	0.995	1.180	2.580	1.545	5.533
1.5	0.6	233.460	1.46E＋06	1.40E＋06	6160.934	1.20E＋10	1.01E＋14	6.570	45.251
	0.9	22.800	6193.040	5673.2	253.903	2.90E＋06	1.47E＋09	5.851	38.156
	1.5	4.780	96.660	73.811	15.757	3905.260	194472.180	4.317	24.135
	3	1.520	5.040	2.729	1.832	26.720	184.560	2.386	10.194
	5	0.880	1.480	0.705	0.818	3.280	9.160	1.241	5.407
3	0.6	728.260	1.46E＋07	1.40E＋07	1.97E＋04	3.78E＋11	1.00E＋16	6.577	45.315
	0.9	48.300	2.87E＋04	2.64E＋04	557.764	2.90E＋07	3.18E＋10	5.853	38.174
	1.5	7.760	249.360	189.14	24.871	16071.560	1278495.840	4.306	24.007
	3	2.160	9.240	4.574	2.161	63.480	576.840	2.429	10.596
	5	1.160	2.280	0.934	0.822	6.080	20.760	1.403	6.328

Parameter estimation

Point estimation is a very important and critical estimation method, in this section, we will apply both classical and non-classical methods of estimation. First, we will apply the maximum likelihood estimation (MLE), and then we will apply the Bayesian estimation method.

Maximum likelihood method

Now we are talking about the first classical method which is the MLE. Let be a random sample of size from the DMOITL distribution. The log-likelihood equation of the vector are given by By differentiating Eq. (17), we can acquire the non-linear likelihood equations with respect to the parameters , and , respectively: and where and .

We use a nonlinear optimization algorithm like the Newton Raphson method because these equations are cannot be solved explicitly.

Bayesian estimation

Bayesian estimation is one of the most important and accurate methods of estimation. In Bayesian estimation the parameters is considered as a random variable that is distributed with a certain distribution. We assign a prior believe about the parameter by using a prior distribution for the two parameters. The capacity to integrate previous information into study helps make the Bayesian technique very valuable for reliability assessment, since one of the primary challenges involved with reliability analysis is data scarcity. For the and parameters of DMOITL distribution are distributed with gamma prior distributions, where and are non-negative values. The and parameters as independent joint prior density functions can be expressed as follows: The joint posterior density function of is derived from likelihood function of DMOITL distribution and joint prior density (20). Under the symmetric loss functions, most of the Bayesian inference procedures have been developed squared-error loss function is commonly symmetric loss function. The Bayes estimators of , say based on squared error loss function is given by and It is noticed that the integrals are given by (22), (23) are not possible to derive explicitly. As a consequence, we estimate the value of integrals in (22), (23) using the Markov Chain Monte Carlo (MCMC) approach. Many studies used MCMC techniques such as Almetwally et al. [26], [27], Basheer et al. [28], Almongy et al. [5], [29], and Bantan et al. [30]. For more reading about Covid papers see [31].

Gibbs sampling and the more generic Metropolis within Gibbs samplers are significant sub classes of Markov chain Monte Carlo (MCMC) techniques. The Metropolis–Hastings (MH) and Gibbs sampling techniques are the two most often used instances of the MCMC method. The MH method, like acceptance–rejection sampling, thinks that a candidate value from a proposal distribution can be produced for each iteration of the algorithm. The MH algorithm, similar to acceptance–rejection sampling, believes that for each iteration of the algorithm, a candidate value from a proposal distribution can be produced. To generate random samples of conditional posterior densities from the DMOITL distribution, we employ the MH within the Gibbs sampling steps: and

Confidence intervals

In this section, we introduce the construction of confidence intervals with two different methods to estimate the unknown parameters of the DMOITL distribution, which are asymptotic confidence interval (ACI) in MLE and credible confidence interval in MCMC of , and .

Asymptotic confidence intervals

Using the asymptotic normal distribution of the MLE is the most popular method to set confidence bounds for the parameters. Fisher information matrix is constructed of the negative second derivatives of the natural logarithm of the likelihood function evaluated at in connection to the asymptotic variance–covariance matrix of the MLE of the parameters. Suppose the asymptotic variance–covariance matrix of the parameter vector is where

Highest posterior density

This method is similar to the ACI for more information see Chen and Shao [32]

The HPD intervals: Chen and Shao [32] discussed this technique to generate the HPD intervals of unknown parameters of the benefit distribution. In this study, samples drawn with the proposed MH algorithm should be used to generate time-lapse estimates. For example, using the MCMC sampling outputs and the percentile tail points, a HPD interval with two points for 2th parameters of the DMOITL distribution can be generated. According to [32], the BCIs of the parameters of DMOITL distribution can be obtained through the following steps:

Arrange , and as and , where denotes the length of the generated of MH algorithm.

The symmetric credible intervals of are obtained as: and .

Simulation analysis

In this part of the paper, we made a simulation study to assess the performance of the distribution by varying the values of the actual values for both parameters and observing the effect of this change on the accuracy of estimation for both methods. The Monte-Carlo simulation process is used in this section to compare the conventional estimation methods: MLE, and Bayesian estimation methods under the square error loss function based on MCMC, for the estimation of DMOITL distribution parameters by R software. Monte-Carlo experiments are carried out on the basis of 10000 randomly generated DMOITL distribution samples, where represents the DMOITL for various parameter actual values such as: Case 1: with different . Case 2: with different . Case 3: with different , and different sample sizes , and ).

Concluding remarks on the simulation results

The Table 2, Table 3, Table 4 summarize the simulation findings for the methodologies provided in this work for estimating parameters of the DMOITL distribution using point estimate and interval estimation. . It is necessary to compare the different point estimating approaches by calculating the Bias, MSE, and lower and higher confidence intervals. These tables let the following conclusions to be drawn:

Table 2

This table contains the simulation results when .

α=0.5			MLE					Bayesian
ϑ	n		Bias	MSE	Lower	Upper	CP	Bias	MSE	Lower	Upper
0.3	20	α	0.8620	2.4017	0.0178	3.9020	97.53%	0.2985	0.2243	0.2859	1.3868
		ϑ	0.1935	0.1140	0.0522	1.0392	97.53%	0.0815	0.0207	0.1684	0.5787
	50	α	0.4020	0.3688	0.0071	1.7968	95.35%	0.2370	0.1358	0.4226	1.3368
		ϑ	0.1023	0.0263	0.1546	0.6500	94.77%	0.0508	0.0084	0.2506	0.5054
	100	α	0.2920	0.1576	0.2581	1.3258	97.56%	0.1413	0.0392	0.4913	0.9333
		ϑ	0.0813	0.0151	0.1980	0.5647	97.56%	0.0317	0.0030	0.2683	0.4231
0.6	20	α	1.3819	4.9814	0.0555	5.3188	94.40%	0.3939	0.3226	0.3111	1.7934
		ϑ	0.4590	0.5431	0.0717	2.1897	94.50%	0.1344	0.0733	0.3954	1.2206
	50	α	0.7927	1.2555	0.1260	2.8456	95.50%	0.4212	0.3186	0.4389	1.6293
		ϑ	0.2909	0.1878	0.2608	1.5209	95.50%	0.1298	0.0488	0.4668	1.0960
	100	α	0.6414	0.6138	0.2592	2.0237	95.70%	0.3738	0.2057	0.4863	1.4197
		ϑ	0.2505	0.1073	0.4368	1.2642	95.60%	0.1156	0.0289	0.5192	0.9901
1.5	20	α	1.8144	5.1585	0.0383	6.1462	94.82%	0.5440	0.5169	0.3543	1.9763
		ϑ	0.9485	3.0928	0.2920	5.6788	95.30%	0.2369	0.3062	0.9455	2.7790
	50	α	1.0254	4.2201	0.0722	6.2294	95.00%	0.7297	0.7411	0.5189	2.1631
		ϑ	0.8153	2.0490	0.9912	4.3157	96.00%	0.3279	0.2524	1.2299	2.6062
	100	α	0.8617	4.4798	0.3870	4.3363	95.30%	0.8238	0.8336	0.7096	2.1376
		ϑ	0.7035	1.3941	1.4189	3.6503	94.40%	0.3886	0.2402	1.3436	2.4744

Table 3

This table contains the simulation results when .

α=1.5			MLE					Bayesian
ϑ	n		Bias	MSE	Lower	Upper	CP	Bias	MSE	Lower	Upper
0.3	20	α	0.3343	0.8797	0.1113	3.5572	97.58%	0.2802	0.6264	1.1133	4.5505
		ϑ	0.0330	0.0142	0.1083	0.5578	96.36%	0.0413	0.0106	0.2401	0.5078
	50	α	0.3167	0.4993	0.5632	3.0702	97.56%	0.2771	0.3696	0.8375	3.0974
		ϑ	0.0152	0.0059	0.1653	0.4650	92.68%	0.0068	0.0052	0.2494	0.4362
	100	α	0.2047	0.2058	0.9010	2.5083	97.50%	0.1360	0.1084	0.9815	2.4394
		ϑ	0.0188	0.0025	0.2266	0.4110	92.50%	0.0133	0.0023	0.2689	0.4068
0.6	20	α	1.3934	4.6094	0.1310	6.0964	96.50%	0.8600	1.8300	1.0674	4.5782
		ϑ	0.1828	0.1063	0.2533	1.3123	96.60%	0.0966	0.0353	0.4543	1.0085
	50	α	0.8413	1.5968	0.4923	4.1903	95.60%	0.6660	1.0122	1.2146	3.5180
		ϑ	0.1052	0.0363	0.3940	1.0164	94.20%	0.0632	0.0128	0.5188	0.8513
	100	α	0.6879	0.9146	0.8852	3.4907	94.70%	0.4863	0.6577	1.3120	2.7546
		ϑ	0.0864	0.0200	0.4669	0.9060	94.90%	0.0446	0.0060	0.5317	0.7638
1.5	20	α	2.4625	6.4564	0.1744	10.6688	95.60%	1.3049	3.1513	1.1146	5.2440
		ϑ	0.6681	0.9558	0.7683	3.5678	95.90%	0.2077	0.1706	1.1420	2.4016
	50	α	1.7995	4.1735	0.2161	8.3829	94.50%	1.4620	3.2965	1.4594	5.2826
		ϑ	0.5615	0.5397	1.1325	2.9904	94.00%	0.2388	0.1269	1.3017	2.2931
	100	α	1.1558	3.8434	1.3621	6.7541	95.80%	1.4137	2.6133	1.7603	4.5595
		ϑ	0.5339	0.3926	1.3907	2.6770	95.50%	0.2382	0.0965	1.4013	2.1529

Table 4

This table contains the simulation results when .

α=3			MLE					Bayesian
ϑ	n		Bias	MSE	Lower	Upper	CP	Bias	MSE	Lower	Upper
0.3	20	α	0.1802	0.6961	1.5780	4.7825	93.06%	0.1463	0.5937	2.0143	4.4308
		ϑ	0.0134	0.0051	0.1750	0.4519	96.53%	0.0131	0.0042	0.2641	0.4025
	50	α	0.0123	0.3528	1.8278	4.1968	93.10%	0.0143	0.3097	2.0341	4.0315
		ϑ	0.0117	0.0017	0.2324	0.3910	93.10%	−0.0006	0.0014	0.2802	0.3994
	100	α	0.1360	0.1126	2.5247	3.7473	93.55%	0.0935	0.1034	2.5911	3.7802
		ϑ	0.0032	0.0006	0.2555	0.3510	96.77%	0.0017	0.0005	0.2850	0.3773
0.6	20	α	0.8831	3.5314	0.6303	7.1359	96.60%	0.7310	2.7925	2.0505	6.8045
		ϑ	0.0556	0.0296	0.3363	0.9750	96.50%	0.0550	0.0158	0.4897	0.8786
	50	α	0.3696	0.7566	1.8255	4.9136	93.80%	0.2795	0.5826	2.3802	5.7343
		ϑ	0.0227	0.0084	0.4481	0.7973	95.00%	0.0313	0.0049	0.5146	0.7578
	100	α	0.5062	0.7676	2.1039	4.9085	95.30%	0.4280	0.4437	2.5958	4.8926
		ϑ	0.0321	0.0055	0.5005	0.7636	94.70%	0.0254	0.0048	0.5556	0.7244
1.5	20	α	1.5541	4.0322	0.8703	8.2378	96.70%	1.0166	2.7427	2.0872	7.7519
		ϑ	0.3843	0.3551	0.9914	2.7773	95.80%	0.1754	0.1075	1.2135	2.2337
	50	α	1.0485	3.8009	2.1835	6.7862	97.50%	0.9916	2.0432	2.7603	6.0531
		ϑ	0.2765	0.1541	1.2300	2.3230	94.80%	0.1581	0.0625	1.3345	2.0548
	100	α	0.9740	1.5538	2.9366	5.5441	94.90%	0.8126	0.9637	2.9286	4.0197
		ϑ	0.2935	0.1305	1.3804	2.2066	94.50%	0.1427	0.0390	1.4226	1.9322

As rises, the Bias and MSE of the DMOITL distribution drop.

Bias and MSE for and parameters grow as increases.

As the value of grows, the Bias and MSE values for the and parameters decrease.

Bayesian estimation is the best approach for estimating the parameters as it provides the smallest MSE and Bias and also has the shorties confidence interval

Using Bayesian estimation, the MLE ACI confidence interval for parameters of the DMOITL distribution has the smallest confidence interval.

Data analysis

In this part of the paper, we used two real data sets as an application on the superiority of the distribution The DMOITL distribution is fitted to more notable fields of Covid-19 with diverse countries such as Italy, Puerto Rico, and Singapore in this part. We compare the fits of the discrete Buur (DB) [Krishna and Pundir [17]] model, discrete Weibull (DW) [Nakagawa and Osaki [16]], discrete inverse Weibull (DIW) [Jazi et al. [33]], Poisson, negative binomial (NB), discrete alpha power inverse Lomax (DAPIL) [Almetwally and Ibrahim [34]], discrete Lindley (DLi) [Gómez-Déniz and Calderín-Ojeda [18]], and DITL models in Table 5, Table 6, Table 7.

Table 5

MLE, CvM, AD, KS and AIC for different alternative models of DMOITL distribution: Puerto Rico.

	α	ϑ	λ	CvM	AD	KS	AIC
DMOITL	116654.6968	2.5126		0.0711	0.4019	0.0758	487.0636
DBuur	16.5248	0.9886		0.1237	0.6821	0.4977	607.7753
DW	0.9999	1.5876		12.5759	76.0974	0.9937	487.1050
DIW	0.0000	0.8312		0.1745	0.9684	1.0000	510.3790
NB	0.1339			0.1951	0.5969	0.3400	884.7561
Poisson			245.8474	0.3611	0.9437	0.4713	4067.0956
DAPL	1.3874	1.4894	3.145E−25	0.1771	0.9839	0.2582	512.5820
DITL	0.2175			0.1270	0.7001	0.4852	596.0123

Table 6

MLE, CvM, AD, KS and AIC for different alternative models of DMOITL distribution: Italy.

	α	ϑ	λ	CvM	AD	KS	AIC
DMOITL	3065.8285	3.3396		0.0681	0.3680	0.0715	475.1201
DBuur	16.2005	0.9795		0.1821	1.0424	0.4526	623.6174
DW	0.9983	1.9497		19.9308	121.8355	0.9900	476.0156
DIW	0.0000	1.4558		0.2991	1.7367	1.0000	495.8519
NB	0.7295			0.0846	0.3810	0.2351	608.7627
Poisson	22.6230	22.6230		0.0918	0.4535	0.2674	700.5536
DAPL	0.0020	1.4589	1.08E−13	0.2188	1.2647	0.1265	486.1749
Dli	0.9202			0.0940	0.4254	0.1464	481.7783
DITL	0.4203			0.1592	0.9048	0.4320	604.7508

Table 7

MLE, CvM, AD, KS and AIC for different alternative models of DMOITL distribution: Singapore.

	α	ϑ	λ	CvM	AD	KS	AIC
DMOITL	1013.6834	3.0476		0.1607	0.9056	0.0792	1853.6278
DBuur	93.7454	0.9963		0.3048	2.0831	0.4334	2376.1589
DW	0.9958	1.7313		80.6484	483.5314	0.992744	1860.2878
DIW	3.52E−18	1.45944		0.631332	4.141622	1	1921.321
NB	0.9222366			0.32625	1.948125	0.31406	2790.6446
Poisson		20.40545		0.356531	2.158016	0.323133	2921.4866
DAPL	0.0037	2.4879	2.93E−07	0.3847	2.6199	0.0847	1879.7269
Dli	0.9124			0.2174	0.9964	0.1324	1870.9463
DITL	0.4421			0.2266	1.5589	0.4168	2311.8193

Table 5, Table 6, Table 7 provide values of Cramér–von Mises (CvM), Anderson–Darling (AD), Kolmogorov–Smirnov (KS) and Akaike information criterion (AIC) statistics for the all models fitted based on three real data sets. These tables also include the MLE of the parameters for the models under consideration. Fig. 3, Fig. 5, Fig. 7 show the fitted DMOITL, PMF, CDF, PP-plot, and QQ-plot of the three data sets, respectively. These statistics show that among all fitted models, the DMOITL distribution has the lowest CvM, AD, KS, and AIC values. Using alternative data, Table 8 presented MLE and Bayesian estimation methods for parameters of the DMOITL distribution. Figs. 4, 6, 8 show convergence plots of MCMC for parameter estimates of DMOITL distribution for different data set.

Fig. 3

Plots of estimated pmfs of distributions for Data set of Puerto Rico.

Fig. 5

Plots of estimated pmfs of distributions for Data set of Italy.

Fig. 7

Plots of estimated pmfs of distributions for Data set of Singapore.

Table 8

MLE and Bayesian estimation method for parameters of DMOITL distribution using different data.

		MSE		Bayesian
		estimate	SE	estimate	SE
Puerto Rico	α	168573.6494	0.0079	168573.6490	0.0044
	ϑ	2.6321	0.0636	2.3089	0.0492
Italy	α	3065.8285	0.0011	3065.8281	0.0010
	ϑ	3.3396	0.0887	3.3342	0.0116
Singapore	α	1013.6834	0.0188	1013.6817	0.0159
	ϑ	3.0476	0.0473	3.0455	0.0163

Fig. 4

Convergence plots of MCMC for parameter estimates of DMOITL distribution for data set of Puerto Rico.

Fig. 6

Convergence plots of MCMC for parameter estimates of DMOITL distribution for data set of Italy.

Fig. 8

Convergence plots of MCMC for parameter estimates of DMOITL distribution for data set of Singapore.

Firstly: This is a COVID-19 data set from Puerto Rico that spans 38 days, from February 26 to April 4, 2021. This data set is comprised of newly reported instances on a daily basis. The data are as follows: 100, 311, 114, 253, 287, 151, 30, 102, 199, 261, 305, 185, 120, 68, 46, 356, 160, 235, 193, 216, 67, 69, 332, 212, 330, 295, 227, 145, 78, 260, 399, 268, 595, 447, 170, 365, 510, 881.

Secondly: This is a 61-day COVID-19 data set from Italy, recorded between 13 June and 12 August 2021. This data set is comprised of newly reported instances on a daily basis. The data are as follows: 52, 26, 36, 63, 52, 37, 35, 28, 17, 21, 31, 30, 10, 56, 40, 14, 28, 42, 24, 21, 28, 22, 12, 31, 24, 14, 13, 25, 12, 7, 13, 20, 23, 9, 11, 13, 3, 7, 10, 21, 15, 17, 5, 7, 22, 24, 15, 19, 18, 16, 5, 20, 27, 21, 27, 24, 22, 11, 22, 31, 31.

Thirdly: This is a 242-day COVID-19 data set from Singapore, recorded between 20 November 2020 and 19 July 2021. This data set is comprised of newly reported instances on a daily basis. The data are as follows: 4, 4, 5, 12, 5, 18, 7, 5, 4, 6, 8, 5, 10, 2, 9, 3, 13, 5, 13, 12, 6, 6, 8, 8, 7, 5, 16, 12, 24, 9, 17, 19, 10, 29, 21, 13, 14, 10, 5, 5, 13, 27, 30, 30, 33, 35, 24, 28, 31, 33, 23, 29, 42, 22, 17, 38, 45, 30, 24, 30, 14, 30, 40, 38, 15, 10, 48, 44, 14, 25, 34, 24, 58, 29, 29, 19, 18, 22, 25, 26, 24, 22, 11, 15, 12, 18, 9, 14, 9, 1, 11, 11, 14, 12, 11, 10, 4, 7, 10, 13, 12, 11, 12, 8, 23, 19, 9, 13, 13, 13, 6, 10, 8, 10, 8, 17, 12, 11, 9, 15, 15, 17, 12, 12, 13, 15, 17, 12, 23, 12, 21, 26, 34, 26, 43, 18, 10, 17, 24, 35, 21, 26, 32, 20, 25, 14, 27, 16, 34, 39, 23, 20, 14, 15, 24, 39, 23, 40, 45, 12, 23, 35, 24, 34, 39, 17, 17, 16, 18, 25, 20, 28, 19, 25, 16, 34, 52, 31, 49, 28, 38, 38, 41, 40, 29, 25, 36, 30, 26, 24, 30, 33, 25, 23, 18, 31, 45, 13, 18, 20, 14, 9, 4, 13, 9, 18, 13, 25, 14, 24, 27, 16, 21, 11, 16, 18, 22, 23, 20, 17, 14, 9, 10, 16, 10, 10, 7, 11, 13, 10, 12, 16, 10, 6, 8, 26, 26, 60, 48, 61, 68, 92.

Concluding remarks on the data analysis

After applying the three data sets on the proposed distribution and observing the results in Table 5, Table 6, Table 7 that provide values of Cramér–von Mises (CvM), Anderson–Darling (AD), Kolmogorov- Smirnov (KS) and Akaike information criterion (AIC) statistics for the all models fitted based on three real data sets we found that our proposed distribution is the best model as it has the lowest value of AIC and KS values. By referring to these values, we can make sure that our proposed distribution is superior among all its competitors

Existence and uniqueness for the log-likelihood

We sketched the log-likelihood for each parameter as shown in Fig. 9, Fig. 10, Fig. 11 by fixing one parameter and varying the other. The figures show that the three data sets behaves very well, as we can see that the two roots of the parameters are global maximum, and also by differentiating the log-likelihood with respect to each parameters, we found that the function is a decreasing function and it intersects the -axis in a single point which is the root of the parameter, and that assures that the roots are unique

Fig. 9

Existence and uniqueness for the log-likelihood for data set of Puerto Rico.

Fig. 10

Existence and uniqueness for the log-likelihood for data set of Italy.

Fig. 11

Existence and uniqueness for the log-likelihood for data set of Singapore.

Conclusion

In this paper, we introduce Discrete Marshall–Olkin Inverted Topp- Leone distribution which is called DMOITL. We derived its statistical properties. We made the point and interval estimation by classical and Bayesian estimation methods for the DMOITL unknown parameters and . We conducted simulation analysis using the R package to differentiate the performance of different estimation methods. We deduced that the Bayesian method is very efficient than the classical method as it gets more efficient results through the values of the MSE and the length of the confidence interval as it is always shorter and the MSE is always smaller. In order to prove the superiority and applicability of the proposed distribution, we made a data analysis through the COVID-19 data. We used three data sets in three different countries thought different intervals of time, and by referring to the results in Table 5, Table 6, Table 7 that provide values of Cramér–von Mises (CvM), Anderson–Darling (AD), Kolmogorov- Smirnov (KS) and Akaike information criterion (AIC) statistics for the all models fitted based on three real data sets we found that our proposed distribution is the best model as it has the lowest value of AIC and KS values.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.