The Discrete Type-II Half-Logistic Exponential Distribution with Applications to COVID-19 Data

1. College of Statistical & Actuarial Sciences, University of the Punjab, Pakistan; ahsanshani36@gmail.com 2. School of Statistics, Minhaj University Lahore, Pakistan; ayeshababar19@gmail.com 3. Department of Statistics, Lahore College for Women University, Lahore Pakistan; sharqa.hashmi@gmail.com 4. Department of Mathematics, College of Science & Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia; ashalghamedi@kau.edu.sa 5. Department of Statistics, Mathematics and Insurance, Benha University, Benha 13511, Egypt; ahmed.afify@fcom.bu.edu.eg


Introduction
In late 2019, a novel coronavirus disease (COVID- 19) was first reported in China and has been announced as an epidemic by the World Health Organization (WHO) (Lee et al., 2020). The epidemic has mostly been controlled in China since March 2020 but continues to inflict public health and socioeconomic situations in all other countries of the world. One of the major reasons for controlling the disease is China's strategy of effective use of its health care system and publicity of awareness programs among people which played a vital role in the control of the COVID-19 pandemic. However, the major source of its rapid spread is human-to-human contact.
It is well recognized that the life duration in the real world is related to continuous non-negative lifetime distributions. However, it is sometimes uneasy to obtain the samples from a continuous distribution. The observed data obtained are discrete because they are usually measured at only a finite number of decimal places and can not assume all points within an interval. When measures are taken on a continuous (ratio or interval) scale, the discrete distributions are more appropriate for such observations. Therefore, it is rational to assume that these observations are from a discretized distribution which is constructed from a continuous distribution same (Chakraborty, 2015).
During the last few decades, many continuous lifetime distributions have been proposed and studied. However, research work on discrete distributions is not widely addressed comparatively to continuous distributions. The discretization of continuous lifetime models has been applied to derive discrete lifetime distributions. The discretization of continuous distributions keeps similar functional form of the survival function (SF), as well as resulting in many reliability properties which remain the same (Nakagawa and Osaki, 1975).
Recently, the methods of generating discrete analogues of continuous distributions have been considered by several authors. For example, the infinite series discretization method (Good, 1953;Kulasekera and Tonkyn, 1992;Kemp,1997;Sato et al., 1999), survival discretization approach has an interesting feature of keeping the original functional form of SF (Nakagawa and Osaki, 1975), hazard function discretization approach (Stein, 1984), compound two-phase method (Chakraborty, 2015), reversed hazard function discretization method (Ghosh et al., 2013).
Some notable recent proposed discrete distributions include discrete Weibull (Nakagawa and Osaki, 1975), discrete skew-laplace , discrete-laplace  The main objective of this article is to provide a new flexible two-parameter discrete model, called the discrete type-II half-logistics exponential (DTIIHLE) distribution using the survival discretization approach. The DTIIHLE distribution can be utilized to model over-dispersed count data sets. Its hazard rate function (HRF) can be decreasing or unimodal. We derive some of its properties in explicit forms such as the quantile function (QF), moments and probability generating function (PGF). The two parameters are estimated via the maximum likelihood (ML) and a simulation study is conducted to explore the performance of the ML estimators. The importance of the newly DTIIHLE distribution is illustrated by analyzing two real-life COVID-19 data sets which represent the number of COVID-19 deaths in Pakistan and Saudi Arabia.
The rest of the article is structured as follows. In Section 2, the DTIIHLE distribution is defined with some plots of its probability mass function (PMF) and HRF. Some properties of the DTIIHLE distribution are provided in Section 3. The ML approach is adopted to estimate the DTIIHLE parameters in Section 4. Simulation results are conducted to explore the behavior of the introduced estimators in Section 5. To validate the use of DTIIHLE distribution in fitting real-life count data, two sets of data from medicine field are fitted in Section 6. Finally, some conclusions are presented in Section 7.

The DTIIHLE Distribution
The SF of type II half-logistic exponential (TIIHLE) distribution (Elgarhy et al., 2019) takes the form where is scale and is the shape parameters.
The probability density function of the TIIHLE distribution is The Discrete Type-II Half-Logistic Exponential Distribution with Applications to COVID-19 Data 923 A discrete analog of any continuous random variable can be obtained using different discretization approaches. A review on such discretization techniques can be explored in Chakraborty (2015). The most common discretization method is the one preserving the functional form of the SF. Let be a continuous random variable (RV) with SF ( ). The corresponding PMF of a discrete RV reduces to To this end, we apply this discretization method of the continuous TIIHLE distribution to generate the corresponding DTIIHLE model which is defined by the PMF where − = and 0 < < 1.
The corresponding cumulative distribution function (CDF), ( ) = ( ≤ ), of DTIIHLE distribution takes the form From Equation (5), one can easily derive the QF as follows The SF and HRF of the DTIIHLE model are given as and The reverse HRF and the second rate of failure of the DTIIHLE model are defined by The recurrence relation which can be used to generate probabilities from the DTIIHLE distribution has the form The PMF plots for different values of its parameters are presented in Figure 1. The HRF plots are displayed in Figure  2. The plots reveal that its PMF can be unimodal, as well as its HRF can be decreasing or unimodal.

The PGF and moments
The Discrete Type-II Half-Logistic Exponential Distribution with Applications to COVID-19 Data 925 The PGF of the DTIIHLE distribution follows as Differentiating G ( ) with respect to and setting = 1, we obtain the mean of DTIIHLE distribution as Again differentiate ′ ( ) with respect to (wrt) and setting = 1, we obtain On differentiating Gʹʹx (Z) wrt Z and setting Z=1, we have On differentiating Gʹʹʹx (Z) wrt Z and putting Z=1, we get Moments about the origin can be calculated using the factorial moments as
The dispersion index (DI) is defined by = σ 2 / . Table 1 shows descriptive measures of the DTIIHLE distribution for different parameter values. One can note that the skewness decreases as the value of the shape parameter increases. If the value of the DI is greater than 1, then the proposed distribution is applicable for over-dispersed data.

Parameter Estimation
Let 1 , 2 , 3 , … , be a random sample of size from the DTIIHLE model. Then, the log-likelihood function is given by The first derivatives wrt and are The ML estimates (MLEs) of and can be obtained using numerical methods.

Simulation Study
A comprehensive simulation study has been conducted by generating 10,000 samples of various sample sizes from the DTIIHLE distribution. Particularly, we generate the samples using the following combination of parameters ( , ) i.e., The average estimates (MLEs), mean square errors (MSEs), and convergence probabilities are listed in Table 2. The MLEs are quite stable and very close to the true values of the parameters. The MLEs are consistent as shown from Table 2.
The MLEs, standard errors (SE) and 95% confidence intervals (C.I.) for the estimates are listed in Tables 3  and 5 for the two data sets, respectively. Some goodness-of-fit measures including log-likelihood (ℓ), AIC, BIC and KS statistic are presented in Tables 4 and 6 for the respective two data sets. From Tables 4 and 6, it is observed that the DTIIHLE distribution outperforms all other fitted models in analyzing number of deaths in Pakistan and Saudi Arabia. It can provide the best fit to the analyzed data among all other competitive distributions. Figures 3 and 4 display the PP plots for all the competitive distributions for the two data and they support the findings in Tables 4 and 6.

Conclusion
In this article, a two-parameter discrete distribution is proposed to model COVID-19 new cases in Pakistan and Saudi Arabia, called the discrete Type-II half-logistics exponential (DTIIHLE) distribution. Several mathematical properties of the DTIIHLE model are discussed. Its parameters have been estimated by using the maximum likelihood approach. A simulation study was carried out to check the performance of parameters based on MSEs and CP. The DTIIHLE model is utilized to model two real-life data sets about the number of COVID-19 deaths in Pakistan and Saudi Arabia due to COVID-19. The newly DTIIHLE model is important to elaborate on the existing discrete distributions in the literature. It has the lowest goodness-of-fit measures values among all discrete competing models. Hence, the proposed model is best among competitive distributions.