key: cord-0710087-bet8g9iq authors: Mohamed, Hanem; Mousa, Salwa A.; Abo-Hussien, Amina E.; Ismail, Magda M. title: Estimation of the Daily Recovery Cases in Egypt for COVID-19 Using Power Odd Generalized Exponential Lomax Distribution date: 2021-09-06 journal: Ann DOI: 10.1007/s40745-021-00336-x sha: 020642b30248286266a71c1a1b88b2b5de9bf496 doc_id: 710087 cord_uid: bet8g9iq Covid-19 has become an important topic this days, because of its bad effect in many fields such as Economics, industrial and commerce. In this paper, Covid-19 will be studied statistically point of view depending on the recovery cases in the Arab Republic of Egypt in the interval of (20 March to 20 August 2020). A power odd generalized exponential Lomax distribution has been considered. Some mathematical properties of the distribution are studied. The method of maximum likelihood and maximum product of spacings are used for estimating the model parameters. Also 95% asymptotic confidence intervals for the estimates of the parameters are derived. A simulation study was conducted to evaluate the numerical behavior of the estimates. The proposed methods are utilized to find estimates of the parameters of power odd generalized exponential Lomax distribution for the recovery cases of corona virus in Egypt. Coronavirus disease is an infectious disease caused by a newly discovered coronavirus. Most people infected with the COVID-19 virus will experience mild to moderate respiratory illness and recover without requiring special treatment. The rest of the paper is unfolded as follows: organized as follows: In Sect. 2 The density function of the POGEL distribution is derived. The main descriptive properties are introduced in Sect. 3 . In Sect. 4 some of the special cases are obtained. The ML and the MPS of the parameters are discussed in Sect. 5. A simulation study is tabulated and discussed in Sect. 6 . A real life application is presented in Sect. 7 while some concluding remarks are given in Sect. 8. [20] introduced an extension of Lindley distribution by using this transformation x = t δ hence; it is of interest to know what would be the distribution of similar power transformation of odd generalized exponential Lomax distribution by using the transformation. Based on the transformation x = t δ family, the proposed distribution is derived by replacing x = t δ in (1) as follows The resulting distribution will be referred to as the power odd generalized exponential Lomax distribution (POGEL). Then the cdf of the distribution is as follows: This Section provides some properties of the POGEL distribution. I. Main properties of the POGEL a. The survival function denoted by S(t) , is given by: (3) , t > 0, , , , > 0, > 1. , t > 0, , , , > 0, > 1. b. The hazard rate function, h(t), is given by: c. The reversed hazard rate function, r(t), is given by: d. The cumulative hazard rate function, H(t), is given by: e. Quantiles and median of the POGEL distribution The quantile function t q , is given by: In particular when q = 0.5 the median of the POGEL distribution is given by: And the Inter-Quantile Range (IQR) which is defined as the difference between the third quartile and the first quartile can be expressed as: f. Some useful expansion for POGEL distribution An expansion for pdf is derived. and substituting from (12) into (3), to get Using binomial expansion of 1 + t − 1 j we obtain Substituting from (14) into (13), we get g. The r th moment is given by: The r th non-central moment of POGEL distribution is given by (11) Then, the r th moment of POGEL under the condition ( j + r + 1) < (k + 1) is derived as follows: Depending on Eq. (16), the basic statistical properties of POGEL are. The mean and the variance, of the POGEL distribution are, respectively, given by: The moment generating function is given by: i. Order statistics Let T (1∶n) , T (2∶n) , … , T (n∶n) denote the order statistics obtained for a random sample t 1 , t 2 , … , t n from POGEL distribution with cdf (4) and pdf (3) . The pdf of r th order statistics is defined by: Using binomial expansion Substituting (3) and (4) in (20) , as follows: In particular, the pdf of the smallest order statistics is obtained by substituting r = 1 in (22) as follows: Also, the pdf of largest order statistics is obtained by substituting r = n in (22) as follows: (21) f r,n (t; ) = 1 B(r, n − r + 1) . (23) The pdf curves of POGEL distribution are plotted in Fig. 1 for some selected values of the parameters. (λ = 3, γ = 3, β = 3, δ = 2, = 2), (λ = 2, γ = 2.5, β = 1.5, δ = 1, θ = 2.5), (λ = 0.3, γ = 0.25, β = 1.2, δ = 1, θ = 1.5), (λ = 1.5, γ = 1.5, β = 2, δ = 3, θ = 1.5). Figure 1 shows that: • The f(t) curves of the POGEL are more flexible for changing values of the parameters. • The f(t) curves take various shapes such as symmetrical, right-skewed, reversed J-shaped and unimodal. The h(t) curves of two POGEL populations are plotted in Fig. 2 . The first population is when < 1, (λ = 0.25, γ = 0.25, β = 2, δ = 1, θ = 1.5), (λ = 0.25, γ = 0.25, β = 1.2, δ = 1, θ = 1.5), The second population is when γ > 1 , (λ = 2, γ = 3, β = 1.3, δ = 2, θ = 3),(λ = 2, γ = 3, β = 0.9, δ = 1.5, θ = 3). Figure 2 show that: • The h(t) curves of the POGEL are more flexible for changing values of the parameters. • The h(t) curves take different shapes such as constant, increasing, decreasing, and reversed J shape. This fact implies that the POGEL can be very useful for fitting data sets with various shapes. The importance of POGEL distribution is that it contains several special cases (sub-models), by using (3) as follows: The odd generalized exponential Lomax (OGEL) introduced by [19] is a special case from POGEL distribution, when = 1 which is given in Eq. (1). (II) Odds generalized exponential power Lomax distribution The odds generalized exponential power Lomax distribution (OGEPL) introduced by [21] when = 1 in (3) is a special case from POGEL with the following pdf: (III) Odd exponential Lomax distribution Odd exponential Lomax distribution (OEL) introduced by [9] when = = 1 in (3) is a special case from POGEL with the following pdf: The exponential distribution (E) with two parameters introduced by [15] when = = = 1 in (3) is a special case from POGEL with the following pdf: (V) Generalized exponential distribution The generalized exponential distribution (GE) introduced by [16] when (3) is a special case from POGEL with the following pdf: In this Section, the ML and the MPS methods are discussed to obtain the estimator of parameters of the POGEL under complete samples. The ML is used to estimate the unknown parameters of the POGEL distribution based on complete samples [22] . Let t 1 , ..., t n be a random sample of size n from POGEL, with parameters = ( , , , , ) , the likelihood function of the density is given by, Then, the log likelihood function, denoted Ln The log-likelihood (30) can be maximized numerically using the R (optim function), for interval estimation of the model parameters, it requires the 4 × 4 observed information matrix I(ω) = for = (λ, γ, β, δ, θ) . Under standard regularity conditions, the multivariate normal N m (ω) 0, (ω) −1 distribution can be used to construct approximate confidence intervals for the parameters. Here, I(̂) is the total observed information matrix evaluated at ̂ . [see "Appendix A"]. One of the most common methods for estimating the parameters of a distribution is the ML method. Although this method is consistent, asymptotically efficient, it was found to be unbounded and inefficient in the estimation in various cases, such as involving certain mixtures of continuous distributions, heavy-tailed distributions and J-shaped distributions [23]. The MPS method was introduced by [24] as an alternative to ML for the estimation of parameters of continuous univariate distributions. The MPS estimators are consistent, asymptotically normal and efficient. Suppose that an ordered random sample t 1 , ..., t n drawn from POGEL distribution with parameters = ( , , , , ) and cdf (8) the spacing is constructed as: To estimate the unknown parameters, the product spacings is defined and the geometric mean of spacings is maximized as follows: Then, by taking the logarithm of G: In this study the maximization of the quantity in (33) is defined as: Substitute (4) in (33) the function H is given by: Taking the partial derivative of (35) with respect to = ( , , , , ) and equating to zero where, [23] Showed that maximizing H as a method of parameter estimation is as efficient as ML estimation, and the MPS estimators are consistent under more general conditions than the ML estimators. The MPS method shows asymptotic properties like the ML estimators [23]. Introduced the variance covariance matrix of the MPS estimators. Therefore, the asymptotic properties of MPS can be used to construct the asymptotic confidence intervals for the parameters [25] . Let I ̂ _ is the observed Fishers information matrix it can be defined as: So on the basis of these derivatives the information matrix I ̂ can be obtained. The approximate (1 − b) 100% confidence intervals for the parameters λ, γ, β, δ and θ are given. In this Section a simulation study is introduced to illustrate the theoretical results considering ML and MPS methods on the basis of generated data from POGEL distribution by taking the parameter θ as known for all methods of estimation. For each method of estimation, initial parameter values and sample sizes, the estimates, mean square error (MSE), relative bias (RB) and asymptotic confidence intervals (ACI) are calculated using the following formulae: (1) (46) Var ̂,̂,̂,̂ . 3.1073) . The results of the simulation study are illustrated in Tables 1 and 2 . From these tables, it is noticeable that: • As expected the MSE, RB and ACI decreased when n increased. • The MSE of the MPS estimates is less than the MSE of the ML estimates for all parameters and sample sizes except for the parameters β and δ at n = 50 in Table 1 and for the parameters λ and β at n = 50 in Table 2 . • The RB of the MPS estimates is less than the RB of the ML estimates for all parameters and sample sizes except for the parameters β at n = 50 in Table 1 and for the parameter λ at n = 50 in Table 2 . • As expected the performance of the MPS estimates is appropriate than the ML estimates. In this Section, the COVID-19 data set is analyzed to illustrate the flexibility of POGEL distribution. The data set is taken from the MINISTRY of HEALTH reports in EGYPT referred to daily recovery cases of a random sample of 154 days in the interval of (20 March to 20 August 2020) of coronavirus patients in the ARAB REPUBLIC of EGYPT these data are given as. Table 3 that the data set is also right-skewed and platykurtic. Figure 3 presents the sequence time for the recovery cases of COVID-19. Figure 3 show that. • The curve take various shapes such as increasing, fixed and highly increasing. • As expected increasing in injuries cases lead to increasing in recovery cases. Before analyzing this data set, the scaled-TTT plot can be used to verify our distribution validity [26] . It allows identifying the shape of the h(t) graphically. The empirical scaled-TTT plot of the COVID-19 data set is shown in Fig. 4 . This Figure indicates that the TTT plot is convex then concave which indicates an upside down bathtub hazard rate. It verifies our distribution validity. This data will be studied from two ways, firstly comparing the two methods of estimation. Secondly illustrate the importance and flexibility of the POGEL distribution with its sub-models [Generalized Exponential (GE), Odd Exponential Lomax. (OEL), Odd Generalized Exponential Lomax (OGEL) and Odd Generalized Exponential Power Lomax (OGEPL) distributions)]. The plot of empirical cdf of the COVID-19 data set is displayed in Fig. 4 for the ML and MPS methods of estimations. Table 4 . Presents the parameter estimation for the ML and MPs methods with its standard errors (SEs). Table 5 give the ML estimates and the corresponding standard errors (SEs) in parentheses of the parameters for all fitted models and the numerical values the Akaike information criterion (AIC), the consistent Akaike information criterion (CAIC), Bayesian Information Criterion (BIC), Anderson Darling (A) and Cramervon Mises (C) for POGEL and its sub models OEPL, OGEL, OEL and GE. For the COVID-19 data set, the -2ln L statistic for GE and OEL against POGEL is (4.48, 6.13) respectively. Therefore, there is a significant difference between GE, OEL and POGEL. Moreover, the values of the statistics AIC, CAIC and BIC are smaller for Fig. 5 The pdf with the value of the estimated parameters and cdfs of the POGEL model and other fitted models for the COVID-19 data set the POGEL distribution which mean that the POGEL distribution is a "appropriate" fit for the COVID-19 data, however a comparison of POGEL, OEPL and OGEL distributions shows that no significant difference between this distributions based on the values of AIC, CAIC and BIC which mean that the POGEL, OEPL and OGEL distributions are slightly appropriate for the COVID-19 data. The fitted cdfs for the COVID-19 data which supported this results are displayed in Fig. 5. In this paper, Covid-19 studied statistically point of view depending on a probability distribution called Power Odd Generalized Exponential-Lomax distribution (POGEL). Some of its statistical properties were introduced. The estimation of the unknown model parameters was done with the maximum likelihood and maximum product spacing methods, with numerical guarantees on their behavior via a simulation study. The POGEL and some of its sub models are good fit for the ARAB REPUBLIC of EGYPT COVID-19 data. From the estimated parameters of the POGEL it can predict that the curve of the daily recovery cases will be decreasing compared to daily cases which is consistent with the curve of the POGEL which takes the reversed J-shape. The POGEL distribution with an increasing but concave hazard rate best describes statistically how the cases respond to treatment. The concave shape is encouraging in the sense that the increase is happening at a decreasing rate. This points to a somewhat success in the interventions by government. The hazard rate is still increasing, and the number of recovery cases has not been stopped yet despite the number of the patients are on the way to decrease. 1. The first-order partial derivatives and the derivation of the elements of the fisher information matrix of ML Calculating the first-order partial derivatives with respect to = ( , , , , ) and equating them to zero, we get the following nonlinear equations: Taking the second partial derivatives for I( , , , , ) as follows: where Funding No funding was involved in the present work. The authors did not receive support from any organization for the submitted work. The authors have no financial or proprietary interests in any material discussed in this article. Data and code availability statement All the data and codes used in this study, as well as, the supplementary material can be made available from the corresponding author, upon reasonable request. Involvement of human participant and animals this article does not contain any studies with animals or humans performed by any of the authors. Information about informed consent No informed consent was required as the studies does not involve any human participant. Ethic statements All authors consciously assure that for the manuscript fulflls the following statements: 1) This material is the authors' own original work, which has not been previously published elsewhere. 2) The paper is not currently being considered for publication elsewhere. 3) The paper refects the authors' own research and analysis in a truthful and complete manner. 4) The paper properly credits the meaningful contributions of co-authors and co-researchers. 5) The results are appropriately placed in the context of prior and existing research. The authors declare no conflict of interest. Monitoring novel corona virus (COVID-19) infections in india by cluster analysis Outbreak prediction of COVID-19 for dense and populated countries using machine learning Introduction to business data mining Optimization based data mining: theory and applications Culture vs policy: more global collaboration to effectively combat COVID-19. The Innovation What are the underlying transmission patterns of COVID-19 outbreak an age-specific social contact characterization Statistical modeling of COVID-19 pandemic stages worldwide Using generalized logistics regression to forecast population infected by Covid-19 Exponential Lomax distribution Exponentiated Weibull Lomax: properties and estimation The odd Lomax generator of distributions: properties, estimation and applications The generalized odd inverted exponential-G family of distributions: properties and applications The odd log-logistic Lindley-G family of distributions: properties, Bayesian and non-Bayesian estimation with applications The odd dagum family of distributions: properties and applications Generalized exponential distributions Generalized exponential distributions: Statistical Inferences Business failures; another example of the analysis of failure data A new method for generating families of distributions The Odd generalized exponential Lomax distribution properties and estimation Power Lindley distribution and associated inference On odds generalized exponential-power Lomax distribution Maximum product spacings method for the estimation of parameters of linear regression Estimating parameters in continuous univariate distributions with a shifted origin An alternative to maximum likelihood based on spacings How to identify a bathtub hazard rate where where (A.13)