Estimation method of mixture distribution and modeling of COVID-19 pandemic

: The mathematical characteristics of the mixture of Lindley model with 2-component (2-CMLM) are discussed. In this paper, we investigate both the practical and theoretical aspects of the 2-CMLM. We investigate several statistical features of the mixed model like probability generating function, cumulants, characteristic function, factorial moment generating function, mean time to failure, Mills Ratio, mean residual life. The density, hazard rate functions, mean, coefficient of variation, skewness, and kurtosis are all shown graphically. Furthermore, we use appropriate approaches such as maximum likelihood, least square and weighted least square methods to estimate the pertinent parameters of the mixture model. We use a simulation study to assess the performance of suggested methods. Eventually, modelling COVID-19 patient data demonstrates the effectiveness and utility of the 2-CMLM. The proposed model outperformed the two component mixture of exponential model as well as two component mixture of Weibull model in practical applications, indicating that it is a good candidate distribution for modelling COVID-19 and other related data sets.


Introduction
In the early days of statistics, mixture models, specifically finite mixture models, were employed to simulate a variety of events, and their use has grown through time. In many scenarios, available data can be seen as a mixture of two or more distributions. We can merge statistical distributions using this notion to create a new one. Finite mixture models are useful in a variety of domains, including biology, engineering, genetics, healthcare, business, marketing, real life, and social sciences. The basic concept behind mixture models is to combine two or more models by adjusting proportions to produce a novel model with new attributes. As a result, it's crucial to investigate the statistical features of the proposed mixture model and use appropriate methods to estimate its unknown parameters. Finite mixture densities can be used to model data from populations known or suspected to contain a number of separate subpopulations. Most commonly used are mixture densities with Gaussian components, but mixtures with other types of component are also increasingly used to model, for example, survival times. Mixing distributions have been studied by several writers, including [1][2][3][4][5]. The classical features of the mixture of Burr XII and Weibull distribution were investigated by Muhammad and Muhammad [6]. Sultan et al. [7] suggested a 2-Component Mixture of Inverse Weibull models (2-CMIWD) and investigated some of its features using density and hazard function graphs. To examine the hybrid of two inverse Weibull distributions, Jiang et al. [8] focused at the forms of the PDF and hrfs as well as graphical approaches. The following are several authors who deal with mixture modeling in different practical problems: Mohammadi et al. [9], Ateya [10], Mohamed et al. [11], and Sindhu et al. [12]. Some other relevant studies are [13][14][15][16][17][18][19].
Because of its practical application, the Lindley model, which belongs to the family of exponential models, is important. Lindley model is useful for modelling many types of life time and reliability data. The Lindley distribution has captivated the curiosity of scholars in recent years. The generalized Lindley (GL), model was introduced by Zakerzadah and Dolati [20], who studied its statistical content and capabilities. A new class of GL models was suggested by Oluyede and Yang [21] and Nadarajah et al. [22]. The researchers [23] developed the Lindley model to illustrate the distinction between Fiducial and subsequent models in the perspective of Fiducial and Bayesian statistics. Furthermore, [24] discusses the statistical features of Lindley models, demonstrating that this model is a superior model for particular application than other models like the exponential model. When modelling various lifetime data sets, Shanker, et al. [25] used the Lindley model. Mazucheli and Achcar [26] demonstrated that the Lindley model may be used to describe strength data effectively, and they recommended it as a suitable alternative to the exponential and Weibull distributions. Furthermore, by adding another shape parameter to the model and naming it a power Lindley model, [27] developed a new extension of the Lindley model. [28,29] investigated a mixture of Lindley models from different perspectives.
Al-Moisheer et al. [30] examined mixture of Lindley models and used ML and the generalized method of moments to evaluate the unknown parameters of the mixture model. Besides that, it is interesting to compare the MLE method to other estimation techniques such as least-squares estimation (LSE), weighted least-squares estimation (WLSE), and other methods of estimation. In the literature, there are various estimating methods for parametric distributions, some of which have been widely investigated from a theoretical perspective. It is worth mentioning, too, that in the case of small n the maximum likelihood method frequently fails. As a consequence, new estimation techniques have recently been suggested. The usefulness of estimating methods varies depending on the user and the application area. For example, even though the moment estimator does not have a closed form expression, it may be preferable to utilize it. The goal of this paper is to provide framework for selecting the optimum estimation technique for the 2-Component Mixture of Lindley Model (2-CMLM) distribution that would be useful to professional statisticians. In this study, we use least square estimation (LSE) and weighted least square estimation (WLSE), in addition to MLE, to estimate the 2-Component Mixture of Lindley Model (2-CMLM). In the literature, analyses of estimation methods for other distributions have been examined, for example, [31][32][33][34][35].
This study has two key objectives: The first is to demonstrate how various frequentist estimators of the proposed distribution perform for different sample sizes and different parametric values. The second step is to investigate some additional model attributes and demonstrate that the distribution outperforms its competitor mixed model with two real data sets.

The 2-component mixture of Lindley model (2-CMLM)
A random variable T is said to have a finite mixture of Lindley model with 2-component (2-CMLM) if it's PDF and CDF can be composed as:

Mode
The mode of the 2-CMLM ( )  is obtained by solving the following nonlinear equation with respect to t :
( ) ( ) For the determination of t  (median) from Eq (7) computational algorithms like Newton-Raphson techniques can be used.
Various graphs of ( )

m th moments about origin
For a random variable , T the th m moments about the origin of a 2-CMLM ( )  are as follows: The mean of the PDF of the 2-CMLM ( )  is: while the variance is given by In particular first four moments about origin The Coefficient of Variation ( ) The

Cumulants
The characteristic function (CF), (22), the CF can be determined as where 1 i =− is the complex unit.

Cumulant Generating Function (CGF)
The cumulant generating function (CGF) is log

Probability Generating Function (PGF)
In Eq (22), we can get the PGF by substituting  with ( ) ln  as follows:

Reliability measures
The reliability function /survival function and failure rate /hazard rate function are used to classify lifespan models in reliability theory. A ratio of the lifespan model to the reliability function is the hazard rate function. If the dependability function's value is lower, it indicates that the item or component has a shorter lifespan, then the hazard rate will be larger, which means the likelihood of failure will be higher. On the other hand, a higher reliability function value means a lower hazard rate, which means a lesser risk of failure. The reliability properties of 2-CMLM ( )  are now being investigated.

Reliability function
The reliability function /survival function ( )

Hazard function
The following is the description of the failure rate function

Mills Ratio
Mills Ratio is a unique technique to describing reliability because of its connection to failure rate.

Cumulative hazard rate function
The cumulative hazard rate function of 2-CMLM ( ) It is a measure of risk: the higher the

Reversed hazard rate function
The ratio between the life likelihood function and its distribution function is defined as the reversed hazard rate of a random life.

Mean Time to Failure (MTTF)
The expected (or average) time for which the device functions satisfactorily is given by the mean time to failure (MTTF). If 2-CMLM ( )  then reliability function is used to express MTTF, which is as follows:

Mean Residual Life (MRL)
Reliabilists, statisticians, survival analysts, and others have investigated the mean residual lifetime (MRL). It has given many of valuable results. The remaining lifetime after t for a component or system of age t is random. The mean residual life or mean remaining life is the expected value of this random residual lifetime and is denoted by where ( ) Rt is given in Eq (28).

Estimation inference via simulation
Several statistical characteristics of the 2-CMLM ( )  are contributed to this section, considering that parametric vector  is unknown. The assessment of parametric vector  is carried out by the three well known estimation methods such as maximum likelihood estimation, Least square Estimation (LSE) and Weighted Least square Estimation (WLSE). From now, 12

Maximum likelihood estimation (MLE)
The most widely known approach of parameter estimate is the maximum likelihood method. The method's popularity is due to its numerous desired qualities, such as consistency, normality and asymptotic efficiency. Let 12 , ,..., As a result, solving this nonlinear system of equations gives the MLE. Although these equations cannot be analytically solved, we use statistical software through iterative approach like Newton method or fixed point iteration methods can be used to solve them.

Least square estimators (LSE)
For estimating unknown parameters, the ordinary least square approach is well-known [36]. The least square estimators of 12

Weighted Least Squares Estimators (WLSE)
Consider the weighted function below (see [37])  One can also get these estimators by solving:

Simulation study
We use the simulation to analyze various estimating strategies that were discussed in subsection 4.1-4.3. As a result, we execute some Monte Carlo simulations with various mixing proportions  The simulation's random samples are generated as described in the next stage. 2. Using the R uniform generator (runif), create one variate u from the uniform distribution (0,1).      • The estimated bias of parameters 12 ,,    , decreases as n increases under all estimation approaches. • From Figure 9 for parametric Set-I, we can see that the estimated bias of parameters 1  and  is over-estimated in all three estimation methods while 2  is under estimated in MLE. • From Figure 13 for parametric Set-III, we can see that the estimated bias of parameters 1  and  is under-estimated in LSE estimation method and 1  over-estimated in WLSE estimation method while 2  is over-estimated in all three estimation methods. • The estimated bias of parameters 12 ,,    is over-estimated in both estimation methods for parametric Set-II (see Figure 11).
• In terms of bias, generally the performances of the MLE, is the good (see Figures 9, 11and 13).
• Furthermore, Figures 10, 12 and 14 show that the MSE for MLE, LSE and WLSE estimate methods reduces as n increases, satisfying the consistency criterion. • Under all estimation procedures, the difference between estimates and assumed parameters decreases to zero as sample size increases. • When the sample size approaches infinity, MLE estimation is often stronger in terms of bias and MSE when compared to alternative estimation techniques for all stated parameter values (see Figures 9-14). The basic conclusion from the previous figures is that as the sample size grows the estimated bias and MSE graphs for parameters 12 ,  and  eventually approach zero for all estimation methods. This validates the accuracy of these estimation approaches, as well as the numerical computations for the 2-CMLM ( )  parameters.

Applications to COVID-19 data
The major purpose of the 2-CMLM ( )  distribution's derivation is to employ it in data analysis purposes, which makes it valuable in a variety of domains, notably those dealing with lifetime analysis. This section demonstrates how the 2-CMLM ( )  works by applying the suggested model to real-world data. This aspect is demonstrated here by comparing two sets of data from COVID-19 pandemic outbreaks. [38][39][40][41][42][43]  the AIC (Akaike information criterion), the BIC (Bayesian information criterion), and the CAIC (Corrected Akaike information criterion) are some of the discriminatory measures/goodness-of-fit (GoF) incorporated in these criteria. The best model for the real data set might be the one with the lowest values of the above-mentioned measures.

DataSet-1:
The data represents a COVID-19 data belong to Italy of 59 days that is recorded from 27 February to 27 April 2020. This data formed of rough mortality rate. This data set can be accessed at https://covid19.who.int/.

DataSet-2:
We investigate the survival times of people in China who have been infected with the COVID-19 virus. The data set under consideration represents patient survival times from the moment they were admitted to the hospital until they deceased. https://www.worldometers.info/coronavirus/ can be used to access the data set. This data is used in [44]. ( )  provides a very good fit for these data, as seen in the Tables. According to dataset one, 2-CMWM has the smallest -LL, as well as the smallest AIC, the BIC, and the CAIC. But if we consider the mixture of two parsimonious models 2-CMLM perform well. The best distribution for fitting the dataset II is 2-CMLM, as seen in Table 2 because the 2-CMLM model has the smallest AIC, the BIC, and the CAIC even though -LL is little bit high as compare to 2-CMWM but most Goodness-of-Fit measures are in favor of 2-CMLM model. So, the best distribution for fitting the dataset II is 2-CMLM, as seen in Figure 15.  Tables 1 and 2. Figures 17 and 18 show the profiles of the log-likelihood function (PLLF) based on data sets.

Conclusions
We studied two component mixture of Lindley models in this study using three estimate techniques: MLE, LSE, and WLSE. Further, some additional statistical and reliability properties of the two Lindley mixture model were obtained, like central moments, Cumulants, Cumulant Generating Function, Probability Generating Function, Factorial Moment Generating Function, Coefficient of variation, skewness and kurtosis, Mills Ratio, Reversed Hazard Rate Function, Mean Time to Failure, and Mean Residual Life. A simulation study was conducted using 1000 replications to explore and compare the performance of the estimation techniques. As a consequence, we found that the ML technique outperformed the others in terms of accuracy and consistency when estimating model unknown parameters. Moreover, to demonstrate the usefulness of the underlying mixture model, we used some real dataset. We demonstrated that the Lindley mixture model is suitable and effective for data modelling, and that it outperforms the exponential mixture model, using two real datasets.