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

1.   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.

2.   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:

and

where $\breve{\Delta} = \left({{\vartheta _1}, \, {\vartheta _2}, \, \pi } \right)$ are the positive scale parameters, while $\pi$ is positive mixing parameter.

Figure 1 shows several graphs of $f\left({\left. t \right|\breve{\Delta} } \right)$ and both density for various parameter values. The PDF demonstrates how the parametric vector $\left({\breve{\Delta} } \right)$ alters the density of 2-CMLM$\left({\breve{\Delta} } \right)$. We may point out that the values for parameters were chosen randomly until a variety of shapes could be captured. The 2-CMLM$\left({\breve{\Delta} } \right)$ can be right skewed as shown in Figure 1.

2.1.   Mode

The mode of the 2-CMLM $\left({\breve{\Delta} } \right)$ is obtained by solving the following nonlinear equation with respect to $t$:

2.2.   Median

The median of 2-CMEM $\left({\breve{\Delta} } \right)$ is presented here. Suppose that $F\left({\left. t \right|\breve{\Delta} } \right)$ be the CDF of 2-CMLM $\left({\breve{\Delta} } \right)$ Model at 0.5^th quantile ${Q_{0.5}}$. Then by solving the following non-linear equation for $t,$ one may get the median $\left({{t^ * }} \right)$.

For the determination of ${t^ * }$ (median) from Eq (7) computational algorithms like Newton-Raphson techniques can be used.

Various graphs of $h\left({\left. t \right|\, \breve{\Delta} } \right)$ for various parameter values are shown in Figure 2. The density of 2-CMLM$\left({\breve{\Delta} } \right)$ is influenced by the parametric vector$\left({\breve{\Delta} } \right)$, as shown in the Figure 2. It's worth noting that parameter values were chosen at random until a range of shapes could be captured. The HRF of each component distribution shows growing activity, while the HRF of 2-CMLM $\left({\breve{\Delta} } \right)$ shows obvious increasing and decreasing behavior, as shown in the figure.

Graphs of the mean of 2-CMLM$\left({\breve{\Delta} } \right)$for various parameter values may be found in Figure 3. It is worth noting that parameter values were chosen at random until a wide range of shapes could be captured. The mean of each component distribution, as well as 2-CMLM$\left({\breve{\Delta} } \right)$, shows a decreasing and constant pattern. Also the 3D Variations of Mean of 2-CMLM $\left({\breve{\Delta} } \right)$ is observed in Figure 4 and mean decrease as ${\vartheta _2}$ increases, as illustrated in this graph.

2.3.   m^th moments about origin

For a random variable $T,$ the ${m^{th}}$ moments about the origin of a 2-CMLM $\left({\breve{\Delta} } \right)$ are as follows:

The mean of the PDF of the 2-CMLM $\left({\breve{\Delta} } \right)$ is:

while the variance is given by

In particular first four moments about origin

and the central moments of the 2-CMLM $\left({\breve{\Delta} } \right)$ are:

The Coefficient of Variation $\left({{{\breve{\varphi } }_{CV}}} \right)$, Skewness $\left({{{\breve{\Psi } }_{Sk}}} \right)$ and the Kurtosis $\left({{{\breve{\psi } }_K}} \right)$ of the 2-CMLM $\left({\breve{\Delta} } \right)$ are:

and

The graphs of Coefficient of Variation of 2-CMLM$\left({\breve{\Delta} } \right)$ for various parameter values are shown in Figure 5. It's interesting to note that parameter values were randomly chosen until a range of shapes could be captured. The Coefficient of Variation of each component distribution increases and remains constant as the Coefficient of Variation of 2-CMLM $\left({\breve{\Delta} } \right)$ increases.

Figure 6 depicts the graphs of the Coefficient of Skewness 2-CMLM $\left({\breve{\Delta} } \right)$ for various parameter values. As seen in Figure 6, the Skewness Coefficient of each component distribution and 2-CMLM$\left({\breve{\Delta} } \right)$ grows and remains constant. Figure 7 shows the increasing and constant behavior of the Coefficient of Kurtosis of each component distribution, as well as the decreasing and increasing behavior of 2-CMLM $\left({\breve{\Delta} } \right)$.

2.4.   Moment Generating Function (MGF)

The MGF of 2-CMLM $\left({\breve{\Delta} } \right)$ is specified as:

2.5.   Cumulants

The characteristic function (CF), $\breve{\xi} \left(\upsilon \right) = E\left[{\exp \left({i\upsilon t} \right)} \right]$ of 2-CMLM $\left({\breve{\Delta} } \right)$ is obtained by substituting $\upsilon$ with '$i\upsilon$' in Eq (22), the CF can be determined as

where $i = \sqrt { - 1}$ is the complex unit.

2.6.   Cumulant Generating Function (CGF)

The cumulant generating function (CGF) is log $\left({\breve{\xi} \left(\upsilon \right)} \right)$

2.7.   Probability Generating Function (PGF)

In Eq (22), we can get the PGF by substituting $\upsilon$ with$\ln \left(\omega \right)$ as follows:

2.8.   Factorial Moment Generating Function (FMGF)

By substituting $\upsilon$ with '$\ln \left({1 + \phi } \right)$' in Eq (22), the FMGF can be determined as

3.   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 $\left({\breve{\Delta} } \right)$ are now being investigated.

3.1.   Reliability function

The reliability function /survival function $R\left({\left. t \right|\, {\text{ }}\breve{\Delta} } \right)$ of 2-CMLM $\left({\breve{\Delta} } \right)$ is.

3.2.   Hazard function

The following is the description of the failure rate function $h\left({\left. t \right|\breve{\Delta} } \right)$ (also known as the hazard rate function) of 2-CMLM$\left({\breve{\Delta} } \right)$

3.3.   Mills Ratio

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

3.4.   Cumulative hazard rate function

The cumulative hazard rate function of 2-CMLM $\left({\breve{\Delta} } \right)$ is

It is a measure of risk: the higher the $H\left({\left. t \right|\, {\text{ }}\breve{\Delta} } \right)$ value, the higher the risk of failure by t-time. It is noted that

Therefore,

3.5.   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.

3.6.   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 $\left({\breve{\Delta} } \right)$ then reliability function is used to express MTTF, which is as follows:

where $R\left(t \right)$ is given in Eq (28). Hence

3.7.   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${\breve{M} _R}\left({\left. t \right|\, \breve{\Delta} } \right)$.

where $R\left({\left. t \right|\, \breve{\Delta} } \right)$ is given in Eq (28).

4.   Estimation inference via simulation

Several statistical characteristics of the 2-CMLM $\left({\breve{\Delta} } \right)$ are contributed to this section, considering that parametric vector $\breve{\Delta}$ is unknown. The assessment of parametric vector $\breve{\Delta}$ 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, ${t_1}, \, {t_2}, ..., \, {t_n}$ represent $n$ observed values from $T$ and their ascending ordering values ${t_{\left(1 \right)}} \leqslant {t_{\left(2 \right)}} \leqslant... \leqslant {t_{\left(n \right)}}.$

4.1.   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 ${t_1}, \, {t_2}, ..., \, {t_n}$ be $n$ observed values from the Eq (2) and $\breve{\Delta}$ be the vector of unknown parameters. The assessments of MLEs of $\breve{\Delta}$ can be provided by optimizing the likelihood function with respect to ${\vartheta _1},$ ${\vartheta _2},$ and $\pi$ given by $L\left({\left. {\bf{t}} \right|\, \breve{\Delta} } \right) = \mathop \prod \limits_{i = 1}^n f\left({{t_i}; \breve{\Delta} } \right)$ or likewise the log-likelihood function for $\breve{\Delta}$ given by

So, by partially differentiating $l\left({\left. {\bf{t}} \right|\, \breve{\Delta} } \right)$ with regard to each of the parameters $\left({{\vartheta _1}, {\vartheta _2}, \pi } \right)$ and setting the findings to zero, the MLEs of the respective parameters are obtained, the likelihood equations are

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.

4.2.   Least square estimators (LSE)

For estimating unknown parameters, the ordinary least square approach is well-known ^[36]. The least square estimators of ${\vartheta _1}, \, {\vartheta _2}$ and $\pi$ denoted by ${\tilde \vartheta _{{1_{LSE}}}}, {\tilde \vartheta _{{2_{LSE}}}}$ and ${\tilde \pi _{LSE}}$, can be obtained by minimizing the function

with respect to ${\vartheta _1}, \, {\vartheta _2}$ and $\pi$ where $F\left({} \right)$ is given by Eq (4). They can be derived in the same way by solving the following nonlinear equations:

and

where

4.3.   Weighted Least Squares Estimators (WLSE)

Consider the weighted function below (see ^[37])

The WLSEs ${\tilde \vartheta _{{1_{WLSE}}}},{\tilde \vartheta _{{2_{WLSE}}}}$ and ${\tilde \pi _{WLSE}}$ can be obtained by minimizing the function

One can also get these estimators by solving:

and

where ${\breve{\Psi } _1}\left({\left. {{t_{(i)}}} \right|\, {\vartheta _1}} \right),$ ${\breve{\Psi } _2}\left({\left. {{t_{(i)}}} \right|\, {\vartheta _2}} \right)$ and ${\breve{\Psi } _3}\left({\left. {{t_{(i)}}} \right|\, \pi } \right)$ are given in Eqs (48-50).

5.   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 $\pi$ and model parameters. Three simulation experiments are used in order to assess the performance of MLE, performance, LSEs and WLSEs of the 2-CMLM $\left({\breve{\Delta} } \right)$ parameters. The bias and MSE measures are used to discuss the precision of the MLEs, LSEs and WLSEs. The efficiency of each parameter estimation approach for the 2-CMLM $\left({\breve{\Delta} } \right)$ model in terms of $n$ is considered. The steps of the simulation algorithm are as follows:

1. By varying the mixing proportion $\pi$ and the model parameters $\left({\pi, \, {\vartheta _1}, \, {\vartheta _2}} \right) = \left({0.4, \, 0.65, \, 0.6} \right), \, \left({0.6, \, 0.25, \, 0.27} \right)$ and $\left({0.9, \, 0.12, \, 0.15} \right)$, generate random samples of sizes $10, \, 13, ..., \, 300$ from the 2-CMLM $\left({\breve{\Delta} } \right).$ 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 $U(0, \, 1).$

3. If $u \leqslant \pi,$ we use the ($rlindley$) function to generate a random variate from the first component, which is a Lindley distribution $\left({{\vartheta _1}} \right)$. If $u > \pi$, the second component, a Lindley distribution $\left({{\vartheta _2}} \right)$, is used to generate a random variate.

4. Begin to (2) until the required sample of size $n$ is obtained.

5. Using 1000 replications, repeat steps 1-4 again. Compute the MLEs, LSEs and WLSEs for the 1000 samples; say ${\breve{\theta } _j}$ for $j = 1, \, 2, ..., \, 1000,$ using the optima function and the Nelder-Mead technique in R to calculate the estimator values. The simulated densities for three parametric sets are shown in Figure 8.

6. Calculate biases and MSEs. These objectives are obtained with the help of the following formulas:

where $\theta = \left({{\vartheta _1}, \, {\vartheta _2}, \, \pi } \right).$

The results of simulation study of this subsection are indicated in Figures 9-14. These empirical findings show that the proposed estimate methods do a good job of estimating the 2-CMLM $\left({\breve{\Delta} } \right)$ parameters. Because the bias tends to zero as $n$ increases, we can deduce that the estimators exhibit the attribute of asymptotic unbiasedness. The mean squared error behavior, on the other hand, indicates consistency because the errors tend to zero as $n$ increases. From Figures 9-14, the following observations can be extracted.

● The estimated bias of parameters ${\breve{\vartheta } _1}, {\breve{\vartheta } _2}, \breve{\pi }$, decreases as $n$ increases under all estimation approaches.

● From Figure 9 for parametric Set-Ⅰ, we can see that the estimated bias of parameters ${\vartheta _1}$ and $\pi$ is over-estimated in all three estimation methods while ${\vartheta _2}$ is under estimated in MLE.

● From Figure 13 for parametric Set-Ⅲ, we can see that the estimated bias of parameters ${\vartheta _1}$ and $\pi$ is under-estimated in LSE estimation method and ${\vartheta _1}$ over-estimated in WLSE estimation method while ${\vartheta _2}$ is over-estimated in all three estimation methods.

● The estimated bias of parameters ${\vartheta _1}, \, {\vartheta _2}, \, \pi$ 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, 11 and 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 ${\vartheta _1}, \, {\vartheta _2}$ and $\pi$ 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 $\left({\breve{\Delta} } \right)$ parameters.

6.   Applications to COVID-19 data

The major purpose of the 2-CMLM $\left({\breve{\Delta} } \right)$ 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 $\left({\breve{\Delta} } \right)$ 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]} also studied the COVID-19 datasets to fit the new distribution. The given data sets are used to compare the fit of the 2-CMLM $\left({\breve{\Delta} } \right)$ to a competing model that is a two component mixture of exponential models (2-CMEM $\left({\breve{\Delta} } \right))$ and two component mixture of Weibull models (2-CMWM $\left({\breve{\Delta} } \right))$ by using R function maxLik (.). We demonstrate that the 2-CMLM $\left({\breve{\Delta} } \right)$ provides great fit to the COVID-19 pandemic lifespan data. The concept "best fit" refers to the proposed model having lower values for the measure chosen for comparison.-Log-likelihood $\left({ - LL} \right)$ 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].

The MLEs of the 2-CMLM$\left({\breve{\Delta} } \right)$ and Goodness-of-Fit measures are provided in Tables 1 and 2. The outcomes of these Tables clearly show that the 2-CMLM$\left({\breve{\Delta} } \right)$ is the best of 2-CMLM$\left({\breve{\Delta} } \right)$ as it has the smaller values of the -LL, AIC, BIC, and CAIC. In comparison 2-CMLM$\left({\breve{\Delta} } \right)$ the 2-CMLM$\left({\breve{\Delta} } \right)$ 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.

Figures 15 and 16 depict probability-probability (P-P) plots for 2-CMLM $\left({\breve{\Delta} } \right)$ and 2-CMEM $\left({\breve{\Delta} } \right)$ and for 2-CMWM $\left({\breve{\Delta} } \right)$ respectively, which support the findings of Tables 1 and 2. Figures 17 and 18 show the profiles of the log-likelihood function (PLLF) based on data sets.

7.   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.

Acknowledgment

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia for funding this work through research groups program under grant number RGP.2/120/42. The authors would like to thank the Editor and anonymous reviewers for their constructive comments that improved the final version of the paper.

Conflict of interest

There is no conflict of interest declared by the authors.