A New Modified Kies Fréchet Distribution: Applications of Mortality Rate of Covid-19.

The purpose of this paper is to identify an effective statistical distribution for examining COVID-19 mortality rates in Canada and Netherlands in order to model the distribution of COVID-19. The modified Kies Frechet (MKIF) model is an advanced three parameter lifetime distribution that was developed by incorporating the Frechet and modified Kies families. In particular with respect to current distributions, the latest one has very versatile probability functions: increasing, decreasing, and inverted U shapes are observed for the hazard rate functions, indicating that the capability of adaptability of the model. A straight forward linear representation of PDF, moment generating functions, Probability weighted moments and hazard rate functions are among the enticing features of this novel distribution. We used three different estimation methodologies to estimate the pertinent parameters of MKIF model like least squares estimators (LSEs), maximum likelihood estimators (MLEs) and weighted least squares estimators (WLSEs). The efficiency of these estimators is assessed using a thorough Monte Carlo simulation analysis. We evaluated the newest model for a variety of data sets to examine how effectively it handled data modeling. The real implementation demonstrates that the proposed model outperforms competing models and can be selected as a superior model for developing a statistical model for COVID-19 data and other similar data sets.

PDF

SF

CHRF

Quantile Density Function

Probability weighted moments

CDF

hrf

QF

MGF

Maximum likelihood Estimation

Maximum product spacing method

Probability Density Function

Survival Function

Probability Weighted Moment

hazard rate function

Weighted Least Square Estimates

Moment Generating Function

Maximum Product Spacing

random variable

Quantile Function

cumulative hazard rate function

Mean square error

Least Square Estimates

Method of Moments Estimates

Extreme Value Theory

Cumulative Distribution Function

L-moments estimation

Introduction

In statistical analysis, extreme value theory (EVT) is very valuable. The EVT was originally related to analyzing the performance of extreme values (EVs). And since EVs have a relatively poor chance of appearing, they may have a very high impact on the observed experiment. Fréchet (F) distribution is the significant model in modelling EVs. The F distribution was originally suggested in [1]. This model is defined in [2] and addressed its broad range of applications in various spheres like accelerated life monitoring, sea waves, horse racing, rainfall, environmental disasters, earthquakes, wind speeds, sea currents, track race records, and so on. More information about the F distribution can be found in the literature; for instance, [3] examined the exponentiated Fréchet model. For relief periods and survival times data, [4] introduced and implemented a new form of the F model. In [5], authors suggested some implementations of the Marshall-Olkin Fréchet distribution.

The CDF and PDF of Fréchet (F) model with scale and shape parameters are

The characteristics of the exponentiated Kies distribution were investigated by Kumar and Dharmaja [6]. For product moments of modified Kies (MKI) model through type II progressive censored sample, and also an approximation of model parameters [7]. Focused on the modified Kies (MKI) model family, in [8], authors proposed a novel family of models. If is the reference CDF for a parameter vector then the MKI family CDF is defined aswhere the parameter vector The PDF of (3) is

In this paper, the three-parameter modified Kies Fréchet (MKIF) distribution, which has several appealing characteristics, is obtained by referring to the distributions earlier. The PDF of the proposed MKIF distribution is extremely versatile, since it can be positive skewed, exponential, or symmetric, allowing for even more tail flexibility. It could model increasing, decreasing, bathtub, and inverted-U hazard rates. Another value of the suggested model is that it has a perfect closed form CDF and is quite simple to handle. Such characteristics make the model strong contender for biomedical life monitoring, actuarial data, and reliability applications.

In the study of any probability distribution, parameter estimation is significant. Because of its appealing properties, MLE is commonly utilized to estimate the parameters of any model. MLEs are unbiased, asymptotically consistent and normally distributed (see [9]). Other estimation techniques developed over time for distributions (see [10], [11], [12], [13], [14], [15], [16], [17]) are dependent on various methodologies, like the methods of L-moments estimation (LME), moments estimation (MOM), least-squares estimation (LSE), probability weighted moment estimation (PWM), weighted least square estimation (WLSE) and maximum product spacing estimation (MPS) and minimum distance estimation. In [18], researchers studied the L-moments and maximum probability approaches for estimating parameters of complementary Beta model. In [19], authors estimated the parameters of generalized power Weibull model employing the MLE and maximum product spacing strategies.

In current article, we present a detailed comparison of three methodologies for estimating unknown parameters of MKIF model, as well as an analysis of the execution of such estimators for different parameter values and sample sizes. We specifically compare MLEs, LSEs, and WLSEs. Theoretically, it is hard to compare the behaviors of various estimation techniques; we conduct detailed simulations study to assess the behaviors of various estimators focused on bias and mean squared error.

Wuhan, China's emerging business zone faced a wave of new coronavirus which killed over one thousand eight hundred people and infected more than one thousand seven hundred people during the first fifty days of epidemic at the end of 2019. This virus was identified as belonging to the coronavirus community. Chinese scholars named the new virus as 2019 new coronavirus. The virus was known SARS-CoV-2, and disease titled COVID-19 by the International Committee on Virus Taxonomy (ICTV) [20], [21], [22]. The inspiration of novel distribution is to model the COVID-19 data. To assess effectiveness of the model techniques, we considered the COVID-19 real data of Canada and the Netherlands. We utilized the COVID-19 daily mortality rate for Canada and Netherlands. Many researchers like Sindhu et al. [23], [24] modeled the COVID-19 data using new models. This study implemented two actual data implementations and concluded from modeling results that recent model is an ideal competitor with some known and popular models like the modified Kies inverted Topp-Leone (MKITL) [25], modified Kies exponential (MKIEx) [26], and Fréchet (F) distribution. In future research, we attempt to address a novel implementation for MKIF distribution based on a trimmed sample (see for more information Sindhu et al. [27], [28]). We strive to develop a new two component mixture of three-parameter modified Kies Fréchet (MKIF) distribution, such as Sindhu et al. [29], [30], [31], [32], [33], [34], [35]. Several authors have done their work on distributions see reference [36], [37], [38].

The remaining part of the current study is is structured as given: The MKIF model is obtained in Section 2. The statistical characteristics of MKIF model are studied in Section 3. In Section 4, we analyze estimation methods for MKIF model. Section 5 provides simulation results for the MKIF model. Two implementations of real data study have given in Section 6 and conclusion is provided in Section 7.

MKIF model specifications

The CDF and PDF of MKIF model are specified as where The hrf i.e. is an ideal mechanism in reliability study. Reliability or survival function is an indicator of capability of appliance to work without failure when placed into operation and it is a non-increasing function. Here, functions of MKIF distribution is

The CHRF is also called the integrated hrf. It is a measure of risk: higher the value, higher the risk of failure by

It is noted that

Therefore,

Just in Fig. 1 and 2 the above-mentioned PDF and hrf demonstrate how the parameters affect the density of MKIF model. We would have to note that the values for parameters have indeed been chosen arbitrarily till we captured a broad range of shapes for the parameters concerned. We note that PDF is right and slightly left-skewed or inverted-U shaped, and slightly symmetrical. It is reversed-J shaped . Fig. 2 provides flexible hazard rate shapes like increasing, U shaped and decreasing.

Fig. 1

Variations of of MKIF along with and .

Fig. 2

Fluctuations of of MKIF along with and .

Useful expansion of MKIF model

This section covers a valuable expansion of PDF and CDF of MKIF distribution. The power series and exponential function could be described as

and

Then, it follows that, using (11), (12) to expand and respectively

If and then power series holds

Applying (13) to expand we obtain the PDF of MKIF in (6) is expressed as with

The MKIF distribution can be presented as a linear mixture of Fréchet (F) models, according to equation (15) As a consequence, the MKIF distribution's properties can be deduced from those of the Fréchet distribution. In the same way, (11), (12) indicate that CDF of MKIF in (5) can be written aswhere and is CDF of Fréchet model with power In addition, the following binomial theorem can be used to extend let be positive integer be then

Therefore, becomes:

Then, applying (12), (13) to expand the in (18) is expressed as

Fig. 1, Fig. 2 display maps of PDF and hrf functions of MKIF model for various values of and respectively. Fig. 2 demonstrates that hrf of MKIF distribution can be rising, decreasing, or inverted U shaped.

Quantile function

The next result can be utilized to simulate values from the MKIF distribution. The QF of is

As a result, the median, as well as the lower and upper quantiles, are calculated as follows:

The accompanying quantile density function is provided by the differentiation of (20)

Skewness and kurtosis based on the quantile function (QF)

Measures identified with moments are a standard procedure to measuring the skewness and kurtosis of a model. These moments, however, do not necessarily occur. Because of this, certain replacements are given by the implementation of the QF. In specific, to measure the skewness, we may utilize the Galton skewness coefficient described as

The following Moors kurtosis coefficient can be used to evaluate the kurtosis.

At different values of with and different levels of Fig. 3 provides maps of the QF and quantile density function. The distribution is found to be U shaped. Fig. 3 display proposed three-dimensional plots of skewness and kurtosis at different levels of As the higher inputs of the parameters and contribute the higher change in median curve. Also median yields lower values when approaches to 0.5. On the other hand significant change in the skewness behavior is noticed along for smaller values of but as increases, it ends up to nearly 0.1. The degree of peakedness of a distribution decreases as increase. In addition it may also be mesokurtic, platykurtic, or leptokurtic.

Fig. 3

Fluctuation of QF and quantile density function of MKIFalong with and q at different levels of with.

Moments and moment generating function (MGF)

The moment of MKIF model can be evaluated directly by extending the PDF of the MKIF as seen in (14): where is the gamma function. The mean of can be obtained by putting in (28). The MGF is widely utilized in characterization of model. The MGF of MKIF model utilizing the Maclaurin series is mentioned as

Probability weighted moments (PWM)

The PWM of MKIF denoted by is formally defined by

The PWM of MKIF can be determined directly by applying the PDF in (14) and using (19) after replacing by , we have

The certain estimation approaches with simulation

There are many ways to evaluate the parameters of models that each of them has its distinctive attributes and strengths. Three of those strategies are presented momentarily in this section and will be graphically, analyzed in simulation study. Here, F(.) is distribution function of MKIF model. Several statistical characteristics of the MKIF distribution are contributed to this section, considering that are unknown. Using three different estimation techniques, we were able to solve the problem of estimating the MKIF distribution parameters. These methods are MLEs, LSEs and WLSEs. From now, represent observed values from and their ascending ordering values

MLE approach

There are many techniques for calculating parameters, but the most widely used is the maximum likelihood method. Let be a random sample from MKIF model with parameters and The likelihood function can be expressed as follows: or likewise the log-likelihood function for and is

We could calculate the MLEs of the parameters and by setting all equations to zero and solving them simultaneously.

Method of ordinary and weighted least squares

The LSEs and WLSEs techniques for estimating unknown parameters are widely recognized [39]. The two techniques for estimating the parameters of MKIF model are discussed here. The LSEs, and can be achieved by minimizing the following function with respect to and where These can be extracted equivalently by solving: and where and

The WLSEs, and can be determined by minimizing the following function, with respect to and respectively.

The estimates can also be obtained by solving: and where where are given in (40–42).

Graphical analysis

We can execute simulation experiments graphically to determine finite sample behaviour of the MLEs, LSEs and WLSEs. The following algorithm has used to make the decision:

1. Generate thousand samples of size from (6). This work is carried out simply by QF and obtained data from uniform distribution.

2. In order, four separate sets of true parameter values and are used.

3. Calculate the estimates for 1000 samples, say for

4. Evaluate biases and MSEs. These objectives are obtained with the help of the following formulas:

where.

5. These steps have repeated for with the mentioned parameters for MLEs, LSEs and WLSEs. The and have been determined. To calculate the value of estimates, we used optim function of R. The outcomes of simulations are indicated in Fig. 5, Fig. 6, Fig. 7, Fig. 8 . These biases vary with respect to in Fig. 5, Fig. 6, Fig. 7, Fig. 8 (Left panels) and MSEs differ with respect to as seen in Fig. 5, Fig. 6, Fig. 7, Fig. 8 (Right panels).

Fig. 5

Fluctuations of Bias and MSE for parametric

Fig. 6

Fluctuations of Bias and MSE for parametric

Fig. 7

Fluctuations of Bias and MSE for parametric

Fig. 8

Fluctuations of Bias and MSE for parametric

We can deduce that the estimators have the property of asymptotic unbiasedness since the bias tends to zero as grows, while the trend in the mean squared error shows consistency since the errors tend to zero as grows.

Final comments on the Simulation Findings

Under all estimation methods, the bias of and reduces as increases.

For the MLEs, the biases of and are generally positive, but negative biases are also noted for under first three parametric sets.

The MLEs are overestimated for and however, it is underestimated for (see left panel of Fig. 5, Fig. 6, Fig. 7, Fig. 8).

The performance of the LSE and WLSE are quite similar for and however, it is slightly different for under Set 1–4 (see left panel of Fig. 5, Fig. 6, Fig. 7, Fig. 8)

Under all the approaches, generally, the least MSE of parameters are observed under MLE (see right panel of Fig. 5, Fig. 6, Fig. 7, Fig. 8).

In most cases, the maximum likelihood method of estimation works better than other approaches in terms of MSE, as seen in right panel of Fig. 5, Fig. 6, Fig. 7, Fig. 8.

In most cases, the WLSE is the next optimum performing estimator, followed by MLE.

The general conclusion from the aforementioned figures is that as the sample size grows, all bias and MSE graphs for all parameters will reach zero. This confirms the accuracy of these estimation methods.

Real data applications

The MKIF model's potentiality for two real datasets is demonstrated in this section. MKIF distribution is collated with other reasonable models, namely: modified Kies inverted Topp-Leone (MKITL) distribution [25], modified Kies exponential (MKEx) [26], and Fréchet distribution. Some goodness-of-fit measures are used to compare the fitted distributions, including the Akaike Information (AIC), Bayesian Information (BIC), CAIC (consistent Akaike information) and HQIC (Hannan-Quinn information) criterions. In general, lower values of these statistics, the stronger the fit to the data. We also evaluate Kolmogorov- Smirnov) statistic along with its P-value (PV) for considered distributions fitted centered on two real datasets. Table 2, Table 3 display the results of these measures. The first dataset includes 36 days of COVID-19 data from Canada, from April 10 to May 15, 2020, as seen at [https://covid19.who.int/]. This data formed of rough mortality rate. In addition, we notice that Almetwally et al. [25] analyzed this dataset to demonstrate the suitability of the new inverted Topp-Leone model. The next set of data is a 30-days COVID-19 data set from the Netherlands, and is obtained between March 31 and April 30, 2020. This data is comprised of an approximate mortality rate and it has also been included by Almongy et al. [40]. To conclude, for the two datasets, the MKIF model shows itself to be the most suitable model, demonstrating its applicability in a realistic environment. The MLE approach has been used to estimate the pertinent parameters of models. The MLEs of the parameters are presented in Table 1, Table 2. As compared to other models used to fit the COVID-19, the MKIF distribution has the largest P-value and the shortest distance of Kolmogorov-Smirnov (K-S), seen in Table 3, Table 4 . The fit empirical, histogram and PP-graph, profile log-likelihood for the MKIF model for COVID-19 datasets from Canada and Netherlands are seen in Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14 . The findings of application 1 and 2 are seen in Table 1, Table 2, Table 3, Table 4 and Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15, Fig. 16 .

Table 2

MLEs and SEs, of considered distribution for Data Set II.

Distributions	MLEs	Standard errors
MKIFy;α,β,η	4.1595366,0.7742977,1.6638938	1.2761030,0.5415642,1.2870155
MKITLα,β	2.3136894,0.4875761	0.34503557,0.02886037
MKIExα,β	1.36564554,0.09433935	0.202082242,0.009222408
Fréchetλ,κ	3.778877,1.550118	0.4732334,0.2014450

Table 3

The goodness-of-fit performance measures for Data I.

Distribution	AIC	CAIC	BIC	HQIC	K-S	PV
MKIFy;α,β,η	104.1101	104.8601	108.8607	105.7682	0.12053	0.6723
MKITLα,β	105.0358	105.3994	108.2028	106.1411	0.14536	0.4323
MKIExα,β	111.5294	111.893	114.6964	112.6347	0.16947	0.2524
Fréchetλ,κ	109.8401	110.2038	113.0072	110.9455	0.17372	0.2274

Table 1

MLEs and SEs, of considered distribution for Data Set I.

Distributions	MLEs	Standard errors
MKIFy;α,β,η	2.948513,1.985900,1.203498	0.2075851,0.7068093,0.4833309
MKITLα,β	3.3555427,0.7285832	0.40463067,0.02748158
MKIExα,β	2.2857088,0.1864468	0.27048389,0.01025465
Fréchetλ,κ	2.704450,3.169386	0.1512232,0.3655088

Table 4

The goodness-of-fit performance measures for Data II.

Distribution	AIC	CAIC	BIC	HQIC	K-S	PV
MKIFy;α,β,η	159.0168	159.9399	163.2204	160.3616	0.07632	0.9893
MKITLα,β	157.2194	157.6639	160.0218	158.1159	0.07813	0.9862
MKIExα,β	159.8504	160.2949	162.6528	160.7469	0.13262	0.6196
Fréchetλ,κ	166.074	166.5185	168.8764	166.9705	0.15126	0.4543

Fig. 10

Fitted PDFs on histogram of dataset I (left) and Fitted CDFs on empirical CDF of dataset I (right).

Fig. 11

Fitted PDFs on histogram of dataset II (left) and Fitted CDFs on empirical of dataset II (right).

Fig. 12

P-P plots of the MKIF distribution for datasets I and II.

Fig. 13

Plotting of profile log-likelihood of MKIF distribution for dataset I.

Fig. 14

Plotting of profile log-likelihood of MKIF distribution for dataset II.

Fig. 15

Plotting of the results given in Table 3.

Fig. 16

Plotting of the results given in Table 4.

Final comments on the two applications

1. According to dataset one, MKIF has the largest P-value, as well as the smallest K-S distance.

2. MKIF has been the best model for fitting the dataset I, as seen in Fig. 9.

Fig. 9

Simulated Distribution of all considered sets.

3. In dataset II, we can notice that MKIF has the largest P-value as well as the smallest K-S distance.

4. The best distribution for fitting the dataset II is MKIF, as seen in Fig. 4.

Fig. 4

Fluctuation of, skewness and kurtosis of MKIFalong and at different level with.

5. Table 3 shows that the MKITL, MKIEx and Frechet distributions all have poor fit for the first data set.

6. Table 4 shows that MKIF is in great agreement with the MKITL.

Concluding remarks

In current article, we suggest a novel three parameter model, entitled three-parameter modified Kies Fréchet (MKIF) model. The MKIF model is more versatile to study lifetime data than other known models. The SF, hrf, CHRF, linear representation of PDF and CDF, QF and moments of the MKIF model are derived. MLE, LSE, and WLSE approaches are compared. We present two executions based on mortality rate of COVID 19, demonstrating that MKIF model is the best model for fitting this type of data among all its competitors. The parameter estimation of MKIF model is derived by MLE, LSE, and WLSE. The performance of the model under different estimation methods is evaluated using simulation study. Two real-life data suggested, model gives a reasonably preferable fit than MKITL, MKIEx and Fréchet distributions.

CRediT authorship contribution statement

Anum Shafiq: Conceptualization, Methodology, Validation, Formal analysis, Investigation. S.A. Lone: Writing - review & editing, Data curation, Formal analysis, Methodology, Validation, Conceptualization. Tabassum Naz Sindhu: Writing - review & editing, Data curation, Formal analysis, Methodology, Validation, Conceptualization. Youssef El Khatib: Supervision, Writing – review editing and funding. Qasem M. Al-Mdallal: Supervision, Writing – review editing and funding. Taseer Muhammad: Writing - review & editing, Data curation, Formal analysis, Methodology, Validation, Conceptualization.

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.