key: cord-0755024-vyelu5b1 authors: Shafiq, Anum; Lone, S.A.; Naz Sindhu, Tabassum; Al-Mdallal, Qasem M.; Muhammad, Taseer title: A New Modified Kies Fréchet Distribution: Applications of Mortality Rate of Covid-19 date: 2021-08-03 journal: Results Phys DOI: 10.1016/j.rinp.2021.104638 sha: dec742ef929f50cd131235d688f469447bc5adb1 doc_id: 755024 cord_uid: vyelu5b1 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. Symbols ( ) f y  PDF ( ) F y  CDF ( ) S y  SF ( ) h y  hrf   H t CHRF ( ; ) Q q  QF ( ; ) Q q   Quantile In statistical analysis, extreme value theory (EVT) is very valuable. The EVT was originally related to 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 models 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 FD (MO-FD) 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   G y  reference CDF for a parameter vector then the MKI family CDF is defined as where 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 [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] The remaining of this paper 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. The CDF and PDF of MKIF model are specified as 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 Just in Figs. 1and 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 along with and 0.8. Fig. 2    This section covers a valuable expansion of PDF and CDF of MKIF distribution. The    power series and exponential function could be described as Then, it follows that, using (11) and (12) to expand and 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) and (12) indicate that CDF of MKIF in (5) can be written as In addition, the following binomial theorem can be used to extend Then, applying (12) and (13) to expand the in (18)   The next result can be utilized to simulate values from the MKIF distribution. The QF of As a result, the median, as well as the upper and lower quantiles, are calculated as follows: The accompanying quantile density function is provided by the differentiation of (20) 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 moment of MKIF model can be evaluated directly by extending the PDF of the r th  MKIF as seen in (14) 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. 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 We could calculate the MLEs of the parameters and by setting all equations to zero , and solving them simultaneously. 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 , The WLSEs, and can be determined by minimizing the following , function, with respect to and respectively. , The estimates can also be obtained by solving:  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. 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 Tables 2 and 3 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 . 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 Figure 4 . 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. Table 3 . Table 4 . In current article, we suggest a novel three parameter model, entitled three-parameter modified Sur la loi de probabilité de lécart maximum Extreme value distributions: theory and applications The exponentiated Fréchet distribution. 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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 n obtained data from uniform distribution.