key: cord-0941220-p2xzar52
authors: Sindhu, Tabassum Naz; Shafiq, Anum; Al-Mdallal, Qasem M.
title: Exponentiated Transformation of Gumbel Type-II Distribution for Modeling COVID-19 Data
date: 2020-10-21
journal: nan
DOI: 10.1016/j.aej.2020.09.060
sha: 75ad7cc0cf003445da55a8743120f9e3b839dcc1
doc_id: 941220
cord_uid: p2xzar52

The aim of this study is to analyze the number of deaths due to COVID-19 for Europe and China. For this purpose, we proposed a novel three parametric model named as Exponentiatedtransformation of Gumbel Type-II (ETGT-II) for modeling the two data sets of death cases due to COVID-19. Specific statistical attributes are derived and analyzed along with moments and associated measures, moments generating functions, uncertainty measures, complete/incomplete moments, survival function, quantile function and hazard function etc. Additionaly, model parameters are estimated by utilizing maximum likelihood method and Bayesian paradigm. To examine efficiency of the ETGT-II model a simulation analysis is performed. Finally, using the data sets of death cases of COVID-19 of Europe and China to show adaptability of suggested model. The results reveal that it may fit better than other well-known models.

The literature includes a variety of models for analyzing the lifetime data. In extreme value (e.v.) study of extreme events, Gumbel model, also recognized as type-1 e.v. model, has gained considerable research attention, in particular over years. Pinheiro and Ferrari [1] investigation can be considered for an overview of the recent innovations and implementations of Gumbel model. There is no doubt that Gumbel type-II model has not been commonly used in statistical modeling until now, and reason might not be far from its lack of data modeling …ts. Traditional probability models are usually criticized for their lack of …t in complex data 1 Corresponding author: e-mail address: q.almdallal@uaeu.ac.ae 1 simulation. Through this context, users of such model in various areas in general, and in particular mathematics and statistics, have been fantastically inspired to establish sophisticated models of probability from basic models. Exponentiated models have been implemented to address issue of lack of …tting typically experienced when using the regular models of probability to model complex data. Findings from this development have always proved to be more accurate than one based on standard models. Exponentiating models are also a strong statistical modelling technique which o¤ers an e¢ cient way to incorporate extra shape parameter to regular model to abtain versatility and robustness. This approach of generalizing models of probability is identi…able to study of Gupta et al. [2] . He presented EE (exponentiated exponential) model via simpli…ed version of traditional exponential model by simply increasing cdf to a power of constant. Since development of EE model, exponentiated models have gained rational features in modelling data from di¤erent complicated phenomena. A large number of regular models of probabilities have its respective exponentiated forms. Gupta et al. [2] proposed Weibull exponentiated (WE) model via generalization of regular Weibull model. Nadarajah and Kotz [3] revised the approach of Gupta et al. [2] and described EF (Exponentiated Frechet) model by generalizing model of Frechet. Nadarajah [4] developed Exponentiated Gumbel (EG) model as a generalization of regular Gumbel model using same approach (Nadarajah and Kotz [3] ). Exponentiated Weibull family model being a generalization of Weibull family model was developed by Mudholkar and Srivastava [5] . Ashour and Eltehiwy [6] established EPL (exponentiated power Lindley) model by generalizing model of power Lindley and many more. Here it should be observed that almost all extensions added extra parameters to base model. Such outcomes manages to complexities in future assumptions. On other hands, additional parameters o¤er more versatility but at same time introduce uncertainty in estimation of parameter(s). Possibly, taking that into consideration, Kumar et al. [7] suggested a DUS transformation, in order to achieve new model. If G (y) be baseline cdf, above mentioned transformation o¤ers a new cdf F (y) as shown below:

and consider exponential model as base model and they named it exponential DUS model.

It had observed that it provides a model containing non-constant hazard rates. The bene…t of using above transformation is that new model retains cheracteristic of being parsimonious in parameter, since it does not include any extra parameters.

Thus, current study is aimed to propose a new model by using DUS transformation by introducing a new extra parameter. And for this purpose we use a traditional Gumbel type-II model [8] [9] [10] [11] [12] as a basic model to a wider distribution class, in order to enhance its e¢ ciency and promote its suitability, in modelling varieties of complex datas. The proposed model is known as the Exponentiated transformation of Gumbel Type-II (ETGT-II).

2

The ETGT-II model can be obtained as follows: Let Y be a r.v. (random variable) with G(y) cdf and let g(y) be respective pdf to be consider as basic model.

and corresponding pdf is

ETGT-II model is more ‡exible, since it includes extra shape parameter to regulate the transformation.

In December, Wuhan City, China con…rmed the …rst case of respiratory disease, pneumonia, with symptoms close to the serious acute 34 respiratory SARS-CoV [13] . COVID-19

symptoms involved fever, shortness of breath, cough and occasional watery diarrhea [14] . In February 2020, 17,238 infections cases of COVID-19 and 361 deaths in China were reported [15] . The objective of current study is to analyze the number of daily deaths due to COVID- 

For illustration, the baseline distribution is assumed to be a Gumbel type-II distribution with parameters ; having pdf as g (yj ; ) = y 1 exp y ; ; y > 0:

and corresponding cdf as G (yj ; ) = exp y ; ; ; y > 0:

Using transformation given in equation (1), the pdf and cdf of proposed model, subsequently   mentioned as Exponentiated transformation of Gumbel Type-II (ETGT-II) 

It is simple and clear to notice that F (yj ; ; ) di¤erentiable and increases in 0 to 1 and lim y!0 F (y) = 0 and lim y!1 F (y) = 1:

The R(y) = P (Y > y) = 1 F (y) determines reliability or survival function of a r.v. Y .

It could be de…ned as the probability that a device does not fail within a certain prescribed time t. The reliability function of ETGT-II is de…ned as R(yj ; ; ) = P (Y > y) = e e e y e 1 :

Hazard rate function 

The odd ratio is de…ned as

Where R y (y) and h y (y) is de…ned in (7) and (8):

The cumulative hrf is de…ned as

Therefore, Fig. 2 shows plots of cdf of ETGT-II distribution. Fig. 3 shows the decreasing and upside-down bathtub pattern of hrfs of ETGT-II distribution. 

The shape of model is a signi…cant attribute as it provides an idea about nature of the model.

For analysis of shapes of hazard rate we have implemented Glaser [16] . He established the

de…nes density function and f 0 (y) de…nes …rst derivative of 5 f (y) in relation to y, then following theorem has been stated. & 0 (y) < 0 for all y > y 0 , then h (y) is uni-modal (called upside bathtub (UBT)).

Proof: For the suggested distribution, we have:

and & 0 (y) = e y y (1 + ) (y ) + [y (1 + ) ] y 2(1+ ) e y :

It can easily be veri…ed that the given two situations may arise:

From (12), we can easily noted that & 0 (y) < 0 for all y > 0: Hence the proposed model has decreasing failure rate and has observed from Fig. 3 .

Also, & 0 (y) > 0 for all y 2 (0; y 0 ), & 0 (y 0 ) = 0; and & 0 (y) < 0 for y 2 (y 0 ; 1); where, y 0 can be attained by solving the below equation

We have drawn hazard function (8) 

Let Y 1 ; Y 2 ; :::; Y n be a r.s. of size n from an absolutely continuous model with f (yj ; ; ) pdf and F (yj ; ; ) cdf. Limiting model of sample maxima Y n;n = max (Y 1 ; Y 2 ; :::; Y n ) is a long stand area in usages of probability and statistics. First, we consider the below asymptotical outcomes for Y n;n (see Arnold et al. [17] ).

For the maximum order statistic Y n;n , here lim n!1 P (Y n;n c n + d n t) = e e t ; 1 < t < 1;

where c n = F 1 1 1 n and d n = 1 nf (cn) ; if 

The next principle results the limiting models of the highest order statistics from ETGT-II model.

Proposition 1: Let Y n;n be the highest order statistics from ET GT II ( ; ; ) model.

Then lim n!1 P (Y n;n c n + d n t) = e e t ; 1 < t < 1;

where c n = F 1 1 1 n and d n = 1 nf (cn) ; if and f (:) ; F 1 (:) ; respectively are given by Equations, (5) and (39).

Proof: For ETGT-II model, we have Hence, the statement follows from (15) and (16) . 

where = n! (m 1)!(n m)! : Thus from (5), (6) and (19) the pdf of Y (m) results as

The cdf of Y (m) is

then cdf of m th order statistic, Y (m) of ETGT-II model is given by

In general, cdfs of Y (n) and Y (1) are given as

Then we obtain from (24) and (25)

where Q (:) is qf of Y: Hence, from (26) and (39), we cannot de…ne qfs of Y (n) and Y (1) in closed-form. In the case of i.i.d. random variables, it is possible to attain an expression for r th ordinary moment of OS, when r < 1: So, as Silva et al. [18] , we can represent r th

where I ¾ j (r) = 1 R 0 ry r 1 [1 F (y)] j dy: Speci…cally, for ETGT-II model, we get

The next outcome shows the r th moment of m th OS Y (m) of pdf (5) can be described as

where | j;m;n = n P j=n m+1

where last integral can be evaluated numerically.

It is of interest to de…ne, for practical purposes, the intrinsic stochastic ordering (SO) of such members according to the parameters when we engage with a general family of models. In this respect, some distributional functions can be used as function of the cdf, hrf, likelihood ratio function. Now, we're concentrating on the order of likelihood ratios de…ned as below.

For r.v. X and Y , we state, X 4 lr Y; if ratio of two respective pdfs a decreasing function in y.

Stochastic ordering of continuous positive r.v. is a signi…cant mechanism for evaluating relative behavior. We must remember certain basic de…nitions. It is assumed that r.v. X is smaller than Y in the

The consequences below are well known, see Ross [19] chapter 9:

The ETGT-II models are ordered with strongest "likelihood ratio"ordering as given below.

Hence it shows that X lr Y; and according to above equation (31) these both are X hr Y;

now di¤erentiate above equation w.r.t y; we obtain d dy

Hence it shows that X lr Y; and according to above equation (31) these both are X hr Y;

X st Y also hold. 

13

After making the transformation t = x using the result (44) (5), it is obtained as follows.

y r dF (yj ; ; ) ; r = 1; 2; :::

The following outcome provides an expression for the r th non-central moment r of Y in terms of gamma function. 

After using Taylor expansion of the function exp f exp f y gg, we have

Let t = (1 + l) y then y 1 dy = dt 1+l , we have after some algebraic manipulation

After integrating, we obtain the …nal result. That completes the proof.

In particular, the mean of Y :

14 and allow the expression of variance of Y :

The r th central moment of the class can be obtained by using Equation (46), as:

Again using the above substitution, we have

The mgf (moment generating function) gives the base for an alternative path to analytical outcomes compared to working speci…cally with pdf and cdf and is broadly used in characterization of distribution. The mgf of ETGT-II distribution may be indicated as M (yj ; ; ) = 1 X q=0 t q q! q (yj ; ; ) ;

The characteristic function of Y can be evaluated as 

After using Taylor expansion, we have

Hence, we obtain

That concludes the objective evidence.

The fgf of ETGT-II model is extracted as

so, we can compose the integral component above as

Which establishes objective result.

The incomplete non-central moments of the distribution play a major role in evaluating inequality, including Lorenz and Bonferroni's income quantiles and curves, that are centered on the incomplete distribution moments. 

After using Taylor expansion of the function e e y , and substitute t = (1 + l) y ; we

where, (a; x) = where (x; a) =

x R a t a 1 e t dt is lower incomplete gamma function.

In predictive inference, it is advantageous in interaction with lifetime distributions to evaluate the conditional moments E (Y r j Y > t)j r=1;2;::: : The r th conditional moment of Y is provided as

The mean deviations of Y about the mean = E(Y ) and the median~ can be stated as 

where F ( ) is speci…ed in (6):

Information generating function, Renyi entropy, Verma, Tsallis and other entropies for the distribution of ETGT-II model are being investigated in this section.

For the ETGT-II model the information generating function for Y is estimated as:

Now making the transformation (! + q) y = t in (74) and after a little simpli…cation we

haveĨ

:

(75)

Entropy is a signi…cant idea in several relevant …elds communications, thermodynamics, information theory, statistical mechanics, topological dynamics, measure-preserving dynamical systems, etc, as a calculation of various characteristics like disorder, energy that cannot produce work, randomness, uncertainty, complexity, etc. There are several concepts of entropy and they are not necessarily perfect for all applications.

The Renyi entropyR ! (Y ) for Y with ETGT-II model is

where f ! (yj ; ; ) = y 1 e y e e y (e 1)

By using the above information, we have

Now substituting, (! + q) y = t; the above integral becomes

After simpli…cation, we have

Finally, Renyi entropy (RE) becomes

It is important to mention that Shannon entropy (SE) of a r:v: Y is obtained as a special case of RE for ! ! 1.

The Verma entropy V ; (Y ) for Y with ETGT-II model is

where f + 1 (yj ; ; ) = y 1 e y e e y (e 1)

It is important to notice that, when ! 1; in (82), it reduces to the RE. On the other hand, if ! 1 and ! 1; in (82), then it approaches to the Shannon entropy. By using the above information, we have

Now substituting, ( + 1 + q) y = t; the above integral becomes

After simpli…cation, we have

(87)

As, f ! (yj ; ; ) and

1 R 0 f ! (yj ; ; ) dy are calculated in (77)-(80) respectively. Therefore, by using these information, T ! (Y ) takes the following form

Classical Shannon Entropy has been generalized in many directions one of them is ! 1 generalized entropy introduced by Mathai and Haubold [20] and is de…ned bỹ

Similar arguments to f 2 ! 1 gives f 2 ! 1 (yj ; ; ) = y 1 e y e e y (e 1)

q! y (2 ! 1 )( +1) e (2 ! 1 +q)y ; (91) by using the above information, we get

Therefore, the …nal form of the above integral is

(93)

Kapur entropyĨ ; (Y ) of Y with ETGT-II model is de…ned as

Similar arguments to f ! incorporating in Eq. (94), we get the following results

6.2.6 !-Entropy

As, f ! (yj ; ; ) and

1 R 0 f ! (yj ; ; ) dy are calculated in (77)-(80) respectively. Therefore, by using these information, H ! (Y ) takes the following form

Three dimenional behavior of the entropies of the ETGT-II model are plotted in Fig. 7 . 

The maximum likelihood estimates (MLEs) are provided by optimizing this equation according to ; and . They are also characterized as the maximum of the log-likelihood function de…ned by l yj ; ; = log L (yj ; ; ) :

The log-likelihood function for the ETGT-II model is provided by the data set y 1 ; :::; y n . l yj ; ; = n log e 1 + n log ( ) + n log ( ) + n log ( ) ( + 1)

By solving the non-linear likelihood equation, we obtain the MLEs of the parameters ;

and obtained by di¤erentiating (100).We obtain the components of score vector ; ; = 

In this section, we continue by presenting estimation of suggested structure parameters by a Bayesian mechanism. We assume the parameters ; , and are random variables. Here, the following independent priors are assumed as gamma ( 1 ; 1 ) ; gamma ( 2 ; 2 ) ; and gamma ( 3 ; 3 ) where i ; and i 2 R + ; i = 1; 2; 3: The joint posterior density of ; , and has the following form

The Bayes estimator (BE) of the parameter is provided under squared error loss function (SELF ) as follows.

in a similar fashion^ BE = E( j x); and^ BE = E( j x): In addition, the Bayes risk is deter- One can notice that, density plots of ; and are slightly symmetrical. In addition, Bayes estimates, 95% Bayesian intervals, posterior variance are also reported in Table 3 .

Furthermore, all Bayes estimates are within the 95% Bayesian intervals. Same behaviour is obtained for the implementation of real life data sets.

We present two examples in this section to explain the e¢ ciency of the proposed model.

We use R software to demonstrate improved e¢ ciency of the ETGT-II model and numerical calculations. We consider the following distributions, for comparative purposes: 11 . PP-graphs for daily deaths due to COVID-19 in Europe.

28 Fig. 12 . PP-graphs for daily deaths due to COVID-19 in China.

We estimate the unknown parameters by maximum likelihood for each model. Table   5 lists the MLEs of the above models with their respective standard errors (evaluated by inverting the information matrices). The calculations were made using the R programming language. Posterior summaries and densities plots of the ETGT-II model for data set I and II are given in Table 8 and plotted in Fig. 15 . 13 . The curves log-likelihood function of ( ; ; ) for data set I.

Fig. 14. The curves log-likelihood function of ( ; ; ) for data set II. 

The authors would like to thanks the reviewers and the editors for their comments that helped to improve the article substantially.

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