key: cord-0823918-itghlf0n
authors: Elbatal, I.
title: A new lifetime family of distributions: Theoretical developments and analysis of COVID 19 data
date: 2021-11-14
journal: Results Phys
DOI: 10.1016/j.rinp.2021.104979
sha: 4ee2460dc203adb4f374304fa1f70c13cefa5729
doc_id: 823918
cord_uid: itghlf0n

In parametric statistical modeling and inference, it is critical to develop generalizations of existing statistical distributions to make them more flexible in modeling real data sets. Thus , this paper contributes to the subject by investigating a new flexible and versatile generalized family of distributions defined from the alliance of the families known as beta-G and Topp-Leone generated (TL-G), inspiring the name of BTL-G family. The characteristics of this new family are studied through analytical, graphical and numerical approaches. Statistical features of the family such as expansion of density function (pdf), cumulative function (cdf), moments (MOs), incomplete moments (IMOs), mean deviation (MDE), and entropy (ENT) are calculated. The model parameters’ maximum likelihood estimates (MaxLEs) and Bayesian estimates (BEs) are provided. Symmetric and Asymmetric Bayesian Loss function have been discussed. A complete simulation study is proposed to illustrate their numerical efficiency. The considered model is also applied to analyze two different kinds of genuine COVID 19 data sets. We show that it outperforms other well-known models defined with the same baseline distribution, proving its high level of adaptability in the context of data analysis.

Several Statisticians have recently shown an interest in providing new produced families of continuous distributions by adding one or more extra shape parameters to the baseline model in order to generate models. These extra form factors increase accuracy and flexibility in data fitting. Among the most well-known generators seem to be: the Marshall Olkin G by Marshall et al. [9] , beta (B) G (BG) by Eugene et al. [1] ,Topp Leone odd Lindley G by Reyad et al. [24] , Topp-Leone (TL) G (TLG) by Rezaei [27] , TL odd log-logistic G by de Brito et al. [25] , B WeibullG by Yousof et al. [26] , The Fréchet TL G by Reyad et al. [24] , the odd Fréchet G by Haq and Elgarhy [28] , exponentiated power generalized Weibull power series family of distributions by Aldahlan et al. [29] , odd generalized NH -G by Ahmed et al. [45] , among others.

One of the continuous distributions that is appealing as a generator is the TL distribution. Topp and Leone [3] suggested this distribution. The characteristics of the TL distribution have been investigated by a number of experts, measures of reliability and stochastic ordering by Ghitany et al. [4] ; Kurtosis's conduct by Kotz and Seier [5] ; MOs of order statistics by Genc [6] ; stress-strength by Genc [7] , among others. Al-Shomrani et al. [8] developed a novel family of distributions dubbed the TLG family of distributions, which is constructed on a one-parameter distribution with bathtub shaped risk rates. The cdf is supplied by H(x; θ, ξ) = (1 − G 2 (x; ξ)) θ , θ > 0, x > 0, (1) J o u r n a l P r e -p r o o f Journal Pre-proof the corresponding pdf is h(x; θ; ξ) = 2θg(x; ξ)G(x; ξ)(1 − G 2 (x; ξ)) θ−1 .

where G(x; ξ) is a baseline cdf, g(x; ξ) is the corresponding pdf, ξ are the parameters specifying the baseline distribution, ξ > 0 and θ > 0. The survival of the T L − G family is given by

The BG family has the coming cdf and pdf

where λ > 0 and α > 0 are two extra parameters that add skewness to the produced distribution and change the tail weight.The pdf for (4) is expressed as follows:

In this article, we create and investigate a novel family of distributions by using two more shape factors in (1) to provide the produced family greater flexibility. We get a new broader family of distributions based on the beta TL G (BTLG) family of distributions by inserting (1) in (4) . The cdf and pdf of BT LG family , respectively, are given by F (x; θ, λ, α, ξ) = I (1−G 2 (x;ξ)) θ (λ, α),

and

f (x; θ, λ, α, ξ) = 2θg(x; ξ)G(x; ξ) B(λ, α)

The BTLG family includes various unique members, and it is crucial in a variety of applications ( reliability, economics, engineering and other areas of research). If X is a random variable with BT L − G pdf (7), we use the notation X ∼ BT L(θ, λ, α, ξ).Some special cases of the new family are listed in the following table 1. Table 1 : Sub families Sub families θ λ α Author T L − G family θ 1 1 Al-Shomrani et al. [8] GT L − G family -1 -New ET L − G family --1 New More paper discussed modeling for Covid-19 spread as Atangana and Araz [30, 33] , Shafiq et al. [31] , Atangana [32] , Hassan et al. [39] , Ibrahim et al. [44] , and Sindhu et al. [34, 35] . Sindhu et al. [43] discussed analysis of the left censored data from the Pareto type II distribution.

The remainder of the paper is laid out as follows: A useful expansion for the BTLG pdf and some special models are introduced in Section 2. Various structural properties including quantile function (QuF), MOs, IMOs , MDEs , residual life (RL) and reversed residual life (RRL) functions are derived in Section 3. The ENT of proposed family are provided in Section 4. The maxLL estimation procedure for the BTL-G parameters is investigated in Section 5. Parameters estimation by Bayesian method is discussed in Section 6. Simulation study is devoted in Section 7. Section 8 illustrates the flexibility and potentiality of the proposed family by means of two applications to real data sets. Finally,Section 9 offers some concluding remarks. The following result investigates useful expansions for F (x) and f (x).If | z |< 1 and b > 0 is a real non-integer ,then the following power series holds.

Using the expansion (8) in (7) , we get

again using the binomial expansion

insert (10) in (6) , the BT L − G density reduces to

Another formula can be extracted from pdf (11) as follows

where d m = bm m+1 and π (δ) (x) = δg(x; ξ)G(x; ξ) δ−1 denotes the exponentiated-G (exp-G) density with power parameter δ.Consequently , Eq (12) represents the BT L − G density as an infinite linear combinations of exp-G densities. Similarly, the cdf of the BT L − G family can also be expressed as a mixture of exp -G cdfs where

where Π (m+1) (x) is the exp-G cdf with power parameter (m + 1).

Many special members of the BTL-G family are of potential interest, for tractability of the related functions and flexibility reasons. Some of them are listed in Table 2 , defined with their cdf and pdf for the sake of place.

J o u r n a l P r e -p r o o f Journal Pre-proof 

The pdf of BT LLo distribution are

The pdf of the BT LE model (for x > 0) are Figure 1 represents the PDF of the BTLE distribution might be right-skewed, symmetric, or decreasing curves. The hazard rate of the BTLE distribution comes in a variety of shapes, including constant, decreasing, and upside down curves, all of which are appealing features for any lifespan model. 

The pdf of the BT LR model (for x > 0) are

This section shows important distributional and structural properties satisfied by the BTL-G family. The conventional MO and MO generating functions of the BTLG family are developed. The varied orders for the MOs are particularly beneficial for determining device anticipated life time, skewness (SK), and kurtosis (KU) in a given collection of observations occurring in reliability applications.

The r th ordinary MO of X can be obtained from (12) as follows

where W r (m+1) denotes the exp-G random variable with power parameter (m + 1)). The measures of SK and KU can be derived from the n th central MOs , say M n (x) of X , where

The BT L − G QuF , say x = Q(u) can be obtained by inverting (6) as follows

where Q G(u) denotes the QuF corresponding to G(x), and G −1 (.) is the inverse of the baseline cdf . One of the earliest SK measures to be suggested is the Bowley SK (Kenney and Keeping, [2] ) defined by

On the other hand , the Moors KU (Moors, [11] ) based on Qus is given by

.

where Q(·) represents the QuF. We used the BTLE distribution as special case of proposed family. Figures  2 and 3 for BTLE distribution show that measures are meaning of this statement sensitive to outliers and they exist even for distributions without MOs. The MO generating function of X can be computed from equation (12) as follows

where M (m+) (t) is the MO generating function of W (m+1) . Consequently, M X (t) can be easily determined from the exp-G generating function. Further, the incomplete moments play an important role for measuring inequality. For example, the first incomplete moment can be used to obtain the formulas of Lorenz and Bonferroni curves.The s th IMOs of X defined by φ s (t) for any real s > 0 can be expressed from (12) as (19) Eq (19) denotes the s th IMOs of W (m+1) . The MDEs about the mean µ = E(X) and the MDEs about the median M are (6) and φ 1 (t) is the first complete MO given by (19) with s = 1. We can determine δ 1 (x) and δ 2 (x) by two techniques , the first can be obtained from (12) 

The r th -order MO of the RL is given by

The mean RL (M RL) of BT L−G family of distributions can be computed by setting r = 1 in the above equation, defined as

The r th order MO of the RRL can be computed by

The mean activity time (M AT ) of the BT LG family of distributions can be determined by setting

J o u r n a l P r e -p r o o f Journal Pre-proof

The Rényi ENT is defined by (ρ > 0, ρ = 1)

Using (7), applying the same procedure of the useful expansion (13) and after some simplifications, we get

Thus Rényi ENT of BT L − G family is

The MaxLEs have appealing features and may be used to create confidence intervals and regions, as well as test statistics. Only complete samples are used to calculate the MaxLEs of the parameters of the BT L − G family of distributions. Assume x 1 , ..., x n be a random sample of size n from the BT L − G distribution given by (6) . Let Φ = (θ, λ, α, ξ) T be q × 1 vector of parameters. The logarithm of likelihood function is

The score vector components ,say, U (Φ) = ∂Ln ∂Φ = ( ∂Ln ∂θ , ∂Ln ∂λ , ∂Ln ∂α , ∂Ln ∂ξ ) T are given by

and

where z (ξ) (.) means the derivative of the function z with respect to U . Setting these equations to zero, U θ = U λ = U α = U ξ k = 0, and solving them simultaneously yields the MaxLE ( Φ) of Φ.

J o u r n a l P r e -p r o o f Journal Pre-proof 6 

As random and parameter uncertainties are represented by a previous joint distribution that is established prior to the data collected on the failure, the Bayesian approach deals with the parameters. The ability to incorporate prior knowledge into research makes the Bayesian method very useful in the analysis of reliability as Sindhu and Atangana [36] , as one of the main problems associated with reliability analysis is the limited availability of data. In the θ, λ, α, and ξ parameters, as prior gamma distributions, we have to use the insightful before. The θ, λ, α, and ξ independent joint prior density function can be written as follows:

where Φ = (θ, λ, α, ξ) T . From the likelihood function and joint prior function, the joint posterior density function of θ, λ, α, and ξ is obtained. The joint posterior of the distribution of BTL-G family can then be written as

Using the most common function for symmetric loss, which is a function for squared error loss. Bayes estimators ofθ,λ,α andξ based on the squared error loss function are defined by the squared error loss function.

In cases when under-estimate is more dangerous than over-estimation, or when positive and negative estimation errors produce distinct effects, under-estimation should be penalized severely. This fine can be obtained by applying Varian's LINEX loss function, which was devised as an asymmetric loss function and is defined as

where is the shape parameter andΦ is estimate of the parameter Φ. When > 0, the loss from overestimation is greater. When < 0, the loss from under-estimation is greater. The LINEX loss function is approximated as a squared loss function for small values of . The risk function associated withΦ under the LINEX loss function is obtained

It should be noted that the integrals given by (27) [37, 38, 41, 42] and Abd El-Raheem et al. [14] . To generate random samples of conditional posterior densities from the BTL-G family, we use the MH within the Gibbs sampling. We need the conditional distribution of posterior as following:

J o u r n a l P r e -p r o o f

and

We present a brief Monte Carlo simulation research in this section to evaluate the MLE and Bayesian for BTLE distribution parameters. Inverting the cdf formula makes it simple to recreate the BTLE distribution.

The inverse cdf is used to generate the random numbers. The simulation results and inverse procedure are acquired using the statistical software R and the (uniroot) library with command stats. where Φ = (θ, λ, α, ξ) T . Table 3 shows the summary measures of the simulated data. The simulation results of the methods mentioned in this study for point estimation are summarized in Tables 3, 4 

The COVID-19 data are provided in this section to evaluate the consistency of the BTLE distribution. We compare the proposed model with other related models such as exponential Lomax (EL) [El-Bassiouny et al. [20] ], Kumaraswamy Weibull (KW) Cordeiro et al. [15] , Kumaraswamy Inverted Topp-Leone (KITL) [Hassan et al. [16] ], Odd Weibull inverse Topp-Leone (OWITL) [Almetwally [17] ], New Exponential-X Fréchet (NEXF) [Alzeley et al. [18] ] and Weibull-Lomax (WL) [Tahir et al. [19] ]. Tables 8 and 10 We conclude that all the models used are suitable for these data, as the P-Value is greater than 0.05, and we can look at the following illustration in two Figures 6 and 7 that makes this conclusion. In addition, J o u r n a l P r e -p r o o f Journal Pre-proof we conclude that, when compared to other models, the BTLE is an appropriate model for these data. Also, the estimated PDF with histogram and estimated CDF with empirical CDF plots of BTLE distribution are shown in Figure 4 and PP-Plot and QQ-Plot of BTLE distribution in Figure 5 . Additionally, we used the total time on test (TTT) plot (Aarset [23] ) to determine the shape of the empirical hazard rare function. The TTT plots of the data I set is given in Figure 8 . Also Figure 8 show box plot of data I set and conclude these don't have outliers. We also note from the third figure 8, that the estimated hazard of BTLE distribution with empirical hazard is increasing, and since the estimated hazard of BTLE distribution with empirical hazard changes almost steadily in the last period, this is due to the inactivity of the virus in this period as it was already. This article proposes the Beta Topp Leone G family as a generalization of the Beta Topp Leone distribution. Quantile function, moments of residual and reversed residual life, moment generation, incomplete moments, and Entropy are some statistical features of the BTL-G Family. This study proposes a novel generalization of exponential, Lomax, and Rayleigh distributions based on the BTL-G Family. Estimation techniques such as Bayesian and ML are discussed. Under asymmetric and symmetric loss function (LINEX and SE), the Bayesian estimator is derived. The numerical analysis is used for the BTLE distribution, which is a subset of the BTL-G Family. The goal of a Monte Carlo simulation study is to see how well estimations perform.

In the vast majority of circumstances, we find that Bayesian estimates outperform identical alternative estimates. Two genuine COVID-19 data sets were collected from Saudi Arabia at separate dates, and they demonstrated that the BTLE distribution is an appropriate model for this data when compared to other competing distributions. Data Availability The data used to support the findings of this study are included in this paper.

The author declare no conflict of interest.

Beta-normal distribution and its applications

Mathematics of Statistics, Part 1

A family of J-shaped frequency functions

On some reliability measures and their stochastic orderings for the Topp Leone distribution

Kurtosis of the Topp-Leone distributions

Moments of order statistics of Topp-Leone distribution

Estimation of P (X > Y ) with Toppâe"Leone distribution

Topp Leone family of distributions: some properties and application

A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families

The Topp Leone odd Lindley-G family of distributions: Properties and applications

A quantile alternative for kurtosis

Applying Transformer Insulation Using Weibull Extended Distribution Based on

Power Weibull Distribution: Different Methods of Estimation Based on Type-I and Type-II Censoring

Accelerated life tests for modified Kies exponential lifetime distribution: binomial removal, transformers turn insulation application and numerical results

The Kumaraswamy Weibull distribution with application to failure data

Kumaraswamy Inverted Topp Leone Distribution with Applications to COVID-19 Data

The Odd Weibull Inverse ToppLeone Distribution with Applications to COVID-19 Data

Statistical Inference under Censored Data for the New Exponential-X Fréchet Distribution: Simulation and Application to Leukemia Data

The Weibull-Lomax distribution: properties and applications

A new inverted topp-leone distribution: applications to the COVID-19 mortality rate in two different countries

A New Transmuted Generalized Lomax Distribution: Properties and Applications to COVID-19 Data

A new extended rayleigh distribution with applications of COVID-19 data

How to identify a bathtub hazard rate

The Topp Leone odd Lindley-G family of distributions: properties and applications

Topp-Leone odd log-logistic family of distributions

The extended odd Fréchet family of distributions: properties, applications and regression modeling

Topp-Leone generated family of distributions: Properties and applications

The odd Fréchet-G family of probability distributions

Exponentiated power generalized Weibull power series family of distributions: Properties, estimation and applications

Modeling third waves of Covid-19 spread with piecewise differential and integral operators: Turkey, Spain and Czechia

A new modified Kies Fréchet distribution: Applications of mortality rate of Covid-19

A novel Covid-19 model with fractional differential operators with singular and non-singular kernels: Analysis and numerical scheme based on Newton polynomial

Modeling and forecasting the spread of COVID-19 with stochastic and deterministic approaches: Africa and Europe

On the analysis of number of deaths due to Covid-19 outbreak data using a new class of distributions

Exponentiated transformation of Gumbel Type-II distribution for modeling COVID-19 data

Reliability analysis incorporating exponentiated inverse Weibull distribution and inverse power law. Quality and Reliability Engineering International

Bayesian estimation for Topp Leone distribution under trimmed samples

Bayesian inference from the mixture of half-normal distributions under censoring

Kumaraswamy Inverted Topp-Leone Distribution with Applications to COVID-19 Data

Alpha Power Weibull Distribution: Different Methods of Estimation Based on Type-I and Type-II Censoring

On the Bayesian analysis of censored mixture of two Topp-Leone distribution

Mixture of two generalized inverted exponential distributions with censored sample: properties and estimation

Analysis of the Left Censored Data from the Pareto Type II Distribution

Parameter Estimation of Alpha Power Inverted Topp-Leone Distribution with Application

Odd Generalized NH Generated Family of Distributions with Application to Exponential Model