Type I Half Logistic Burr X-G Family: Properties, Bayesian, and Non-Bayesian Estimation under Censored Samples and Applications to COVID-19 Data

Statistics Department, Faculty of Science, King AbdulAziz University, Jeddah 21551, Saudi Arabia Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt Department of Statistics, Faculty of Business Administration, Delta University of Science and Technology, Mansoura, Egypt .e Higher Institute of Commercial Sciences, Al Mahalla Al Kubra 31951, Algarbia, Egypt


Introduction
Statistical researchers have been encouraged in recent years to propose new broad families of continuous univariate distributions and to focus their efforts on improving their desired characteristics. For the time being, there is still a need for providing wider classes of distributions in order to provide them with greater flexibility and precision when fitting data. Some of the more recent generators sounding in the literature are the beta-G [1], type I half logistic [2], odd exponentiated half logistic G [3], Marshall-Olkin Burr X-G [4], generalized odd log-logistic-G [5], beta Burr type X − G [6], new generalized odd log-logistic-G [7], generalized Burr X-G [8], type II half logistic [9], the transmuted odd Fréchet-G family in [10], Kumaraswamy-type I half logistic [11], and Burr X-exponential-G [12], among others.
Reference [13] proposed a new simple family of distributions with cumulative distribution function (CDFu) and probability density function (PDFu) using the Burr X as generator; the so-called Burr X − G family is as follows: h BX (x; θ) � 2θg(x; δ) G(x; δ) 3 G(x; δ)e − (G(x;δ)/G(x;δ)) 2 1 − e − (G(x;δ)/G(x;δ)) 2 θ− 1 , (2) where g(x; δ) and G(x; δ) are the PDFu and CDFu of any baseline distribution based on a parameter δ. e type I halflogistic-G (TIHL − G) family is [2] a represented family with an additional positive parameter lambda > 0. e CDFu of the TIHL − G distribution family is e corresponding PDFu is e failure (hazard) rate function is We intend to benefit from the combined features of the Burr X-G and the TIHL-G in this work by introducing a new generated family of distributions known as the type I halflogistic Burr X-G (HLBX -G). We hope that the new family will provide more flexibility and attract a broader range of applications in reliability, engineering, and other research areas.

Useful Expansion
e following results are useful for expansions of f(x) and F(x). If |z| < 1 and b > 0 is a real noninteger, then the following power series holds: When we apply (10) to the final word in (7), we obtain Using (11) in (12), we get e power series expansion of e − (k+1)(G(x;δ)/G(x;δ)) 2 is Mathematical Problems in Engineering By adding (14) to (13), we get Making use of the generalized binomial expansion to (1 − G(x; δ)) − (2m+3) , we can write (16) e HLBX − G density function may be represented as an endless combination of Expo-G density functions by substituting (16) into (15) where and π (2(m+1)+d) (x) � (2(m + 1) + d)g(x)G 2m+d+1 (x) is the expo-G PDFu with power parameter (2(m + 1) + d). As a result, numerous mathematical and statistical features of the HLBX − G distribution are evident from those of the exp-G distribution. Similarly, the HLBX − G family CDFu may be represented as a combination of exp-G CDFus where where Π (2(m+1)+d) (x) is the exp-G cdf with power parameter (2(m + 1) + d).

Some HLBX-G Family Special Models.
We present three submodels of this family based on the baseline distributions: Lomax, exponential, and Rayleigh. ese models' CDFu and PDFu files are given in Table 1 Plots of the HLBXLo densities are represented in Figure 1.
e CDFu and PDFu of the HLBXE model (for x > 0) are Plots of the HLBXE densities are represented in Figure 2.

Half-Logistic Burr X Rayleigh (HLBXR) Distribution
e CDFu and PDFu of the HLBXR model (for x > 0) are Plots of the HLBXR densities are represented in Figure 3.

Fundamental Properties
We looked at the statistical properties of the HLBX − G distribution; Mos, InMos, MeD, Lo, and Bo curves; ReL and RReL functions; and PrWMs in this section.

Moments.
e r th ordinary Mo of X can be obtained from (17) as Mathematical Problems in Engineering 5 where Z (2(m+1)+d) . e exp-G random variable with the power parameter (2(m + 1) + d) is denoted. For ξ > 0, the second formula for the r th moment follows from (17) as , which is numerically calculable in terms of the baseline QuFu, i.e., For most parent distributions, this integration can be calculated numerically. Skewness and kurtosis can be calculated using the n th central Mo, say M n (x) of X, where

Remark 1.
If X have the ordinary Mo in (23), the MoGFu of X can be investigated by using two formulae. e first formula can be computed from equation (17) as where M (2(m+1)+d) (t) is the MoGFu of Z (2(m+1)+d) . As a result, MX(t) may be simply calculated from the exp-G generating function. e following is a second alternative formula that may be obtained from (17): where φ(t, ε) � ε 1 0 u ε− 1 e tQ G (u) du can be computed numerically from the baseline quantile function, i.e., Q G (u) � G − 1 (u). Figure 4 show the mean, variance, skewness, and kurtosis for HLBXE model.

Incomplete Moments.
e MeDs, Bo, and Lo curves, and other applications rely heavily on the first InMos. ese curves have a wide range of uses, including economics, demography, and medicine.
is is obvious not only in econometrics research, but also in other disciplines. For every real s > 0, the s th InMos of X specified by η s (t) may be calculated from (17) as  (6), and η 1 (t) is the first InMo given by (27) with s � 1. We can determine δ μ (x) and δ M (x) by two techniques; the first can be obtained from (17) as e second technique is given by which can be computed numerically and For a positive random variable X, the Lo and Bo curves, for a given probability p, are given by L(

Residual Lives.
e r th order Mo of the ReL is given by Mathematical Problems in Engineering (MReL) of HLBX − G family s can be obtained by setting r � 1 in equation (30), defined as e well-known formula can be used to calculate the r th order Mo of the RReL (or inactivity time): (33)

Probability Weighted
Moments. e (r, s) th PrWMos of the HLBX − G family is given by using equations (6) and (7), and with a little math, we can get where erefore, the (r, s) th PWMs of the HLBX − G family can be expressed as Using (7), applying the same procedure of the useful expansion (17) and after some simplifications, we get where us, Rényi entropy of HLBX − G family is defined as

Statistical Inference under Type II Censored Sample
Reference [16] examined the two most prevalent censoring systems, known as Type I and Type II censoring schemes. In Type II censoring, a life test is stopped after a specific number of failures. n and r are fixed and predefined in this case, while T � xr is a random variable. See [17] for further details.

Maximum Likelihood
Estimation. e MLL has desirable features and may be used to calculate confidence intervals and test statistics. In both the Type II CS and the special case (full sample if r � n ), we compute the MLL estimates (MLE) of the parameters of the HLBX − G family. Let x 1 , x 2 , . . . x r , . . . x n be a n-sample random sample from the HLBX − G distribution provided by (7). We spoke about (n − r) observations, where r is the number of the uncensored items. Let ψ � (λ, θ, δ) T be q × 1 vector of parameters. e likelihood function of HLBX-G family under Type II CS can be written as where and δ k is the k th element of the vector of parameters ψ. e MLE of parameters λ, θ, and δ is obtained by setting U λ � U θ � U δ � 0 and simultaneously solving these equations to produce the MLL estimators. ese equations cannot be solved analytically; however, they can be solved numerically using iterative approaches with statistical software.

Bayesian Estimation.
e prior distribution and the loss function (LoFu) are both used in the Bayesian estimation technique. All parameters are regarded as random variables with a particular distribution known as the prior distribution in this technique. We must select one if no prior knowledge is provided, which is typically the case. We picked independent gamma distributions as our priors since the prior distribution is crucial in parameter estimation. e LoFu, on the other hand, plays an important role in the Bayesian approach. Most Bayesian inference techniques are based on symmetric and asymmetric LoFus. Two of the most frequent symmetric LoFus are the squared error and the linear exponential (Linex) LoFus. e independent joint prior density function of ψ can be written as follows: Reference [18] discussed how to elicit the hyperparameters of the informative priors. From the MLEs (λ B , θ B , δ B ), we will get these beneficial priors by multiplying the estimate and variance by the inverse of the Fisher information matrix (FIM_ij) of ψ, say (λ B , θ B , δ B ). By equating mean and variance of gamma priors, the estimated hyperparameters can be written as a i � ψ i 2 /FIM ii and e joint posterior PDFu of ψ is obtained from LL function and joint prior function: Mathematical Problems in Engineering en the joint posterior of HLBX-G family under Type II CS can be written as e Bayes estimators of ψ, say (λ B , θ B , δ B ) based on squared error LoFu, is given by e Bayes estimates of the unknown parameters psi under the Linex LoFu may be calculated as follows: ψ Linex � (− 1/u)log(E(e − uψ |x)). See, for example, [16,18] for more information on Bayesian estimation. It is worth noting that the integrals (47) cannot be obtained explicitly. As a consequence, we estimate the value of integrals using the Markov Chain Monte Carlo (MCMC) approach.
Gibbs sampling and, more generally, Metropolis within Gibbs samplers are significant MCMC subclasses. Two popular MCMC techniques are the Metropolis-Hastings (MH) algorithm and Gibbs sampling. e MH algorithm, like acceptance-rejection sampling, evaluates whether a candidate value can be created from a proposal distribution throughout each iteration of the algorithm. e following are the MH inside Gibbs sampling stages that we used to produce random samples from conditional posterior densities of the HLBX-G family in a Type II CS: where e MH algorithm (Algorithm 1) generates a sequence of draws from this distribution.

Applications
ree real-world COVID-19 data applications from different countries are presented in this section to test the goodness of the HLBX-G family distributions. e HLBXE, HLBXL, and HLBXR models are compared with other related models such as Weibull-Lomax (WL) [19], Gompertz Lomax (GL) [20], exponentiated power Lomax (EPL) [21], Kumaraswamy exponentiated Rayleigh (KER) [17], Lomax, exponential and Rayleigh distributions. Tables 2-4 show MLE and standard errors (StEr) for all parameter of models. Also, these tables provide Kolmogorov-Smirnov (D1) statistic along with its P value (D2), Cramér-von Mises (D3), and Anderson-Darling (D4) for all models fitted based on three real datasets of COVID-19 data with different countries as Italy, Canada, and Belgium, where these data are formed of drought mortality rate. Furthermore, the histograms of the three datasets are shown in Figures 5-7. e three datasets were obtained from the following electronic address: https://github.com/CSSEGISandData/ COVID-19/. e first set of data represents COVID-19 data belonging to Italy of 172 days, from 1 March to 21 August 2020. e data are as follows: 0.  (1) Start with any initial value ψ (i) l as a length of ψ satisfying π(ψ) > 0. (2) Using the initial value, sample a candidate point ψ * from proposal q(ψ * ).
(2) Different sample sizes n are considered to be 25, 50, and 100.

Simulation Results
In this section, the Monte Carlo simulation procedure is performed for comparison between the MLEs and Bayesian estimation method under square error and Linex LoFus based on MCMC, for estimating parameters of HLBXL distribution as an example of HLBX family distribution, and this is the best distribution according to the above section of the application. We can use a different program to generate these analyses as Mathcad, Mathematica, Maple, and R packages. Algorithm 2 is used for the Monte Carlo simulation experiments.
Algorithm 3 is used for the Monte Carlo simulation of Type II censored sample experiments. We could define the best estimation methods as those that minimise estimate bias and mean squared error (MSE). Tables 5-7 reveal the following observations: (1) As sample size increases, the bias and MSE decrease (2) When the number of failures increases in a Type II CS, the values of the bias and MSE for HLBXL distribution parameters decrease (3) We find that the Bayesian estimates under Linex (2) LoFu perform better than other estimates of HLBXL distribution with respect to MSE and bias (4) We find that the Bayesian method under Linex (2) loss function performs better than other estimations for estimating the parameters of HLBXL distribution with respect to MSE and bias (5) As θ increases and the others are fixed, then the bias and MSE are increasing for θ, β, α, and the bias and MSE are decreasing for estimates (6) As λ increases and the others are fixed, then the bias and MSE are increasing for β, α, and the bias and MSE are decreasing for estimates

Conclusion
A new generalized generator of the half-logistic Burr X-G family was proposed and studied in this paper. Several statistical properties, including QuFu, Mos, InMos, MeD, Lo and Bo curves, and En were derived. e HLBX Lomax, HLBX exponential, and HLBX Rayleigh distributions are discussed. MLL and Bayesian estimation methods were used to estimate the unknown parameters. e HLBXL distribution fits better than the other submodels. To distinguish the performance of estimation methods, a simulation analysis was performed using the R package. For Bayesian estimation, the MCMC method was used. ree real COVID-19 datasets from different countries, including Italy, Canada, and Belgium, were considered. Finally, we plan to use this family to generate new models from the proposed generating family and investigate their statistical properties, as well as investigating the statistical inference of the new models using various methods and demonstrating the importance of the new models using new real datasets [19][20][21][22].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.