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

A new five-parameter transmuted generalization of the Lomax distribution (TGL) is introduced in this study which is more flexible than current distributions and has become the latest distribution theory trend. Transmuted generalization of Lomax distribution is the name given to the new model. This model includes some previously unknown distributions. The proposed distribution's structural features, closed forms for an rth moment and incomplete moments, quantile, and Rényi entropy, among other things, are deduced. Maximum likelihood estimate based on complete and Type-II censored data is used to derive the new distribution's parameter estimators. The percentile bootstrap and bootstrap-t confidence intervals for unknown parameters are introduced. Monte Carlo simulation research is discussed in order to estimate the characteristics of the proposed distribution using point and interval estimation. Other competitive models are compared to a novel TGL. The utility of the new model is demonstrated using two COVID-19 real-world data sets from France and the United Kingdom.


Introduction
Many generators have been studied in recent years by expanding some effective classical distributions. Many applied fields, including dependability, demographics, engineering, economics, actuarial sciences, biological research, hydrology, insurance, medicine, and finance, have employed such created families of distributions for modeling and evaluating lifetime data. However, there are still a lot of realworld data occurrences that do not fit into any of the statistical distributions. Shaw and Buckley [1] introduced a new class of distributions known as transmuted distributions with cumulative distribution function (CDF) as By differentiating equation (1), we get the probability density function (pdf ) as follows: where R(x) and Z(x) are the base distribution's pdf and CDF. ere are various transmuted distributions suggested. Aryal and Tsokos [2] proposed the transmuted Weibull distribution as a new generalization of the Weibull distribution. Merovci [3] devised and explored the varied structural properties of the transmuted Rayleigh distribution. Khan and King [4] obtained the transmuted modified Weibull distribution. e transmuted Lomax distribution was presented by Ashour and Eltehiwy [5]. Transmuted Pareto distribution is introduced by Merovci and Puka [6]. e transmuted generalized linear exponential distribution was introduced by Elbatal et al. [7]; among others. Poboková and Michalková [8] proposed a transmuted Weibull distribution. Ali and Athar [9] have created a new generalized transmuted family of distributions (TD). ey utilized Weibull distribution to generalized transmuted families of distributions (TWDn). e Lomax distribution is a heavy-tail pdf popular in business, economics, and actuarial modeling. In some cases, it is also known as the Pareto Type-II distribution. In the event of a business failure, Lomax used it to fit data. It is essentially a Pareto distribution with a 0-support level. e pdf is as follows: R(x; c; δ) � c δ(1 + cx) − (δ+1) , x ≥ 0, δ and c > 0. (3) e CDF for (3) is where δ and c are the shape and scale parameters, respectively. e CDF and pdf of Gompertz-generalized G-family of distribution are given by Alizadeh et al. [10] as where ϑ and π are extra form parameters that change the tail weights. R(x) and Z(x) are the parent (or baseline) distribution's pdf and CDF, respectively. Now, if the density from (3) and (4) is replaced into (5) and (6), Oguntunde et al. [11] introduced a novel generalization of the Lomax distribution known as the Gompertz Lomax distribution (GoLom) with vector parameters ƛ where ƛ � (δ, c, ϑ, π); the CDF and pdf are x ≥ 0, π, δ, c and ϑ > 0.
e corresponding pdf is created by inserting the densities from (3) and (4) into (6) in the following order: f(x; ƛ) � R(x; ƛ) � ϑδc(1 + cx) δ π− 1 e (ϑ/π) 1− [1+cx] δ π { } ; x ≥ 0, π, δ, c and ϑ > 0. (8) e transmuted generalized Lomax (TGL) distribution is a new five-parameter transmuted generalization of the Lomax distribution presented in this article. is generalization stems from a fundamental motivation as follows: (i) Providing a very flexible life distribution that includes several new existing distributions as submodels (ii) Making a significant difference in data modeling One of the advantages of this distribution is that it works on modeling COVID-19 data. In COVID-19, a new coronavirus disease has expanded worldwide since December 2019, producing over 218 million cases and over 4.5 million deaths, reported by the World Health Organization (WHO).
ere have been about 6.5 million cases of COVID-19 in France as of September 2021, with over 112850 deaths. ere have been about 6.8 million cases of COVID-19 in the United Kingdom as of September 2021, with over 132740 deaths. erefore, we decided to find the best mathematicalstatistical model for modeling the data of the countries of France and the United Kingdom.
ere were also many researchers who worked on finding a model for these data, such as Almetwally [12], Almetwally [13], Almetwally [14], and others. e following is a representation of how this article is structured. In Section 2, we define the new distribution. e new distribution's structural features are discussed in Section 3. e maximum likelihood estimators (MLEs) of parameters under complete and Type-II censored samples are investigated in Section 4. Section 5 describes the various bootstrap confidence intervals. Section 6 describes a Monte-Carlo simulation analysis using entire sample sizes and 2 Computational Intelligence and Neuroscience Type-II censored samples to estimate point and interval estimation of TGL distribution parameters. In Section 7, two real-world data sets are introduced, and at the end of the article, there is a conclusion.
e survival (reliability) function F(x; Z) and the hazard rate function h(x; Z) have the following definitions: For specific parameters selections, the pdf of TGL model is shown in Figure 1.
We can deduct from Figure 1 plots of the TGL distribution's pdf can be unimodal, normal, or right-skewed.
We may derive from Figure 2 that the TGL distribution's hazard function can take the form of a decreasing, increasing, or upside-down shape.

Lemma 1.
e TGL density function's limit is provided as Proof. e density function's conclusion is easy to illustrate (10).
Furthermore, the TGL hazard function's limit as x⟶0 is 0 and x⟶ ∞ is ∞ as shown as follows: is statement is simple to demonstrate.

Statistical Properties
e statistical aspects of the TGL distribution are examined in the following subsections, including moments, mode, quantile function, Rényi entropy, and order statistics.

Computational Intelligence and Neuroscience
First, to obtain I 1 , as a result, we use binomial expansion.
So, I 1 is given by In a similar way, I 2 is as follows: en, μ r ′ can be expressed as  Computational Intelligence and Neuroscience

3.2.
Mode. e mode of the TGL distribution was obtained in this subsection. e ln f(x) is as follows: ]. e mode is the solution of the following equation: 3.3. Quantile and Median. By inverting cdf (9) as follows, the TGL distribution can be easily simulated: on (0, 1), if U follows a uniform distribution, then Special Case β � 1:

Rényi
Entropy. e variance of the uncertainty is measured by the Rényi entropy of a pdf f(x) with random variable X of TGL distribution. e Rényi entropy is defined as for any real parameter ψ > 0 and ψ ≠ 1: We can extract the integrated component using the density function (10) as follows: e binomial expansion is then used as follows: As a result, the TGL distribution's Rényi entropy is 3.5. Order Statistics. In this part, we will look at single-order statistics for the TGL distribution. Let us say x 1 , . . . , x n ; there are n TGL random variables that are both independent and identically distributed. Let x (1) , x (2) , . . . , x (n) stand for the order statistics derived from these n variables. e pdf of the r th order statistic, say f r: n (x), is then calculated as Computational Intelligence and Neuroscience where C r: n � n!/(r − 1)!(n − r)! e binomial expansion is used in this case; then, r th order statistic of TGL distribution is For the TGL distribution, the k th moments of r th order statistics are dy.

(28)
By using the binomial expansion, For the TGL distribution, the k th moments of r th order statistics are

Parameter Estimation
e MLEs of the parameters Z � (δ, c, ϑ, π, β) under complete and Type-II censored samples are investigated in this section. Approximate confidence intervals (ACIs) for unknown values are also calculated using the Fisher information matrix.

MLEs under Complete Sample.
In the statistical literature, various approaches for parameter estimation have been given, with the MLEs method being the most extensively employed. We explore applying MLEs to estimate the parameters of the TGL distribution with a complete sample. If x 1 , x 2 , . . . , x n is a random sample of this distribution of size n with a set of parameter vectors Z � (δ, c, ϑ, π, β), then the log-likelihood function, say ℓ 1 (Z), may be stated as where ]. e partial differential equations ℓ 1 (Z) are calculated as follows: e nonlinear equations are numerically solved to determine ML estimators as zℓ 1 (Z)/zδ � 0, zℓ 1 (Z)/zc � 0, zℓ 1 (Z)/zϑ � 0, and zℓ 1 (Z)/zπ � 0zℓ 1 (Z)/zβ � 0 using an iterative technique.

MLEs under Type-II Censored Sample.
e MLEs of parameters for TGL distribution based on Type-II censored samples are investigated in this subsection. e Fisher information matrix for Type-II censored model is also used to calculate the approximate confidence intervals for the unknown parameters Z � (δ, c, ϑ, π, β). Let x 1 , x 2 , . . . , x n be a random sample of size n, and we only look at the first k-th order statistics based on the Type-II censored sample. Likelihood function in this scenario is of the kind where C is a constant and x 1: k: n , x 2: k: n , . . . , x k: k: n is the data that has been censored. e log-likelihood function ℓ 2 (Z) is possibly written as follows without constant term from (15).
Computational Intelligence and Neuroscience where x i � x i: k: n , i � 1, 2, . . . , k, denotes the time of the k-th failure and x k denotes the time of the k-th failure. e MLEs Z � (δ, c, ϑ, π, β) are the solutions to the following five equations: It is to be noted that equation (16) cannot be solved explicitly. To obtain the MLEs Z � (δ, c, ϑ, π, β), a numerical approach is required, and a numerical technique is needed. We obtain the observed Fisher information matrix since its expectation requires numerical integration. e 5 × 5 observed information or Hessian matrix H(Z) is 8 Computational Intelligence and Neuroscience e Fisher information matrix H(Z) is given by the negative expected of second partial derivatives of (15) for the unknown parameters Z � (δ, c, ϑ, π, β) locally at (δ, c, ϑ, π, β) given in (16). Under some regularity conditions, (δ, c, ϑ, π, β) is approximately normal with mean (δ, c, ϑ, π, β) and covariance matrix H − 1 (Z). Practically, we estimate H − 1 (Z) byH − 1 (δ,c,ϑ,π,β) ; then , e elements of the observed Hessian matrix are computed using an iterative numerical solution method. Now, the ACIs Z � (δ, c, ϑ, π, β) can be obtained as follows: where z α is the 100 α − th percentile of a standard normal distribution.

Simulation Study
In this section, we discuss the Monte Carlo simulation study to estimate point and interval estimation of parameters of TGL distribution based on complete sample sizes and Type-II censored samples. e simulation results are in Tables 2-5

Concluding Remarks of Simulation Results
Tables 2-5 show the simulation results of point and interval estimates of TGL distribution parameters using Type-II censored samples and entire sample sizes. Based on these Tables, the following concluding remakes are noticed.

Data Set (1): COVID-19 of France.
e COVID-19 data in question is from France, and it covers a period of 108 days, from March 1 to June 16, 2021. is data was formed by using daily new deaths (ND), daily cumulative cases (CC), and daily cumulative deaths (CD) as follows:    Table 6, the TGL distribution is fitted to COVID-19 of France country. e TGL model is compared with other competitive models as Mead and Afify [16] proposed the Burr-XII model (KEBXII) with Kumaraswamy exponentiated, Weibull-Lomax (WL) distribution, Odds Exponential-Pareto IV (OEPIV) distribution proposed by Baharith et al. [17], Marshall-Olkin Alpha power Weibull (MOAPW) by Almetwally et al. [18], Marshall-Olkin Alpha power extended Weibull (MOAPEW) by Almetwally [19], Marshall-Olkin alpha power inverse Weibull (MOAPIW) by Basheer et al. [20], Marshall-Olkin alpha power Lomax (MOAPL) by Almongy et al. [21], and Gompertz Lomax (GOLOM) distribution by Oguntunde et al. [11]. According to this result, we note that the estimate of TGL has the best measure where it has the smallest value of Cramer-von Mises (W * ), Anderson-Darling (A * ), and Kolmogorov-Smirnov (KS) statistic along with its P value. e fitted TGL, pdf, CDF, and PP-plot of the data set are displayed in Figure 3.
e COVID-19 data in question is from the United Kingdom and spans 82 days, from May 1 to July 16, 2021. is data is formed by using daily ND, daily CC, and daily CD as follows: In Table 7, the TGL distribution is fitted to COVID-19 of e United Kingdom country. e TGL model is compared with other competitive models as, KEBXII, WL, OEPIV, MOAPW, MOAPEW, and GOLOM distributions. According to this result, we note that the estimate of TGL  Computational Intelligence and Neuroscience 13 has the best measure where it has the smallest value of W * , A * , and KS statistic along with its P value. e fitted TGL, pdf, CDF, and PP-plot of the data set are displayed in Figure 4.

Conclusion
We investigate the so-called five-parameter transmuted generalized Lomax distribution in this study. Lomax and Gompertz Lomax (GoLom) distributions are included in the TGL model. e TGL distribution's structural properties are deduced. e maximum likelihood approach is used to estimate the population parameters based on complete and Type-II censored samples. We discussed the Monte Carlo simulation study to estimate point and interval estimation of parameters of TGL distribution based on complete sample sizes and Type-II censored samples. e proposed distribution was applied to two COVID-19 real-world data sets from France and United Kingdom. We compared a new transmuted generalization of the Lomax distribution (TGL) with KEBXII, WL, OEPIV, MOAPW, MOAPEW, and GOLOM distributions. It was shown to provide a better fit than several other models. We hope that the presented model will be used in a variety of fields, including engineering, survival and lifetime data, meteorology, biology, hydrology, economics (income disparity), and others.

Data Availability
All data used to support the findings of the study are available within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.