Abstract

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.

1. 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 real-world 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 and are the base distribution's pdf and CDF.

There 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. The transmuted Lomax distribution was presented by Ashour and Eltehiwy [5]. Transmuted Pareto distribution is introduced by Merovci and Puka [6]. The 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). They utilized Weibull distribution to generalized transmuted families of distributions (TWDn).

The 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. The pdf is as follows:

The CDF for (3) iswhere and are the shape and scale parameters, respectively.

The CDF and pdf of Gompertz-generalized G-family of distribution are given by Alizadeh et al. [10] aswhere and are extra form parameters that change the tail weights. and 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 where ; the CDF and pdf are

The corresponding pdf is created by inserting the densities from (3) and (4) into (6) in the following order:

The transmuted generalized Lomax (TGL) distribution is a new five-parameter transmuted generalization of the Lomax distribution presented in this article. This 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). There have been about 6.5 million cases of COVID-19 in France as of September 2021, with over 112850 deaths. There have been about 6.8 million cases of COVID-19 in the United Kingdom as of September 2021, with over 132740 deaths. Therefore, we decided to find the best mathematical-statistical model for modeling the data of the countries of France and the United Kingdom. There were also many researchers who worked on finding a model for these data, such as Almetwally [12], Almetwally [13], Almetwally [14], and others.

The following is a representation of how this article is structured. In Section 2, we define the new distribution. The new distribution's structural features are discussed in Section 3. The 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 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.

2. Transmuted Generalized Lomax Model

The TGL distribution and its submodels are shown here. The CDF of the TGL distribution with vector parameters where can be derived by substituting (7) and (8) in (1) and (2) asand its pdf is as follows:

As a result, the pdf (10) is defined as X ∼ TGL . Table 1 lists the TGL distribution's special submodels.

The survival (reliability) function and the hazard rate function have the following definitions:

For specific parameters selections, the pdf of TGL model is shown in Figure 1.

Figure 1 

Plots of the TGL distribution's pdf.

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.

Figure 2 

Plots showing the TGL distribution's hazard function.

Lemma 1. The TGL density function's limit is provided as

Proof. The 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:This statement is simple to demonstrate.

3. Statistical Properties

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

3.1. Moments

The instant near 0 is calculated. We can write as follows from (10).

First, to obtain I₁, as a result, we use binomial expansion.

So, I₁ is given by

In a similar way, I₂ is as follows:

Then, can be expressed as

3.2. Mode

The mode of the TGL distribution was obtained in this subsection. The is as follows:

. The 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 :

3.4. Rényi Entropy

The variance of the uncertainty is measured by the Rényi entropy of a pdf with random variable of TGL distribution. The Rényi entropy is defined as for any real parameter :

We can extract the integrated component using the density function (10) as follows:

The 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 ; there are TGL random variables that are both independent and identically distributed. Let stand for the order statistics derived from these variables. The pdf of the order statistic, say , is then calculated aswhere The binomial expansion is used in this case; then, order statistic of TGL distribution is

For the TGL distribution, the moments of order statistics are

By using the binomial expansion,

For the TGL distribution, the moments of order statistics are

4. Parameter Estimation

The MLEs of the parameters 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.

4.1. 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 is a random sample of this distribution of size with a set of parameter vectors , then the log-likelihood function, say , may be stated aswhere . The partial differential equations are calculated as follows:

The nonlinear equations are numerically solved to determine ML estimators as , , , and using an iterative technique.

4.2. MLEs under Type-II Censored Sample

The MLEs of parameters for TGL distribution based on Type-II censored samples are investigated in this subsection. The Fisher information matrix for Type-II censored model is also used to calculate the approximate confidence intervals for the unknown parameters . Let 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 kindwhere is a constant and is the data that has been censored. The log-likelihood function is possibly written as follows without constant term from (15).where denotes the time of the k-th failure and denotes the time of the k-th failure. The MLEs are the solutions to the following five equations: