A New Flexible Logarithmic-X Family of Distributions with Applications to Biological Systems

Probability distributions play an essential role in modeling and predicting biomedical datasets. To have the best description and accurate prediction of the biomedical datasets, numerous probability distributions have been introduced and implemented. We investigate a novel family of lifetime probability distributions to represent biological datasets in thispaper. The proposed family is called a new ﬂexible logarithmic-X (NFLog-X ) family. The suggested NFLog-X family is obtained by applying the T-X method together with the exponential model having the PDF m ( t ) � e − t . Based on the NFLog-X approach, a three parameters probability distribution, namely, a newﬂexiblelogarithmic-Weibull(NFLog-Wei)distributionisintroduced.Themethodofmaximumlikelihoodestimationisadoptedfor estimatingtheparametersoftheNFLog-X family. In the end, we examine three diﬀerent biological datasets in order to give a thorough numerical research that illustrates the NFLog-Wei distribution. Comparisons are made between the analytical goodness-of-ﬁt metrics of thesuggesteddistribution. Wemadecomparisonwiththe(i)alphapowertransformedWeibull,(ii)exponentiatedWeibull,(iii)Weibull, (iv) ﬂexible reduced logarithmic-Weibull, and (v) Marshall–Olkin Weibull distributions. After performing the analyses, we observe that the proposed method outclassed other competitive distributions.


Introduction
Probability distributions are frequently used to model the lifetime phenomena in applied sectors [1]. In the literature of distributions theory, the most frequently used distributions to model the lifetime phenomena are the exponential (Exp), Rayleigh (Ray), and Weibull (Wei) distributions. However, when the lifetime phenomena are complex, then these probability distributions are not suitable to model and predict the data accurately (Ahmad et al. [2] and Liao et al. [3]). For example, the Exp distribution is concerned with describing data that have a constant HF (hazard function). On the other hand, the Ray distribution is used to model data with an increasing HF. Similarly, the Wei distribution having the Exp and Ray as the special models is one of the popular probability distributions (Sarhan and Zaindin [4] and Huo et al. [5]). e Weibull model offers/provides the features of both the Exp and Ray probability distributions and has widely been used in modeling lifetime phenomena with monotone failure rates. However, when the lifetime phenomena have a monotone (increasing, decreasing, and constant) HF, then, the Weibull distribution is the best choice to use (Almalki and Yuan [6] and Liu et al. [7]).
Recently, Ahmad et al. [21] studied a Z-family by adding a new parameter. We can write the distribution function (DF) F(x; β, λ) of the Z-family through the following equation: such that β > 0 can be considered as an extra parameter. Wang et al. [22] developed another method called, a NG-X (new generalized-X) family by the following DF: where θ > 0. Mohammed et al. [23] proposed another new approach to develop new probability distribution for modeling lifetime events. ey named their proposed method, a NLT-X (new lifetime-X) distributions. e DF F(x; η, λ) of the NLT-X distributions is given by the following equation: with an additional parameter η > 0. We additionally propose a new class of probability distribution in this paper by implementing the T-X method. e new class is called a NFLog-X family of distributions. Using the proposed NFLog-X approach, we can formulate an upgraded version of the Wei distribution whicnh can be presented and dubbed as NFLog-Wei distribution. e proposed NFLog-Wei distribution offers a close fit to the healthcare datasets.

The Proposed Method
Here, we propose a new method to introduce new updated and modified versions of the lifetime distributions. By incorporating the exponential model, having the PDF m(t) � e − t with the T-X method (Alzaatreh et al. [24]), the suggested approach is presented.
Let us assume that we have a RV (random variable), represented by T, considered as a baseline RV with PDF m(t), where T ∈ [π 1 , π 2 ] for − ∞ < π 1 < π 2 < ∞. Let X be another RV with DF K(x; λ). Let suppose G[K(x; λ)] considered as a function in the DF, meeting each of the three requirements outlined below: According to Alzaatreh et al. [24], the DF F(x) of the T-X family is as follows: with PDF given by Now, setting 2 )s) and using m(t) � e − t , exists in (1), we can obtain easilythe DF F(x; δ, λ) of the NFLog-X distributions, represented as below where (d/dx)K(x; λ) � k(x; λ).
In this article, we implement the NFLog-X distributions approach and introduce the NFLog-Wei distribution. Section 4 offers the expression of the DF, PDF, SF, HF, and CHF of the NFLog-Wei distribution.

The Identifiability Property
e identifiability property is a very useful statistical property that ensures precise inferences. Here, we derive the identifiability property of the NFLog-X distributions. Let δ 1 and δ 2 be the two parameters having DFs F(x; δ 1 , λ) and F(x; δ 2 , λ), respectively. e parameter δ is identifiable, if δ 1 � δ 2 . Mathematically, we have Incorporating equation (2) in equation (4), we get Taking square root of equation (5), we get , From equation (6), we can see that δ 1 � δ 2 . erefore, the parameter δ is identifiable.
From Figure 2, we can see that the PDF f(x; δ, λ) of the NFLog-Wei model has four different patterns, including (i) decreasing or reverse in the form of J-shaped but reversed shown in (red curve), (ii) left-skewed (green curve), (iii) right-skewed (black curve), and (iv) symmetrical (blue curve).
Furthermore, the HF h(x; δ, λ) and CHF H(x; δ, λ) of the NFLog-Wei distribution are given by the following equations: respectively. Different plots for the HF h(x; δ, λ) of the NFLog-Wei distribution are provided in Figure 3. e plots of h(x; δ, λ) From Figure 3, we can see that the HF h(x; δ, λ) of the NFLog-Wei distribution has three different patterns, including (i) increasing (red curve), (ii) unimodal (green curve), and (iii) reverse J-shaped (black curve).
Despite the prominent advantages of the NFLog-Wei distribution over the other distributions, the NFLog-Wei model has also disadvantages, for example (i) e NFLog-Wei distribution is a continuous distribution used to evaluate continuous datasets. Consequently, the suggested NFLog-Wei distribution cannot be utilized to assess discrete data sets.
(ii) Because of the NFLog-Weidistribution PDF's complicated structure, the expressions of its estimators cannot be reduced to a simple, closed form easily represented. erefore, the numerical estimates of the estimators can be obtained with the help of computer software. (iii) Due to the complexity of the PDF of the NFLog-Wei distribution, additional computing work is necessary to determine its mathematical features.

Estimation and Simulation
Here, we obtain the MLEs (δ MLE , λ MLE ) of the NFLog-X distributions. In addition, we do provide a comprehensive Monte-Carlo simulation study (MCSS) for assessing the performances of δ MLE and λ MLE .

Estimation.
In the research that has been conducted on the topic, a number of different strategies and procedures for estimating the parameters of probability models have been proposed and put into practice. Among them, the MLE is one of the most usually adopted methods. Here, we implement this method to obtain the δ MLE and λ MLE .

Simulation.
In this second subsection, a comprehensive MCSS is conducted to assess the behaviors of δ MLE and λ MLE of the NFLog-Wei distribution. e RNs (random numbers) are successfully generated from the PDF f(x; δ, λ) via the inverse DF method. e outcome of the simulation are acquired for a total of four groups and sets (Set I, Set II, Set III, and Set IV) of parameters values, given by Set I: α � 0.6, β � 2.3, and δ � 3.5, Set II: α � 1.3, β � 3.8, and δ � 4.6, Set III: α � 3.0, β � 4.0, and δ � 4.5, and Set IV: α � 3.4, β � 2.5, and δ � 3.5.
To check performances of δ MLE and λ MLE , two statistical measures are considered. ese measures include the (i) mean square error (MSE) and (ii) bias. e numerical values of the MSE and bias are, respectively, computed as follows: and 1 n n i�1 (δ − δ). (25) e values of the MSE and bias are also computed for λ. Corresponding to Set I: α � 0.6, β � 2.3, and δ � 3.5 and Set II: α � 1.3, β � 3.8, and δ � 4.6, we can easily see the outcomes resulted from doing the simulation in Table 1. Whereas, in link to Set III: α � 3.0, β � 4.0, δ � 4.5 and Set IV: α � 3.4, β � 2.5, δ � 3.5, the outcomes of the simulation are shown in Table 2.
Based on the findings of the simulation, which are shown in Tables 1 and 2, we can see that as the size of n increases.

Applications
By doing an analysis on three different biomedical datasets, the purpose of this article is to demonstrate the utility of the NFLog-Wei distribution (Table 3).
We compare the NFLog-Wei distribution with the Wei model and three other traditional and new probability distributions, as an example of these distributions, the Marshall-Olkin Weibull (MO-Wei) studied by Marshall and Olkin [25], APT-Wei (alpha power transformed Weibull) proposed by Dey et al. [26], and a flexible reduced logarithmic-Weibull (FRLog-Wei) distribution, introduced by Liu et al. [7]. e DFs of the competitive probability distributions are outlined below: (i) e APT-Wei distribution is obtained by the following equation: where α 1 ≠ 1, α 1 > 0. (ii) e MO-Wei distribution is represented by the equation that is as follows: where c > 0. (iii) e FRLog-Wei distribution is represented by the equation that is as follows: where σ > 0.
To determine the optimum model among the fitted distributions, we consider different goodness-of-fit measures (analytical measures). It is generally agreed that (i) e CM test statistic is computed as follows: (ii) e AD test statistic is calculated as follows: (iii) e KS test statistic is derived as follows: (iv) e AIC is obtained as follows: (v) e BIC is calculated as follows: (vi) e HQIC is obtained as follows:                     Table 4. e numerical values of the analytical measures of the competing probability models are provided in Tables 5 and 6. In justification of the numerical depiction that may be seen in Tables 5 and 6, confirm the best fitting of the NFLog-Wei model to Data 1. Figure 6 presents a graphic representationof the NFLog-Wei model. We can see that the plots shown in Figure 6 also confirm the close-fitting (best fitting) of the NFLog-Wei model to Data 1.
We also applied the NFLog-Wei distribution and other competing probability models to Data 2. Corresponding to this dataset, the numerical values of the competing probability distributions can be easily shown in Table 7. Also, the metrics of the analysis measures of the distributions that were "fitted" are already provided in Tables 8 and 9. e numerical results, in Tables 8 and 9, demonstrate that the NFLog-Wei distribution has the least results of the analytical metrics.
is fact supports the best fitting power of the NFLog-Wei distribution to the guinea pigs infected dataset. In addition, Figure 9 illustrates the NFLog-Wei distribution graphically. e plots in Figure 9 support the close fit (best fit) of the NFLog-Wei distribution to the guinea pig-infected dataset.
Again, we applied the NFLog-Wei distribution and the competing distributions to Data 3. Corresponding to Data 3, the numerical results of the fitted probability models can be found in Table 10. e numerical values of the statistical tests of the competing probability models are given in Tables 11 and 12. In light of the numerical findings presented in the Tables 11 and 12, we can observe that the NFLog-Wei is the best competing probability model for Data 3. To support the best fitting power of the NFLog-Wei distribution to Data 3, a graphical illustration is also presented in Figure 12. e visual illustration provided in Figure 12 supports the best fit capability of the NFLog-Wei distribution to Data 3.

Conclusion
In this study, a novel family of probability models was presented. e proposed family was named a new flexible logarithmic-X family. A subcase of the NFLog-X family was studied in detail. e unknown parameters of the NFLog-X family of distributions were computed using the maximum likelihood method. Furthermore, a MCSS was carried out to assess the performances of δ MLE and λ MLE of the NFLog-X family. Finally, three applications (real-life datasets) to the biomedical datasets were presented to illustrate the potentiality and flexibility of the NFLog-X method.
e comparison of the NFLog-X method was made with the Wei distribution and its three other wellknown distributions including the APT-Wei, FRLog-Wei, and MO-Wei distributions. On the basis of eight analytic metrics, it is demonstrated that the NFLog-Wei distribution is the optimal probability distribution for modeling the medical datasets.
In the future, we are motivated to introduce further flexible forms of the NFLog-X distributions for data modeling in various sectors. We are also motivated to study the bivariate and multivariate extensions of the NFLog-X distributions [27][28][29] and also in the upcoming stage of this research, we will use the newly invented family of distributions in addition to the suggested distribution to analyse the censored sample technique. In order to create randomised censored samples based on the new distribution, we will carry out research on a variety of censoring techniques, including the type-I and type-II censored sample. e scope of our analysis might be increased to encompass the implementation of the suggested model to various accelerated life testing scenarios, such as constant and partially constant tests, and perhaps even outcomes of progressive load accelerated life tests.

Data Availability
In the paper, the datasets are listed.

Conflicts of Interest
It is stated by the authors that they have no competing interests.