A Flexible Extension to an Extreme Distribution

: The aim of this paper is not only to propose a new extreme distribution, but also to show that the new extreme model can be used as an alternative to well-known distributions in the literature to model various kinds of datasets in different ﬁelds. Several of its statistical properties are explored. It is found that the new extreme model can be utilized for modeling both asymmetric and symmetric datasets, which suffer from over- and under-dispersed phenomena. Moreover, the hazard rate function can be constant, increasing, increasing–constant, or unimodal shaped. The maximum likelihood method is used to estimate the model parameters based on complete and censored samples. Finally, a signiﬁcant amount of simulations was conducted along with real data applications to illustrate the use of the new extreme distribution.


Introduction
Data analysis has become of great interest in many fields of science such as health sciences, reliability analysis, industry, environmental studies, and others. The requirement of obtaining suitable models and statistical distributions has become essential, since defining new distributions will enable us to better describe and predict phenomenal and experimental data. See for example [1][2][3][4][5][6][7][8][9], among others.
Recently, several methods of obtaining new distributions from old ones have been developed. Many generalized classes of life time distributions have been discussed in the literature. It has been proven in many papers that the new generalizations are more flexible in modelling and better fit real-life data. These new distributions also have several desirable properties such as the asymptotic behavior of their probability density function and the hazard rate function's monotonicity, which has made them superior to the original distribution. All of this has encouraged authors to work more on developing new lifetime distributions using different generalization methods. Here, we refer to the papers of [10] for the Marshall-Olkin class, [11] for the Beta and Gamma classes, [12] for the odd exponentiated half-logistic-G (OEHL-G) family, [13] for the flexible Weibull class, [14] for the odd log-logistic Lindley class, [15] for the odd Chen class, [16] for the exponentiated odd Chen class, [17] for a new Kumaraswamy generalized class, [18,19] for the extended Gamma and log-Bilal models, respectively, [20] for type I half logistic odd Weibull-G and [21] for the Poisson transmuted-G family, among others.

1.
The PDF, CDF, and HRF can be derived in closed-forms.

2.
The HRF can be constant, increasing, increasing-constant or unimodal shaped, which makes the proposed model able to be used to analyze different types of datasets.

3.
It can be used to model both symmetry and asymmetric datasets. 4.
It has more flexibility as compared to well-known models, especially the IEx model.

5.
It can be used to model platykurtic-shaped data. 6.
It can be used to model over-and under-dispersed data (See Section 6).

The OEHLIEx Distribution
A random variable X is said to have the IEx distribution with parameter β if its CDF G(x; β) = e −β x . Using the CDF of the IEx model in Equation (1), we obtain the CDF of the OEHLIEx distribution, which can be expressed as where α and λ are the positive shape parameters, while β is the positive scale parameter. The corresponding probability density function (PDF) to Equation (2) is The PDF of the OEHLIEx model can be represented as an infinite mixture of an exponentiated IEx (Exp-IEx) distribution: where ϕ k,l = 2αλ (−1) j+k+l (λ(i + j + 1)) k k!(k + l + 1) represents the Exp-IEx density with the power parameter (k + l + 1). The corresponding reliability function to Equation (2) can be obtained as For the proposed model, the hazard rate function (HRF) and its cumulative can be reported, respectively, as and where the cumulative hazard function is the integral of the hazard function. Figure 1 shows the PDF plots and its HRF for different parameter values. It is immediate that the PDF is unimodal shaped and can be used to discuss right-and left-skewed datasets, whereas the HRF can be increasing, constant, increasing-constant, or unimodal shaped. Therefore, the OEHLIEx distribution can be utilized to analyze various types of data in several practical fields. represents the Exp-IEx density with the power parameter ( + + 1) . The corresponding reliability function to Equation (2) can be obtained as For the proposed model, the hazard rate function (HRF) and its cumulative can be reported, respectively, as and where the cumulative hazard function is the integral of the hazard function. Figure 1 shows the PDF plots and its HRF for different parameter values. It is immediate that the PDF is unimodal shaped and can be used to discuss right-and left-skewed datasets, whereas the HRF can be increasing, constant, increasing-constant, or unimodal shaped. Therefore, the OEHLIEx distribution can be utilized to analyze various types of data in several practical fields.

Quantile Function (QF), Skewness, and Kurtosis
The QF of the OEHLIEx distribution is given as follows: if U has a uniform random variable on (0,1), then

Quantile Function (QF), Skewness, and Kurtosis
The QF of the OEHLIEx distribution is given as follows: if U has a uniform random variable on U(0, 1), then Equation (8) can be used to generate a random sample, and the median can be derived at u = 0.5. Moreover, it can be used to obtain the skewness and kurtosis, where skewness = Regarding Tables 1-3, the proposed model is suitable for modelling asymmetric as well as symmetric datasets that are platykurtic shaped.

Incomplete Moments
The sth incomplete moments can be listed as where ω * s (t) = t 0 x s τ k+l+1 (x)dx. Thus, the sth incomplete moments of the OEHLIEx model can be proposed as 3.3. Reliability Function of Linear Consecutive, Parallel, Series, and Bridge Type Network Systems If the random variable X has the OEHLIEx distribution, then the reliability function of the linear consecutive k − out − o f n : F system can be expressed as where * ( ) = ( ) . Thus, the sth incomplete moments of the OEHLIEx model can be proposed as

Reliability Function of Linear Consecutive, Parallel, Series, and Bridge Type Network Systems
If the random variable X has the OEHLIEx distribution, then the reliability function of the linear consecutive − − : system can be expressed as for more details concerning the values of , ( ; , ), and , see [26]. In the special case of the system − − − : , the parallel and series system when = and = 1, respectively.
Consider two systems: one of them is parallel, whereas the other is a series with independent n components. Each component has the OEHLIEx model; thus, the reliability function in the case of the parallel system can be reported as where as the reliability function in the case of the series system can be expressed as In reliability theory, there exists another type of engineering system in the so-called bridge-type network or a complex system, which has many applications in this field. Such systems as these can be evaluated by using many approaches such as conditional probability, a connection matrix, and tree diagrams, as well as cut and tie sets. Assume a bridgetype network consists of five components (A, B, C, D, and E) where each component has the OEHLIEx model, these components can be connected as follows: Consider the previous network in which success requires that at least one of the paths AC, BD, AED, or BEC is good. To evaluate the reliability function of this network, the conditional probability approach has been utilized. The previous network in Figure 2 can be subdivided into two systems, one with E considered bad, i.e., it always failed, and one with E considered good, i.e., it cannot fail. Thus, for more details concerning the values of m, N(j; k, n), and j, see [26]. In the special case of the system k − out − o f − n : F, the parallel and series system when k = n and k = 1, respectively. Consider two systems: one of them is parallel, whereas the other is a series with independent n components. Each component has the OEHLIEx model; thus, the reliability function in the case of the parallel system can be reported as where as the reliability function in the case of the series system can be expressed as In reliability theory, there exists another type of engineering system in the so-called bridge-type network or a complex system, which has many applications in this field. Such systems as these can be evaluated by using many approaches such as conditional probability, a connection matrix, and tree diagrams, as well as cut and tie sets. Assume a bridge-type network consists of five components (A, B, C, D, and E) where each component has the OEHLIEx model, these components can be connected as follows: Consider the previous network in which success requires that at least one of the paths AC, BD, AED, or BEC is good. To evaluate the reliability function of this network, the conditional probability approach has been utilized. The previous network in Figure 2 can be subdivided into two systems, one with E considered bad, i.e., it always failed, and one with E considered good, i.e., it cannot fail. Thus, where F * = 1 − R * represents the unreliability function of a component (*). Since R A = R B = R C = R D = R E , then the reliability R NW can be expressed as Assume four different systems, namely, parallel, serious, linear consecutive, and bridge network; each system consists of 20 components, except the bridge network which consists of five components. Tables 4-6 list some numerical values of the reliability function for these systems using Maple software. System ↓ Parameter → = .
Parallel  Assume four different systems, namely, parallel, serious, linear consecutive, and bridge network; each system consists of 20 components, except the bridge network which consists of five components. Tables 4-6 list some numerical values of the reliability function for these systems using Maple software. Regarding Tables 4-6, it is clear that the reliability of parallel, series, linear consecutive, and bridge network systems increases in two cases, one of them for fixed values of β and λ with α → ∞ , and the other for fixed values of α and λ with β → ∞ . Whereas, the reliability of these systems decreases for fixed values of α and β with λ → ∞.

Entropy
The Rényi entropy of a positive random variable X representing a measure of variation of the uncertainty is defined by where (2ρ − 1) = −1, −2, −3, . . . . and The binomial coefficients will be computed with the negative values using built-in functions in Maple software. The Shannon entropy is a special case of the Rényi entropy when ρ → 1 . Tables 7-9 list some numerical values of entropies using Maple software. Regarding Tables 7-9, it is clear that the entropy increases in two cases, one of them for fixed values of α and λ with β → ∞ , and the other for fixed values of β and λ with α → ∞ . Whereas, the entropy decreases for fixed values of α and β with λ → ∞.

Maximum Likelihood Estimation (MLE) Based on Complete Samples.
In this section, we derive the MLE of the unknown parameters λ, β, and α of the OEHLIEx model based on complete samples. Consider a random sample X 1 , X 2 , . . . , X n from the OEHLIEx model; then, the log-likelihood (LL) function can be expressed as By substituting from Equation (3) into Equation (17), the MLEs of the OEHLIEx parameters can be obtained by maximizing with respect to λ, β, and α. We used R software to obtain the parameters' values. The (1 − δ) 100% confidence intervals (CIs) of the model parameters can be calculated using the following relations: where Z δ 2 is the upper δ 2 th percentile of the standard normal distribution.

MLE Based on Type-II Censored Samples
The likelihood function for a type-II censored sample can be reported as where n represents the number of components, and the experiment is stopped when k items failed. If X 1 , X 2 , . . . , X n represent an independent and identically distributed random sample from the OEHLIEx distribution and X 1 , X 2 , . . . , X k , k ≤ n represent an ordered sample obtained from a type-II right censoring sample, then the log-likelihood (LL * ) function is By substituting from Equations (3) and (5) into Equation (20), the MLEs of the OEHLIEx parameters can be obtained by maximizing results of the equation with respect to λ, β, and α.

Data Analysis and Discussion Based on Complete Samples
In this section, we illustrate the empirical importance of the OEHLIEx distribution using three applications on real data. These data are used to compare the fits of the OEHLIEx distribution with some competitive models such as the inverse exponential (IEx), exponential (Ex), exponentiated half-logistic (EHL), generalized half-logistic (GHL), and normal (N) models. For the comparison of the models, we should use the values of LL, and Kolmogorov-Smirnov (K-S) test with its p-value.
The first data set (I) represents the relief times of twenty patients receiving an analgesic (see, [27]).
The second data set (II) represents the strengths of glass fibers (see, [28]). The third data set (III) represents the failure times (in minutes) for a sample of 15 electronic components in an accelerated life test (see, [29]).
Tables 18-20 list the MLEs with their corresponding standard errors (in parentheses), and goodness-of-fit (GoF) measures for the datasets.   Regarding Tables 18-20, it is clear that the OEHLIEx model is the best model among all tested models. Regarding data set I, it is noted that the TrIEx and EHL models work quite well besides the OEHLIEx model where p-value > 0.05, but we always search for the most fitting model. Thus, we recommend using the OEHLIEx model to analyze data set I. Similarly, for dataset III, it is found that the TrIEx, IEx, Ex, and GHL models work quite well besides the OEHLIEx model, but we also recommend utilizing the OEHLIEx model to analyze these data. Figures 3-5 show the empirical estimated CDF "ECDF", probability-probability (PP), and fitted PDF "FPDF" plots for data sets I, II, and III, respectively, which support the results of Tables 18-20. Moreover, it is noted that the datasets plausibly came from the OEHLIEx model. Regarding Tables 18-20, it is clear that the OEHLIEx model is the best model amo all tested models. Regarding data set I, it is noted that the TrIEx and EHL models wo quite well besides the OEHLIEx model where p-value > 0.05, but we always search for most fitting model. Thus, we recommend using the OEHLIEx model to analyze data se Similarly, for dataset III, it is found that the TrIEx, IEx, Ex, and GHL models work qu well besides the OEHLIEx model, but we also recommend utilizing the OEHLIEx mo to analyze these data.  show the empirical estimated CDF "ECDF", probability-probability (P and fitted PDF "FPDF" plots for data sets I, II, and III, respectively, which support results of Tables 18-20. Moreover, it is noted that the datasets plausibly came from OEHLIEx model.                  Figure 9 shows the total time in test (TTT) plots for data sets I, II, and III. It is clear that the datasets suffer from an increasing hazard rate. Thus, the proposed model can be used to model the HRF for these data sets. Figure 9. The TTT plots for data I "left panel", data II "middle panel", and data III "right panel". Table 21 lists some computational statistics for the three datasets by utilizing the OEHLIEx model.      Figure 9 shows the total time in test (TTT) plots for data sets I, II, and III. It is clear that the datasets suffer from an increasing hazard rate. Thus, the proposed model can be used to model the HRF for these data sets. Figure 9. The TTT plots for data I "left panel", data II "middle panel", and data III "right panel". Table 21 lists some computational statistics for the three datasets by utilizing the OEHLIEx model.   Figure 9 shows the total time in test (TTT) plots for data sets I, II, and III. It is clear that the datasets suffer from an increasing hazard rate. Thus, the proposed model can be used to model the HRF for these data sets.   Figure 9 shows the total time in test (TTT) plots for data sets I, II, and III. It is clear that the datasets suffer from an increasing hazard rate. Thus, the proposed model can be used to model the HRF for these data sets. Figure 9. The TTT plots for data I "left panel", data II "middle panel", and data III "right panel". Table 21 lists some computational statistics for the three datasets by utilizing the OEHLIEx model.  Figure 9. The TTT plots for data I "left panel", data II "middle panel", and data III "right panel". Table 21 lists some computational statistics for the three datasets by utilizing the OEHLIEx model. Based on the model parameters, data sets I and II suffer from under-dispersed phenomena (index of dispersion < 1), whereas data set III suffers from over-dispersion (index of dispersion > 1). Moreover, data sets I, II, and III represent positive-skewed data with a platykurtic shape. The censored data have been obtained from (http://www.biochemia-medica.com, accessed on 6 March 2021). They represent the recovery time of 50 patients suffering from cancer, monthly. The MLEs of the unknown parameters, −LL*, K-S, and p-value for the proposed model are given in Table 22. Depending on −LL*, K-S, and p-values, it is noted that the OEHLIEx model is appropriate to analyze data set IV. The (1 − δ)100% CIs of the parameters α, β, and λ are, respectively, [126.999, 137.201], [0, 0.019], and [0, 0.107]. Figure 10 shows the TTT and PP plots based on the type-II right censored sample. Based on the model parameters, data sets I and II suffer from und phenomena (index of dispersion 1), whereas data set III suffers from ove (index of dispersion > 1). Moreover, data sets I, II, and III represent positivewith a platykurtic shape.

Dataset IV: Analysis and Discussion Based on Type-II Right Censored S
The censored data have been obtained from (http://www.biochemiaaccessed on 6 March 2021). They represent the recovery time of 50 patients su cancer, monthly. The MLEs of the unknown parameters, −LL*, K-S, and pproposed model are given in Table 22.   Figure 10 shows PP plots based on the type-II right censored sample. According to Figure 10, it is observed that the proposed model fits th Moreover, the shape of the HRF increases. Table 23 shows some computatio for data IV utilizing the OEHLIEx model. According to Figure 10, it is observed that the proposed model fits the data well. Moreover, the shape of the HRF increases. Table 23 shows some computational statistics for data IV utilizing the OEHLIEx model. Regarding Table 23, it is noted that the dataset IV suffers from under-dispersed phenomena which are positive skewed and platykurtic shaped.

Conclusions
In this article, a new flexible extension of an extreme distribution with three-parameter has been proposed, which generalized the inverse exponential distribution. The HRF of the new extension can be constant, increasing, increasing-constant, or unimodal shaped. Furthermore, it can be utilized for modelling asymmetric "positive and negative" as well as symmetric datasets and can be used to model over-and under-dispersed data. Thus, the new extension can be used effectively to model different kinds of data in several fields. The model parameters have been estimated utilizing the MLE approach. A simulation has been performed for different samples sizes, and it was found that the MLE technique works quite well for estimating the parameters for datasets considered herein. Finally, four data applications which illustrate the flexibility of the new extension and its excellence over other models have been also analyzed.