Odd Inverse Power Generalized Weibull Generated Family of Distributions: Properties and Applications

Department of Mathematics, College of Science, Jouf University, P. O. Box 848, Sakaka 72351, Saudi Arabia Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia Department of Mathematics, College of Science and Arts, Qassim University, Ar Rass, Saudi Arabia /e Higher Institute of Commercial Sciences, Al Mahalla Al Kubra, Algarbia 31951, Egypt


Introduction
ere are many applications for inverse (I) distributions, including econometrics, biology, life testing, engineering sciences as well as medical investigation and survey sampling concerns. It is also used in financial literature, environmental studies, survival and dependability theory, and other fields. By applying the inverse transformation to well-known random variables that display distinct properties of density and hazard rate shape, several writers have explored (lifetime) phenomena that a noninverted distribution cannot examine, for example, the I exponential model by [1], I Rayleigh distribution by [2], I Lindley distribution by [3], I power Lindley distribution by [4], I Kumaraswamy distribution by [5], Nadarajah-Haghighi distribution by [6], and I Topp Leone by [7], among others. e author in [8] introduced a new three-parameter distribution called I power generalized Weibull distribution. e distribution function (CDF) of I power generalized Weibull (IPGW) distribution is given by where θ > 0 is a scale parameter and α, λ > 0 are shape parameters. e associated density function (PDF) is as follows: g λ,α,θ (z) � θαλz − α− 1 1 + θz − α λ− 1 exp 1 − 1 + θz − α λ . (2) existing ones. Adding one or more form parameters results in these new generators, which improve accuracy and flexibility in modeling for a variety of diverse real-life applications. e most recent families of distributions to appear in the literature are as follows: a method for introducing a parameter into a family of distributions by [9], beta-G by [10], odd Nadarajah-Haghighi-G by [11], the odd Lindley-G by [12], and the odd Fréchet-G by [13], odd generalized exponential-G by [14], exponentiated power generalized Weibull power series family of distributions by [15], and odd generalized NH-G by [16], among others.
Using [17] T-X concept, we create a new broader and more flexible family of distributions known as the odd I power generalized Weibull-G (OIPGW-G) family.
λ | V�λ,α,θ,ξ > 0 and z∈R , (3) where O α ξ (z) � (G(z; ξ)/G(z; ξ)) α . e corresponding PDF is given by A random variable (RV) Z with PDF (4) is now indicated as Z ∼ OIPGW-G(V). e reliability function (RF) and hazard rate function (HRF) can be derived from each other as F V (z) � 1 − F V (z) and τ(z; V) � f V (z)/F V (z). e following is an interpretation of the OIPGW-G family. Let Y be a RV with a continuous G distribution that describes a stochastic system. e probability that a person (or component) will not be working (failure or death) at time z after a lifespan of Y is characterized with (G(z; ξ)/G(z; ξ)). e RV Z represents the variability of this chances of failure, and we suppose that it follows the OIPGW-G family of parameters θ, α, and λ, and the CDF of Z it is possible to write As a result of this, OIPGW-G family is used for the following reasons: (i) Specific models with all sorts of HRFs to be defined (ii) Better matches than other models with the same baseline distribution that has been generated (iii) Make kurtosis more flexible than the baseline model (iv) e pdf can be symmetric or right or left skewed and reversed J shaped. An extremely versatile model, the OIPGW-G family of distributions is able to adapt to a variety of various models when its parameters are altered. e OIPGW-G family of distributions includes the following well-known families as special cases in Table 1.
e quantile function (QF) Q G (u) is given by the relation as follows: Equation (6) can be used in deriving the Bowley skewness coefficient and the Moors kurtosis coefficient. e following is how the rest of this article is organized. e linear form of the PDF and CDF for the new family is expressed in Section 2. Section 3 contains several special models of the OIPGW-G family. Section 4 discusses the new family's structural characteristics as moments (Mos), incomplete Mos (IMos), mean deviations (MDes), Bonferroni (Bon) and Lorenz (Lor) curves, moment generating function (MoGF), and Probability Weighted Moments (PrWMos). Also, some numerical analyses for the mean M(Z), variance Var(Z), coefficient of skewness CS(Z), coefficient of kurtosis CK(Z), and coefficient of variation CV(Z) are discussed in the same section. Section 5 computes entropies of various sorts and also some numerical values of different entropies for specific selected parameter values for the odd inverse power generalized Weibull exponential model. In Section 6, parameters are estimated using three different techniques as the maximum likelihood (ML), least square (LS), and weighted LS (WLS) techniques. Many bivariate and multivariate type models have also been examined in Section 7. In Section 8, two applications to real-world data sets illustrate the empirical significance of the odd inverse power generalized Weibull exponential model. At the end of the paper, there are conclusions.

Important Representation
We give a helpful linear form for the OIPGW-G PDF in this section. If |z 1 /z 2 | < 1 and ∇ > 0 is a real noninteger, then the next power series (PS) expansions hold.

Some Special Models of the OIPGW-G Family
In this part, we described four submodels of the OIPGW-G family of distributions. ere is no doubt about it, and the PDF (4) will be the most tractable when the CDF G(z) and PDF g(z) have easy-to-understand analytic expressions. When we start with the baseline distributions: uniform (U), exponential (Ex), Weibull (W), and Rayleigh (R), we get at four submodels of this family. Table 2 shows the CDF and PDF of various baseline models. Figures 1 and 2 represent the plots of PDFs and the HRFs for the models which are reported in Table 2. From Figure 1, we can note that the PDFs can be right skewed and symmetric with "unimodal" and "bimodal" shapes. e HRFs can be "constant," "decreasing," "increasing," "increasing-constant," "upside-down-constant," and "decreasingconstant."

Ordinary Moments and Incomplete Moments Functions.
e r th ordinary Mos of Z, say μ / r , is driven from (14) as where Z m+1 has a power parameter of (m + 1) and represents the exp-G random variable. It is possible to obtain the Mathematical Problems in Engineering where It is possible to determine the MDes, Bon, and Lor curves using the first IMos. It is highly important in economics and dependability as well as in demography as well as in the fields of insurance and medical. Not only in econometrics but also in many other fields as well, this is apparent. e s th IMos of Z defined by υ s (t) for any real s > 0 can be investigated from (14) as (20) Equation (20) denotes the s th incomplete moments of Z m+1 . As well as providing significant information on population characteristics, the MDes have been used to income fields and property in the field of economics for a z HRF HRF  (3), and υ 1 (t) is the first IMo given by (20) with s � 1, where We can determine δ μ (z) and δ M (z) by two techniques; the first can be obtained from (14) as e second technique is given by can be calculated numerically and Q G (u) � G − 1 (u; φ). e Lor and Bon curves, for a given probability p, are given by Tables 3  and 4  e first formula of MoGF may be computed as follows from equation (14): can be easily determined from the exp-G MoGF. A second alternative formula can be computed from (14) as

Probability Weighted Moments.
e (s, r) th PrWMos of the OIPGW-G family is from (3) and (4), and after some algebra, we get where erefore, the (s, r) th PrWMos of the OIPGW-G family can be expressed as us, the (s, r) th PrWMos of Z is

Entropies
is section is dedicated to obtain the expression for different entropy measures of the OIPGW-G family. e Rényi entropy (RéE), presented by [18], is defined by      Mathematical Problems in Engineering Using (4), applying the same method of the useful expansion (14) and after some algebra, we get where us, Ré entropy of OIPGW-G family is given by where e Tsallis entropy (TE) measure (see [19]) is defined by e Havrda and Charvat entropy (HaChE) measure (see [20]) is defined by e Arimoto entropy (ArE) measure (see [21]) of OIPGW-G is defined by  Tables 5-8.  From Tables 3-8, we can note that the values of entropies can be negative or positive.

Statistical Inference
e parameters of the OIPGW-G are estimated using various methods including ML, LS, and WLS methods of estimation.

Ordinary and Weighted Least Square Estimators.
Suppose z 1 , . . . , z n is a random sample from OIPGW-G with corresponding ordered sample of z (1) , . . . , z (n) . e mean and variance of OIPGW-G are independent of unknown parameter and are as follows: E(F(Z (i) )) � (i/(n + 1)) and var(F(Z (i) ) � (i(n − i + 1)/(n + 1) 2 (n + 2)), where F(Z (i) ) is cdf of OIPGW-G with be the i th order statistic. en, LS estimators are obtained by minimizing the SSE: with respect to Ω. e WLS estimators of Ω can be obtained by minimizing the following expression:    (n + 1) 2 (n + 2) i(n − i + 1) with respect to Ω.

Simulation Outcomes.
Here, we come up with a numerical study to compare the behavior of different estimates. We generate 1000 random samples of size n � 50, 100, and 200 from the OIPGWEx distribution from the following equation: ree sets of the parameters are assigned. e MLE, LSE, and WLSE of α, θ, λ, and β are determined by using MATHCAD (14). en, the estimates of all methods and their mean square errors (MSEs) are documented in Tables 9-11.

Copula under the OIPGWEx Model
e fact that C · (u, v) is a straightforward function of the U marginal CDFs, F(u) and G(v), is a property shared by all of the probability distributions in this section. ese sorts of joint models are referred to as "Copulas (Co)." Copula techniques are widely used in insurance, econometrics, and finance.
ey are multivariate distributions with uniform marginals on the interval I (0,1) � (0, 1). Using FGM Co, modified FGM Co, Clayton Co, and Rényi's entropy, we construct several new bivariate type OIPGWEx (BvOIPG-WEx) models. Additionally, the MvOIPGWEx type of the multivariate KBX is discussed. ese new models, on the other hand, may be the subject of future research efforts. (44)

e Bivariate OIPGWEx Extension via Clayton Copula.
Bivariate extension through Clayton Copula is a weighted variant of the Clayton Co, which has the form Similarly, the multivariate OIPGWE extension can be ob-

Modeling Failure (Relief ) Times.
e failure time data are the first data set. Recent analyses of these data were conducted by [27]. Table 12 lists the MLEs. Table 13 lists C 1 , C 2 , C 3 , C 4 , A · , C · , K. S., and P. V. e total time test (TTT) plot for the relief time data, as well as the related box plot, is shown in Figure 7. Based on Figure 7, the HRF of the relief times is "increasing HRF," and these data have only one EV observation. Figure 8 gives the E-PDF, E-CDF, E-HRF, and P-P plot for relief time data. Figure 9 gives Kaplan-Meier (KM) survival plot for relief time data.
Based on Table 13, we have come to the conclusion that the OIPGWEx model is much better than the Ex, OLEx, MOEx, MEx, LBHEx, GMOEx, BEx, MOKwEx, KwE, BXEx, and KwMOEx models with C 1 � 38.74, C 2 � 42.7, C 3 � 41.41, C 4 � 39.52, A · � 0.178, C · � 0.0317, K. S � 0.0999, and P. V. � 0.9884, so the OIPGWEx model is an excellent alternative to these competitive distributions in modeling relief times data set. According to Figures 8 and 9, the OIPGWEx distribution provides adequate fits to the empirical functions.

Modeling Survival times.
e survival data are the second data set. Recent analysis of these data was conducted by [29]. Table 14 lists the MLEs. Table 15 lists the C 1 , C 2 , C 3 , C 4 , A · , C · , K. S., and P. V. Figure 10 gives the TTT plot along with the corresponding box plot for the survival time data.
Based on Figure 10, the HRF of the survival times is "increasing HRF," and these data have only four EV observations. Figure 11 gives the estimated PDF (E-PDF), E-CDF, E-HRF, and P-P plot survival time data. Figure 12 gives the Kaplan-Meier survival plot survival times data. Based on Table 15, we have come to the conclusion that the OIPGWEx model is far superior to the Ex, OLEx, MOEx, MEx, LBHEx, GMOEx, BEx, MOKwEx, KwEx, BXEx, and KwMOEx models with C 1 � 206.88, C 2 � 215.98, C 3 � 207.47,  Table 13: C 1 , C 2 , C 3 , C 4 , A · , C · , K. S., and P. V. for the first data.

Conclusions
A novel flexibly generated family of distributions was developed and investigated in this study which is called OIPGW-G family. e new proposed family contains many new models, and its density can be right skewed and symmetric with unimodal and bimodal shapes.
e new HRF of the new models can be "constant," "decreasing," "increasing," "increasing-constant," "upside-down-constant," and "decreasing-constant." Some of the mathematical properties of the new family are computed. Numerical calculations for the expected value, skewness, variance, and kurtosis are computed. Different types of entropies are calculated. Some numerical values of RéE, QE, HaChE, and ArE for some selected values of parameters for the OIPG-WEx model are computed. Estimation of OIPGW-G parameters is performed by ML, LS, and WLS estimation methods. Some bivariate and multivariate OIPGWEx type models have been also derived. Two genuine data sets are used to demonstrate the family's utility and versatility. e OIPGWEx model is far superior to the Ex, OLEx, MOEx, MEx, LBHEx, GMOEx, BEx, MOKwEx, KwEx, BXEx, and KwMOEx models in modeling the two data sets according to the C 1 , C 2 , C 3 , C 4 , A · , C · , K. S., and P. V. statistics. In the future, we are planning to use this family to generate new statistical models and study its structural properties.

Data Availability
e data used in the study are included within the article.