On the Extended Generalized Inverted Kumaraswamy Distribution

In this work, we provide a new generated class of models, namely, the extended generalized inverted Kumaraswamy generated (EGIKw-G) family of distributions. Several structural properties (survival function (sf), hazard rate function (hrf), reverse hazard rate function (rhrf), quantile function (qf) and median, sth raw moment, generating function, mean deviation (md), etc.) are provided. The estimates for parameters of new G class are derived via maximum likelihood estimation (MLE) method. The special models of the proposed class are discussed, and particular attention is given to one special model, the extended generalized inverted Kumaraswamy Burr XII (EGIKw-Burr XII) model. Estimators are evaluated via a Monte Carlo simulation (MCS). The superiority of EGIKw-Burr XII model is proved using a lifetime data applications.


Introduction
Study of data is the most important and fundamental topic in statistics. e probability distributions help in the characterization of the variability and uncertainty prevailing in data by identifying the patterns of variation. e objective of statistical modeling is to develop appropriate probability distributions that adequately explain a data set generated by surveys, observational studies, experiment, etc.
In this context, there have been fundamental and significant thriving in probability distribution theory via the introduction of new generalized families of distributions, and several techniques to develop new distributions have been proposed. Some well-known systems of distributions are the beta generalized family of distributions by Eugene et al. [1], gamma generalized family by Zografos and Balakrishnan [2], Kumaraswamy generalized class of distributions by Cordeiro and de Castro [3], McDonald generalized family by Al-Sarabia [2012], gamma generalized family of distributions (type 2) by Ristic and Balakishnan [4], gamma generalized family (type 3) by Torabi and Hedesh [5], transformed-transformer (T-X) family by Alzaatreh et al. [6], logistic generalized family of distributions by Torabi and Montazeri [7], Weibull generalized class by Bourguignon et al. [8], Lomax generalized family of distributions by Cordeiro et al. [9], logistic X by Tahir et al. [10], odd generalized exponential family (OGE-G) by Tahir et al. [11], Garhy generalized class by Elgarhy et al. [12], Kumaraswamy-Weibull generalized family of distributions by Hassan and Elgarhy [13], exponentiated Weibull generalized family by Hassan and Elgarhy [14], additive Weibull generalized family by Hassan and Hemeda [15], type II half logistic generalized class by Hassan et al. [16], Zubair-G family of distributions by Ahmad [17], generalized inverted Kumaraswamy (GIKw) generated class by Jamal et al. [18], exponentiated Kumaraswamy-G class by Silva et al. [19], and type II Kumaraswamy half logistic family by El-Sherpieny and Elsehetry [20]. e inverted distributions are applied in various spheres of life including life testing, biology, environmental science, engineering sciences, and econometrics. Al-Fattah et al. [21] proposed the inverted Kumaraswamy (IKw) model via Y � 1/X − 1 transformation, when X has a Kumaraswamy distribution. Iqbal et al. [22] further generalized the model via transformation T � X c to introduce the IKw distribution and proposed the generalized inverted Kumaraswamy (GIKum) distribution with respective cdf and pdf: where α > 0, β > 0, c > 0 are the shape parameters, and x > 0. Let s(t) denote the expression for pdf of some random variable (rv), T ∈ [a, b], where − ∞ ≤ a < b < ∞, and consider D[W(x)] is some function of cdf of another rv, say X; the T-X family can be defined as where D[W(x)] satisfies the following: (2) D[W(x)] is differentiable and monotonically nondecreasing function.
We give a new G class, the extended generalized inverted Kumaraswamy generated (EGIKw-G) family, considering s(t) to be GIKum and using the generator (W λ (x, ϑ)/1 − W λ (x, ϑ)) as D[W(x)] in (2) in order to obtain the distributions which show higher flexibility compared with other commonly used standard distributions; see [23,24]. For W(x) some baseline cdf, the expression for the cdf of EGIKw-G class is or equivalently where α > 0, β > 0, λ and c > 0 are extra positive parameters which offer the skewness, hence promoting the tails weight variation, and ϑ denotes baseline parametric space. For the conditions on baseline distributions, a detailed note can be found in Alzaatreh et al. [6]. In the following section, the pdf, reliability measures, and qf are explored. In Section 3, four special submodels of EGIKw-G class are discussed. In Section 4, several useful properties of the suggested class are provided. In Section 5, MCS study and MLEs are considered to verify the convergence properties. In Section 6, the practical importance of considered G class is examined through real-word data.

Density and Reliability Measures
In this part of paper, we offer a brief discussion on some of the other basic functions related to the EGIKw-G class of models including the pdf, the sf, the hrf, the rhrf, and the cumulative hazard rate function (chrf) which have an important role in reliability theory. If X follows EGIKw-G class (4), then its pdf is e expressions for the sf, the hrf, the rhrf, and the chrf are given by respectively. e EGIKw-G class can be easily simulated through inverting (4) as follows: let u be a standard uniform rv, rv; the inverse cdf or qf is given by solving Furthermore, median, three quartiles, and seven octiles can be, respectively, obtained by Q(0.5), Q(0.5); and O j � Q(j/8), j ∈ (1, 2, 3, 4, 5, 6, 7). e qf is useful for evaluating some crucial properties including skewness, kurtosis, and central probabilistic results. e Bowley skewness is given by For some baseline distribution W(x) when the resulting EGIKw-G distribution is symmetric, right skewed, and left skewed, we have S k � 0, S k > 0, and S k < 0, respectively. A measure of kurtosis, the Moors kurtosis (see, e.g., Moors [25]), is given as e tail of the EGIKw-G distribution becomes heavier as K um increases, provided that W(x), α, β, c, and λ remain unchanged.
Note that the EGIKw-G class of models outlined above reduces to generalized inverted Kumaraswamy generated (EGIKw-G) class proposed by Jamal et al. [18], for c � 1, and when c � 1, λ � 1, the exponentiated-G class given by Cordeiro et al. [26] is obtained. Hence, parameter c offers more flexibility to the extremes for the density function curves, and therefor new G class becomes more suitable for data sets which exhibit heavy tail. For every generated model, "W" and " w" represent baseline cdf and pdf, respectively.

Special Models
e EGIKw-G density function (4) offers high flexibility in tails along with promoting variation in tail weights to extremes of specific model. In this section, we provide four of many possible submodels under EGIKw-G class offering a more better fit to the data. For brevity, in the remainder of this paper, we shall comment in detail on only four of the most impotent EGIKw-G distributions, namely, EGIKw-Normal, EGIKw-Fréchet, EGIKw-Uniform, and EGIKw-Burr XII distributions.

EGIKw-Burr XII Distribution.
e Burr XII pdf and cdf   Computational Intelligence and Neuroscience e corresponding cdf takes the following form: A rv X with the above pdf is denoted as X ∼ EGIKw − BurrXII(α, β, c, λ, ψ, ξ). Figure 4 displays some interesting shapes of EGIKw-Burr XII pdf and hrf. It is obvious from these plots that great flexibility is achieved with the proposed models.

Structural Properties of EGIKw-G Family of Distributions
In this part of article, we provide some useful expressions for EGIKw-G class including explicit expansions of density and cumulative distribution function, r th moment, m d, moment generating function (mgf), and pdf of order statistics.

Expansions for EGIKw-G cdf and pdf. We express
EGIKw-G cdf and pdf in terms of finite (or infinite) weighted sums of exponentiated-G cdf and pdf, respectively. Consider the EGIKw-G cdf given by (4) For d > 0 real noninteger and |y| < 1, the power series representations are For d > 0 integer value, Using the series expansions given above, the EGIKw-G distribution function (4) is rewritten as where reveals that EGIKw-G pdf can be written in baseline pdf as a multiple of its cdf's power series. Otherwise, in case of c to be a real noninteger, the W(x) λ(cj+k) in (22) can have following form Using the binomial expansion for [1 − W(x)] l , we obtain Using (24) into (23), we have Further, (4) is rewritten as where ∞ l�r t i,j,k,l,r is sum in constants. e expansion (27) holds for all real noninteger c values. It should be noted that EGIKw-G cdf can also be provided in the form of exponential-G cdf as where V r (x) � W(x) r denotes exponential-G cdf, where r is power parameter. e corresponding results for EGIKw-G pdf are obtained by differentiating (22) for c > 0 integer and by (27) and (28) for c > 0 real noninteger value, respectively, as is exponential-G density function having parameter (r + 1). Equation (31) expresses EGIKw-G density in terms of exponential-G densities. Equations (29)-(31) are among main results from this section.

Moments.
Moments play a crucial role in studying some important characteristics (tendency, dispersion, skewness, kurtosis, etc.) of a distribution. e p th EGIKw-G moment can be given as weighted sum in probability weighted moments (PWMs) of order (p, q) of the parent distribution. Let X and Y, respectively, come from EGIKw-G and baseline G distribution. We can write p th raw moment for X in terms of (p, q) th PWM ( where τ p,λ(cj+k)− 1 , is the (p, λ(cj + k) − 1) th PWM of baseline distribution and l i,j,k ″ is defined in (29). For c > 0 noninteger, we can write where z ⌣ r is from (30) and τ p,r denotes (p, r) th PWM of baseline distribution. Hence, moments for any EGIKw-G model can be calculated using baseline PWMs.
Computational Intelligence and Neuroscience 7 Furthermore, μ p ′ can be obtained using baseline qf, Q(u) � W − 1 (u) � x. For c > 0 integer, from (22), and for c > 0 noninteger, from (30), we, respectively, obtain Using u � W(x u ) in the above expressions, we have respectively. Moreover, we can also provide the EGIKw-G moments in the form of exponential-G moments. Let X r+1 be an exponential-G rv with cdf, V r+1 (x) � W(x) r , and pdf, v r+1 (x) � (r + 1)w(x)W(x) r , and (r + 1) be the power parameter, so Hence, we have where z r ″ is defined in (31). us, EGIKw-G moments can be written as function of baseline exponential-G moments.

Moment Generating Function.
Let X ∼ EGIKw-G (α, β, c, λ). We consider various expressions of mgf for X as where μ p ′ � E(X p ) is the p th EGIKw-G noncentral moment. Another representation of M(t), when c > 0 integer, is derived from (29) as where the function φ(t, λ(cj + k) − 1) � exp(tx)w(x)W(x) λ(cj+k)− 1 dx is obtained using baseline qf as For c > 0 noninteger, using (30) we also have and the function φ(t, r) � exp(tx)w(x)W(x) r dx is easily deduced from baseline qf as Another representation for M(t) for c > 0 noninteger is obtained from (31) as where M r+1 (t) is mgf of X ∼ exponential-G(r+1) rv. Hence, M(t) of any EGIKw-G model can be determined from the corresponding exponential-G mgf.

Mean Deviations.
e m d of a population measures its amount of scattering. For a rv X having pdf f(x) and cdf F(x), the md about mean and md about median are, respectively, written as δ μ (X) and δ M (X) and are, respectively, given by where μ 1 ′ is the first ordinary moment, F(μ 1 ′ ) is from (4), M is median obtained from (7) for u � (1/2), and T(z) � z − ∞ xf(x)dx represents 1 st incomplete moment. Using parent qf, two additional expressions for T(x) are derived. Firstly, when c > 0 integer, For c > 0 real noninteger, we have where l i,j,k ″ , z ⌣ r are defined in (29) and (30), respectively. Another useful expression for T(z) is obtained from exponential-G distribution as where z r ″ is defined by (31).

Rényi
Hence, Equivalently depending on the parent qf, where In this section, (50) and (51) are main results.

Stress-Strength
Reliability. e reliability measure of industrial components has crucial role especially in engineering. e reliability of a product is the probability that it will do its intended job up to a specific time, given that it is operating under normal conditions. e component fails when X 2 (random stress) placed on it exceeds X 1 (random strength), and for X 1 > X 2 it will work satisfactorily. us, R � P(X 2 < X 1 ) measures the component's reliability (Kotz et al. [28]). Let X 1 and X 2 be independent rv, rv; let X 1 be an EGIKw-G rv with f 1 (x), (5), and parameters α 1 , β 1 , c 1 , λ 1 ; and let X 2 be a rv with cdf F 2 (x), (4), and parameters α 2 , β 2 , c 2 , λ 2 with common baseline parametric space ϑ.
en, R is obtained as Alternatively, with the change of rv, X � Q 1 (u), Computational Intelligence and Neuroscience where Q 1 (u) is qf from (7) corresponding to f 1 (x). Interestingly, we see that R is independent of W(x), the baseline distribution. Additionally, various different forms will be yielded by using linear expression. One form is derived for c 1 , c 2 > 0 integers by using Similar expressions can be obtained for the case c 1 , c 2 > 0 nonintegers. As usual, when α 1 � α 2 , β 1 � β 2 , c 1 � c 2 , λ 1 � λ 2 , i.e., corresponding to the identically distributed case, we have R � (1/2).

Lorenz L(p) and Bonferroni B(p) Curves.
e Lorenz curve for c > 0 integer, is given as follows: Equivalently based on parent qf and in the form of exponential-G distribution, we have respectively. e corresponding expressions for Bonferroni curve are, respectively, given by (58)-(60) as (60) Similar expressions can be obtained using (30) for the case of c > 0 noninteger.

Moments of Residual Life Function.
In reliability theory and life testing problems, residual life has an important role. e n th moment is provided by

(61)
Similarly, n th residual moment of a rv having EGIKw-G distribution for c > 0 integer and for c > 0 noninteger is obtained by inserting pdf of (29) and (30) in the above expression, respectively, as Equivalently depending upon the parent qf, we have 10 Computational Intelligence and Neuroscience An alternative representation can be derived from exponential-G distribution as 4.9. Order Statistics. Order statistics are useful in detection of outliers and robust statistical estimation, characterization of probability distributions, reliability analysis, analysis of censored samples, etc. Let X 1 , X 2 , . . . , X n be n rv from the EGIKw-G distribution. Let X (1) , X (2) , . . . , X (n) denote the order statistics. e density of i th ordered value is where B(., .) is expression for beta function. We offer the pdf of EGIKw-G order statistics in the form of baseline pdf as multiple of W(x). Replacing (27) in the above expression yields Let us consider where c 0,z � (s 0 ) z , c t,z � (ts 0 ) − 1 t m�1 [m(z + 1) − t]s m c t− m,z (Gradshteyn and Ryzhik [1]). Hence, we have (29) for c > 0 integer and with (30) for c > 0 noninteger, we, respectively, obtain Clearly, the above equations can be given in the form of exponential-G densities as (70) Equations (70) for c > 0 integer and (71) for c > 0 noninteger immediately yield the pdf of EGIKw-G order statistics as a function of exponential-G pdf,s. Hence, the corresponding moments can be provided in the form of baseline PWMs for c > 0 integer and for c > 0 noninteger, respectively, by Depending upon the parent qf for c > 0 integer and for c > 0 noninteger, we, respectively, obtain Computational Intelligence and Neuroscience us, the mgf and other properties for EGIKw-G order statistics can also be obtained likewise.

Monte Carlo Simulation
In this part, we examined the usefulness of MLEs for EGIKw-Burr XII (a special model from the family) parameters, through an extensive numerical investigation. Average bias (AB) and root mean square error (RMSE) are considered to evaluate the performance of estimators for varying n, s. e qf given by (7) Tables 1 and 2. From the results, it is clear that as n increases, the RMSE for estimators on the average decreases. It is also observed that for all four sets, the AB showed decreasing pattern as n increases. us, MLE method performs quite well in parameter estimation of proposed G class.

Application
In this part of work, we use EGIKw-Burr XII distribution for cancer patients' data to illustrate the merit of GIKw-Burr XII model compared to the generalized inverted Kumaraswamy    e key statistics of data are offered in Table 3. Furthermore, the TTT-transform curve is depicted by Figure 5, which suggests an upside down bathtub or unimodal failure rate and, therefore, indicates that the EGIKw-Burr XII distribution is suitable for fitting this data set. Table 4 gives MLEs and standard error (SE) (within parentheses) results.
e computed goodness-of-fit (gof) results are provided in Table 5. Histograms with estimated pdf plot, cdf plot, QQ-plot, and PP-plot of the EGIKw-Burr XII and other distributions are provided in Figures 6-9, respectively. It is clear from these results that EGIKw-Burr XII model with six parameters offers a better fit than other distributions.

Conclusions
In this work, a four-parameter generated class of models, EGIKw-G class, is proposed. Submodels of the proposed class, namely, the EGIKw-Normal, EGIKw-Fréchet, EGIKw-Uniform, and the EGIKw-Burr XII distributions, are discussed. Various properties including sf, hrf, rhrf, qf and median, s th raw moment, mgf, md, Rényi entropy, reliability parameter, Lorenz and Bonferroni curves, residual lifetime, and distribution of order statistics are presented. Particular attention is given to EGIKw-Burr XII distribution. A MCS is presented to investigate the performance of AB and RMSE of MLEs. A real application is provided to check the usefulness Computational Intelligence and Neuroscience of EGIKw-G class and its performance compared to other well-known distributions. e gof measures used all revealed that the novel model performed better than its counterparts [29,30].
Data Availability e data are included in the paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest.