Abstract
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 , hazard rate function , reverse hazard rate function , quantile function and median, raw moment, generating function, mean deviation , etc.) are provided. The estimates for parameters of new G class are derived via maximum likelihood estimation 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 . The superiority of EGIKw-Burr XII model is proved using a lifetime data applications.
1. Introduction
Study of data is the most important and fundamental topic in statistics. The probability distributions help in the characterization of the variability and uncertainty prevailing in data by identifying the patterns of variation. The 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].
The 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 = to introduce the IKw distribution and proposed the generalized inverted Kumaraswamy (GIKum) distribution with respective and :where are the shape parameters, and . Let denote the expression for of some random variable , , where , and consider is some function of of another , say X; the T-X family can be defined aswhere satisfies the following:(1).(2) is differentiable and monotonically nondecreasing function.(3) as , as .
We give a new G class, the extended generalized inverted Kumaraswamy generated (EGIKw-G) family, considering to be GIKum and using the generator as in (2) in order to obtain the distributions which show higher flexibility compared with other commonly used standard distributions; see [23, 24]. For some baseline , the expression for the of EGIKw-G class is
or equivalentlywhere are extra positive parameters which offer the skewness, hence promoting the tails weight variation, and 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 , reliability measures, and 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.
2. 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 , the , the , the , and the cumulative hazard rate function which have an important role in reliability theory. If X follows EGIKw-G class (4), then its is
The expressions for the , the , the , and the are given byrespectively. The EGIKw-G class can be easily simulated through inverting (4) as follows: let be a standard uniform , ; the inverse or is given by solving as
Furthermore, median, three quartiles, and seven octiles can be, respectively, obtained by , ; ; and . The is useful for evaluating some crucial properties including skewness, kurtosis, and central probabilistic results. The Bowley skewness is given by
For some baseline distribution when the resulting EGIKw-G distribution is symmetric, right skewed, and left skewed, we have = 0, , and , respectively. A measure of kurtosis, the Moors kurtosis (see, e.g., Moors [25]), is given as
The tail of the EGIKw-G distribution becomes heavier as increases, provided that 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 , and when , the exponentiated-G class given by Cordeiro et al. [26] is obtained. Hence, parameter 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, “” and “” represent baseline and , respectively.
3. Special Models
The 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.
3.1. EGIKw-Normal Distribution
The EGIKw-Normal is obtained from (5) for and , so
and the iswhere , and ; and , respectively, denote the standard normal and . The in the above expression is EGIKw-N, e.g., , it reduces to standard EGIKw-N distribution. The and plots of EGIKw-N model are depicted in Figure 1. As given in Figure 1(b), the gives increasing, inverted bathtub, or decreasing (reversed-J) shapes.
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3.2. EGIKw-Fréchet Distribution
The Fréchet and for x 0, , respectively. Correspondingly, the EGIKw-Fréchet is
The iswhere . For , we obtain the extended generalized inverted Kumaraswamy inverse exponential distribution. Figure 2(a) indicates that the EGIKw-Fréchet offers various interesting shapes. Figure 2(b) reveals that the model can also offer various shapes including decreasing, increasing, J, revered-J, and bathtub shapes.
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3.3. EGIKw-Uniform Distribution
The EGIKw-U is obtained from (5), taking , where , as follows:
The is
A , say , with above model is given as . For , we have standard EGIKw-Uniform model. Figure 3 illustrates shapes of and for the EGIKw-Uniform model. The plot in Figure 3(a) offers a variety of shapes. Moreover, it is obvious from Figure 3(b) that this model can accommodate constant, decreasing, and unimodal .
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3.4. EGIKw-Burr XII Distribution
The Burr XII and are and , respectively. Hence, the EGIKw-Burr XII is
The corresponding takes the following form:
A X with the above is denoted as . Figure 4 displays some interesting shapes of EGIKw-Burr XII and . It is obvious from these plots that great flexibility is achieved with the proposed models.
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4. 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, moment, , moment generating function , and of order statistics.
4.1. Expansions for EGIKw-G and
We express EGIKw-G and in terms of finite (or infinite) weighted sums of exponentiated-G and , respectively. Consider the EGIKw-G given by (4)
For real noninteger and , the power series representations are
For integer value,
Using the series expansions given above, the EGIKw-G distribution function (4) is rewritten aswhere . For any integer value of , index i is stopped at , ; for an integer , the index stops at , ; and for an integer value of , index k is stopped at , . Thus, (22) reveals that EGIKw-G can be written in baseline as a multiple of its ’s power series. Otherwise, in case of to be a real noninteger, the in (22) can have following form
Using the binomial expansion for , we obtain
Further, (4) is rewritten aswhere . Replacing by in (22), we havewhere is sum in constants. The expansion (27) holds for all real noninteger values. It should be noted that EGIKw-G can also be provided in the form of exponential-G aswhere denotes exponential-G, where is power parameter. The corresponding results for EGIKw-G are obtained by differentiating (22) for integer and by (27) and (28) for real noninteger value, respectively, aswhere for , ; is exponential-G density function having parameter . Equation (31) expresses EGIKw-G density in terms of exponential-G densities. Equations (29)–(31) are among main results from this section.
4.2. Moments
Moments play a crucial role in studying some important characteristics (tendency, dispersion, skewness, kurtosis, etc.) of a distribution. The EGIKw-G moment can be given as weighted sum in probability weighted moments (PWMs) of order of the parent distribution. Let and , respectively, come from EGIKw-G and baseline G distribution. We can write raw moment for in terms of PWM (, ) of . For integer, we havewhere , is the PWM of baseline distribution and is defined in (29). For noninteger, we can writewhere is from (30) and denotes PWM of baseline distribution. Hence, moments for any EGIKw-G model can be calculated using baseline PWMs.
Furthermore, can be obtained using baseline , . For integer, from (22), and for noninteger, from (30), we, respectively, obtain
Using in the above expressions, we haverespectively. Moreover, we can also provide the EGIKw-G moments in the form of exponential-G moments. Let be an exponential-G with , , and , , and (r + 1) be the power parameter, so
Hence, we havewhere is defined in (31). Thus, EGIKw-G moments can be written as function of baseline exponential-G moments.
4.3. Moment Generating Function
Let EGIKw-G . We consider various expressions of for aswhere is the EGIKw-G noncentral moment. Another representation of , when integer, is derived from (29) aswhere the function is obtained using baseline as
For noninteger, using (30) we also have
and the function is easily deduced from baseline as
Another representation for for noninteger is obtained from (31) aswhere is of exponential-G(r+1). Hence, of any EGIKw-G model can be determined from the corresponding exponential-G .
4.4. Mean Deviations
The of a population measures its amount of scattering. For a X having f(x) and F(x), the about mean and about median are, respectively, written as and and are, respectively, given bywhere is the first ordinary moment, is from (4), is median obtained from (7) for , and represents incomplete moment. Using parent , two additional expressions for are derived. Firstly, when integer,For real noninteger, we havewhere are defined in (29) and (30), respectively. Another useful expression for is obtained from exponential-G distribution aswhere is defined by (31).
4.5. Rényi Entropy
Entropies of any , say , are measures of diversity of uncertainty. These measures have been used in various fields including engineering, physics, and economics. Rényi entropy is the most popular measure of entropy and is given as (Rényi [27])
Hence,
Equivalently depending on the parent ,where . In this section, (50) and (51) are main results.
4.6. Stress-Strength Reliability
The reliability measure of industrial components has crucial role especially in engineering. The 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. The component fails when (random stress) placed on it exceeds (random strength), and for it will work satisfactorily. Thus, measures the component’s reliability (Kotz et al. [28]). Let and be independent , ; let be an EGIKw-G with , (5), and parameters ; and let be a with , (4), and parameters with common baseline parametric space . Then, is obtained as
Alternatively, with the change of , ,where is from (7) corresponding to . Interestingly, we see that is independent of W(x), the baseline distribution. Additionally, various different forms will be yielded by using linear expression. One form is derived for integers by usingwhere , . Thus,
Similar expressions can be obtained for the case nonintegers. As usual, when , i.e., corresponding to the identically distributed case, we have .
4.7. Lorenz and Bonferroni Curves
The Lorenz curve for integer, is given as follows:
Equivalently based on parent and in the form of exponential-G distribution, we haverespectively. The corresponding expressions for Bonferroni curve are, respectively, given by (58)–(60) as
Similar expressions can be obtained using (30) for the case of noninteger.
4.8. Moments of Residual Life Function
In reliability theory and life testing problems, residual life has an important role. The moment is provided by
Similarly, residual moment of a having EGIKw-G distribution for integer and for noninteger is obtained by inserting of (29) and (30) in the above expression, respectively, as
Equivalently depending upon the parent , we have
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 be from the EGIKw-G distribution. Let denote the order statistics. The density of ordered value iswhere is expression for beta function. We offer the of EGIKw-G order statistics in the form of baseline as multiple of . Replacing (27) in the above expression yields
Let us considerwhere (Gradshteyn and Ryzhik [1]). Hence, we have
with . Using (68) in (65) with (29) for integer and with (30) for noninteger, we, respectively, obtain
Clearly, the above equations can be given in the form of exponential-G densities as
Equations (70) for integer and (71) for noninteger immediately yield the of EGIKw-G order statistics as a function of exponential-G ,s. Hence, the corresponding moments can be provided in the form of baseline PWMs for integer and for noninteger, respectively, by
Depending upon the parent for integer and for noninteger, we, respectively, obtain
Thus, the and other properties for EGIKw-G order statistics can also be obtained likewise.
5. Estimation
We employ for estimating unknown parameters of EGIKw-G distribution. Let be p-dimensional baseline parametric vector. Consider ’s , with each coming from a EGIKw-G model. The log-likelihood is obtained from (5) as follows:
The components of , the score vector, are
By solving , we obtain the MLEs .
6. 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 and root mean square error are considered to evaluate the performance of estimators for varying , s. The given by (7) with Burr XII as baseline model was considered for generating EGIKw-Burr XII . The simulation was repeated 2,000 times for varying samples. Four different parametric values, , were considered. The , and values for different , s are presented in Tables 1 and 2. From the results, it is clear that as increases, the for estimators on the average decreases. It is also observed that for all four sets, the showed decreasing pattern as increases. Thus, method performs quite well in parameter estimation of proposed G class.
7. 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 (GIKw-Burr XII) by Jamal et al. (2019), the exponentiated Kumaraswamy Burr XII (EKwBXII) distributions by Paranaiba et al. (2013), the inverse Weibull Burr XII (IW-Burr XII) Model by Amal et al. (2018), the exponentiated Kumaraswamy Burr XII (EK-Burr XII) distribution by Silva et al. (2019), the exponentiated Weibull distribution (EWD) by Nassar and Fissa (2003), the generalized inverse Weibull distribution (GIWD) by De Gusmao et al. (2011), the modified extension of exponential (MEXED) distribution by El-Damcese and Ramadan (2015), and the well-known Burr XII distribution.
For each considered model, we obtain the estimates using MLE method and adopt the minimum value of -log(likelihood) at MLE denoted by , Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion (CAIC), Hannan–Quinn Information Criterion (HQIC), Anderson-Darling statistics, Cramér–von Mises statistics, and Kolmogorov–Smirnov (K-S) tests. Data is about remission times of 128 bladder cancer patients in months from Lee and Wang (2003) and is provided as follows:
6.94, 8.66, 0.08, 2.09, 3.48, 4.87, 13.11, 23.63, 0.20, 9.02, 13.29, 0.40, 2.23, 3.52, 4.98, 6.97, 2.26, 3.57, 5.06, 7.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 26.31, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 79.05, 4.18, 5.34, 7.59, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 1.26, 2.83, 4.33, 5.49, 7.66, 4.34, 5.71, 7.93, 11.79, 11.25, 17.14, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 1.40, 3.02, 18.10, 1.46, 4.40, 5.85, 8.26, 3.36, 6.93, 8.65, 11.98, 19.13, 3.36, 6.76, 12.07, 21.73, 2.07, 12.63, 22.69.
The 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 and standard error (SE) (within parentheses) results. The computed goodness-of-fit results are provided in Table 5. Histograms with estimated pdf plot, 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.
8. 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 , , , and median, raw moment, , , 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 is presented to investigate the performance of and of . A real application is provided to check the usefulness of EGIKw-G class and its performance compared to other well-known distributions. The measures used all revealed that the novel model performed better than its counterparts [29,30].
Data Availability
The data are included in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R299), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.