Alpha Power Inverted Kumaraswamy Distribution: Definition, Different Estimation Methods and Application

In this study, an alpha power inverted Kumaraswamy distribution having three shape parameters is obtained by applying the alpha power transformation to the inverted Kumaraswamy distribution. Then, its survival and hazard rate functions are expressed in closed forms. Some of its submodels and limiting cases are provided as well. Its parameters are estimated by using the maximum likelihood, maximum product of spacings, and least squares methods. A Monte-Carlo simulation study is conducted to show the performances of the considered estimation methods. An application to a real data set including values of breaking stress of carbon fibers is provided to illustrate an implementation of the proposed distribution and its modeling capability. The results show that alpha power inverted Kumaraswamy distribution can be an alternative to the its rivals.


Introduction
Probability distribution functions are usually used for modeling data from various fields. Some of the probability distributions are identified with the relevant field of science. For example, the Kumaraswamy distribution introduced by Kumaraswamy (1980) has been generally known for its applications in the hydrology. After with Jones' (2009) detailed work on the Kumaraswamy distribution, it has become more familiar to statisticians. Recently, AL-Fattah et al.
(2017) presented a transformation of the Kumaraswamy distribution, called as inverted Kumaraswamy (IKum) having a longer right tail when compared with the other well-known distributions. They stated that it allows more accurate predictions of rare events existing in the right tail of the distribution, i.e., long term reliability predictions. Later on, Abu-Moussa and El-Din (2018) obtained the maximum likelihood (ML) and Bayesian estimates for parameters of the IKum distribution under the progressive type-II censoring scheme. Aly and Abuelamayem (2020) derived the multivariate IKum distribution and used the ML and Bayesian methods to estimate its parameters. Hameed et al. (2020) conducted a study to estimate the stress strength reliability of a component under the IKum distribution. Recently, Bagci et al. (2021) used the IKum distribution to model the wind speed data and estimated its parameters by using the maximum product of spacings (MPS) and least squares (LS) methods.
A random variable Z following the IKum distribution, i.e., Z ∼IKum(β, λ), has the probability density function (pdf) Alpha power inverted Kumaraswamy distribution f Z (z; β, λ) = βλ(1 + z) −(λ+1) 1 − (1 + z) −λ β−1 ; z > 0, β > 0, λ > 0 (1) and the cumulative distribution function (cdf) Here, β and λ are the shape parameters. Statistical properties of the IKum distribution are provided by AL-Fattah et al. There are several methods for extending/generating statistical distributions; see Lee et al. (2013). These methods are generally used for making an existing distribution more flexible in terms of modeling capability. Recently, Mahdavi and Kundu (2017) used the alpha power transformation (APT) for generating a probability distribution. The APT method is conceptually based on adding a parameter to the distribution function to bring more flexibility to it. The APT method has some advantages of being easy applicability. Let F (t) be a cdf and f (t) be a pdf of a random variable T . The APT of F (t) and f (t) for t ∈ R are respectively. See Mahdavi and Kundu (2017) for the details.
Notice that the APT family of distributions is reparemetrized version of the "exp-G" family of distributions introduced by Barreto-Souza and Simas (2014) and the equivalent "truncated-exponential skew-symmetric" family of distributions obtained by Nadarajah et al. (2014) as stated in Jones (2018). Beside with keeping mind this fact, for the purpose of being parallel to the recent literature, the APT terminology will be used in the rest of this study.
In the literature, the APT method has been applied to many different distributions. The rest of this study is organized as follows. In Section 2, the pdf, cdf, and some statistical inferences of the APIK distribution are obtained. In Section 3, sub-models and limiting cases of the APIK distribution, and also distributions obtained by applying different variable transformations are derived. Afterward, in Section 4, the ML, MPS, and LS estimation for parameters of the APIK distribution are given, and then the Monte-Carlo simulation results for comparing the performances of these methods are provided. In Section 5, a real data set is modeled by using the APIK distribution and some of its strong alternatives to compare their modeling performances with each other. Finally, the last section reserved for some concluding remarks.

The APIK Distribution
In this section, the APIK distribution is obtained by applying the APT method to the IKum distribution.
Definition 2.1. A random variable X following the APIK distribution, i.e., X ∼APIK(α, β, λ), has the cdf and the pdf Definition 2.2. Let X ∼APIK(α, β, λ), then it has the survival function and the hazard rate function (hrf) The pdf, cdf and hrf of the APIK distribution, given in (3), (4) and (5), respectively, are plotted for different parameters settings in Figure 1(a), Figure 1 It is clear from the Figure 1(a) and Figure 1(c) that the pdf and hrf of the APIK distribution can be monotone decrease or monotone increase-decrease for the different parameter settings, respectively.
Alpha power inverted Kumaraswamy distribution Proof. It follows from due to the fact that lim Due to the fact given in Proposition 2.1, Definition 2.3 is obtained only for the APIK distribution, i.e., α = 1, for the sake of brevity.
Definition 2.3. The quantile function of the APIK distribution, i.e., inverse of the cdf of the APIK distribution, is Hence, the median of the APIK distribution is

Related Distributions
The APIK distribution includes some well-known distribution as a sub-model. It also converges to the some other distributions as a limiting case under different variable transformations. Here, referred distributions are given briefly.
It should be mentioned that following distributions are obtained by using the variable transformations and some limiting cases which were already defined by AL-Fattah et al. (2017).

Sub-models of the APIK distribution
The sub-models of the APIK distribution are obtained for different parameters settings as given below.

Transformations
A random variable Y has the following pdf and also its submodels for the certain parameter settings.

Limiting Distributions
The random variable Y has the following pdf for the certain parameter settings.

Parameter estimation
In this section, the ML, MPS and LS estimation methods are used for estimating the parameters α, β and λ of the APIK distribution.

ML estimation
Let x 1 , x 2 , . . . , x n be a random sample from the APIK distribution. Then, the log-likelihood (ln L) function of it is The likelihood equations and Alpha power inverted Kumaraswamy distribution are obtained by taking partial derivatives of the ln L function given in (6) with respect to the parameters α, β and λ, and equating them zero. The ML estimates of the parameters α, β and λ are obtained by solving the likelihood equations (7) -(9), simultaneously.

MPS estimation
The MPS method is based on maximizing a geometric mean of spacing. Therefore, the MPS estimates of α, β and λ of the APIK distribution are the points in which the objective function D α, β, λ; x Note that x 0 = 0 and x (n+1) = ∞ since the support for F X (x) is the positive real line. Therefore, F X (x 0 ; α, β, λ) ≡ 0 and F X (x n+1 ; α, β, λ) ≡ 1. See Arslan and Oncel (2017), Volovskiy and Kamps (2020), and references given them for further information.

LS estimation
In the LS method, it is aimed to minimize a sum of squares of the differences between theoretical and expected quantiles with respect to the parameters of interest, i.e., α, β and λ. Therefore, the LS estimates of the unknown parameters α, β and λ of the APIK distribution are the points in which the objective function S α, β, λ; x Note that optimization tools "fminsearch", "fminunc" or "ga" which are available in software MATLAB2015b can be used to find the ML, MPS, and LS estimates of the parameters α, β, and λ. See also Arslan

Monte-Carlo simulation
In this subsection, the Monte-Carlo simulation is conducted to compare performances of the ML, MPS, and LS methods in estimating the parameters α, β and λ of the APIK distribution. The bias, variance and mean squared eror (MSE) criteria are used in the comparisons. The simulated values of the MSE criterion are calculated by using the equalities The joint efficiency criterion, i.e., Def, is also used for the parameters α, β and λ. The simulated values of the bias, variance, and MSE criteria are calculated for 1000 runs in different sample sizes n = 50, 100, and n = 300 with the certain parameters settings. The simulation outcomes are given in Table 1 and summarized as follows.
i. When α = 0.5, β = 0.9, λ = 2.5: The MPS method gives the smallest bias values for α in all sample sizes. The LS method results larger bias values than the ML. In terms of the MSE criterion, the LS method produces the smallest values.
For the β, the ML, MPS, and LS methods have small biases for all sample sizes, however the ML method has the smallest and the MPS has the largest bias values for n = 100. In terms of the MSE criterion, the MPS method gives the smallest values for n = 50 and n = 100.
Concerning the λ, the ML method results the smallest bias values for n = 100 and n = 300, however the LS produces the largest bias values. In terms of the MSE criterion, the MPS method gives the smallest values for n = 50 and n = 100.
Overall, the MPS method is the best for n = 50 and 100 when the Def criterion taken into account.
ii. When α = 0.9, β = 1.5, λ = 0.5: For the λ, the MPS method gives the largest bias values for n = 100 and n = 300, however it has the smallest variance values for all sample sizes. In terms of the MSE criterion, the MPS method is one step ahead of the ML and LS since it produces the smallest MSE values for all sample sizes.
Overall, the MPS, LS and ML methods are preferable for n = 50, n = 100, and n = 300, respectively, when the Def criterion taken into account.
iv. When α = 2.0, β = 1.5, λ = 1.1: The ML method produces the smallest bias values for all sample sizes. The MPS method gives larger bias values than the LS, except n = 300. In terms of the MSE criterion, the LS method is preferable over the ML and MPS.
Concerning the β, the LS method results the smallest bias values while the ML and MPS methods have also small bias for n = 100 and n = 300. When the MSE criterion is taken into account, the MPS shows better performance than the ML and LS since it produces the smallest MSE values for all sample sizes.
For the λ, the ML, MPS, and LS methods produce small bias values for all sample sizes. However, the MPS method gives the smallest variance and the MSE values for all sample sizes.
Overall, the MPS method is one step ahead of the ML and LS since it produces the smallest Def values for all sample sizes.
Alpha power inverted Kumaraswamy distribution

Application
In this section, an application to a real data set in which contains values of breaking stress of carbon fibers (in GPa) is provided. The data set, given in Table 2, is obtained from Nichols and Padgett (2006) who conducted a study on developing control chart when process measurements follow the Weibull distribution. Recently, Jamal et al. (2019) modeled the breaking stress of carbon fibers data by using the Weibull, IKum, GIKw and GIKw-Weibull (GIKw-W) distributions. They showed that the GIKw-W distribution is preferable over the Weibull, IKum, and GIKw distributions in modeling the breaking stress of carbon fibers data when some goodness-of-fit measures are taken into account.
In this study, the APIK distribution is used for modeling the breaking stress of carbon fibres data; see Table 2. Beside, the GIKw-W and MOEIK distributions, which can be a strong alternative to the APIK distribution, are considered in the application. The IKum distribution is also included into the application to make comparisons complete. Modeling performances of the IKum, APIK, MOEIK, and GIKw-W distributions are compared by using well-known information criteria such as value of the ln L, Akaike Information Criterion (AIC), corrected AIC (AICc), and goodness-of-fit measures the Kolmogorov-Smirnov (KS), root-mean-squared error (RMSE) and coefficent of determination (R 2 ). As it is known, higher values of the R 2 and ln L, and lower values of the AIC, AICc, KS, and RMSE indicate better fit.
The parameters α, β, and λ of the APIK distribution are estimated by using the ML, MPS, and LS methods as given in Section 4. For estimating the unknown parameters of the IKum, MOEIK, and GIKw-W distributions, the ML method is considered. The optimization tool "fminunc" available in software MATLAB2015b is utilized in estimating the unknown parameters of the IKum, APIK, MOEIK, and GIKw-W distributions. The corresponding results are given in Table 3. Also, values of the information criteria and goodness-of-fit measures for the IKum, APIK, MOEIK, and GIKw-W distributions are provided in Table 4. Furthermore, the pdf and cdf fitting plots of the IKum, APIK, MOEIK, and GIKw-W distributions are shown in Figure 2(a) and 2(b) for an illustration, respectively.   In comparisons for the modeling performances of the APIK, MOEIK, GIKw-W, and IKum distributions, it is seen that the APIK distribution has the smallest AIC and AICc values. Note that the APIK and MOEIK distributions have three parameters in the meanwhile the GIKw-W distribution has five parameters.
When the goodness-of-fit measures are taken into account to compare the modeling performances of the APIK, MOEIK, GIKw-W, and IKum distributions, the APIK distribution has the smallest KS and RMSE values. Also, the GIKw-W distribution comes after the APIK distribution, since they have more or less the same RMSE and R 2 values.
Overall, in terms of the information criteria AIC and AICc, the APIK distribution is preferable over the IKum, MOEIK, and GIKw-W distributions. Also, the APIK distribution shows better modeling performance than the IKum, MOEIK, and GIKw-W distributions when the KS criterion is considered. Although the APIK distribution has fewer number of parameters than the GIKw-W distribution, it stands ahead of the GIKw-W distribution in modeling the breaking stress of carbon fibres data. To sum up, it is shown that the APIK distribution can be an alternative to the GIKw-W and MOEIK distributions, since it stands out in much more criteria.

Conclusion
In this study, the APIK distribution is obtained by using the APT method. Then, the sub-models and related distributions of the APIK distribution are obtained as well. To the best of the Authors' knowledge, some of its sub-models obtained in this study have not been introduced yet. The parameters of the APIK distribution are estimated by using the ML, MPS, and LS methods. The Monte-Carlo simulation study is conducted to show efficiencies of the considered estimation methods in estimating the parameters α, β, and λ of the APIK distribution. The real data set including values of breaking stress of carbon fibers is used to show the modeling capability of the APIK distribution. Also, the MOEIK and GIKw-W distributions, which can be considered strong alternatives to the APIK, are included in the application to make the study complete. As it can be seen from Table 4 that the APIK distribution has better modeling performance when compared with its rivals in many criteria. Overall, based on the results obtained in this study, the Authors hope that the APIK distribution can be useful for engineering studies as well as many other fields.