Improved Estimation of the Inverted Kumaraswamy Distribution Parameters Based on Ranked Set Sampling with an Application to Real Data

: The ranked set sampling (RSS) methodology is an effective technique of acquiring data when measuring the units in a population is costly, while ranking them is easy according to the variable of interest. In this article, we deal with an RSS-based estimation of the inverted Kumaraswamy distribution parameters, which is extensively applied in life testing and reliability studies. Some estimation techniques are regarded, including the maximum likelihood, the maximum product of spacing’s, ordinary least squares, weighted least squares, Cramer–von Mises, and Anderson–Darling. We demonstrate a simulation investigation to assess the performance of the suggested RSS-based estimators via accuracy measures relative to simple random sampling. On the basis of actual data regarding the waiting times between 65 consecutive eruptions of Kiama Blowhole, additional conclusions have been drawn. The outcomes of simulation and real data application demonstrated that RSS-based estimators outperformed their simple random sampling counterparts signiﬁcantly based on the same number of measured units.


Introduction
In some investigations, cost-effective sampling is a key concern, particularly when the measurement of the characteristic of interest is expensive, uncomfortable, or time-consuming. In order to increase the accuracy achieved per unit of the sample, the ranked set sampling (RSS) approach is an excellent tool for achieving observational economy. Reference [1] proposed the RSS technique as an alternative to the frequently used simple random sample (SRS) methodology for increasing the efficiency of the sample mean. In RSS, the population is divided into q sets of q units each by randomly selecting q 2 units from it. Without taking actual measurements, the q units in each set are ranked with respect to the study variable. For actual quantification, the lowest-ranked unit from the first set of q units is chosen. The second smallest ranked unit is measured from the second set of q units. The process is repeated; say h times until the greatest rated unit from the last set is determined. To obtain an RSS sample of size n • = qh, the entire process can be replicated a number of times, say h times. The following matrix notation is considered to express the RSS design.  The mathematical foundation of RSS was initially created in reference [2]. In statistical inference, the parametric estimate approach employing the RSS sampling strategy is of utmost importance. A large number of studies have recently been conducted on the issue of RSS-based estimate for a variety of parametric models. For example, refeference [3] used the data from the RSS technique to calculate the population variance. In subsequent years, RSS was employed to estimate the statistical distribution's involved parameters. Reference [4] used RSS to estimate the exponential distribution parameters, while reference [5] investigated the estimator of the Cauchy distribution's location parameter. Reference [6] looked at estimating normal and exponential distribution parameters based on RSS. Reference [7] investigated estimators for the logistic distribution's location and scale parameters, whereas reference [8] considered the RSS to get the geometric parameter estimators. Reference. [9] succeeded in estimating the parameters of the half-logistic distribution, while reference. [10] investigated estimators of logistic distribution parameters using RSS. Reference [11] achieved estimating modified Weibull distribution parameters. Reference [12] used RSS to derive maximum likelihood (ML) and Bayesian estimators for generalized exponential model. Reference [13] handled with parameter estimators of Pareto distribution using RSS. The parameter estimator of the Rayleigh distribution was regarded in [14] using different methods of estimations and ranking designs. Within the framework of RSS, reference [15] examined the approach of the ML for the shape and scale parameters of the generalized Rayleigh distribution. Reference [16] considered estimation of the new Weibull-Pareto distribution parameters using RSS. Reference [17] discussed the parameter estimators of Zubair Lomax distribution using RSS. Reference [18] investigated efficient estimation of the generalized quasi-Lindley distribution parameters under RSS. For recent results and references, see [19][20][21][22][23][24][25][26][27].
The Kumaraswamy distribution (KD) was offered in [28], which is one of the most significant lifetime distributions with a range of [0,1]. It cannot, however, be utilized for most lifetime data sets that theoretically have limitless support. It is regarded as a viable alternative to beta distribution since they both share the same characteristics such as being uni-modal, decreasing, increasing, or constant. The KD's probability density function (PDF) with two positive shape parameters is defined by: Reference [29] proposed the inverted KD distribution (IKD) with the goal of providing a new flexible lifetime distribution for analyzing real data sets in the best situation. The following are indeed the principles of the IKD: It's the distribution of the random variable Z = 1/X − 1, where X is a random variable following the KD in Equation (1). The PDF and cumulative distribution function (CDF) of the IKD are, respectively, characterized by: F(z; τ, υ) = 1 − (1 + z) −τ υ ; z, τ, υ > 0. Figure 1 presents some possible PDF plots of the IKD for some selected distribution parameters. It is clear that the distribution is skewed to the right. Mathematics 2022, 10, x FOR PEER REVIEW 3 of 21 Figure 1 presents some possible PDF plots of the IKD for some selected distribution parameters. It is clear that the distribution is skewed to the right. τ → ∞ According to reference [29], the IKD has a long right tail, and when compared to other distributions, the IKD gives optimistic forecasts of uncommon events happening in the right tail. To our knowledge, there have been no published works that have utilized the RSS to estimate IKD parameters. In this study, we assumed that the ranking is perfect and we focus on several classical estimations of the IKD parameters using RSS and SRS. The considered methods are the ML, maximum product of spacing (MPS), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CvM), and Anderson-Darling (AD). A simulation study compares the suggested RSS based estimators to the basic SRS based on some criteria measures.
The following is the paper's configuration: we provide the ML estimators (MLEs) and MPS estimators (MPSEs) of the IKD parameters in Section 2. Section 3 obtains the IKD parameter estimators using LS and WLS approaches. Using the AD and CvM methodologies, we obtain the IKD parameter estimators in Section 4. Sections 5 and 6 describe, respectively, simulation research as well as its application to real-world situations, and comparing RSS estimators to SRS equivalents. The paper comes to a close with main findings in Section 7.

ML and MPS Methods of Estimation
The ML and MPS estimation methods are examined in this section for estimating the IKD parameters. First, we'll go through the RSS framework methodology, and then obtain the estimators from SRS.

ML Estimators
denote cth order statistics (OS) in the tth cycle, where h is the number of cycles and q is the set size. The data in RSS are all mutually independent, and the data are distributed identically for each = 1, 2, . . . , . The distribution of the cth data, for each = 1, 2, . . . , , is identical to the distribution of the cth OS of the random sample X1,X2,...,Xq, that is; Substituting Equations (2) and (3) in Equation (4), then we have: The CDF (3) comprises several well-known distributions, including (i) the Lomax distribution for υ = 1, (ii) the beta type II (inverted beta) distribution for τ = 1, (iii) loglogistic distribution for τ = υ = 1, (iv) the inverse Weibull distribution as υ → ∞, and the generalized exponential distribution when τ → ∞. According to reference [29], the IKD has a long right tail, and when compared to other distributions, the IKD gives optimistic forecasts of uncommon events happening in the right tail.
To our knowledge, there have been no published works that have utilized the RSS to estimate IKD parameters. In this study, we assumed that the ranking is perfect and we focus on several classical estimations of the IKD parameters using RSS and SRS. The considered methods are the ML, maximum product of spacing (MPS), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CvM), and Anderson-Darling (AD). A simulation study compares the suggested RSS based estimators to the basic SRS based on some criteria measures.
The following is the paper's configuration: we provide the ML estimators (MLEs) and MPS estimators (MPSEs) of the IKD parameters in Section 2. Section 3 obtains the IKD parameter estimators using LS and WLS approaches. Using the AD and CvM methodologies, we obtain the IKD parameter estimators in Section 4. Sections 5 and 6 describe, respectively, simulation research as well as its application to real-world situations, and comparing RSS estimators to SRS equivalents. The paper comes to a close with main findings in Section 7.

ML and MPS Methods of Estimation
The ML and MPS estimation methods are examined in this section for estimating the IKD parameters. First, we'll go through the RSS framework methodology, and then obtain the estimators from SRS.

ML Estimators
Let {Z ct , c = 1, 2, . . . , q, t = 1, 2, . . . , h} denote cth order statistics (OS) in the tth cycle, where h is the number of cycles and q is the set size. The data in RSS are all mutually independent, and the data are distributed identically for each c = 1, 2, . . . , q. The distribution of the cth data, for each c = 1, 2, . . . , q, is identical to the distribution of the cth OS of the random sample X 1 , X 2 , . . . , X q , that is; Substituting Equations (2) and (3) in Equation (4), then we have: The likelihood function of IKD using RSS {Z ct , c = 1, 2, . . . , q, t = 1, 2, . . . , h}, where n • = qh is the sample size, is as follows: The log-likelihood function of IKD, denoted by l * SS is as follows: where ξ ct = (1 + z ct ). The MLEs of τ, and υ, denoted byτ ML , andυ ML , of the IKD using RSS are produced by solving the nonlinear equations ∂l * SS /∂τ = 0, and ∂l * SS /∂υ = 0.
Let Z 1 , Z 2 , . . . , Z n • be a SRS of size n • from a IKD with PDF (2) and CDF (3). The log-likelihood function, say l * SRS , for τ and υ is given by: The MLEs for τ and υ, say τ ML and υ ML , are given as the solution of the following equations: ∂l * SRS /∂τ = 0, and ∂l * SRS /∂υ = 0.
Using numerical technique in Equation (7), we obtain the MLEs of τ and υ, denoted by τ ML and υ ML .

LS and WLS Methods
The LS and WLS methods for estimating unknown parameters are well established in reference [31]. Here, the LS estimators (LSEs) and WLS estimators (WLSEs) of τ, and υ are examined using SRS and RSS design. We'll go over the RSS framework technique first, and then obtain these estimators using SRS.

AD and CvM Methods
We employ two estimating methods that are based on the minimization of two well-known goodness-of-fit statistics. The two techniques are the CvM and AD, both of which are based on the difference between the CDF and empirical distribution function estimations. The CvM estimators (CvMEs) and AD estimators (ADEs) of the IKD are explored using SRS and RSS designs.

Numerical Evaluation
A simulation study is undertaken here to evaluate the performance of the estimation methods under SRS and RSS. The random samples were produced using IKD for different values of υ and τ. The following is the simulation algorithm:

2.
Select q random samples via SRS each set of size q from the IKD, then rank the units within each set.

3.
Choose a sample for actual quantification by including the smallest ranked unit in the first set, the second smallest ranked unit in the second set, the process continues in this way until the largest ranked unit is selected from the last set. 4.
Repeat steps 2 and 3 for h cycles to obtain a sample of sizes n • = qh, where the set size q = 3, 4, 5 with cycles number h = 5 and 10. Obtain the ML estimates under RRS and SRS, respectively, by solving numerically Equations (6) and (7). 8.
The MPS estimates of τ, and υ, are produced by solving Equations (9) and 10 for RSS, while solving Equations (14) and (15) in case of SRS. 9.
The LS and WLS estimates of τ, and υ, are computed by solving numerically Equations (18) The PE, MSE, and the Eff are all presented in Tables 1-12. One can draw the following conclusions from the simulation findings.


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details :   83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9  12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data.
All estimates based on RSS are more efficient than their competitors based on SRS in most of the situations (see Tables 9-12).


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details :   83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9 12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details: 83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9 12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted As the cycles number increase, the Eff value of estimates increases in most of the cases (Tables 9-12).


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details :   83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9 12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data.
In most of the cases, the MSEs of all estimates decrease as n • increases in both SRS and RSS (see Figures 3 and 4).


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.


The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details: 83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9 12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data.


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.


The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details: 83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9 12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data.


The MSEs of υ and τ estimates for all methods of estimation in RSS are smaller than the others via SRS (Figures 6 and 7).


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.


The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts The MSEs of υ and τ estimates for all methods of estimation in RSS are smaller than the others via SRS (Figures 6 and 7).


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.


The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A


The MSEs of υ and τ estimates for all methods of estimation in RSS are smaller than the others via SRS (Figures 6 and 7).    Efficiency of all estimates increases as the set size increases (Figures 8 and 9).


The MSEs of υ and τ estimates for all methods of estimation in RSS are smaller than the others via SRS (Figures 6 and 7).    Efficiency of all estimates increases as the set size increases (Figures 8 and 9).


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details: Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data. Figure 8 shows that AD is the most Eff method for υ estimates at (υ, τ) = (0.5, 1.2), and h = 10 for different set sizes. Also, we conclude that the MPS is the least Eff method for υ estimates at (υ, τ) = (0.5, 1.2), and h = 10 for different set sizes.


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details: Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data.


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details :   83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9 12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According Efficiency of all estimates increases as the set size increases (Figures 8 and 9).


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details: Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data. Based on the aforementioned theoretical results, actual data sets are checked using the RSS and SRS sampling techniques. The SRS and RSS estimators from the IKD are shown in Tables 13 and 14 for various set sizes under five and ten cycles utilizing different estimating techniques. The R-package is used to generate the RSS and SRS observations.


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details: Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data. Based on the aforementioned theoretical results, actual data sets are checked using the RSS and SRS sampling techniques. The SRS and RSS estimators from the IKD are shown in Tables 13 and 14 for various set sizes under five and ten cycles utilizing different estimating techniques. The R-package is used to generate the RSS and SRS observations.
The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.


The MSE always decreases as sample size increases, indicating that the estimates are all consistent.  The estimates get more accurate as the sample size increases, indicating that they are asymptotically unbiased.  In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 hours' worth of eruption data collected. These are the data set's details :   83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9 12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data. Based on the aforementioned theoretical results, actual data sets are checked using the RSS and SRS sampling techniques. The SRS and RSS estimators from the IKD are shown in Tables 13 and 14 for various set sizes under five and ten cycles utilizing different estimating techniques. The R-package is used to generate the RSS and SRS observations. In general, for fixed υ and τ as the sample size increases the suggested estimates of υ and τ approach their real values.

Real Data Application
In order to demonstrate the utility of the suggested estimators, a real data set was taken into consideration and carefully thoroughly explained in this part. The information is based on the times between 64 consecutive eruptions of Kiama Blowhole in 1998. A popular tourist destination is the Kiama Blowhole, which is around 120 km south of Sydney. The water is forced into a cliff's gap by the ocean's surging. The water then bursts forth via an opening, typically dousing everyone close. Since 12 July 1998, there have been 1340 h' worth of eruption data collected. These are the data set 's details:   83  51  87  60  28  95  8  27  15  10   18  16  29  54  91  8  17  55  10  35   47  77  36  17  21  36  18  40  10  7   34  27  28  56  8  25  68  146  89  18   73  69  9  37  10  82  29  8  60  61   61  18  169  25  8  26  11  83  11  42   17  14  9  12 Using the Kolmogorov-Smirnov (K-S) test, the data set is checked for such a fitted model and the estimates are obtained using the ML method. With a p-value (PV) of 0.591, the K-S distance is 0.409. As a result, it is obvious that the IKD is a suitable model for fitting these data. Data's estimated PDF and CDF are displayed in Figure 10. According to this graph, the IKD seems to be a suitable model for fitting the data. Based on the aforementioned theoretical results, actual data sets are checked using the RSS and SRS sampling techniques. The SRS and RSS estimators from the IKD are shown in Tables 13 and 14 for various set sizes under five and ten cycles utilizing different estimating techniques. The R-package is used to generate the RSS and SRS observations.  Based on the aforementioned theoretical results, actual data sets are checked using the RSS and SRS sampling techniques. The SRS and RSS estimators from the IKD are shown in Tables 13 and 14 for various set sizes under five and ten cycles utilizing different estimating techniques. The R-package is used to generate the RSS and SRS observations. We considered the K-S test for quantifying the distance between the empirical distribution function of the real data and the CDF using the estimators' parameters in each design, based on the choices of q and h, in order to demonstrate the superiority of RSS over the SRS using various estimation methods considered in this study. Be aware that we have substituted the K-S for the mean squared in this case. Obviously, estimators that outperform the other competitors have larger PVs (greater than 5%) and lower K-S values. Remember that for data, the MLEs based on SRS are regarded as the true population parameters. The SRS design is considered for this dataset and for each estimation method, where the estimators are obtained using a sample of size qh = 40. Using the RSS with sample of sizes q = 4 and h = 10 is considered for calculating the estimators. We compare the SRS and RSS designs in terms of the K-S distance value and PVs results given in Table 15, and the corresponding fittings are displayed in Figures 11 and 12. Due to the smallest values of the K-S and the corresponding largest PVs, the RSS is more efficient than the SRS based on the same number of measured units for all estimators. we have substituted the K-S for the mean squared in this case. Obviously, estimators that outperform the other competitors have larger PVs (greater than 5%) and lower K-S values.
Remember that for data, the MLEs based on SRS are regarded as the true population parameters. The SRS design is considered for this dataset and for each estimation method, where the estimators are obtained using a sample of size qh = 40. Using the RSS with sample of sizes q = 4 and h = 10 is considered for calculating the estimators. We compare the SRS and RSS designs in terms of the K-S distance value and PVs results given in Table 15, and the corresponding fittings are displayed in Figures 11 and 12. Due to the smallest values of the K-S and the corresponding largest PVs, the RSS is more efficient than the SRS based on the same number of measured units for all estimators. Figure 11. Plots of the estimated PDFs under SRS and RSS designs for the dataset for n º = qh = 40. Figure 11. Plots of the estimated PDFs under SRS and RSS designs for the dataset for n º = qh = 40.

Conclusions
This article considered the problem of RSS-based estimation for the IKD parameters using some estimation techniques; such as, the maximum likelihood, the maximum product of spacing, the ordinary and weighted least squares, the Cramer-von Mises, and the Anderson-Darling. With the aid of a simulation study and the use of a real dataset, the performance results of the proposed estimators are contrasted with those of their SRS equivalents based on the same number of measured units. The numerical simulation results show that for all outcomes shown in the tables with the same sample sizes, the proposed RSS estimators are superior to their SRS counterparts in terms of the smallest MSE. The estimates are asymptotically unbiased because their accuracy improves with sample

Conclusions
This article considered the problem of RSS-based estimation for the IKD parameters using some estimation techniques; such as, the maximum likelihood, the maximum product of spacing, the ordinary and weighted least squares, the Cramer-von Mises, and the Anderson-Darling. With the aid of a simulation study and the use of a real dataset, the performance results of the proposed estimators are contrasted with those of their SRS equivalents based on the same number of measured units. The numerical simulation results show that for all outcomes shown in the tables with the same sample sizes, the proposed RSS estimators are superior to their SRS counterparts in terms of the smallest MSE. The estimates are asymptotically unbiased because their accuracy improves with sample size. The results of the real dataset also showed that the RSS design is superior to the SRS design due to the largest values for its P-values. As future research the IKD parameters can be estimated using other modifications of RSS as modified robust extreme ranked set sampling [32], neoteric RSS [33] and varied RSS [34]. Also, the process performance index of the IKD can be obtained in future [35,36].