Reliability Estimation of Inverse Lomax Distribution Using Extreme Ranked Set Sampling

Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, Jordan Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 11432, Saudi Arabia Department of Statistics and Operations Research, King Saud University, Riyadh 11451, Saudi Arabia


Introduction
The inverse Lomax (ILo) distribution is considered as the reciprocal of the Lomax distribution. In some situations, it is a good alternative to the famous distributions like gamma, inverse Weibull, and Weibull. It has varied applications in modelling several types of data, including economics and actuarial sciences (see [1]). It has an application in geophysical databases [2]. The ILo distribution has an important application in reliability analysis [3]. Statistical inference for this distribution has been discussed by several researchers (see, for example, [4,5]). In the present work, the ILo distribution is taken under the stress strength (S-S) model associated with any system that depends on different sampling schemes. The cumulative distribution function (cdf) of the ILo distribution with shape parameter ω and scale parameter ρ is specified by the following: The probability density function (pdf) of the ILo distribution is as follows: The RSS was first introduced in [6] as a sampling scheme. The RSS scheme is used in situations when it is difficult and expensive to measure a large number of elements, but visually (without inspection) ranking some of them is easier and cheaper. This sampling design is both a cost-effective and powerful alternative to the commonly used SRS. This scheme involves randomly selecting m 1 sets (each of size m 1 elements) from the study population. The elements of each set are ordered with respect to the variable of the study by any negligible cost method or visually without measurements. Finally, the a th minimum from the a th set, a = 1, 2 ⋯ , m 1 , is specified for measurement. The obtained sample is called a RSS of set size m 1 . The whole procedure can be repeated q times to yield a RSS of size m • = qm 1 . The mathematical theory of the RSS method has been provided in [7]. Studies on RSS scheme have been proposed by several authors (see, for example, [8][9][10][11][12][13][14][15]).
Several modifications of the RSS have been proposed to improve the efficiency of the estimators. Herein, we are interested in the RSS and ERSS, presented in [16]. The ERSS procedure involves randomly selecting m 1 sets (each of size m 1 elements). The elements of each set are ordered with respect to variable of the study by visual inspection or any other cost free method. For an odd set size (OSZ), we select from the first ððm 1 − 1Þ/2Þ samples the smallest ranked unit, from the other ððm 1 − 1Þ/2Þ the largest ranked unit, and for the last sample select the median of the sample for actual measurement. For even set size (ESZ), we chose from ðm 1 /2Þ samples the smallest ranked unit and from the other ðm 1 /2Þ samples the largest ranked unit for actual measurement. This procedure can be repeated q times to obtain m 1 q units from ERSS data.
The S-S reliability R = P ðY < XÞ is the probability of the system working when a strength X is greater than a stress Y. So, the system will stop working when the applied stress is greater than its strength. Thus, the parameter R is a measure of a system's reliability, which has many applications in physics, engineering, genetics, psychology, and economics. There is an extensive literature on estimating R based on SRS (see, for instance, [17][18][19][20][21][22][23][24]). However, in recent years, statistical inferences about the S-S model based on the RSS method have been considered by several researchers. Reference [25] discussed estimation of S-S reliability for exponential populations. Reference [26] proposed three estimators of R when X and Y are independent exponential populations. References [11,27] discussed the estimation of the S-S model when Y and X are two independent Burr type XII distribution under several modifications of the RSS method. Estimation of the S-S model for Weibull and Lindley distributions has been discussed, respectively, in [28,29]. Reference [30] obtained a reliability estimator of R for the exponentiated Pareto distribution under the RSS scheme.
The S-S model is one of the important approaches in reliability analysis. The S-S model can be used to solve a variety of engineering problems, such as determining whether a building's strength should be subjected to the design earthquake, whether a rocket motor's strength should be greater than the operating pressure, and comparing the strength of different materials. The ILo is one of the distributions which is used quite effectively for modelling the strength of data used in economics, geography, actuarial, and medical fields. It has been discovered to be very flexible in analyzing situations with a realized nonmonotonic failure rate, which has wide applications in modelling life components. The RSS method and its modifications are frequently employed to gather samples that are more representative of the underlying population, when sampling units are expensive and difficult to measure but easy and inexpensive to arrange according to the variable of interest. In this method, ranking can be done using expert opinions, auxiliary variables, or any other low-cost approach. Statistical inference on the S-S model, based on the RSS scheme and its variations, has recently gotten a lot of attention. Due to the importance of the ILo distribution in reliability research, we propose to evaluate the reliability estimator of the S-S model where the strength X~ILoðρ, ωÞ and stress Y~ILoðρ, φÞ are both independent. Under SRS, RSS, and ERSS methods, the maximum likelihood (ML) estimators of R are derived. Based on the ERSS scheme, we get the ML estimator of R when both X and Y populations have similar or dissimilar set sizes. We evaluate the accuracy of estimators using absolute biases (ABs), mean squared errors (MSEs), and relative efficiencies (REs) in a simulated exercise. The remainder of this essay is structured in the following manner. In Section 2, we extract R's expression and use SRS to calculate R's ML estimator. In Section 3, the RSS is used to obtain an estimator for the S-S model. Section 4 presents reliability estimators of the S-S model using ERSS methodology. A numerical analysis is included in Section 5. Finally, in Section 6, we bring the paper to a close.

Estimator of R Using SRS
In this section, we derive the expression of R as well as obtain its ML estimator. Assuming that the strength X and stress Y are independently distributed random variables with the same scale parameter, where X~ILoðρ, ωÞ and Y~ILoðρ, φÞ, the system's reliability with stress variable Y and strength variable X is given by the following: The strength-stress parameter R given in (3) depends on the shape parameters ω and φ. Let X 1 , X 2 , ⋯, X n • be a SRS of size n • from the ILoðρ, ωÞ, and Y 1 , Advances in Mathematical Physics with a common scale parameter. The log-likelihood of the observed sample is given by The partial derivatives of ℓ with respect to ρ, φ, and ω are, respectively, given by Setting Equations (5)-(7) with zero and solving numerically, we get the ML estimators of ρ, φ, and ω, say b ρ, b φ, and b ω. After that, the ML estimator of R, sayR, is obtained as follows:R

Estimator of R Using RSS
We derive the reliability estimator when the random samples of strength X~ILoðρ, ωÞ and stress Y~ILoðρ, φÞ are observed from the RSS design. Let fX aðaÞe , a = 1, 2, ⋯, m 1 , e = 1, 2, ⋯, q x g be a RSS of size n • = m 1 q x for X where X aðaÞe is the a th order statistics of size m 1 of the e th cycle. Similarly, let fY b ðbÞg , b = 1, 2, ⋯, m 2 , g = 1, 2, ⋯, q y g be a RSS method of size m • = m 2 q y , where m 2 is the set size and q y is the number of cycles. For simplified forms, we use the notations X ae and Ybg instead of the notations XaðaÞe and YbðbÞg, respectively, for easy understanding and the simplicity. The pdf of Xa e and Ybg are given, respectively, by The likelihood function, say ℓ 1 , based on RSS is given by The ML estimators of ω, φ, and ρ are the solutions of the following equations: As can be seen, we use iterative approaches to solve Equations (11)-(13) because there are no explicit solutions. As a result, the ML estimator of S-S reliability is obtained based on the invariance property of ML estimators.

Estimator of R Using ERSS
In this section, we obtain the ML estimator of R when strength X and stress Y have an ILo distribution under the ERSS design. In these respects, the reliability estimator is considered in two cases when both X and Y distributions have similar or dissimilar set sizes. We derive the reliability estimator when the random samples of strength X~ILoðρ, ωÞ and stress Y~ILoðρ, φÞ are observed from ERSS.

4.1.
Estimator of R = P ½Y OSZ < X OSZ . Herein, we derive the reliability estimator when the observed data of strength X 3 Advances in Mathematical Physics and stress Y populations are drawn from the ERSS scheme with OSZ. Suppose that fX að1Þ e ; a = 1, 2, and v = ½ðm 1 + 1Þ/2 are the ERSS scheme drawn from X~ILoðρ, ωÞ with sample size m 1 q x , where m 1 is the set size and q x is the number of cycles. Let X að1Þe , X m 1 ðvÞe , and X aðm 1 Þe are the smallest, median, and largest order statistics from the a th set of size m 1 of the e th cycle, respectively. The observed ERSS with OSZ (for one cycle) is presented in Table 1.
The pdfs of the smallest, median, and largest order statistics from the a th set of size m 1 of the e th cycle are defined, respectively, as follows.
Similarly, assume that fY bð1Þ where g = 1, 2, ⋯, q y and u = ½ðm 2 + 1Þ/2 are the ERSS drawn from Y~ILoðρ, φÞ with a sample size m 2 q y , where m 2 is the set size and q y is the number of cycles. Let Y bð1Þg , Y m 2 ðuÞg , and Y bðm 2 Þg are the smallest, median, and largest order statistics from the b th set of size m 2 of the g th cycle, respectively. The pdfs of the smallest, median, and largest order statistics from the b th set of size m 2 of the g th cycle are defined, respectively, as follows: The likelihood function, say ℓ 2 , based on ERSS method with OSZ is given by the following.

Estimator of
Herein, we derive the reliability estimator when the observed data of strength X and stress Y distributions are drawn from the ERSS method with ESZ. Let fX að1Þe ; a = 1, 2, ⋯, cg ∪ fX aðm 1 Þe = c + 1, ⋯, m 1 g where e = 1, 2, ⋯, q x and c = ½m 1 /2 are the ERSS with ESZ drawn from X~ILoðρ, ωÞ with sample size m 1 q x . Let X að1Þe and X aðm 1 Þe are the smallest and largest order statistics from the a th set of size m 1 of the e th cycle, respectively. The observed ERSS with ESZ (for one cycle) is represented in Table 2.
The pdfs of X að1Þe and X aðm 1 Þe from the a th set of size m 1 of the e th cycle are defined in (14) and (16). Similarly, let fY bð1Þg ; b = 1, 2, ⋯, dg ∪ fY bðm 2 Þg , b = d + 1, ⋯, m 2 g, where g = 1, 2, ⋯, q y and d = ½m 2 /2 are the ERSS with ESZ drawn from Y~ILoðρ, φÞ with sample size m 2 q y . The pdfs of Y bð1Þg and Y bðm 2 Þg from the b th set of size m 2 of the g th cycle are defined in (17) and (19). The likelihood function, say ℓ 3 , based on ERSS with ESZ, is given by the following: The ML estimators of ω, φ, and ρ are the solutions of the following likelihood equations: Observed

Advances in Mathematical Physics
Setting Equations (25)- (27) with zero and solving numerically, we obtain the ML estimators of ω, φ, and ρ. Consequently, the S-S reliability estimator is provided using (3).
Suppose that fY bð1Þg ; b = 1, 2, ⋯, dg ∪ fY bðm 2 Þg , b = d + 1, ⋯, m 2 g, where g = 1, 2, ⋯, qy and d = ½m 2 /2 are the ERSS with ESZ drawn from Y~ILoðρ, φÞ with sample size m 2 q y , where the density function of Y bð1Þg and Y bðm 2 Þg are obtained in Equations (17) and (19). Hence, the likelihood function, say ℓ 4 , in this case, is given by the following: The partial derivatives of ω and φ are provided in (21) and (26). The partial derivative of ρ is given by The parameter estimators of ω, φ, and ρ are the solutions of the Equations (21), (26), and (29), and after setting them to zero, the S-S reliability estimator is obtained consequentially from (3).

4.4.
Estimator of R = P½Y OSZ < X ESZ . Here, we obtain the S-S reliability estimator when the observed samples of strength X are drawn from ERSS with ESZ, while observed samples of stress Y are drawn from ERSS with OSZ.

Numerical Representation
This section introduces some simulations to assess how well the ML estimation of the S-S reliability function worked based on the proposed sampling scheme. A comparison is made between different estimates based on SRS, RSS, and ERSS methods. The following is a full description of the simulated experiment. (v) A numerical technique is utilized to obtain the ML of parameters and consequently the reliability estimate using the three sampling strategies (vi) The performance of the S-S reliability estimates for the three sampling strategies is evaluated using ABs, MSEs, and REs measures (vii) The AB is defined as: (viii) Three REs of reliability estimatesR are provided and defined as follows: (ix) Tables 3-6 describe the reliability estimatesR, ABs, and MSEs based on SRS, RSS, and ERSS schemes. The REs ofR based on ERSS and RSS with respect to SRS and RSS for various sample sizes are presented in Tables 3-6   Tables 3-6 and Figures 1-6 show the following numerical results:  Table 3 (iv) Except for ðm 1 , m 2 Þ = ð2, 3Þ, ð4, 3Þ, ð7, 7Þ, the MSEs ofR based on RSS scheme are more efficient than the corresponding via ERSS at true value R = 0:833 (Table 4) (v) At true value R = 0:714, the MSEs ofR based on RSS scheme are more efficient than the corresponding via ERSS except for ðm 1 , m 2 Þ = ð2, 2Þ, ð5, 5Þ as seen in Table 5 (vi) At actual value R = 0:625, the MSEs ofR based on RSS scheme are more efficient than the corresponding ERSS, except for ðm 1 , m 2 Þ = ð2, 2Þ, ð3, 3Þ, ð4, 3Þ (see Table 6)         9 Advances in Mathematical Physics (xiv) Figure 6 illustrates that at ðm 1 , m 2 Þ = ð4, 3Þ, the RE 2 of the ERSS scheme is more efficient than those via RSS and SRS schemes with the exception of true value R = 0:714 (xv) The MSEs of the S-S reliability estimate in all schemes decrease as the actual value of R increases in most of the cases (see Figures 1-6).

Conclusions
This article tackles the estimation of the S-S reliability R = P½Y < X when the strength X and stress Y are independent inverse Lomax distributed random variables. Maximum likelihood estimators of R are computed using the SRS, RSS, and ERSS schemes. The reliability estimator is computed in four situations using ERSS design. Simulation 10 Advances in Mathematical Physics research is conducted to evaluate the performance of the proposed estimates. From the simulation outcomes, it is observed that the MSEs of reliability estimates based on SRS data are bigger than the comparable based on RSS and ERSS data, respectively. In most cases, at R = 0:909, the MSEs of reliability estimates under ERSS are the shortest when compared to similar estimators based on RSS and SRS data. The efficiency of all estimates improves as the actual value of reliability increases in almost all cases. This study showed that the reliability estimates based on RSS are more efficient than those based on ERSS and SRS. For most actual values of R, the reliability estimate via the ERSS technique is more efficient than those under RSS and SRS for small even set sizes. In some cases, estimates of reliability obtained by ERSS are more efficient than those obtained through RSS and SRS designs. In a future work, one may consider the problem of estimating R based on double extreme ranked set sampling [31], modified robust extreme ranked set sampling [32], stratified quartile ranked set sampling [33], and multistage percentile and quartile ranked set samples methods [34,35].

Data Availability
There is no data is included in the paper.

Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.