A New Estimation Study of the Stress-Strength Reliability for the Topp–Leone Distribution Using Advanced Sampling Methods

Faculty of Science, Department of Statistis, King Abdulaziz University, Jeddah, Saudi Arabia Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt &e Higher Institute of Commercial Sciences, Al Mahalla Al Kubra 31951, Algarbia, Egypt Department of Statistics, &e Islamia University of Bahawalpur, Punjab 63100, Pakistan Universit ́e de Caen, LMNO, Campus II, Science 3, 14032, Caen, France Institute of Numerical Sciences, Kohat University of Science & Technology, Kohat 26000, Pakistan


Introduction
First and foremost, the simplified version of the Topp-Leone (TL) distribution is a bounded support one-parameter J-shaped distribution. It possesses a bathtub-shaped hazard rate (see [1]). For these reasons, it is very effective for modeling life-time tests. Mathematically, it is defined by a one-parameter probability density function (PDF) and a one-parameter cumulative distribution function (CDF) given, respectively, by Hereafter, the TL distribution will be sometimes denoted by TL(α) in order to highlight the parameter.
In recent years, the TL distribution and its model have obtained a lot of attention in the literature. For instance, the authors in [2] proposed an extension of the TL distribution. e authors in [3] examined several features of the TL distribution, including the failure rate (classical and reversed) and the mean residual life time. A bivariate generalization of the TL distribution was introduced in [4]. Two-side based version of the TL distribution was proposed in [5]. Estimation of the stress-strength (SS) model was discussed based on censored samples in [6]. e author in [7] derived some moments properties of order statistics associated to the TL distribution. Using record values, the authors in [8] discussed Bayesian and non-Bayesian estimation methods for the unique parameter of the TL distribution. e author in [9] examined admissible minimax estimates of the TL distribution.
On the other hand, the notion of ranked set sampling (RSS) has been presented in [10] as a sampling scheme for data collection. is scheme is a useful technique when it is hard to measure a large number of elements, but it is easier visually (without inspection) to rank some of them. e RSS scheme has evolved into a complex strategy for enhancing mean estimation accuracy. e mathematical foundations for RSS design have been provided in [11,12].
e RSS scheme has been effective in applied sciences, with impressive wins in agriculture and ecology. e selection of a RSS of size n entails the drawing of n random samples, each containing n units. Judgment is used to rank the n units in each sample. e first sample is used to measure the smallest rank unit, and the second sample is used to measure the second smallest rank unit, and so on. When the unit with rank n is chosen from the nth set, the first cycle is finished. It is worth noting that the n observations are dispersed randomly. Because n in the RSS is usually low, the following approach is used to create a sample of size (ns) by repeating its times or cycles.
Reference [13] suggested the median ranked set sampling (MRSS) scheme as another contribution to the RSS scheme. e MRSS method is easy to apply since only the middle of the sample is considered. e following is an outline of the MRSS method: We choose n 2 units randomly by using the simple random sampling (SRS) from the population, the n 2 randomly chosen units are divided into n sets, each of size n, and the units within each set are ranked according to a variable of interest. If the set size n is odd, the ((n + 1)/2)th lowest ranked unit in each set, which is the set's median, should be chosen for measurement. Furthermore, if n is even, the (n/2)th smallest lowest ranked units from the first n/2 sets should be chosen, followed by the ((n + 2)/2)th lowest ranked units from the second n/2 sets. To obtain ns units, the cycle can be repeated s times.
For several years, the SS models have attracted the attention of significant statisticians due to their applicability in distinct fields like economics, engineering, medicine, and quality control. To depict the life of a component with unpredictable strength and random stress, the SS model is utilized. e chance that the system will be able to sustain the stress placed on it is the reliability of the system, denoted by R.
at is, R � P(Y < X), where X and Y are continuous random variables (RVs) that are independent, modeling the random strength and stress, respectively. Applications of the SS model in medicine and engineering were provided in [14]. An outstanding review of the growth of the SS model up to 2003 was conducted in [15]. e problem of estimating R has been intensively used in the statistical literature under various sampling schemes. In this work, we give a review of RSS schemes; for instance, the authors in [16,17] concerned with estimating R when X and Y are independent RVs with exponential distributions under the RSS. e authors in [18,19] handled with the same problem but RR with Burr type XII distributions under some forms of RSS schemes. Also, the authors in [20] discussed the estimation of R when X and Y are independent RVs with Weibull distributions under some versions of RSS. In this setting, Lindley distributions were supposed in [21], and exponentiated Pareto distributions were discussed in [22], both based on RSS. Using MRSS, the authors in [23] considered this estimation problem with generalized inverse exponential distributions.
Although no author has yet examined the estimation of R for the TL model via ranking schemes and their modifications in the literature, we would like to point out that reference [6] contributed to this problem using censored schemes. So, in this manuscript, we use two different ranking schemes, namely, MRSS and RSS, as well as the traditional complete scheme, i.e., SRS, to converge on the estimation of R when X ∼ TL(α) and Y ∼ TL(β) are independent. When both X and Y are selected from the same sample schemes (SRS, RSS, and MRSS), we get the ML estimator of R. Also, we obtain the ML of R when both X and Y are selected from various sampling schemes (MRSS with odd size or even sample size). To demonstrate the theoretical results, a simulation study is provided. e following is a summary of the manuscript. When the SRS is considered, Section 2 calculates the ML estimator of R. When the RSS is considered, the ML estimator of R is used in Section 3. When observed data from both X and Y are chosen from the MRSS with an even set size (MRSSE) or MRSS with an odd set size (MRSSO) or vice versa, Section 4 deals with the ML estimator of R. In Section 5, the simulation findings are discussed. In the last section, we wrap up the paper.

ML Estimation of R Based on SRS
Let X ∼ TL (α) and Y ∼ TL (β) be independent. en, with a standard integral development, the measure R of the SS model for the TL distribution is basically given by To get the ML estimator of R, we need to get the ML estimators for the unknown distribution parameters first. In this regard, suppose that X 1 , X 2 , . . ., X n * is a classical random sample from the distribution TL (α) and Y 1 , Y 2 , . . ., Y m * is a classical random sample from the distribution TL (β). e related observations are denoted as small x 1 , x 2 , . . ., x n * and y 1 , y 2 , . . ., y m * , and this is consistent across all the paper. e joint likelihood function for the observed samples is as follows: e corresponding joint log likelihood (LL) function is obtained as follows: ln ℓ � 2 ln n * + 2 ln m * + n * ln α + m * ln β +(α − 1) As usual, the ML estimators of α and β are produced by the maximization of LL with respect to the parameters, which is equivalent to differentiating the function in equation (4) and equating with zero, and we obtain anks to the so-called invariance property, we deduce the ML estimator of R, denoted by R; it is derived by directly substituting equation (5) in equation (2).

ML Estimation of R Based on RSS
where n is the set size and s x is the number of cycles, and that {Y j(j)h , j � 1,2. . ., m; h � 1,2, . . .s y ) is a RSS observed from the distribution TL (β), with sample size m * � ms y ,where m is the set size and s y is the number of cycles. In this setting, the following is the likelihood function ℓ 1 of the observed samples: e LL function of ℓ 1 is as follows: Scientific Programming With regard to α and β, we have Because the system "Equations (8) and (9) equal to zero" is difficult to solve analytically, we use an iterative technique to evaluate the ML estimators. en, owing to the invariance property, the ML estimator of R follows by inserting these estimators into Equation (2).

ML Estimation of R Based on MRSS
In this section, we investigate the ML estimator of R based on MRSS in four situations. In the first and second situations, we estimate R � P(Y MRSSO < X MRSSO ) and R � P(Y MRSSE < X MRSSE ), i.e., when both stress and strength are of the same size. When both X and Y have different set sizes, we estimate R in the third and fourth situations.

Estimation with an Odd Set Size.
Suppose that X i( g )e where i � 1,. . .,n, e � 1,. . .,s x , and g � [n + 1/2] is the MRSS with sample size n * � ns x , where n is the set size and s x is the number of cycles, selected from the distribution TL (α), and that Y j(k)h where j � 1,. . .,m, h � 1,. . .,s y , and k � [m + 1/2] is the MRSS with sample size ms y , where m is the set size and s y is the number of cycles, selected from the distribution TL (β). e associated likelihood function ℓ 2 for the observed samples is given by e LL function of ℓ 2 is as follows: Again, the ML estimators of α and β are derived by maximizing ln ℓ 2 . In this regard, the first partial derivatives of ln ℓ 2 with respect to the parameters are By solving the system "Equations (12) and (13) equal to zero," we get the ML estimators of α and β. Hence, reliability estimator of R is produced by immediately by substitution of parameter's estimators in equation (2).

Scientific Programming
where We solve "Equations (16) and (17) equal to zero" to take out the ML estimators. Further, we insert the obtained ML estimators in Equation (2) to have an estimator of R.

Estimation with an Odd Strength and Even Stress Sizes.
Here, we get the ML estimator of R under the following configuration: based on the MRSSO, we select observed samples of X from the distribution TL(α); based on the MRSSE, we select observed samples of Y from the distribution TL(β).
Suppose that X i( g )e , where i � 1,. . .,n, e � 1,. . .,s x , g � [(n + 1)/2], with sample size ns x , where n is the odd set size and s x is the number of cycles, is the observed MRSSO from the distribution TL(α), and that where the PDFs of X i( g )e , Y j( v )h , and Y j(v+1)h are as defined before. e LL of the observed data is Maximizing ln ℓ 4 with respect to α and β, we obtain the ML estimators. e partial derivatives of ln ℓ 4 with respect to the parameters are given by 6 Scientific Programming ere appear to be no closed form solutions to Equations (20) and (21) when equal with zero. As a result, numerical techniques are used to find the solution.
e ABs and MSEs of the reliability estimates based on the SRS, RSS, and MRSS are summarized in Tables 1-4. ey also show the efficiency of the RSS and MRSS-based reliability estimates with respect to the SRS, as well as the efficiency of the MRSS with respect to the RSS, for different sample sizes and distribution parameters.
We can deduce the following from Tables 1-4: (i) For all scenarios, the reliability estimates under the RSS scheme outperform the corresponding ones under the SRS (see Tables 1-4) (ii) e reliability estimates under the MRSS scheme outperform the corresponding ones under SRS in most situations, except when X has an odd set size and Y has even set sizes at R � 0.6 (see Tables 1-4) (iii) e reliability estimates under the MRSS scheme outperform the corresponding ones under the RSS in most situations, except at (n, m) � (2, 2) where R � 0.6 and at (n, m) � (4, 4) where R � 0.833 (see Table 1) (iv) e reliability estimates under the MRSS scheme outperform than the corresponding ones under the RSS in most situations, except at (n, m) � (5, 5), where R � 0.714 (see Table 2) (v) e reliability estimates under the RSS scheme outperform the corresponding ones under the MRSS for all cases (see Table 3) (vi) e reliability estimates under the RSS scheme outperform the corresponding ones under the MRSS for all cases for (n, m) � (2, 3) except at R � 0.833 while the reliability estimates under the MRSS scheme outperform the corresponding ones under the RSS for all of the situations at (n, m) � (4, 3) except R � 0.9 (see Table 4)

Conclusions
In this manuscript, we have examined the estimation of the unknown reliability measure R � P [Y < X], assuming that X and Y are modeled by independent identically distributed RVs from the TL distribution. We obtain the ML estimator of R in the setting of the SRS or RSS. In the MRSS design, we establish the reliability estimator of R in four situations. In the first and second situations, we obtain the reliability estimator when both X and Y have the same set size. e reliability estimator is derived in the third and fourth situations when the observed samples from the stress distribution have the MRSSO and the observed samples from the strength distribution have the MRSSE, and vice versa. We check the performance of different estimates through numerical studies.
Data Availability e data are fully included in the article or the mentioned references.