Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1803 https://internationalpubls.com An Improved Generalized Class of Estimator for Finite Population Mean in Stratified Systematic Sampling using Auxiliary Information Anjali Bhardwaj1*, Manoj K. Srivastava2, Namita Srivastava3, Udita Gupta1 1Department of Statistics, Research Scholar, Dr. Bhimrao Ambedkar University, Agra (U.P), 282004, India 2Vice- Chancellor, Shaheed Mahendra Karma Vishwavidyalaya, Bastar, Dharmpura, Jagdalpur, 494001, India 3Department of Statistics, Professor & HoD, St. John college, Agra (U.P), 282002, India Corresponding author, Email address: dranjalibhardwaj0506@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: This paper introduces, the generalized class of exponential-type estimator in stratified systematic sampling scheme is proposed for estimating the population mean οΏ½Μ…οΏ½ of the study variable 𝑦 using auxiliary information π‘₯. Theoretical expressions for bias and mean square error (𝑀𝑆𝐸) are derived up to the first order of approximation. A simulation study, along with analysis of three real data sets, is conducted to evaluate the performance of these estimators, where the percent relative efficiency (𝑃𝑅𝐸) is considered as a performance criterion. The result indicate that the proposed estimator is outperforms the traditional mean, product and regression estimators in term of efficiency. The simulation study was performed using R software. Keywords: Efficiency, Exponential estimator, 𝑃𝑅𝐸, Stratified sampling, Systematic sampling. 1. Introduction In statistical surveys, when subpopulations within a larger population differ, it is beneficial to sample each subpopulation (or stratum) independently. Stratification involves dividing the population into homogeneous subgroups before sampling. These strata must be mutually exclusive, meaning each element belongs to only one stratum, and collectively exhaustive, ensuring no population element can be excluded. Simple random sampling or systematic sampling is applied within each stratum. This approach typically enhances the representativeness of the sample by reducing sampling error. Systematic sampling, first explored by Madow and Madow [14], is commonly used in surveys of finite populations. This method involves selecting sample members from a larger population based on a random starting point and a fixed periodic interval. Typically, every "nth" individual is chosen from the population for inclusion in the sample population. Despite the fixed interval, systematic sampling is still considered random, provided that the starting point is chosen randomly and the interval is predetermined. Systematic sampling has the advantage of selecting the whole sample with just one random starting point. In addition to its simplicity, which is of considerable importance, this method often yields more efficient estimators compared to simple random sampling or stratified random sampling for certain types of population, as noted by (Cochran [7], Gautschi [9], Hajeck [11]). mailto:dranjalibhardwaj0506@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1804 https://internationalpubls.com The primary limitation of the ratio and product estimators is that of having efficiency not exceeding that of the regression estimator. Consequently, many researchers have focused on modifying these estimators to develop more efficient alternatives estimators. Notable contributions in this area come from; Singh and Vishwakarma [20,21], Sharma and Tailor [17], Onyeka [16], Tailor [29], Choudhury and Singh [4], Khare and Sinha [12], Singh and Audu [23]. Clement [5] addressed this issue by using systematic sampling and proposed a calibration ratio-type estimator under a stratified systematic sampling framework. Chaudhary and Dutta [3] addressed this issue by using some calibration estimators of finite population mean under stratified systematic sampling in the presence of non- response. In the following sections, we have introduced a generalized class of exponential-type estimator for the population mean in stratified systematic sampling, utilizing information from a single auxiliary variable for the first time. The mean square error (𝑀𝑆𝐸) expression for the proposed exponential estimator has been derived. Additionally, an empirical comparison between the proposed estimator and traditional estimators is provided, based on both simulation studies and real data analysis. 2. Notation [estimation procedure] Consider a finite population Ω of 𝑁 elements. suppose Ω = (Ω1, Ω2, … , Ω𝑁) is divided into of 𝐿 strata with π‘β„Ž units in the β„Žπ‘‘β„Žstratum from which a systematic random sample of size π‘›β„Ž is taken, and a random sample of π‘›β„Ž unit is selected from the first π‘˜β„Ž units, and then every π‘˜β„Ž π‘‘β„Ž subsequent unit is included in the sample of size π‘›β„Ž. where π‘˜β„Ž = π‘β„Ž π‘›β„Ž , π‘˜β„Ž a positive integer, and this method provides π‘˜β„Ž samples, each of size π‘›β„Ž. Both the study variate (𝑦) and auxiliary variate (π‘₯) are observed for each and every unit selected in the sample. Subsequently, the above scheme is stratified systematic sampling and the notations are defined as follows: The total population size be 𝑁 = βˆ‘ π‘β„Ž 𝐿 β„Ž=1 and sample size 𝑛 = βˆ‘ π‘›β„Ž 𝐿 β„Ž=1 , respectively. Associated with the π‘–π‘‘β„Ž element of the β„Žπ‘‘β„Ž stratum are π‘¦β„Žπ‘– and π‘₯β„Žπ‘– with π‘₯β„Žπ‘– > 0 being the covariate; where π‘¦β„Žπ‘– is the 𝑦 value of the π‘–π‘‘β„Ž element in stratum β„Ž, and π‘₯β„Žπ‘– is the π‘₯ value of the π‘–π‘‘β„Ž element in stratum β„Ž, β„Ž = 1,2, … . , 𝐿 and 𝑖 = 1,2, … , π‘β„Ž. For the β„Žπ‘‘β„Ž stratum, let π‘Šβ„Ž = π‘β„Ž 𝑁 be the stratum weight and π‘“β„Ž = π‘›β„Ž π‘β„Ž , the sample fraction. Let the β„Žπ‘‘β„Ž stratum means of the study variable 𝑦 and auxiliary variable π‘₯. οΏ½Μ…οΏ½β„Ž = βˆ‘ π‘¦β„Žπ‘– π‘›β„Ž π‘›β„Ž 𝑖=1 and οΏ½Μ…οΏ½β„Ž = βˆ‘ π‘₯β„Žπ‘– π‘›β„Ž π‘›β„Ž 𝑖=1 are the unbiased estimators of stratified systematic sample means corresponding to the population means οΏ½Μ…οΏ½β„Ž = βˆ‘ π‘¦β„Žπ‘– π‘β„Ž π‘β„Ž 𝑖=1 and οΏ½Μ…οΏ½β„Ž = βˆ‘ π‘₯β„Žπ‘– π‘β„Ž π‘β„Ž 𝑖=1 of (𝑦, π‘₯ ) respectively. Based on π‘›β„Ž observations. Let οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 be the mean of systematic sample in the β„Ž stratum, then the estimate of the sample mean and population mean οΏ½Μ…οΏ½ in stratified systematic sampling scheme is given by [Cochran (1997)] as: �̅�𝑠𝑑.𝑠𝑦𝑠 = βˆ‘ π‘Šβ„Ž 𝑛𝑖 𝑖=1 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 , �̅�𝑠𝑑.𝑠𝑦𝑠 = οΏ½Μ…οΏ½ = βˆ‘ π‘Šβ„Ž 𝑛𝑖 𝑖=1 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 The following error terms are also defined. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1805 https://internationalpubls.com βˆˆπ‘¦= οΏ½Μ…οΏ½β„Ž.π‘ π‘¦π‘ βˆ’ οΏ½Μ…οΏ½β„Ž οΏ½Μ…οΏ½β„Ž , ∈π‘₯= οΏ½Μ…οΏ½β„Ž.π‘ π‘¦π‘ βˆ’ οΏ½Μ…οΏ½β„Ž οΏ½Μ…οΏ½β„Ž οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 = οΏ½Μ…οΏ½(1 + πœ–π‘¦), οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 = οΏ½Μ…οΏ½(1 + πœ–π‘₯) Such that 𝐸(βˆˆπ‘¦) = 𝐸(∈π‘₯) = 0 Where �̅�𝑠𝑑.𝑠𝑦𝑠 and �̅�𝑠𝑑.𝑠𝑦𝑠 are usual unbiased estimators of population mean οΏ½Μ…οΏ½, οΏ½Μ…οΏ½ respectively. 𝐸(βˆˆπ‘¦ 2 ) = βˆ‘ πœƒβ„ŽπœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 = 𝑉20 𝐿 β„Ž=1 ; 𝐸(∈π‘₯ 2) = βˆ‘ πœƒβ„ŽπœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 = 𝑉02 𝐿 β„Ž=1 𝐸(βˆˆπ‘¦βˆˆπ‘₯) = βˆ‘ πœƒβ„ŽβˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πœŒπ‘¦π‘₯πΆβ„Žπ‘¦πΆβ„Žπ‘₯ = βˆ‘ πœƒβ„ŽβˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘¦π‘₯ = 𝑉11 𝐿 β„Ž=1 𝐿 β„Ž=1 where, πΆβ„Žπ‘¦π‘₯ = πœŒβ„Žπ‘¦π‘₯πΆβ„Žπ‘¦πΆβ„Žπ‘₯ , and πœƒβ„Ž = ( π‘β„Žβˆ’1 π‘β„Žπ‘›β„Ž ) Let π‘†β„Žπ‘¦ 2 = 1 (π‘β„Žβˆ’1) βˆ‘ (π‘¦β„Žπ‘– βˆ’ οΏ½Μ…οΏ½β„Ž)2π‘β„Ž 𝑖=1 and π‘†β„Žπ‘₯ 2 = 1 (π‘β„Žβˆ’1) βˆ‘ (π‘₯β„Žπ‘– βˆ’ οΏ½Μ…οΏ½β„Ž)2π‘β„Ž 𝑖=1 represent the population variances of the study variable and the auxiliary variable respectively, with the corresponding population covariance π‘†β„Žπ‘¦π‘₯ = 1 (π‘β„Žβˆ’1) βˆ‘ (π‘¦β„Žπ‘– βˆ’ οΏ½Μ…οΏ½β„Ž)(π‘₯β„Žπ‘– βˆ’ οΏ½Μ…οΏ½β„Ž)π‘β„Ž 𝑖=1 . Also πΆβ„Žπ‘¦ 2 = π‘†β„Žπ‘¦ 2 οΏ½Μ…οΏ½2 and πΆβ„Žπ‘₯ 2 = π‘†β„Žπ‘₯ 2 οΏ½Μ…οΏ½2 are the population coefficient of variation of the variate (𝑦, π‘₯) respectively. here, πœŒβ„Žπ‘¦ βˆ— = [1 + (π‘›β„Ž βˆ’ 1)πœŒβ„Žπ‘¦], πœŒβ„Žπ‘₯ βˆ— = [1 + (π‘›β„Ž βˆ’ 1)πœŒβ„Žπ‘₯] where, πœŒβ„Žπ‘₯ = 𝐸(π‘₯β„Žπ‘–βˆ’οΏ½Μ…οΏ½β„Ž)(π‘₯β„Žπ‘– β€² βˆ’οΏ½Μ…οΏ½β„Ž) 𝐸(π‘₯β„Žπ‘–βˆ’οΏ½Μ…οΏ½β„Ž)2 ; πœŒβ„Žπ‘¦ = 𝐸(π‘¦β„Žπ‘–βˆ’οΏ½Μ…οΏ½β„Ž)(π‘¦β„Žπ‘– β€² βˆ’οΏ½Μ…οΏ½β„Ž) 𝐸(π‘¦β„Žπ‘–βˆ’οΏ½Μ…οΏ½β„Ž)2 And πœŒβ„Žπ‘¦π‘₯ = 𝐸(π‘₯β„Žπ‘–βˆ’οΏ½Μ…οΏ½β„Ž)(π‘¦β„Žπ‘–βˆ’οΏ½Μ…οΏ½β„Ž) √𝐸(π‘₯β„Žπ‘–βˆ’οΏ½Μ…οΏ½β„Ž)2(π‘¦β„Žπ‘–βˆ’οΏ½Μ…οΏ½β„Ž)2 be the usual population correlation coefficient between (𝑦, π‘₯) respectively. 𝐢𝑠𝑑.𝑠𝑦𝑠 = 𝐢π‘₯ = βˆ‘ π‘Šβ„ŽπΆβ„Žπ‘₯ 𝐿 β„Ž=1 : represent the population coefficient of variation of 𝑋 for the β„Žπ‘‘β„Ž stratum, 𝛽2𝑠𝑑.𝑠𝑦𝑠 = 𝛽2π‘₯ = βˆ‘ π‘Šβ„Žπ›½2π‘₯(β„Ž) 𝐿 β„Ž=1 : represent the population coefficient of kurtosis of 𝑋 for the β„Žπ‘‘β„Ž stratum, 𝛽1𝑠𝑑.𝑠𝑦𝑠 = 𝛽1π‘₯ = βˆ‘ π‘Šβ„Žπ›½1π‘₯(β„Ž) 𝐿 β„Ž=1 : represent the population coefficient of skewness of 𝑋 for the β„Žπ‘‘β„Ž stratum, πœŒπ‘¦π‘₯𝑠𝑑.𝑠𝑦𝑠 = πœŒπ‘¦π‘₯ = βˆ‘ π‘Šβ„ŽπœŒβ„Žπ‘¦π‘₯ 𝐿 β„Ž=1 : represent the population correlation coefficient between π‘Œ and 𝑋 for β„Žπ‘‘β„Ž stratum 3. Existing estimators In the context of stratified systematic sampling, several existing estimators of the population mean used in survey sampling are presented, along with their adaptations for stratified systematic sampling using auxiliary information. Their corresponding variance or mean square error (𝑀𝑆𝐸) expressions are also provided. 1. The usual sample mean estimator in stratified systematic sampling is defined as given by οΏ½ΜƒοΏ½0𝑠𝑑.𝑠𝑦𝑠 = οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 … (1) The variance of οΏ½ΜƒοΏ½0 is given by π‘‰π‘Žπ‘Ÿ(οΏ½ΜƒοΏ½0𝑠𝑑.𝑠𝑦𝑠) = 𝑀𝑆𝐸(οΏ½ΜƒοΏ½0𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2 βˆ‘ πœƒβ„ŽπœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2𝐿 β„Ž=1 = οΏ½Μ…οΏ½2𝑉20 … (2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1806 https://internationalpubls.com 2. The stratified systematic version of Cochran [6] classical ratio estimator for population mean is: �̃�𝑅.𝑠𝑑.𝑠𝑦𝑠 = οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 ( οΏ½Μ…οΏ½ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 ) … (3) The 𝑀𝑆𝐸 of estimator is given by 𝑀𝑆𝐸(�̃�𝑅.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2 βˆ‘ πœƒβ„Ž 𝐿 β„Ž=1 [πœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 + πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 βˆ’ 2βˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘¦π‘₯] = οΏ½Μ…οΏ½2[𝑉20 + 𝑉02 βˆ’ 2𝑉11] .. (4) 3. The stratified systematic version of Murthy [15] classical product estimator for population mean is: �̃�𝑃.𝑠𝑑.𝑠𝑦𝑠 = οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ ) … (5) The 𝑀𝑆𝐸 of estimator is given by 𝑀𝑆𝐸(�̃�𝑃.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2 βˆ‘ πœƒβ„Ž 𝐿 β„Ž=1 [πœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 + πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 + 2βˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘¦π‘₯] = οΏ½Μ…οΏ½2[𝑉20 + 𝑉02 + 2𝑉11] .. (6) 4. The stratified systematic version of regression estimator for population mean is: �̃�𝑅𝑒𝑔.𝑠𝑑.𝑠𝑦𝑠 = οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) … (7) The 𝑀𝑆𝐸 of estimator is given by 𝑀𝑆𝐸(�̃�𝑅𝑒𝑔.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2(𝑉20𝑉02βˆ’π‘‰11 2 ) 𝑉02 … (8) Equation (8) will become 𝑀𝑆𝐸(�̃�𝑅𝑒𝑔.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2𝑉20(1 βˆ’ πœŒβ„Žπ‘¦π‘₯ 2 ) … (9) 5. The stratified systematic version of Bahl and Tuteja [2] ratio-type and product-type exponential estimators for population mean is: �̃�𝑅𝐸.𝑠𝑑.𝑠𝑦𝑠 = οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp ( οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 ) … (10) �̃�𝑃𝐸.𝑠𝑑.𝑠𝑦𝑠 = οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp ( οΏ½Μ…οΏ½β„Ž.π‘ π‘¦π‘ βˆ’οΏ½Μ…οΏ½ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠+οΏ½Μ…οΏ½ ) … (11) The 𝑀𝑆𝐸 of estimators �̃�𝑅𝐸 and �̃�𝑃𝐸 are respectively given by 𝑀𝑆𝐸(�̃�𝑅𝐸.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2 βˆ‘ πœƒβ„Ž [πœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 + πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 4 βˆ’ βˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘¦π‘₯]𝐿 β„Ž=1 = οΏ½Μ…οΏ½2 [𝑉20 + 𝑉02 4 βˆ’ 𝑉11] .. (12) 𝑀𝑆𝐸(�̃�𝑃𝐸.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2 βˆ‘ πœƒβ„Ž [πœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 + πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 4 + βˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘¦π‘₯]𝐿 β„Ž=1 = οΏ½Μ…οΏ½2 [𝑉20 + 𝑉02 4 + 𝑉11] … (13) 6. The stratified systematic version of difference type estimator for population mean is: �̃�𝐷.𝑠𝑑.𝑠𝑦𝑠 = [𝑄1οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄2(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)] … (14) Where 𝑄1and 𝑄2 are unknown constants. The optimal values of 𝑄1and 𝑄2 are given as 𝑄1 = 𝑉02 [𝑉20𝑉02βˆ’π‘‰11 2 +𝑉02] ; 𝑄2 = βˆ‘ �̅�𝑉11 οΏ½Μ…οΏ½[𝑉20𝑉02βˆ’π‘‰11 2 +𝑉02] 𝐿 β„Ž=1 The 𝑀𝑆𝐸 of estimator is given by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1807 https://internationalpubls.com 𝑀𝑆𝐸(�̃�𝐷.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2[𝑉20𝑉02βˆ’π‘‰11 2 ] [𝑉20𝑉02βˆ’π‘‰11 2 +𝑉02] … (15) Equation (15) can also be written as 𝑀𝑆𝐸(�̃�𝐷.𝑠𝑑.𝑠𝑦𝑠) = βˆ‘ οΏ½Μ…οΏ½2πœƒβ„ŽπœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 (1βˆ’πœŒβ„Žπ‘¦π‘₯ 2 ) [1+πœƒβ„ŽπœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 (1βˆ’πœŒβ„Žπ‘¦π‘₯ 2 )] 𝐿 β„Ž=1 = βˆ‘ οΏ½Μ…οΏ½2𝑉20(1βˆ’πœŒβ„Žπ‘¦π‘₯ 2 ) [1+𝑉20(1βˆ’πœŒβ„Žπ‘¦π‘₯ 2 )] 𝐿 β„Ž=1 … (16) 7. The stratified systematic version of Singh and Vishwakarma [22] ratio-product estimator for population mean is: �̃�𝑅𝑃.𝑠𝑑.𝑠𝑦𝑠 = �̅�𝑠𝑑.𝑠𝑦𝑠 [𝑄3 ( οΏ½Μ…οΏ½ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 ) + (1 βˆ’ 𝑄3) ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )] … (17) Where 𝑄3 is suitably constant 𝑄3 = 1 2 ( 𝑉11 𝑉02 + 1) The 𝑀𝑆𝐸 of estimator is given by 𝑀𝑆𝐸(�̃�𝑅𝑃.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2 βˆ‘ πœƒβ„ŽπœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 (1 βˆ’ πœŒβ„Žπ‘¦π‘₯ 2 )𝐿 β„Ž=1 = οΏ½Μ…οΏ½2𝑉20(1 βˆ’ πœŒβ„Žπ‘¦π‘₯ 2 ) … (18) 8. The stratified systematic version of Yadav and Kadilar [31] Exponential ratio-type estimator for population mean is: �̃�𝐸𝑅.𝑠𝑑.𝑠𝑦𝑠 = 𝑄4οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp ( οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 ) … (19) Where π‘˜π‘œ is suitably constant 𝑄4 = βˆ‘ [1+ 3 8 𝑉02βˆ’ 1 2 𝑉11] [1+𝑉20+𝑉02βˆ’2𝑉11] 𝐿 β„Ž=1 𝑀𝑆𝐸(�̃�𝐸𝑅.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2 βˆ‘ [1 βˆ’ (1+ 3 8 πœƒβ„ŽπœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 βˆ’ 1 2 πœƒβ„ŽβˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘¦π‘₯) 2 (1+πœƒβ„Ž{πœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 +πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 βˆ’2βˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘¦π‘₯}) ]𝐿 β„Ž=1 = οΏ½Μ…οΏ½2 [1 βˆ’ (1+ 3 8 𝑉02βˆ’ 1 2 𝑉11) 2 (1+𝑉20+𝑉02βˆ’2𝑉11) ] … (20) 9. The stratified systematic version of Singh, R. et al. [24] exponential estimator for population mean is: �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 = οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝛼(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛼(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽 ] … (21) The 𝑀𝑆𝐸 of estimator is given by 𝑀𝑆𝐸(�̃�𝐸.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½2 4 βˆ‘ πœƒβ„Ž [4πœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 + 𝛿2πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 βˆ’ 4π›ΏβˆšπœŒβ„Žπ‘¦ βˆ— πœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘¦π‘₯]𝐿 β„Ž=1 = οΏ½Μ…οΏ½2 4 [4𝑉20 + 𝛿2𝑉02 βˆ’ 4𝑉11] … (22) Where 𝛿 = 𝛼�̅� 𝛼�̅�+𝛽 10. The stratified systematic version of Grover and Kaur [10] a generalized class of ratio type exponential estimators for population mean is: �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 = {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝛼(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛼(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽 ] … (23) Where 𝑄4and 𝑄5 are unknown constants. The optimal values of 𝑄1and 𝑄2 are given as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1808 https://internationalpubls.com 𝑄5 = βˆ‘ 𝑉02[𝛿2𝑉02βˆ’8] 8[𝑉11 2 βˆ’π‘‰20𝑉02βˆ’π‘‰02] 𝐿 β„Ž=1 ; 𝑄6 = οΏ½Μ…οΏ½[𝛿3𝑉02 2 βˆ’π›Ώ2𝑉02𝑉11+4𝛿𝑉20𝑉02βˆ’4𝛿2𝑉11 2 βˆ’4𝛿𝑉02+8𝑉11] 8οΏ½Μ…οΏ½[𝑉20𝑉02βˆ’π‘‰11 2 + 𝑉02] The MSE of estimator is given by 𝑀𝑆𝐸(�̃�𝐺.𝑠𝑑.𝑠𝑦𝑠) = βˆ‘ οΏ½Μ…οΏ½2 64 [64 βˆ’ 16𝛿2πœƒβ„ŽπœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 βˆ’ (𝛿2πœƒβ„ŽπœŒβ„Žπ‘₯ βˆ— πΆβ„Žπ‘₯ 2 βˆ’8) 2 1+πœƒβ„ŽπœŒβ„Žπ‘¦ βˆ— πΆβ„Žπ‘¦ 2 (1βˆ’πœŒβ„Žπ‘¦π‘₯ 2 ) ]𝐿 β„Ž=1 = οΏ½Μ…οΏ½2 64 [64 βˆ’ 16𝛿2𝑉02 βˆ’ 𝑉02(𝛿2𝑉02βˆ’8) 2 𝑉02(1+𝑉20)βˆ’π‘‰11 2 ] … (24) Table 1. Some members of our proposed and existing estimator 𝛼 𝛽 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 1 0 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) ] 1 1 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2 ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2 ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2 ] 1 𝐢π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] 1 πœŒπ‘¦π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] 𝐢π‘₯ πœŒπ‘¦π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] πœŒπ‘¦π‘₯ 𝐢π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] 1 𝛽1π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] 𝛽1π‘₯ 𝐢π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1809 https://internationalpubls.com 𝐢π‘₯ 𝛽1π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] 𝛽1π‘₯ πœŒπ‘¦π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] πœŒπ‘¦π‘₯ 𝛽1π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] 1 𝛽2π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ (οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) (οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] 𝛽2π‘₯ 𝐢π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝐢π‘₯ ] 𝐢π‘₯ 𝛽2π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝐢π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝐢π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] 𝛽2π‘₯ πœŒπ‘¦π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2πœŒπ‘¦π‘₯ ] πœŒπ‘¦π‘₯ 𝛽2π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) πœŒπ‘¦π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] 𝛽2π‘₯ 𝛽1π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝛽2π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽2π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽1π‘₯ ] 𝛽1π‘₯ 𝛽2π‘₯ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] {𝑄5 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄6(οΏ½Μ…οΏ½ βˆ’ οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)} exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝛽1π‘₯(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛽1π‘₯(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽2π‘₯ ] 4. The proposed estimator Motivated by Koyuncu [13] we propose the following estimator in stratified systematic sampling. �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 = {𝑄7 οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 + 𝑄8 ( οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠 οΏ½Μ…οΏ½ )} exp [ 𝛼(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠) 𝛼(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½β„Ž.𝑠𝑦𝑠)+2𝛽 ] … (25) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1810 https://internationalpubls.com Here 𝑄7 and 𝑄8 are appropriate constants that should be selected to minimize the mean square error (𝑀𝑆𝐸) of the estimator �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠. It can be observed that the proposed estimator �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 encompasses all the aforementioned estimators for various values of 𝛼 and 𝛽. In this context, 𝑄𝑖 where (𝑖 = 7,8) represents the constants for reducing the bias within this class of estimators, and 𝛼 and 𝛽 have been previously explained. In Table 1, we present specific members of both the proposed and existing classes of estimators that utilize different combination of 𝛼 and 𝛽. By expressing equation (25) in terms of 𝑒′𝑠, we obtain �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 = {𝑄7οΏ½Μ…οΏ½(1 + πœ–π‘¦) + 𝑄8 ( οΏ½Μ…οΏ½(1+πœ–π‘₯) οΏ½Μ…οΏ½ )} exp [ 𝛼(οΏ½Μ…οΏ½βˆ’οΏ½Μ…οΏ½(1+πœ–π‘₯)) 𝛼(οΏ½Μ…οΏ½+οΏ½Μ…οΏ½(1+πœ–π‘₯))+2𝛽 ] �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 = {𝑄7οΏ½Μ…οΏ½(1 + πœ–π‘¦) + 𝑄8(1 + πœ–π‘₯)} exp [ βˆ’π›ΌοΏ½Μ…οΏ½πœ–π‘₯ 2(𝛼�̅�+𝛽)+π›ΌοΏ½Μ…οΏ½πœ–π‘₯ ] 𝛿 = 𝛼�̅� 𝛼�̅�+𝛽 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 = {𝑄7οΏ½Μ…οΏ½(1 + πœ–π‘¦) + 𝑄8(1 + πœ–π‘₯)} exp [ βˆ’π›Ώπœ–π‘₯ 2 (1 + π›Ώπœ–π‘₯ 2 ) βˆ’1 ] �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 = {𝑄7οΏ½Μ…οΏ½(1 + πœ–π‘¦) + 𝑄8(1 + πœ–π‘₯)} (1 βˆ’ 1 2 𝛿 ∈π‘₯+ 3 8 𝛿2 ∈π‘₯ 2) �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 = {𝑄7οΏ½Μ…οΏ½ (1 + πœ–π‘¦ βˆ’ π›Ώπœ–π‘₯ 2 + 3 8 𝛿2 ∈π‘₯ 2βˆ’ 𝛿 2 πœ–π‘¦πœ–π‘₯) + 𝑄8 (1 + πœ–π‘₯ βˆ’ π›Ώπœ–π‘₯ 2 + 3 8 𝛿2 ∈π‘₯ 2βˆ’ π›Ώβˆˆπ‘₯ 2 2 )} … (26) Subtracting οΏ½Μ…οΏ½ from both the sides of equation (26) (�̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 βˆ’ οΏ½Μ…οΏ½) = οΏ½Μ…οΏ½(𝑄7 βˆ’ 1) + 𝑄7οΏ½Μ…οΏ½ (πœ–π‘¦ βˆ’ π›Ώπœ–π‘₯ 2 + 3 8 𝛿2 ∈π‘₯ 2βˆ’ 𝛿 2 πœ–π‘¦πœ–π‘₯) + 𝑄8 (1 + πœ–π‘₯ βˆ’ π›Ώπœ–π‘₯ 2 + 3 8 𝛿2 ∈π‘₯ 2βˆ’ π›Ώβˆˆπ‘₯ 2 2 ) … (27) By taking the expectation of both sides in equation (27), we derive the bias of the estimator �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠, up to the first order of approximation as follows: 𝐡(�̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠) = οΏ½Μ…οΏ½(𝑄7 βˆ’ 1) + 𝑄7 �̅�𝛿 2 ( 3 4 𝛿𝑉02 βˆ’ 𝑉11) + 𝑄8 (1 + 𝛿𝑉02 2 ( 3𝛿 4 βˆ’ 1)) … (28) Now, by squaring both sides of equation (27) and taking the expectation, we obtain the 𝑀𝑆𝐸 of the estimator to the first order of approximation as follow: 𝑀𝑆𝐸(�̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠) = [οΏ½Μ…οΏ½2 + 𝑄7 2οΏ½Μ…οΏ½2(1 + 𝑉20 + 𝛿2𝑉02 βˆ’ 2𝛿𝑉11) + 𝑄8 2(1 + {𝛿 βˆ’ 1}2𝑉02) βˆ’ 2𝑄7οΏ½Μ…οΏ½2 {1 + 3𝛿2 8 𝑉02 βˆ’ 𝛿 2 𝑉11} βˆ’ 2𝑄8οΏ½Μ…οΏ½ {1 + 𝛿 2 ( 3𝛿 4 βˆ’ 1) 𝑉02} + 2𝑄7𝑄8οΏ½Μ…οΏ½{1 + (𝛿 βˆ’ 1)(𝛿𝑉02 βˆ’ 𝑉11)}] … (29) Optimality condition for the proposed class of estimator: To explore the optimal conditions for this proposed class of estimator, let us consider: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1811 https://internationalpubls.com πœ•π‘€π‘†πΈ(�̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠) πœ•π‘„7 = 0 ; πœ•π‘€π‘†πΈ(�̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠) πœ•π‘„8 = 0 So that 𝑄7(π‘œπ‘π‘‘) = [𝑉02βˆ’π‘‰11][8+𝛿2(π›Ώβˆ’1)𝑉02βˆ’4𝛿] 8[𝑉20+𝑉02βˆ’2𝑉11+(π›Ώβˆ’1)2(𝑉20𝑉02βˆ’π‘‰11 2 )] … (30) 𝑄8(π‘œπ‘π‘‘) = οΏ½Μ…οΏ½[𝑉20+ 𝛿𝑉20𝑉02 2 ( 3𝛿 4 βˆ’1)βˆ’( 𝛿 2 +1)𝑉11βˆ’ 𝛿 2 (π›Ώβˆ’1)𝑉11 2 βˆ’ 𝛿𝑉02 2 ( 𝛿2𝑉02 4 βˆ’1)+ 𝛿2𝑉02𝑉11 8 (𝛿+1)] [𝑉20+𝑉02βˆ’2𝑉11+(π›Ώβˆ’1)2(𝑉20𝑉02βˆ’π‘‰11 2 )] … (31) Substituting the value of 𝑄7(π‘œπ‘π‘‘) and 𝑄8(π‘œπ‘π‘‘) into equation (29) yields the optimal 𝑀𝑆𝐸 of the estimator as follows: 𝑀𝑆𝐸(�̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠) π‘œπ‘π‘‘ = οΏ½Μ…οΏ½2[( 𝛿 2 βˆ’1) 2 (𝑉20𝑉02βˆ’π‘‰11 2 )+ 𝛿2𝑉02{𝑉11 2 ( 𝛿2 2 +1βˆ’ 3𝛿 2 )βˆ’π‘‰20𝑉02( 3𝛿 4 βˆ’1) 2 } 4 βˆ’ 𝛿4𝑉02 2 32 ( 𝛿2𝑉02 2 +𝑉11)] [𝑉20+𝑉02βˆ’2𝑉11+(π›Ώβˆ’1)2(𝑉20𝑉02βˆ’π‘‰11 2 )] … (32) 5. Empirical study 5.1 Real population To assess the theoretical findings and empirically evaluate the efficiency and optimality of the proposed exponential-type estimator (�̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠) in comparison to other estimators. In this study, we have used three different real populations that we have collected from various sources, along with a simulation study. Population 1: The first population is sourced from Model Assisted Survey Sampling by β€œCarl-Erik Sarndal, Bengt Swensson, Jan Wretman” [ Appendix-B β€œThe MU284 Population”] This dataset includes 125 municipalities, where the total population in (1985) is used as study variable 𝑦 and the number of Conservative seats in municipal council serves as auxiliary variables π‘₯. Now the entire population of 125 municipalities are divided into 5 strata based on geographic region indicator, we have, Strata Geographic region indicator Total region 1. 1 25 2. 2 25 3. 3 25 4. 4 25 5. 5 25 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1812 https://internationalpubls.com Table 2. Present data statistics for population (1) Stratu m π‘β„Ž π‘›β„Ž οΏ½Μ…οΏ½β„Ž οΏ½Μ…οΏ½β„Ž 𝐢π‘₯(β„Ž) 𝐢𝑦(β„Ž) 𝜌π‘₯(β„Ž) πœŒπ‘¦(β„Ž) πœŒπ‘¦π‘₯(β„Ž) 𝛽1π‘₯(β„Ž) 𝛽2π‘₯(β„Ž) 1 25 5 62.44 16 0.1908 0.7881 0.856 0.034 0.677 1.141 4.270 2 25 5 34.72 9.72 0.1701 0.2586 0.898 0.846 0.900 1.059 3.307 3 25 5 21.04 9 0.0836 0.0916 0.815 0.442 0.856 0.953 4.004 4 25 5 34.32 12.4 0.1511 0.2842 0.902 0.3 0.514 0.561 2.694 5 25 5 45.88 11.32 0.1300 0.5173 0.896 0.254 0.671 0.798 3.092 οΏ½Μ…οΏ½ = 39.68 οΏ½Μ…οΏ½ = 11.68 𝐢π‘₯ = 0.1451 πœŒπ‘¦π‘₯ = 0.724 𝛽1π‘₯ = 0.902 𝛽2π‘₯ = 3.473 Population 2: The second population is sourced from Theory and Analysis of Sample Survey DESIGNS - Second Edition by β€œDaroga Singh, F S Chaudhary” [chapter-4] This dataset comprises 70 villages in India, along with their 1981 population and cultivated area (in acres). In this study, the cultivated area (in acres) from (1985) is used as study variable 𝑦, while the population of the village serves as the auxiliary variables π‘₯. Now the entire population of 70 villages is divided into two equal strata. Table 3. Present data statistics for population (2) Stratu m π‘β„Ž π‘›β„Ž οΏ½Μ…οΏ½β„Ž οΏ½Μ…οΏ½β„Ž 𝐢π‘₯(β„Ž) 𝐢𝑦(β„Ž) 𝜌π‘₯(β„Ž) πœŒπ‘¦(β„Ž) πœŒπ‘¦π‘₯(β„Ž) 𝛽1π‘₯(β„Ž) 𝛽2π‘₯(β„Ž) 1 35 7 1355.91 2238.8 0.9186 0.6410 0.717 0.911 0.832 1.504 5.420 2 35 7 609.51 1272.25 0.5496 0.2842 0.352 0.863 0.526 2.371 8.404 οΏ½Μ…οΏ½ = 982.71 οΏ½Μ…οΏ½ = 1755.52 𝐢π‘₯ = 0.7341 πœŒπ‘¦π‘₯ = 0.679 𝛽1π‘₯ = 1.937 𝛽2π‘₯ = 6.912 Population 3: The third population is sourced from Multivariate Statistical Methods- A Primer 4rd Edition by β€œBryan F. J. Manly, Jorge A. Navarro Alberto” [ Chapter-1] The considered data relates to total 50 body measurements of female sparrows. we consider the total length as study variable 𝑦 and length of beak and head as auxiliary variables π‘₯. Now the whole population of 50 body measurements of female sparrows is divided into two equal strata. we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1813 https://internationalpubls.com Table 4. Present data statistics for population (3) Stratu m π‘β„Ž π‘›β„Ž οΏ½Μ…οΏ½β„Ž οΏ½Μ…οΏ½β„Ž 𝐢π‘₯(β„Ž) 𝐢𝑦(β„Ž) 𝜌π‘₯(β„Ž) πœŒπ‘¦(β„Ž) πœŒπ‘¦π‘₯(β„Ž) 𝛽1π‘₯(β„Ž) 𝛽2π‘₯(β„Ž) 1 25 5 157.12 31. 43 0.0236 0.0209 0.928 0.355 0.682 0.291 2.194 2 25 5 159.04 31.44 0.0273 0.0243 0.899 0.223 0.595 0.477 2.564 οΏ½Μ…οΏ½ = 158.08 οΏ½Μ…οΏ½ = 31.44 𝐢π‘₯ = 0.0255 πœŒπ‘¦π‘₯ = 0.639 𝛽1π‘₯ = 0.384 𝛽2π‘₯ = 2.379 Table 5. The 𝐏𝐑𝐄𝐬 of various estimators with respect to οΏ½ΜƒοΏ½πŸŽ.𝐬𝐭.𝐬𝐲𝐬 , are calculated for different combination of 𝛂 and 𝛃 using population (1). Estimators 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 οΏ½ΜƒοΏ½0.𝑠𝑑.𝑠𝑦𝑠 100.0 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,0) 127.19 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,0) 871.28 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,0) 3758.62 �̃�𝑅.𝑠𝑑.𝑠𝑦𝑠 156.56 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,1) 124.90 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,1) 843.15 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,1) 3248.26 �̃�𝑃.𝑠𝑑.𝑠𝑦𝑠 61.47 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 124.91 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 843.28 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 3250.60 �̃�𝑅𝑒𝑔.𝑠𝑑.𝑠𝑦𝑠 185.08 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 125.49 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 850.05 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 3368.46 �̃�𝑅𝐸.𝑠𝑑.𝑠𝑦𝑠 127.19 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 123.22 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 824.97 �̃�𝐴𝐡.𝑆𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 2946.07 �̃�𝑃𝑒.𝑠𝑑.𝑠𝑦𝑠 78.11 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 125.91 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 855.02 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 3456.95 �̃�𝐷.𝑠𝑑.𝑠𝑦𝑠 715.58 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 125.10 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 845.52 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 3289.16 �̃�𝑅𝑃.𝑠𝑑.𝑠𝑦𝑠 185.08 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 126.15 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 858.01 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 3511.12 �̃�𝐸𝑅.𝑠𝑑.𝑠𝑦𝑠 579.44 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 122.41 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 816.86 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 2817.80 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 125.32 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 848.04 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 3333.11 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 124.39 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 837.44 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 3151.2 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 120.58 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 800.02 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 2563.41 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 126.91 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 867.65 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 3689.64 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 114.88 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 758.86 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 2004.35 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 126.68 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 864.65 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 3633.40 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1814 https://internationalpubls.com �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 118.82 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 785.63 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 2358.43 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 126.47 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 861.93 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 3583.00 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 98.46 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 716.07 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 1401.76 Table 6. The 𝐏𝐑𝐄𝐬 of various estimators with respect to οΏ½ΜƒοΏ½πŸŽ.𝐬𝐭.𝐬𝐲𝐬 are calculated for different combination of 𝛂 and 𝛃 using population (2). Estimators 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 οΏ½ΜƒοΏ½0.𝑠𝑑.𝑠𝑦𝑠 100.0 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,0) 247.04 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,0) 396.02 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,0) 435.44 �̃�𝑅.𝑠𝑑.𝑠𝑦𝑠 149.88 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,1) 246.98 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,1) 395.84 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,1) 434.92 �̃�𝑃.𝑠𝑑.𝑠𝑦𝑠 20.19 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 246.99 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 395.89 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 435.06 �̃�𝑅𝑒𝑔.𝑠𝑑.𝑠𝑦𝑠 257.30 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 247.00 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 395.90 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 435.09 �̃�𝑅𝐸.𝑠𝑑.𝑠𝑦𝑠 247.04 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 246.98 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 395.86 �̃�𝐴𝐡.𝑆𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 434.96 �̃�𝑃𝑒.𝑠𝑑.𝑠𝑦𝑠 40.76 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 246.97 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 395.83 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 434.88 �̃�𝐷.𝑠𝑑.𝑠𝑦𝑠 301.11 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 246.93 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 395.68 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 434.43 �̃�𝑅𝑃.𝑠𝑑.𝑠𝑦𝑠 257.30 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 247.02 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 395.95 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 435.25 �̃�𝐸𝑅.𝑠𝑑.𝑠𝑦𝑠 325.30 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 246.88 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 395.56 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 434.06 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 247.02 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 395.96 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 435.26 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 246.87 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 395.53 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 433.95 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 246.64 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 394.83 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 431.85 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 247.03 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 396.00 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 435.39 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 246.49 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 394.41 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 430.57 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 247.03 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 396.00 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 435.39 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 246.45 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 394.28 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 430.19 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 247.02 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 395.97 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 435.30 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1815 https://internationalpubls.com �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 246.03 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 395.40 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 433.58 Table 7. The 𝐏𝐑𝐄𝐬 of various estimators with respect to οΏ½ΜƒοΏ½πŸŽ.𝐬𝐭.𝐬𝐲𝐬 are calculated for different combination of 𝛂 and 𝛃 using population (3). Estimators 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 οΏ½ΜƒοΏ½0.𝑠𝑑.𝑠𝑦𝑠 100.0 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,0) 157.06 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,0) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,0) 400.90 �̃�𝑅.𝑠𝑑.𝑠𝑦𝑠 60.11 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,1) 159.46 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,1) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,1) 377.28 �̃�𝑃.𝑠𝑑.𝑠𝑦𝑠 16.95 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 157.13 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 400.25 �̃�𝑅𝑒𝑔.𝑠𝑑.𝑠𝑦𝑠 167.48 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 158.64 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 385.39 �̃�𝑅𝐸.𝑠𝑑.𝑠𝑦𝑠 157.06 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 159.68 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 167.54 �̃�𝐴𝐡.𝑆𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 192.39 �̃�𝑃𝑒.𝑠𝑑.𝑠𝑦𝑠 36.31 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 157.17 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 399.88 �̃�𝐷.𝑠𝑑.𝑠𝑦𝑠 167.52 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 158.03 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 391.39 �̃�𝑅𝑃.𝑠𝑑.𝑠𝑦𝑠 167.48 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 157.24 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 399.21 �̃�𝐸𝑅.𝑠𝑑.𝑠𝑦𝑠 157.20 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 166.07 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 167.55 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 228.72 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 160.83 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 363.44 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 158.55 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 386.27 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 162.13 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 167.57 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 349.92 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 157.09 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 400.63 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 128.64 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 167.52 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 191.22 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 157.75 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 394.19 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 164.11 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 167.57 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 327.80 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 157.48 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 167.58 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 396.83 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 166.42 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 167.56 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 295.59 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1816 https://internationalpubls.com 5.2 Simulation To extend the findings of the numerical study, we conducted a simulation study using a hypothetically generated normal population. The previous section clearly demonstrated the superior effectiveness of the proposed estimator compared to competing estimators. This simulation study aims to assess the efficiency of the proposed estimator over existing ones for the stratified systematic sampling technique, utilizing auxiliary variable 𝑋. An artificial dataset was generated following the procedure outlined by Tracy et al. [30], as shown in Table 1 (Appendix 1). A bivariate normal population was artificially generated using R software, with a population size of 1,800, divided into three strata of equal sizes: 𝑁1 = 600, 𝑁2 = 600 and 𝑁3 = 600 respectively. The sample sizes 𝑛1 = 100, 𝑛2 = 100 and 𝑛3 = 100 were drawn from these strata using systematic sampling, following the proportional allocation method. The correlation coefficients between the study and auxiliary variables for each stratum were set as πœŒπ‘¦π‘₯1= 0.8, πœŒπ‘¦π‘₯2= 0.7, πœŒπ‘¦π‘₯3= 0.9, respectively. The standard deviations were fixed at 𝑆π‘₯1 = 4.7, 𝑆π‘₯2= 6.2, 𝑆π‘₯3= 8.4 and 𝑆𝑦1, 𝑆𝑦2, 𝑆𝑦3= 4.8 for each stratum. This process was repeated 10,000 times independently, generating 10,000 samples of size 100,100,100 units have been drawn from each stratum from given a population. To conduct the simulation study, the procedure using R-Language software is outlined as follows: 1. Sample selection- A bivariate stratified systematic random sample of size 𝑛𝑖 from π‘–π‘‘β„Ž stratum, with βˆ‘ 𝑛𝑖 = 𝑛 is selected from the above artificially generated population. the sample sizes 𝑛𝑖′𝑠 are chosen by proportional allocation method. 2. Estimation - The 𝑀𝑆𝐸 is calculated for the repeated samples across different estimators, with the entire procedure is repeated 10,000 times and obtain 10,000 values i.e οΏ½Μ…οΏ½ for calculating the PREs. 3. Mean Square Error - The MSE of the estimators is calculated using as 𝑀𝑆𝐸 = βˆ‘ (οΏ½ΜƒοΏ½π‘–βˆ’οΏ½Μ…οΏ½)10,000 𝑖=1 10,000 2 . Where, �̃�𝑖 = οΏ½ΜƒοΏ½0.𝑠𝑑.𝑠𝑦𝑠, �̃�𝑅.𝑠𝑑.𝑠𝑦𝑠 , �̃�𝑃.𝑠𝑑.𝑠𝑦𝑠….., �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,0) ,….., �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) , �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,0) ,…., �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) , �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,0) ,….., �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) . 4. Percent Relative Efficiency - The PRE of the estimators is calculated as 𝑃𝑅𝐸(�̃�𝑖𝑖 , οΏ½ΜƒοΏ½0.𝑠𝑑.𝑠𝑦𝑠) = π‘£π‘Žπ‘Ÿ(οΏ½ΜƒοΏ½0.𝑠𝑑.𝑠𝑦𝑠) 𝑀𝑆𝐸(�̃�𝑖) Γ— 100 �̃�𝑖 = οΏ½ΜƒοΏ½0.𝑠𝑑.𝑠𝑦𝑠, �̃�𝑅.𝑠𝑑.𝑠𝑦𝑠 , �̃�𝑃.𝑠𝑑.𝑠𝑦𝑠….., �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,0) ,….., �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) , �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,0) ,…., �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) , �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,0) ,….., �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) Simulation results for the PREs of the proposed estimator w.r.t existing estimators across different strata, are provided in Table (8). Estimators 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 Estimator 𝑃𝑅𝐸 οΏ½ΜƒοΏ½0.𝑠𝑑.𝑠𝑦𝑠 100.0 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,0) 177.33 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,0) 1124.77 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,0) 5112.90 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1817 https://internationalpubls.com �̃�𝑅.𝑠𝑑.𝑠𝑦𝑠 374.03 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,1) 176.82 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,1) 1216.96 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,1) 4857.57 �̃�𝑃.𝑠𝑑.𝑠𝑦𝑠 43.78 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 177.32 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 1216.60 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝐢π‘₯) 4890.82 �̃�𝑅𝑒𝑔.𝑠𝑑.𝑠𝑦𝑠 76.11 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 176.83 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 1216.96 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,πœŒπ‘¦π‘₯) 4854.97 �̃�𝑅𝐸.𝑠𝑑.𝑠𝑦𝑠 177.05 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 136.42 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 1247.22 �̃�𝐴𝐡.𝑆𝑑.𝑠𝑦𝑠 (𝐢π‘₯,πœŒπ‘¦π‘₯) 2436.44 �̃�𝑃𝑒.𝑠𝑑.𝑠𝑦𝑠 63.57 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 177.32 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 1216.60 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝐢π‘₯) 4890.82 �̃�𝐷.𝑠𝑑.𝑠𝑦𝑠 1187.49 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 176.58 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 1216.96 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽1π‘₯) 4843.78 �̃�𝑅𝑃.𝑠𝑑.𝑠𝑦𝑠 1287.53 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 177.32 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 1216.60 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝐢π‘₯) 4903.78 �̃�𝐸𝑅.𝑠𝑑.𝑠𝑦𝑠 155.01 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 128.75 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 1251.34 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽1π‘₯) 2124.20 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 176.99 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 1216.60 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,πœŒπ‘¦π‘₯) 4874.21 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 176.58 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 1216.96 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽1π‘₯) 4842.74 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 174.35 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 1219.09 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (1,𝛽2π‘₯) 4661.24 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 177.33 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 1216.60 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝐢π‘₯) 4898.57 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 109.70 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 809.20 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝐢π‘₯,𝛽2π‘₯) 1828.42 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 177.24 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 1216.60 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,πœŒπ‘¦π‘₯) 4898.01 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 174.34 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 1219.09 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (πœŒπ‘¦π‘₯,𝛽2π‘₯) 4658.87 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 177.46 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 1216.60 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽2π‘₯,𝛽1π‘₯) 4883.17 �̃�𝐸.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 175.31 �̃�𝐺.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 1218.38 �̃�𝐴𝐡.𝑠𝑑.𝑠𝑦𝑠 (𝛽1π‘₯,𝛽2π‘₯) 4736.37 6. Results and Discussion As discussed earlier, we utilized three real datasets and conducted a simulation study to evaluate the efficiency of the proposed generalized class of exponential-type estimator, along with other existing estimators, within the stratified systematic sampling scheme using information from a single auxiliary variable 𝑋. The proposed estimator and its adapted version in stratified systematic sampling were compared with respect to their percent relative efficiencies (𝑃𝑅𝐸𝑠). Tables 2-4 provide the data descriptions, while Tables 5-7 present the 𝑃𝑅𝐸𝑠 results for the real datasets. It was observed that the 𝑃𝑅𝐸𝑠 of the proposed estimator vary with different choices of 𝛼 and 𝛽. Additionally, the proposed Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1818 https://internationalpubls.com estimator proved to be more precise than other existing estimators in both the real datasets and the simulation study, based on 𝑃𝑅𝐸𝑠. Conclusion In this paper, we introduced an improved generalized class of exponential-type estimator for estimating the finite population mean under the stratified systematic sampling scheme. Auxiliary variable information is used to enhance precision. The mean squared error (𝑀𝑆𝐸) of the proposed exponential estimator was derived using the Taylor series. An empirical study, based on both simulated and real datasets, was conducted to assess the efficiency of the proposed estimator. The 𝑀𝑆𝐸 and percent relative efficiency (𝑃𝑅𝐸) were employed to compare the precision of the proposed estimator with that of the usual ratio, product, regression, and other existing estimators. The study demonstrated that the proposed improved class of exponential-type estimator yields better results compared to other existing estimators, making it a reliable and more precise option for practical use. REFERENCES [1] Ahmad S, Hussain S, Aamir M, Yasmeen U. (2021). 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Journal of Statistical Theory and Practice. 6: 274– 285. [29] Swain A-K-P-C. (1964). The use of systematic sampling in ratio estimate. Journal of the Indian Statistical Association. 2(213): 160-164. [30] Tailor R. (2012). An almost unbiased ratio-cum-product estimator of population mean using known coefficients of variation of auxiliary variables. International Journal of Statistics and Economics. 8(12): 70-85. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1820 https://internationalpubls.com [31] Tracy D-S, Singh S, Arnab R. (2003). Note On Calibration in Stratified and Double Sampling. No. 12-001 Survey Methodology Statistics Canada Catalouge. 29(1): 99-104. [32] Yadav S-K, and Kadilar C. (2013). Efficient family of exponential estimators for the population mean. Hacettepe Journal of Mathematics and Statistics. 42(6): 671-7. Appendix -1 Table 1. Parameters and distributions of study and auxiliary variables Population Parameters and distributions of the study Variable Parameters and distributions of the auxiliary variable Population β„Ž = 1,2,3 𝑓(π‘¦β„Žπ‘– βˆ— ) = 1 Ξ“(1.5) π‘¦β„Žπ‘– βˆ—1.5βˆ’1π‘’βˆ’π‘¦β„Žπ‘– βˆ— 𝑓(π‘₯β„Žπ‘– βˆ— ) = 1 Ξ“(0.3) π‘₯β„Žπ‘– βˆ—0.3βˆ’1π‘’βˆ’π‘₯β„Žπ‘– βˆ— Table 2. Properties of strata Strata Study variable Auxiliary variable Stratum 1 𝑦1𝑖 = 50 + 𝑦1𝑖 βˆ— π‘₯1𝑖 = 15 + √(1 βˆ’ 𝜌π‘₯𝑦1 2 )π‘₯1𝑖 βˆ— + 𝜌π‘₯𝑦1 ( 𝑆1π‘₯ 𝑆1𝑦 ) 𝑦1𝑖 βˆ— Stratum 2 𝑦2𝑖 = 150 + 𝑦2𝑖 βˆ— π‘₯2𝑖 = 100 + √(1 βˆ’ 𝜌π‘₯𝑦2 2 )π‘₯2𝑖 βˆ— + 𝜌π‘₯𝑦2 ( 𝑆2π‘₯ 𝑆2𝑦 ) 𝑦2𝑖 βˆ— Stratum 3 𝑦2𝑖 = 100 + 𝑦3𝑖 βˆ— π‘₯3𝑖 = 200 + √(1 βˆ’ 𝜌π‘₯𝑦3 2 )π‘₯3𝑖 βˆ— + 𝜌π‘₯𝑦3 ( 𝑆3π‘₯ 𝑆3𝑦 ) 𝑦3𝑖 βˆ— Appendix- 2 N1<-600;N2<-600;N3<-600 n1<-100;n2<-100;n3<-100 N<-1800 R1<-0.8;R2<-0.7;R3<-0.9 Sx1<-4.7;Sx2<-6.2; Sx3<-8.4 Sy1<-4.8;Sy2<-4.8;Sy3<-4.8 set.seed(12345) Y_1<-rgamma(N1,1.5,1)+50 y1<-Y_1[order(Y_1)] Y_2<-rgamma(N2,1.5,1)+100 y2<-Y_2[order(Y_2)] Y_3<-rgamma(N3,1.5,1)+150 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1821 https://internationalpubls.com y3<-Y_3[order(Y_3)] X_1<-15+sqrt(1-R1^2)*rgamma(N1,0.3,1)+R1*(Sx1/Sy1)*Y_1 x1<-X_1[order(X_1)] X_2<-100+sqrt(1-R2^2)*rgamma(N2,0.3,1)+R2*(Sx2/Sy2)*Y_2 x2<-X_2[order(X_2)] X_3<-200+sqrt(1-R3^2)*rgamma(N3,0.3,1)+R3*(Sx3/Sy3)*Y_3 x3<-X_3[order(X_3)] Df1<-cbind(x1,y1);Df2<-cbind(x2,y2);Df3<-cbind(x3,y3) #====mean W<-c(0.3333333,0.3333333,0.3333333) W1<-(N1/N);W2<-(N2/N);W3<-(N3/N) Xbar<-228.716;Ybar<-101.4751 d<-0.8126446 d1<-0.9999706 d2<-0.8126207 d3<-1.415815 d4<-0.9998337 d5<-0.9999706 d6<-0.8124975 d7<-(-0.8689623) d8<-189.65 a<-1 b<-(N*Xbar) #====== set.seed(12345) mst<-NA;yst<-NA;xst<-NA;yr<-NA ;myr<-NA;yre<-NA;yp<-NA;myp<-NA;myre<-NA;yrt<- NA;myrt<-NA;ypt<-NA;mypt<-NA;yd<-NA;myd<-NA;yrp<-NA;myrp<-NA;yer<-NA;myer<- NA;ye<-NA;mye<-NA;yg<-NA;myg<-NA;yA<-NA;myA<-NA for (i in 1:10000) { m1<-c(sample(1:600,100,replace=F)) m2<-c(sample(1:600,100,replace=F)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1822 https://internationalpubls.com m3<-c(sample(1:600,100,replace=F)) ma1<-Df1[m1,] head(ma1) ma2<-Df2[m2,] head(ma2) ma3<-Df3[m3,] head(ma3) maa1<-as.data.frame(ma1) maa2<-as.data.frame(ma2) maa3<-as.data.frame(ma3) y11<-mean(maa1[,2]) y22<-mean(maa2[,2]) y33<-mean(maa3[,2]) x11<-mean(maa1[,1]) x22<-mean(maa2[,1]) x33<-mean(maa3[,1]) yst[i]<-W[1]*y11+W[2]*y22+W[3]*y33 mst[i]<-(yst[i]-Ybar)^2 xst[i]<-W[1]*x11+W[2]*x22+W[3]*x33 yr[i]<-yst[i]*(Xbar/xst[i]) myr[i]<-(yr[i]-Ybar)^2 yp[i]<-yst[i]*(xst[i]/Xbar) myp[i]<-(yp[i]-Ybar)^2 yre[i]<-yst[i]+(d*(Xbar-xst[i])) myre[i]<-(yre[i]-Ybar)^2 yrt[i]<-yst[i]*(exp((Xbar-xst[i])/(Xbar+xst[i]))) myrt[i]<-(yrt[i]-Ybar)^2 ypt[i]<-yst[i]*(exp((xst[i]-Xbar)/(xst[i]+Xbar))) mypt[i]<-(ypt[i]-Ybar)^2 yd[i]<-(d1*yst[i])+(d2*(Xbar-xst[i])) myd[i]<-(yd[i]-Ybar)^2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1823 https://internationalpubls.com yrp[i]<-yst[i]*((d3*(Xbar/xst[i]))+((1-d3)*(xst[i]/Xbar))) myrp[i]<-(yrp[i]-Ybar)^2 yer[i]<-d4*yst[i]*(exp((Xbar-xst[i])/(Xbar+xst[i]))) myer[i]<-(yer[i]-Ybar)^2 ye[i]<-yst[i]*(exp((a*(Xbar-xst[i]))/((a*(Xbar+xst[i]))+2*b))) mye[i]<-(ye[i]-Ybar)^2 yg[i]<-((d5*yst[i])+(d6*(Xbar-xst[i])))*(exp((a*(Xbar-xst[i]))/((a*(Xbar+xst[i]))+2*b))) myg[i]<-(yg[i]-Ybar)^2 yA[i]<-((d7*yst[i])+(d8*((xst[i])/Xbar)))*(exp((a*(Xbar-xst[i]))/((a*(Xbar+xst[i]))+2*b))) myA[i]<-(yA[i]-Ybar)^2 } mse1<-mean(mst) MSE1<-1/10000*sum(myr);MSE2<-1/10000*sum(myp);MSE3-1/10000*sum(myre);MSE4<- 1/10000*sum(myrt);MSE5<-1/10000*sum(mypt);MSE6<-1/10000*sum(myd);MSE7<- 1/10000*sum(myrp);MSE8<-1/10000*sum(myer);MSE9<-1/10000*sum(myg);MSE10<- 1/10000*sum(myA);MSE11<1/10000*sum(mye) pre1<-(mse1/MSE1) *100; pre2<-(mse1/MSE2)*100;………….. pre11<-(mse1/MSE11)*100