EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5922 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generalized Class of Estimators for Median Estimation Using Auxiliary Information Sohaib Ahmad1,∗, Saadia Masood2, Manahil SidAhmed Mustafa3, Elsiddig Idriss Mohamed3, Elfarazdag M. M. Hussein3 1 Department of Statistics, Abdul Wali Khan University, Mardan, Pakistan 2 Department of Mathematics and Statistics, PMAS University of Arid Agriculture, Rawalpindi, Pakistan 3 Department of Statistics, Faculty of Science, University of Tabuk, Tabuk, Kingdom of Saudi Arabia Abstract. In this article, we propose a comprehensive class of estimator for estimation of popu- lation median under simple random sampling. To boost the efficiency of an estimator we utilize the auxiliary information. The numerical expression of the bias and mean squared error are con- sequent up to the first order of approximation. The proposed and existing estimators have been evaluated via real data sets and their performances were assessed using measurements of minimal mean square error and maximum percentage relative efficiency. The study showed that compared to some adopted existing estimators in this study, the proposed class of estimators performed bet- ter and efficient. We also visualize all the estimators using results of MSE and PRE. According to the results, the suggested estimator is superior to the adopted existing estimators. The significance and potential applications of our proposed class of estimators are highlighted by these results. 2020 Mathematics Subject Classifications: 62D05, 62f10 Key Words and Phrases: Median estimation, visualization, auxiliary information, MSE, effi- ciency 1. Introduction The accuracy of parameter estimates for a population can be greatly enhanced with the incorporation of auxiliary infromation during the selection or estimation process, or both. Although many determinations have been made to progress estimator precision, estimat- ing the population mean remains a persistent challenge in sampling. When the auxiliary information is taken into account, a huge number of alternatives become accessible. In ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5922 Email addresses: sohaib ahmad@awkum.edu.pk (S. Ahmad), saadia.masood@uaar.edu.pk (S. Masood), msida@ut.edu.sa (M. S. Mustafa), eidriss@ut.edu.sa (E. I. Mohamed), e.hussein@ut.edu.sa (E. M. M. Hussein) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 2 of 17 order to deal with situations involving a large number of covariates, numerous estima- tors have been developed, consisting of incorporating aspects from ratio, product, and regression estimators. The variance, the coefficient of variation, and the kurtosis have all been used extensively by researchers for parameter estimation in populations. For this to work, we need to draw from a representative subset of the population. The population of interest is chosen with the help of simple random sampling. Using the ratio, product, or regression estimation approaches requires advanced knowledge of the population con- straints of the auxiliary information. By adjusting the appropriate auxiliary information, many authors have provided alternative estimators. It is generally agreed that estimators who use auxiliary information in survey sampling can yield implicitly better estimates than those who do not. Sampling is a technique for efficiently and effectively collecting information about a population in a way that maximizes the precision of estimates with minimal outlay of resources. In order to make more precise estimates, sampling is done to gain insight into a population’s characteristics with minimal effort. Since these methods yield reliable estimates, we employ the mean and standard least-squares approaches to estimate population parameters. As a consequence, the data may not follow a normal distribution but exhibit severely skewed distributions (for example wages and consumption). Due to the mean sensitivity with outliers, it cannot be used to reliably compute these quantities. Therefore, the me- dian is unaffected by outliers and extreme values, it can be used as a measure of center tendency. Attempts to provide a novel method for obtaining a reliable conclusion in such cases are notoriously challenging for academics. The use of auxiliary information in median estimation depends on the idea that related variables can enhance target variable estima- tion accuracy. Considering auxiliary information helps to both strengthen and stabilize the estimation results particularly when the original data has limited quantity or missing values. Through these methods the estimator accounts for new information facilitating both variance reduction and improve efficiency. A regression estimator can use auxil- iary variables to recognize how the target variable associates with covariates thus enabling better median predictions. From a theoretical point of view the addition of auxiliary infor- mation enhances two important statistical criteria consisting of efficiency and consistency. The estimation method which takes auxiliary information into consideration delivers lower MSE results than methods working with observed data only. Some conditions allow auxil- iary information to create estimators that deliver robust results particularly when outliers occur. Several researchers have recommended estimators, either by making changes to the already existing estimators or by building completely novel estimators. Some notable work by these authors includes [3, 4, 5], [7–15], [16, 17, 18, 19, 20, 21, 22, 23] and [24]. The main purpose of this work is developed a new generalized class of estimators using auxiliary information for achieving precise median estimation. This work increases the efficiency of median estimation performance in multiple statistical models through deliberate use of auxiliary information. The study investigates both consistency and unbiasedness as statistical properties of the proposed estimators. The paper performs an analysis of new estimators versus standard methods while demonstrating their use in practical situations S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 3 of 17 to explain enhanced accuracy potential in estimation. This paper makes a significant contribution toward enhancing median estimation accuracy through the use of auxiliary information. The proposed estimators provide flexibility which enables them to process diverse data types so they can be utilized throughout economic sectors and medical practices and social science applications. Median estimation improvements through auxiliary data lead to enhanced decision-making in both policy analysis and medical diagnostics along with precise statistical analysis of other fields. The paper includes a theoretical evaluation of these generalized estimators’ statistical characteristics which includes unbiasedness and consistency and efficiency to prove their validity. The research has major implications because it enhances both statistical methods and field applications of median estimation techniques in empirical situations. To account for situations in which the population distribution is not normally distributed, we generate estimators for estimation of population median in the current study. The remaining of the article is arranged as follows: Section 2, include the procedures and materials. The literature review of the available estimator for estimate of median under simple random sample is given in Section 3. In Section 4, we provide the suggested estimator. The numerical study is given in Section 5. In Section 6, the article findings and conclusions are outlined. 2. Methods and materials Consider a population Ω = (Ω1,Ω2, . . . ,ΩN ) consisting of N divergent units. Let Yi and Xi be the ith standards of the study and auxiliary variables. Consider a sample of n is chosen from Ω. The sample and population medians of the study and the auxiliary variable are denoted by My, Mx, and M̃y and M̃x with probability density functions of fy(My) and fx(Mx). The correlation coefficient between M̃y and M̃x are represented by ρyx and is defined as ρyx(M̃y, M̃x) = 4P11(y, x) − 1, where P11 = (y ≤My ∩ x ≤Mx). To obtain the bias and MSE we used the following error terms: ξ0 = ( M̃y−My My ) and ξ1 = (M̃x−Mx Mx ), E(ξ20) = λC2 My, E(ξ21) = λC2 Mx, E(ξ0ξ1) = λCMyx, where CMy = [Myfy(My)]−1 , CMx = [Mxfx(Mx)]−1 , CMyx = [ρyxCMyCMx] , λ ( 1 4 ( 1 n − 1 N ) ) 3. Literature review In this section, we have discussed existing estimators for population median which are given by: (i) The [9] recommended a usual median estimator, which is given by: M̃U = M̃y (1) S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 4 of 17 The variance of M̃U is given by: V ar(MU ) = λM2 yC 2 My (2) (ii) The [12] recommended ratio estimator for median, is given by: M̃R = M̃y ( Mx M̃x ) (3) The bias and MSE of MR are given by: Bias(MR) = λMy[C2 My − CMyx], and MSE(M̃R) ≈ λM2 y [C2 My + C2 Mx − 2CMyx]. (4) (iii) The [14] suggested the usual difference estimator M̃D, which is given by: M̃D = M̃y + d1(Mx − M̃x) (5) The variance of (M̃D): V (M̃D) = λM2 yC 2 My(1 − ρ2yx) (6) (iv) The exponential ratio estimator developed by [24] is given by: M̃E = M̃yexp ( Mx − M̃x Mx + M̃x ) , (7) The bias and MSE of (M̃E), are given by: Bias(M̃E) ∼= λMy [ 3 4 C2 Mx − 1 2 CMyx ] , MSE(M̃E) ∼= λM2 y [ C2 My + 1 4 C2 Mx − CMyx ] . (8) (v) The [13] proposed the difference-in-difference estimator, which is given by: M̃RD1 = d2M̃y + d3(Mx − M̃x), (9) the values of (d2) and (d3) are given by: d2 = 1 1 + λC2 My(1 − ρ2yx) S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 5 of 17 d3 = MyCMyρyx MxCMx{1 + λC2 My(1 − ρ2yx)} . The minimum MSE of (M̃RD1) is given by: MSE(M̃RD1)min = λM2 yC 2 My(1 − ρ2yx) [1 + λC2 My(1 − ρ2yx)] (10) (vi) The [8] recommended the following difference-in-ratio type estimator, which is given as: M̃RD2 = (d4M̃y + d5(Mx − M̃x)) ( Mx M̃x ) (11) The (d4) and (d5) are constants, which are given by: d4 = [ 1 − λC2 Mx 1 − λC2 Mx + λC2 My(1 − ρ2yx) ] , d5 = My Mx [ MyCMyρyx MxCMx{1 + λC2 My(1 − ρ2yx)} ] . Substituting the values of (d4) and (d5), are given by: MSE(M̃RD2) = λM2 yC 2 My(1 − ρ2yx)(1 − ρ2yx) [(1 − λC2 My) + λC2 My(1 − ρ2yx)] (12) (vii) The [15] given the generalized class of estimator: M̃s = M̃y exp ( a(Mx − M̃x) a(Mx + M̃x) + 2b ) (13) The properties of (M̃S) is given by: Bias(M̃s) = My [ 3 8 θ2CMy2 − 1 2 θCMyx ] , MSE(M̃s) = M2 y 4 [ 4CMy2 + θ2C2 Mx − 4θCMyx ] (14) where θ = ( aMx aMx+b ) S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 6 of 17 4. Suggested generalized class of estimator When the data is collected with the help of survey sampling, it is possible that the data follow a normal distribution. In order to get reliable estimates of population parameters, we employ the mean and standard least-squares methods. Sometimes the acquired data may not normally distributed (in terms of wage, consumption, etc.), but follow extremely skewed distributions. In such scenario, it is not reliable to calculate quantities like mean etc. In such circumstances, the median is used, because it is unaffected by outliers and extreme values. Median can be used as a measure of center tendency. By taking motivation from [5], we suggested the resulting improved class of estimators for population median. This work makes an essential contribution to statistical estimation because it introduces a new approach to median estimation through generalized class of estimators using auxiliary information. The framework developed by this research enables the integration of auxil- iary data with the retention of essential median estimator features including robustness and consistency. A new approach enhances precision in estimation because researchers need exact median results in economics and medical research and social science stud- ies especially when data is restricted. The paper verifies the statistical aspects of the proposed estimators while confirming both theoretical and practical proof for their ap- plication. This research develops an innovative statistical tool that gives practitioners and researchers improved ways to increase the accuracy of their analytical work through performance comparison with traditional methods. M̃GP = ψ1My ( 1 4 ( Mx M̄x + M̄x Mx ) × ( exp ( Mx − M̄x Mx + M̄x ) + exp ( M̄x −Mx Mx + M̄x ))) + ψ2 ( Mx − M̄x ) exp ( α(Mx − M̄x) α(Mx + M̄x) + 2β ) (15) ψ1 and ψ2 are the constants. After simplification of M̃GP , we have M̃GP = [ ψ1My(1 + ξ0) ( 1 + 5 8 ξ21 ) − ψ2Mxξ1 ] [ 1 − 1 2 θξ1 + 3θ2 8 ξ21 ] (16) Expanding (16), we get M̃GP−My = My+My [ (ψ1 − 1) + ψ1 { ξ0 − 1 2 θξ1 − 1 2 θξ0ξ1 + 1 8 (5 + 3θ2)ξ21 } − ψ2R{ξ1 − 1 2 θξ21} ] (17) From (17), the bias of M̃GP is given by: Bias(M̃GP ) = My [ (ψ1 − 1) + ψ1 { 1 8 (5 + 3θ2)CMx− 1 2 θCMyx } + ψ19Rλ 1 2 θC2 Mx ] (18) Squaring (17) and taking expectations: MSE(M̃GP ) = M2 y [1 + ψ2 1A11 + ψ2 2B11 − 2ψ1C11 − 2ψ2D11 + 2ψ1ψ2E11], (19) S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 7 of 17 where A11 = 1+λ [ CMy + ( 5 4 + θ2 ) C2 Mx − 2θCMyx ] , B11 = R2λ[C2 Mx], C11 = 1+λ [( 5 + 3θ2 8 ) C2 Mx − 1 2 θCMyx ] , D11 = Rλ [ θC2 Mx 2 ] , E11 = Rλ[θC2 Mx − CMyx]. Differentiate (19) w.r.t ψ1 and ψ2, the values of ψ1 and ψ2 as given by: ψ1(opt) = B11C11 −D11E11 A11B11 − E2 11 , and ψ2(opt) = A11D11 − C11E11 A11B11 − E2 11 . Putting the optimum values of ψ1(opt) and ψ2(opt) in (19), we get the minimum MSE of M̃GP as given by: MSE(M̃GP )min ≈M2 y [ 1 − A11D 2 11 +B11C 2 11 − 2C11D11E11 A11B11 − E2 11 ] . (20) 5. Numerical study To determine how effective our suggested class of estimators is, we perform a mathematical analysis with some actual data. We compare the effectiveness of our suggested class to that of existing estimators by using a percentage relative efficiency (PRE). The theoretical expression for PRE is given by: PRE = V ar(M̃U ) MSE(M̃i) × 100 Where (i=R, D, E, RD1, RD2, S, GP) Population 1: [Source: [14] ] Y = Fish caught in 1995, X = fish caught in 1994. N = 69, n = 17, My = 2068, Mx = 2011, fmy = 0.00014, fmx = 0.00014, ρyx = 0.1505. Population 2: [Source: [2] ] Y = U.S export in Singapore X = money supply in Singapore. N = 67, n = 23, My = 4.8, Mx = 7, fmy = 0.48294, fmx = 0.343079, ρyx = 0.61194. Population 3: [Source: [1] ] Y = Oil price from 1996 to 2017 X = oil price in preceding from 1996 to 2017. N = 1134, n = 210, My = 48.5500, Mx = 48.4900, fmy = 0.00754, fmx = 0.00754, ρyx = 0.99530. Population 4: [Source: [6] ] Y = Master degree in 2007 X = Master degree in 2006. S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 8 of 17 N = 51, n = 11, My = 25.8000, Mx = 25.6000, fmy = 0.07280, fmx = 0.00754, ρyx = 0.99530. Table 1: Members of the recommended class of estimators α β M̃S M̃GP 1 Cx M̃S1 M̃GP1 1 β2(x) M̃S2 M̃GP2 β2(x) Cx M̃S3 M̃GP3 Cx β2(x) M̃S4 M̃GP4 1 ρyx M̃S5 M̃GP5 Cx ρyx M̃S6 M̃GP6 ρyx Cx M̃S7 M̃GP7 β2(x) ρyx M̃S8 M̃GP8 ρyx β2(x) M̃S9 M̃GP9 1 NMx M̃S10 M̃GP10 Table 2: MSE using Population 1 Estimators V alues M̃S M̃GP M̃U 565443.600 627047.800 373018.400 M̃R 988372.800 626625.000 373125.800 M̃D 552636.100 627371.200 372936.300 M̃E 627420.200 626097.100 372981.000 M̃RD1 489395.200 627404.400 372927.900 M̃RD2 480458.300 627415.800 372925.000 624987.400 373542.300 627418.100 372924.400 622322.700 374222.000 564223.900 398752.300 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 9 of 17 Table 3: PRE using Population 1 Estimators Values M̃S M̃GP M̃U 100 90.175 151.586 M̃R 57.209 90.236 151.542 M̃D 102.317 90.129 151.619 M̃E 90.121 90.312 151.601 M̃RD1 115.539 90.124 151.622 M̃RD2 117.688 90.122 151.623 90.472 151.373 90.122 151.624 90.860 151.098 100.216 141.803 Table 4: MSE using Population 2 Estimators Values M̃S M̃GP M̃U 0.0306078 0.0198950 0.019076 M̃R 0.0229662 0.019150 0.019068 M̃D 0.019146 0.019498 0.019073 M̃E 0.019658 0.019477 0.019039 M̃RD1 0.019130 0.020011 0.019076 M̃RD2 0.019130 0.020535 0.019078 0.020052 0.019077 0.019430 0.019073 0.019431 0.019059 0.030343 0.0190869 Table 5: PRE using Population 2 Estimators Values M̃S M̃GP M̃U 100 153.846600 160.449400 M̃R 133.273200 159.825500 160.519400 M̃D 159.864600 156.972900 160.470600 M̃E 155.701300 157.144600 160.758400 M̃RD1 159.997400 152.952900 160.444700 M̃RD2 159.997600 149.047300 160.427900 152.638500 160.443100 157.528400 160.475800 157.519800 160.588600 100.871100 160.360200 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 10 of 17 Table 6: MSE using Population 3 Estimators Values M̃S M̃GP M̃U 17.06228000 4.79872700 0.09149240 M̃R 0.16061000 5.56053900 0.09486919 M̃D 0.16000850 4.39771000 0.08942814 M̃E 4.33531100 5.23074000 0.09153061 M̃RD1 0.15999760 4.50701300 0.09001304 M̃RD2 0.15999750 4.39849600 0.08943241 4.80085300 0.09150277 4.35807100 0.09150277 5.56585900 0.09150277 17.047300 0.09150277 Table 7: PRE using Population 3 Estimators Values M̃S M̃GP M̃U 100 355.5584 18648.8400 M̃R 10623.4200 306.8457 17985.0500 M̃D 10663.3600 387.9809 19079.3100 M̃E 393.5652 326.1924 18641.0600 M̃RD1 10664.0800 378.5717 18955.3400 M̃RD2 10664.0900 387.9116 19078.4000 355.4009 18646.7300 391.5098 19125.6100 306.5524 17981.0000 100.0879 15439.0300 Table 8: MSE using Population 4 Estimators Values M̃S M̃GP M̃U 3.363368 1.497556 0.02741737 M̃R 0.3659373 1.840453 0.0276705 M̃D 0.02953252 1.47938 0.02740078 M̃E 1.476601 1.751034 0.02785551 M̃RD1 0.02953121 1.533249 0.02744888 M̃RD2 0.02953121 1.625763 0.02752447 1.497648 0.02741745 1.48423 0.02740524 1.841761 0.02767128 3.319765 0.02803544 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 11 of 17 Table 9: PRE using Population 4 Estimators Values M̃S M̃GP M̃U 100 224.5904 12267.29 M̃R 919.1105 182.7468 12155.07 M̃D 11388.6900 227.3499 12274.72 M̃E 227.7777 192.079 12074.34 M̃RD1 11389.2000 219.3621 12253.21 M̃RD2 11389.3000 206.8794 12219.56 224.5767 12267.25 226.6069 12272.72 182.617 12154.72 101.3134 11996.84 Figure 1: MSE of median estimators using population 1 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 12 of 17 Figure 2: PRE of median estimators using population 1 Figure 3: MSE of median estimators using population 2 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 13 of 17 Figure 4: PRE of median estimators using population 2 Figure 5: MSE of median estimators using population 3 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 14 of 17 Figure 6: PRE of median estimators using population 3 Figure 7: MSE of median estimators using population 4 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 15 of 17 Figure 8: PRE of median estimators using population 4 S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 16 of 17 6. Conclusion In this article, we have suggested an improved generalized class of estimator for estimation of population median under simple random sampling. We examined the suggested estimator’s bias and MSE up to the first approximation order. On four distinct datasets, the suggested generalized class of estimators is contrasted with existing estimators. Table 1 contains the members of the suggested class of estimators using different choices of α and β. Tables 2–9 show the mean squared error (MSE) and the PRE for the existing and the suggested class of estimators for four real data sets, respectively. The MSEs of various suggested and competing estimators for real Data sets I and IV are depicted graphically in Figures 1, 3, 5, and 7. The Data sets I to IV are shown graphically in Figures 2, 4, 6, and 8, which show the PREs of several suggested and competing estimators. When compared to other estimators, the proposed one has the best practical efficiency based on the results of real data sets, with the lowest MSE and maximum PRE. Using auxiliary information for population mean, distribution function and population proportion are discussed in detail by [21-24]. This work can be easily extended to estimate population median under systematic random sampling. Further extension of the current work is to develop improved estimators for estimation of median using neutrosophic data. This work contributes its main innovation through the creation of a generalized class of estima- tors for median estimation which used auxiliary information effectively. This estimation method differs from standard median methods because it uses observed data and external information sources while related variables contribute to enhanced estimation precision. This work presents an adaptable framework that uses auxiliary data to deliver improved median estimations throughout diverse complex situations when observed data exists with defects or incompleteness. The pa- per thoroughly examines the statistical properties of these new estimators including unbiasedness, consistency along with efficiency in order to prove their theoretical validity. These estimators find practical applications across economics together with healthcare and social sciences to support crucial decision-making processes whenever accurate median estimation becomes essential. These findings establish better statistical inference approaches that demonstrate substantial progress in robust estimation research. Competing interests The authors declare no competing interests. References [1] Aamir, M., Shabri, A., & Ishaq, M. (2018). Improving forecasting accuracy of crude oil prices using decomposition ensemble model with reconstruction of IMFs based on ARIMA model. Malaysian Journal of Fundamental and Applied Sciences, 14, 471-483. [2] Aczel, A. D. (1996). Complete business statistics. Irwin Professional Publishing. [3] Aladag, S., & Cingi, H. (2015). Improvement in estimating the population median in simple random sampling and stratified random sampling using auxiliary information. Communications in Statistics-Theory and Methods, 44, 1013-1032. [4] Bedi, P. K. (1996). Efficient utilization of auxiliary information at estimation stage. Biometrical Journal, 38, 973-976. [5] Baig, A., Masood, S., & Ahmed Tarray, T. (2020). Improved class of difference-type estimators for population median in survey sampling. Communications in Statistics-Theory and Methods, 49, 5778-5793. S. Ahmad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5922 17 of 17 [6] Bandyopadhyay, A., Singh, G. N., & Das, P. (2016). Estimation of population median in presence of non-response under two-phase sampling. Sri Lankan Journal of Applied Statistics, 17. [7] Enang, E. I., Etuk, S. I., Ekpenyong, E. J., & Akpan, V. M. (2016). An alternative exponential estimator of population median. International Journal of Statistics and Economics, 17, 85-97. [8] Gupta, S., Shabbir, J., & Ahmad, S. (2008). Estimation of median in two-phase sampling using two auxiliary variables. Communications in Statistics-Theory and Methods, 37, 18151822. [9] Gross, S. (1980). Median estimation in sample surveys. In Proceedings of the Section on Survey Research Methods (Vol. 1814184). Alexandria, VA: American Statistical Association. [10] Jhajj, H. S., Kaur, H., & Walia, G. (2014). Efficient family of ratio-product type estimators of median. Model Assisted Statistics and Applications, 9, 277-282. [11] Irfan, M., Javed, M., Shongwe, S. C., Zohaib, M., & Haider Bhatti, S. (2021). Estimation of population median under robust measures of an auxiliary variable. Mathematical problems in Engineering, 2021, 1-14. [12] Kuk, A. Y., & Mak, T. K. (1989). Median estimation in the presence of auxiliary information. Journal of the Royal Statistical Society: Series B (Methodological), 51, 261-269. [13] Rao, T. J. (1991). On certail methods of improving ration and regression estimators. Com- munications in Statistics-Theory and Methods, 20, 3325-3340. [14] Singh, S. (2003). Advanced Sampling Theory With Applications: How Michael”” Selected”” Amy (Vol. 2). Springer Science & Business Media. [15] Singh, R., Chauhan, P., Sawan, N., & Smarandache, F. (2007). Improvement in estimating the population mean using exponential estimator in simple random sampling. Auxiliary Information and a priori Values in Construction of Improved Estimators, 33. [16] Shabbir, J., Gupta, S., & Hussain, Z. (2015). Improved estimation of finite population median under two-phase sampling when using two auxiliary variables. Scientia Iranica, 22, 1271-1277. [17] Ahmad, S., Hussain, S., & Ahmad, S. (2021). Finite population distribution function esti- mation using auxiliary information under simple random sampling. Statistics, Computing and Interdisciplinary Research, 3, 29-38. [18] Subzar, M., Lone, S. A., Ekpenyong, E. J., Salam, A., Aslam, M., Raja, T. A., & Almutlak, S. A. (2023). Efficient class of ratio cum median estimators for estimating the population median. Plos one, 18, e0274690. [19] Shabbir, J., & Gupta, S. (2017). A generalized class of di fference type estimators for popu- lation median in survey sampling. Hacettepe Journal of Mathematics and Statistics, 46, 1015- 1028. [20] Hussain, I., Qureshi, M., Ismail, M., Iftikhar, H., Zywio lek, J., & López-Gonzales, J. L. (2024). Optimal features selection in the high dimensional data based on robust technique: Application to different health database. Heliyon, 10. [21] Ahmad, S., Qureshi, M., Iftikhar, H., Rodrigues, P. C., & Rehman, M. Z. (2025). An improved family of unbiased ratio estimators for a population distribution function. AIMS Mathemat- ics, 10, 1061-1084. [22] Kumar, A., & Siddiqui, A. S. (2024). Enhanced estimation of population mean using simple random sampling. Research in Statistics, 2, 2335949. [23] Bhushan, S., & Kumar, A. (2023). Evaluating the performance of logarithmic type estimators using auxiliary attribute. Life Cycle Reliability and Safety Engineering, 12, 285-292. [24] Bahl, S and Tuteja, R. (1991). Ratio and Product type exponential estimators. Journal of Information of Optimization Sciences, 12: 159-164.