A NEW CLASS OF RATIO ESTIMATORS FOR POPULATION MEAN USING AUXILIARY PARAMETERS: AN APPLICATION TO REAL-WORLD DATA

GANITA,Vol.75(1), 2025, 259-267 A NEW CLASS OF RATIO ESTIMATORS FOR POPULATION MEAN USING AUXILIARY PARAMETERS: AN APPLICATION TO REAL-WORLD DATA GAGAN KUMAR, DUSHYANT TYAGI, GAUTAM GUPTA and SURENDRA KUMAR Abstract This study introduces a naive general family of ratio estimators that leverage various auxiliary measures to estimate the population mean of a study variable under a simple random sampling without replacement framework. The paper examines specific cases that incorporate auxiliary information such as the coefficient of variation, median, and quartile deviation. The bias and Mean Square Error (MSE) of the introduced estimators are retained to a first-order approximation. Furthermore, theoretical conditions for comparing the efficiency of the introduced estimators with existing ones are provided, and their performance is validated using real-world data. Numerical analysis demonstrates that the proposed estimators are more efficient than other ratio-based estimators, appealing for the use in various areas of applications of the real world, which includes agriculture, biological sciences, defence, economics, mathematical sciences. management, medical sciences, social sciences etc. 2010 Mathematics subject classification: 62D05; 94A20. Keywords and phrases: Study Variable, Auxiliary Parameter, Median, Bias, Mean Square Error. 1. Introduction An alternative to thorough enumeration that saves time, money, and labour is sampling. The sample mean y¯ is most appropriate for the population mean Y¯ since the associated statistics is the best estimator for the parameter being studied. When estimating the parameters of the primary variable Y, auxiliary variable X is quite important. For higher population mean estimate, ratio type estimators are employed if it has a positive correlation with the primary variable. In contrast, when X and Y have a negative correlation, product type estimators are employed to improve the estimation of Y¯ . Sampling strategies are frequently used when working with huge populations or when it is impractical to obtain data from every unit. By looking at a portion of the population, these techniques offer a useful way to investigate the properties of a target variable. Population parameters are usually estimated using sample statistics; for instance, the sample mean y¯ is a commonly used estimator for the population mean Y¯ due to its objectivity, despite the possibility of substantial variance. Much effort has gone into creating more effective estimators for Y¯ by enhancing current techniques. Utilizing auxiliary data to improve the accuracy of sample-based population parameter estimations is the main goal of many of these initiatives. Many authors from all over the globe used the auxiliary information in different forms for enhanced estimation of Y¯ and suggested various efficient estimators of Y¯ . 260 G Kumar et al. Cochran [1] introduced the classical ratio estimator, utilizing information on Xthat is highly positively correlated with Y. Srivastava [2] proposed an improved estimator for by incorporating X information in sample surveys. Sisodia ¯ Yand Dwivedi [3] developed a modified ratio estimator for ¯ Yusing the known coefficient of variation (CV) of X. Rao [4] explored methods to efficient estimation of ¯ Ythrough ratio and regression estimators. Singh et al. [5] proposed a new estimator for ¯ Yby utilizing the coefficient of kurtosis, while Upadhyaya and Singh [6] worked on an elevated estimator for ¯ Yby combining the known CV and coefficient of kurtosis of X. Singh and Tailor [7] enhanced the ratio estimator ¯ Yfor using the known correlation coefficient, Kadilar and Cingi [8] focused on developing improved ratio estimators for better estimation of ¯ Y. In double sampling, Singh and Vishwakarma [9] and [10] created an effective variation of the product and ratio estimators and in sample surveys, certain estimators of the finite population mean use auxiliary data. Yadav et al. [11] introduced modified ratio estimators leveraging non-traditional auxiliary parameters for efficient estimation of ¯ Y. Khoshnevisan et al. [12] proposed a general family of ratio estimators for ¯ Y, identifying various estimators as members of their family. Koyuncu and Kadilar [13] designed efficient estimators for ¯ Y, while Yan and Tian [14] improved the estimation of ¯ Yusing the known coefficient of skewness of X. Subramani and Kumarapandiyan [15] proposed an efficient estimator for ¯ Yby utilizing known quartile information of X. Jeelani et al. [16] developed new ratio estimators for Y, combining the coefficient of skewness and quartile deviation of X. Finally, Subramani [17] introduced a general family of estimators for based on auxiliary parameters. Abid et al. [18] proposed enhanced ratio estimators for ¯ Yby employing unconventional measures of dispersion, while Subzar et al.[19] developed an efficient class of ratio estimators for ¯ Yusing known auxiliary parameters. Unal and Kadilar [20], Singh and Usman [21] introduced improved estimators for ¯ Ythat accounted for non-response scenarios by incorporating auxiliary parameters. Priam [22] proposed a generalized family of estimators for the efficient estimation of ¯ Y. Yadav et al. [23] focused on efficient estimation of average paddy production using robust measures, and Yadav et al.[24] introduced a naive class of estimators aimed at improving the estimation of average peppermint yield through known auxiliary information (X). Zaman et al. [25] explored robust ratiotype estimators for ¯ Yin simple random sampling (SRS) using simulation studies. Ali et al. [26] proposed an advanced class of estimators for ¯ Yleveraging known auxiliary parameters, while Singh et al.[27] introduced a refined family of estimators utilizing Xinformation for improved estimation of ¯ Y. Vishwakarma [28] Weighted technique for estimating the finite population mean in the presence of auxiliary data. Numerous other researchers have also contributed to the development of enhanced estimators for ¯ Yby utilizing known auxiliary parameters. The remainder of this paper is organized into various sections. The remainder of the work is divided into many areas, such as the theoretical efficiency comparison, the proposed estimator, the review of estimators, the numerical analysis, and the comparison of the results. 2. Review of Estimators Cochran [1] utilized the auxiliary information and suggested the usual ratio estimator of ¯ Yas, t0=¯y ¯ X ¯x! A New Class of Ratio Estimators for Population Mean.... 261 The MSE of tRfor an approximation of order one is given by, MS E(tR)=λ¯ Y2(C2 y+R2C2 x−2RCyx) (2.1) Where, λ=1 n −1 N,¯ Y=1 N N X i=1 Yi,¯ X=1 N N X i=1 Xi,S2 y=1 N−1 N X i=1 (Yi−¯ Y)2, S2 x=1 N−1 N X i=1 (Xi−¯ X)2,Syx =1 N−1 N X i=1 (Yi−¯ Y)(Xi−¯ X) Sisodia and Dwivedi [3] used the CV of Xand suggested the following estimator of ¯ Y as, t1=¯y ¯ X+Cx ¯x+Cx! The MSE of t1is given by, MS E(t1)=λ¯ Y2(C2 y+R2 1C2 x−2R1Cyx) (2.2) Where, R1=¯ X ¯ X+Cx Singh at al. [5] used coefficient of kurtosis of Xand given an estimator of ¯ Yas, t2=¯y ¯ X+β2 ¯x+β2! MS E(t2)=λ¯ Y2(C2 y+R2 2C2 x−2R2Cyx) (2.3) Where, R2=¯ X ¯ X+β2 Upadhyaya and Singh [6] proposed two estimators of ¯ Yby using Cxand β2as, t3=¯y ¯ Xβ2+Cx ¯xβ2+Cx!,t4=¯y ¯ XCx+β2 ¯xCx+β2! MS E(ti)=λ¯ Y2(C2 y+R2 iC2 x−2RiCyx),i=3,4 (2.4) Where, R3=¯ Xβ2 ¯ Xβ2+Cx,R4=¯ XCx ¯ XCx+β2 Zaman at el.[29] introduced a class of estimators of ¯ Yas, t5=¯y ¯ X+β2/G(x) ¯x+β2/G(x)! Where, G(x) is the function of X. Some of the special cases of t5are, t5(1) =¯y ¯ X+β2/Cx ¯x+β2/Cx!,t5(2) =¯y ¯ X+β2/Md ¯x+β2/Md!,t5(3) =¯y ¯ X+β2/QD ¯x+β2/QD ! Where Mdis the median and QD is the quartile deviation of X. The MSE of t5(1) are given by, MS E(t5(i))=λ¯ Y2(C2 y+R2 5(i)C2 x−2R5(i)Cyx),i=1,2,3 (2.5) Where, R5(1) =¯ X ¯ X+β2/Cx,R5(2) =¯ X ¯ X+β2/Md,R5(3) =¯ X ¯ X+β2/QD , 262 G Kumar et al. 3. Proposed estimator Motivated by Zaman et al. [29], many other authors in the literature and thinking of that many estimators will be the particular cases of the proposed class of estimator, we have introduced a generalized estimator for estimation of ¯ Yas, tp=¯y ¯ X+β2/G(x) ¯x+β2/G(x)!α (3.1) Where, αis the scalar to be obtained such that MSE of tpis least. To study the large sampling property of tp, the following approximations are used, ¯y=¯ Y(1 +e0),¯x=¯ X(1 +e1), with E(e0)=E(e1)=0 and E(e2 0)=λC2 y E(e2 1)=λC2 x,E(e0e1)=λCyx Representing tppusing e0and e1, we obtain, tp=¯ Y(1 +e0) ¯ X+β2/G(x) ¯ X(1 +e1)+β2/G(x)!α =¯ Y(1 +e0) ¯ X+β2/G(x) ¯ Xe1+¯ X+β2/G(x)!α =¯ Y(1 +e0)         1 ¯ Xe1 ¯ X+β2/G(x)+1         α =¯ Y(1 +e0)(1+θe1)−α where, θ=¯ X ¯ X+β2/G(x) =¯ Y(1 +e0) 1−αθe1+α(1 +α) 2e2 1 −...! tp=¯ Y"1+e0−αθe1−αθe0e1+α(1 +α) 2e2 1 −...# Subtracting ¯ Yon both sides of above equation, we have, tp−¯ Y=¯ Y"e0−αθe1−αθe0e1+α(1 +α) 2e2 1 −...#(3.2) We have bias of tpas, Bias(tp)=λ¯ Y"α(1 +α) 2C2 x−αθCyx#(3.3) Squaring on both sides of (3.2) and for order one, we have, MS E(tp)=¯ Y2Ehe2 0+α2θ2e2 1 −2αθe0e1+...i(3.4) A New Class of Ratio Estimators for Population Mean.... 263 Putting values of different expectation, we get the MSE of tp, as, MS E(tp)=λ¯ Y2hC2 y+α2θ2C2 x−2αθCyxi(3.5) Differentiating MS E(tp) with respect to αand putting it equal to zero, we get, ∂ ∂α MS E(tp)=0 which gives, α=Cyx θC2 x =αopt (3.6) Putting the value of αopt in (3.5), we get the minimum value of MS E(tp) as, MS Emin(tp)=λ¯ Y2C2 y− C2 yx C2 x(3.7) 4. Efficiency Comparison The estimator tpis better than tRunder the condition if, MS Emin(tp)−MS E(tR)>0, or, C2 y− C2 yx C2 x−hR2C2 x−2RCyxi>0 The tpis more efficient than[3]t1if, MS Emin(tp)−MS E(t1)>0, or, C2 y− C2 yx C2 x−hR2 1C2 x−2R1Cyxi>0 The tpis better than Singh et al. [5] estimator t2under the condition if, MS Emin(tp)−MS E(t2)>0, or, C2 y− C2 yx C2 x−hR2 2C2 x−2R2Cyxi>0 The tpis more efficient than Upadhyaya and Singh [6] estimator ti,i=3,4 , if, MS Emin(tp)−MS E(ti)>0,i=3,4 or, C2 y− C2 yx C2 x−hR2 iC2 x−2RiCyxi>0 The tpis better than Zaman et al. [29] estimator ti,i=1,2,3, under the condition if, 264G Kumar et al. MS Emin(tp)−MS E(ti)>0,i=1,2,3 or, C2 y− C2 yx C2 x−hR2 5(i)C2 x−2R5(i)Cyxi>0 5. Numerical Study To verify the theoretical results, we have considered the following population, given in Kadilar and Cingi [8]. The parameters of the above population are presented in Table-1. The MSE and Percentage Relative Efficiency (PRE) of the estimators with Table 1. Parameters of the population N=80 n=20 ¯ Y=51.8264 ¯ X=11.2646 ρ=0.9413 Sy=18.3569 Cx=0.7500 Sx=8.4542 Cy=0.3542 β2=2.866 β1=1.05 T M =9.318 respect to tRare presented in Table-2. The graphs and MSE and PRE are presented in Table 2. MSE and PRE of the estimators S. No. Estimator MSE PRE 1. tr2565510.488 100.00 2. t12551970.93 100.53 3. t22554353.47 100.44 4. t32551970.931 100.53 5. t42552963.345 100.49 6. t5(1) 2544185.217 100.84 7. t5(2) 2542989.056 100.89 8. t5(3) 2542988.851 100.89 9. topt 1458856.246 175.86 Figure-1and Figure-2respectively. 6. Results For better population mean estimate, we presented a generalized class of ratio estimators in this work. To a first-order approximation, the bias and MSE are obtained. Efficiency requirements are determined after a theoretical comparison of the introduced estimators’ performance with that of the current estimators. A real dataset is used to validate these theoretical conclusions. It is evident from Table-2 that the MSE of the estimators in competition lie in the interval [2542988.851, 2565510.488] while the MSE of introduced estimator is 1458856.246 and the PRE of competing estimators lie in the interval [100.00, 100.89] while the PRE of the proposed estimator is 175.86. These results have also been shown in the form of graphs in Figure-1 and Figure-2 respectively. A New Class of Ratio Estimators for Population Mean.... 265 Figure 1. MSE of different estimators Figure 2. PRE of the estimators with respect to tR 7. Conclusion The suggested estimator is the most efficient of the compared estimators, as indicated in Table 2, since it obtains the lowest MSE and the highest PRE. As a result, the introduced estimator can be used successfully for precise estimation in a variety of applications. Further it is also to mention that in future direction, some more modified estimators with greater efficiency may be created and may be applied in various application areas. For instance, the average production may be estimated having information on the area, the factory average production may be estimated using units as auxiliary information, the current year average may be estimated by using previous year information etc. . 266G Kumar et al. Acknowledgement The authors are thankful to the learned referee for valuable suggestions, leading to improvement of the paper. References [1] Cochran, W.G., Sampling Techniques, Third Edition, Wiley Eastern Limited. (1940). [2] Srivastava, S.K., An estimator using auxiliary information in sample surveys, Calcutta Statistical Association Bulletin, 16, (1967), 121-132. [3] Sisodia, B.S. and Dwivedi, V.K., A modified ratio estimator using coefficient of variation of auxiliary variable, Journal of Indian Society Agricultural Statistics, 33, (1981), 13–18. [4] Rao, T. J., On certain methods of improving ratio and regression estimators, Communication in Statistics Theory and Method, 20, (1991), 3325-3340. [5] Singh, H.P. 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Gagan Kumar, Department of Economics, Govt. PG College Musafirkhana, Amethi 227813, India e-mail: gkb[email protected] Dushyant Tyagi, Department of Economics, Rajdhani College (University of Delhi), New Delhi 110015, India e-mail: [email protected] Gautam Gupta, Department of Sociology, Dr. Ambedkar Govt. PG College Unchahar, Raibareli 229406, India, e-mail: [email protected] Surendra Kumar, Department of Mathematics, Mahamaya Govt. Degree College Mahona, Lucknow 226203 India e-mail: [email protected]