Estimation of Population Mean Using Known Median and Co-Efficent of Skewness

The present paper deals with two modified ratio estimators for estimation of population mean of the study variable using the linear combination of the known population values of the Median and the Co-efficient of Skewness of the auxiliary variab le. The biases and the mean squared errors of the proposed estimators are derived and are compared with that of existing modified ratio estimators for certain natural populations. Further we have also derived the conditions for which the proposed estimators perform better than the existing modified rat io estimators. From the empirical study it is also observed that the proposed modified ratio estimators perform better than the existing modified ratio estimators.


Introduction
The sampling theory describes a wide variety of techniques for using au xiliary information to obtain mo re efficient estimators like Ratio, Product and Regression estimators for the estimation of the mean of the study variable Y. Rat io estimators, imp roves the precision of estimate of the population mean of a study variable by using prior information on au xiliary variable X which is positively correlated with the study variable Y. Over the years the ratio method of estimation has been extensively used because of its intuitive appeal and the computational simp licity. When the population parameters of the auxiliary variable X such as Population Mean, Co-efficient of Variation, Co-efficient of Kurtosis, Co-efficient of Skewness, Median are known, a number of mod ified estimators such as modified rat io estimators, mod ified product estimators and modified linear regression estimators are proposed in the literature. Before discussing further about the modified rat io estimators and the proposed modified ratio estimators the notations to be used in this paper are described below: · N − Population size · n − Sa mp le size · f = n/N, Samp ling fraction · Y − Study variable · X − Auxiliary variable · X � , Y � − Population means · x � , y � − Sample means · S X , S y − Population standard deviations · C X , C y − Co-efficient of variat ions · ρ − Co-efficient of correlation ( N−1 )( N−2 ) S 3 , Co -efficient of skewness of the auxiliary variab le , Co-efficient of kurtosis of the auxiliary variable · M d −Median of the auxiliary variab le · B ( . ) − Bias of the estimator · MSE ( . ) − Mean squared error of the estimator The Ratio estimator for estimating the population mean Y � of the study variable Y is defined as The lists of modified ratio estimators together with their biases, mean squared errors and constants available in the literature are classified into two classes namely Class 1, Class 2 and are given respectively in Table 1 and Table 2 in the Appendix.
It is to be noted that "the existing mod ified ratio estimators" means the list of modified ratio estimators to be considered in this paper unless otherwise stated. It does not mean to the entire list of mod ified ratio estimators available in the literature. For a mo re detailed discussion on the ratio estimator and its modifications one may refer to Cochran [1], Kadilar and Cingi [2,3], Koyuncu and Kadilar [4], Murthy [5], Prasad [6], Rao [7], Singh [9], Singh and Tailor [10,12], Singh et.al [11], Sisodia and Dwivedi [13], Subraman i and Ku marapandiyan [14,15], Upadhyaya and Singh [16], Yan and Tian [17] and the references cited there in. Co-Efficent of Skewness The modified ratio estimators given in Tab le 1 and Table  2 are b iased but have minimu m mean squared errors compared to the classical ratio estimator. The list of estimators given in Table 1 and Table 2 uses the known values of the parameters like X � , C x , β 1 , β 2 , ρ, M d and their linear co mbinations. Recently Yan and Tian [17] have used Coefficient of Skewness for the estimation of population meanHowever, it seems, no attempt is made to use the linear co mb ination of known values of the Median and Co-efficient of Skewness of the auxiliary variable to improve the ratio estimator. The points discussed above have motivated us to introduce two modified rat io estimators using the linear co mb ination of the known values of Median and Co-efficient of Skewness of the auxiliary variable. When the population med ian and coefficient of skewness are unknown the proposed estimators can be modified using their respective estimates i.e samp le med ian, and sample coefficient of skewness obtained fro m the sample. The proposed estimators can be applicable in the following practical situation.
1. A national park is partit ioned into N units.
• Y = the number of animals in the i th unit • X = the size of the i th unit 2. A certain city has N bookstores.
• Y = the sales of a given book title at the i th bookstore • X = the size of the i th bookstore 3. A forest that has N trees.
• Y = the volu me of the tree • X = the diameter o f the tree

Proposed Modified Ratio Estimators
In this section, we have suggested two mod ified ratio estimators using the linear co mb ination of Median and Co-efficient of Skewness of the auxiliary variable. The proposed modified ratio estimators for estimating the population mean Y � together with the first degree of approximation, the biases and mean squared errors and the constants are given below:

Efficiency Comparison
For want of space; for the sake of convenience to the readers and for the ease of comparisons, the modified rat io estimators given in Tab le 1, Table 2 are represented into two classes as given below. Further it is to be noted that the proposed estimator Y � � p 1 is compared with the modified ratio estimators listed in Class 1 whereas the proposed estimator Y � � p 2 is co mpared with the modified rat io estimators listed in Class 2.
Class 1:The biases, the mean squared errors and the constants of the modified ratio type estimators Y � � 1 to Y � � 9 listed in the Table 1 are represented in a single class (say, Class 1), wh ich will be very much useful for comparing with that of proposed modified rat io estimators and are given below:

Class 2:
The biases, the mean squared errors and the constants of the 11 modified ratio estimators Y � � 1 to Y � � 11 listed in the Table 2 are represented in a single class (say, Class 2), wh ich will be very much useful for comparing with that of proposed modified rat io estimators and are given below: As derived earlier in section 2, the biases, the mean squared errors and the constants of two proposed modified ratio estimators are given below: Fro m the expressions given in (4) and (6) we have derived the conditions for which the proposed estimator Y � � p 1 is more efficient than the existing modified ratio estimators given in Class 1, Y � � i ; i = 1, 2, 3, . . . , 9 and are given below.
; i = 1, 2, 3, . . . , 9(8) Fro m the expressions given in (5) and (7) we have derived the conditions for which the proposed estimator Y � � p 2 is more efficient than the existing modified ratio estimators given in Class 2, Y � � j ; j = 1, 2, 3, . . . , 11 and are given below: 11 (9)   The performances of the proposed modified ratio estimators are assessed with that of existing modified rat io estimators listed in Table 1 and Tab le 2 for certain natural populations. In this connection, we have considered two natural populations for the assessment of the performances of the proposed modified ratio estimators with that of existing modified ratio estimators. The population 1 is taken fro m Singh and Chaudhary [8] g iven in page 177 and population 2 is taken fro m Murthy [5] g iven in page 228. The population parameters and the constants computed from the above populations are given below:

Numerical Study
The constants of the existing and proposed modified ratio estimators for the above populations are given in the Table 4 and Table 5: The biases of the existing and proposed modified ratio estimators for the above populations are given in the Table 6  and Table 7: The mean squared errors of the existing and proposed modified ratio estimators for the above populations are given in the Table 8 and Table 9:  Table 6 and Table 7, it is observed that the bias of the proposed modified ratio estimator Y � � p 1 is less than the biases of the existing modified ratio estimators Y � � i ; i = 1, 2, 3, . . . , 9 given in Class 1 and the bias of the proposed modified ratio estimator Y � � p 2 is less than the biases of the existing modified rat io estimators Y � � j ; j = 1, 2, 3, . . . , 11 given in Class 2. Similarly fro m the values of Table 8 and Table 9, it is observed that the mean squared error of the proposed modified rat io estimator Y � � p 1 is less than the mean squared errors of the existing modified rat io estimators Y � � i ; i = 1, 2, 3, . . . , 9 given in Class 1 and the mean squared error of the proposed modified rat io estimator Y � � p 2 is less than the mean squared erro rs of the e xisting modified rat io estimators Y � � j ; j = 1, 2, 3, . . . , 11 given in Class 2.

Conclusions
In this paper we have proposed two modified ratio estimators using linear co mb ination of Median and Co-efficient of Skewness of the auxiliary variable. The biases and mean squared errors of the proposed estimators are obtained and compared with that of existing modified ratio estimators. Further we have derived the conditions for which the proposed estimators are more efficient than the existing mod ified ratio estimators. We have also assessed the performances of the proposed estimators for so me known populations and observed that the biases and mean squared errors of the proposed estimators are less than the biases and mean squared errors of the existing modified ratio estimators.
Hence we strongly reco mmend the proposed modified estimators over the existing modified ratio estimators for the use of practical applicat ions.