A Generalized Multivariate Ratio and Regression Type Estimator for Population Mean Using A Linear Combination of Two Auxiliary Variables

In this paper, we propose generalized multivariate ratio and regression type estimators of a finite population mean using a linear combination of two auxiliary variables and obtain the expressions for biases and mean square errors for the proposed estimators. The conditions under which the proposed estimators are more efficient then the relevant estimators have been obtained. An empirical study has been done for the support of the problem.


Introduction
An important objective in any statistical estimation procedure is to obtain the estimators of parameters of interest with more precision.It is also well known that incorporation of more information in the estimation procedure yields better estimators, provided the information is valid and proper.Information on variables correlated with the study variable is known as auxiliary information that may be utilized either at preparation stage or at design stage or at the estimation stage to arrive at improved estimator compared to those, not utilizing such auxiliary information.For example, if the study variable is the quantity of fruits produced in each plot, then auxiliary variable can be the area of the each plot or the production of fruit in the same plot in previous year, another auxiliary variable can be the number of workers in each plot.Use of such auxiliary information is made through the ratio and regression method of estimation to obtain improved estimators of population mean.The ratio method of estimation uses the auxiliary information to improve the precision which results in improved estimators when the regression of the study variable on the auxiliary variable is linear and passes through origin.When the regression of on is linear, it is not necessary that the line should always pass through origin.Under such condition, it is more appropriate to use the regression type estimators.The use of auxiliary information in sample surveys is widely studied in the books written by Cochran (1977) and Sukhatme, Sukhatme and Asok (1984).Further, the use of supplementary information provided by auxiliary variables in survey sampling was discussed by several authors (Upadhyaya et. al, 1999;Kadilar et. al, 2005Kadilar et. al, , 2006Kadilar et. al, , 2007Kadilar et. al, , 2009;;Bacanli et. al, 2008;Gupta et. al, 2007;Al-Omari et. al, 2009;Tailor et. al, 2011;Khare et. al, 2011Khare et. al, , 2015) ) In this paper, we propose generalized multivariate ratio and regression type estimators using a linear combination of two auxiliary variables.The expressions for biases and mean square errors of the proposed estimators are obtained and a comparison of the proposed estimators has been made with the relevant estimators.

Materials and Methods
The classical ratio and regression estimators for the population mean ̅ of the study variables using an auxiliary variable whose population mean ̅ is known are given by and where, ̅ ∑ , ̅ ∑ , is the regression coefficient of on , is the number of the units in the sample (Cochran 1977), and are the population mean squares of and respectively.is the population covariance between and (Cochran, 1977).The bias and mean square errors of the are given by where, is an arbitrary constant.Lu J (2013) proposed the multivariate chain ratio type estimator and regression type estimator using a linear combination of two auxiliary variables which is given as follows: and where, is an arbitrary constant, ̅ ̅ ̅ and ̅ ̅ ̅ , , The mean square errors and the bias of the estimators and are given by where, Differentiating ( 9) with respect to and separately, equating them to zero and solving the equations, we get Similarly, differentiating (11) with respect to , equating it to zero and solving the equation, we get (15) Putting the optimum value in the and , we have Now putting the optimum values and in equation ( 9) and ( 11) respectively, we have , Serals (1967) defined an estimator ̅ ̅ , which has ̅ ̅ for the value of ( ) ; where The ̅ is given by where, and is the coefficient of variation of

The Proposed Estimators and Their Mean Square Error (MSE)
We propose a generalized multivariate ratio type estimator and regression type estimator ) for the population mean ̅ in simple random sampling using a linear combination of two auxiliary variables, which are given as follows and where, is an arbitrary constant, ̅ ̅ ̅ and , and are weights that satisfy the condition and The expressions for the MSE of the proposed estimators up to the term of order are given as follows: where, Differentiating ( 23) with respect to and separately, equating them to zero and solving the equations, we get Similarly, Differentiating (24) with respect to , and equating it to zero and solving the equation, we get (28) Putting the optimum value in the and , we have, Now putting the optimum values and in equation ( 23) and ( 24) respectively.
Many estimators turn out as special cases of which are given as follows: (i) Putting in (20) then the proposed estimator reduces to Searls' estimator ̅ .

(ii)
If =1 and then the proposed estimator reduces to

Comparison of the Proposed Estimators with the Relevant Estimators
On comparing the with ̅ and other relevant estimators, we get certain conditions under which the proposed estimators have less mean square errors than the relevant estimators and the conditions are as given below: Putting the value of in (44), we get Putting the value of in (48), we get (50) It has also been seen that the determinant of the matrix of second order derivative of with respect to is negative for the optimum values of The minimum value of { } for the optimum value of are given as follows: { } (51) Neglecting the term of order , we have { } (52)

Determination of the Sample Size for a Specified Variance
Another aspect for choosing the sample size so that the available resources are used in an effective way is to minimize the cost of the survey for a specified variance.Let be the variance of the estimator fixed in advance, then we have, The total cost apart from overhead cost is minimized by obtaining the optimum value of and for specified precision .For this purpose, we defined a function which is given as follows: ) where, is the Lagrange's multiplier.After differentiating with respect to and equating it to zero, we get, √ Putting the value of in (53), we get (56) Putting the value of in (55), we get (57) It has also been seen that the determinant of the matrix of second order derivative of with respect to is negative for the optimum values of which shows the solution for given by (55).We can obtain the value of for which the estimator { } attains the variance with expected cost given by IASSL ISSN 2424-6271 29

Empirical Study
The data on physical growth of 100 fish (Laengelmavesi on Lake, near Tampere, Finland) has been taken under study.Our goal is to estimate the average weight of fish and it is also well understood that incorporation of more information in the estimation procedure yields better estimators.So for the purpose, the study variable , auxiliary variable and the additional auxiliary variable are taken as follows: -weight (in g) of the fish, -Length from the nose to the beginning of the tail (in cm), -Length from the nose to the notch of the tail (in cm).Since the auxiliary variables are positively correlated with the weight of the fish, the information on these two auxiliary variables will yield better estimators.The values of parameters of the population are given as follows: ̅ ̅ ̅ From Table 1, we see that the proposed estimator has minimum mean square error than ̅ and other special cases of the estimators.The proposed estimator has minimum mean square error than ̅ and other special cases of the estimators.Also the proposed estimator has minimum mean square error than Further, we observe that the special estimator has minimum mean square error than the estimators and .Also the estimator has minimum mean square error than  From table 2, we observe that the proposed estimator is more efficient than ̅ and other special cases of the estimators.The proposed estimator is more efficient than ̅ and other special cases of the estimators.Also the proposed estimator has minimum mean square error than .Further, we observe that the special estimator has minimum mean square error than the estimators and .Also the estimator has minimum mean square error than  ISSN 2424-6271 IASSL The special estimator has minimum mean square error than the estimators and .Also the estimator has minimum mean square error than Further, we observe that when the sample size increases, the mean square errors of the estimators decrease.

Conclusions
In this work, we have proposed generalized multivariate chain ratio and regression type estimators with a linear combination of two auxiliary variables and using known coefficient of variation of study variable.The conditions under which the proposed estimators have minimum mean square errors are mentioned in the section 5.A generalization can be made with more than two variables also.Here, we conclude that the information on coefficient of variation of study variable is fruitful in increasing the precision of the estimators compared to those, not utilizing such information.However, these values depend on the population parameters where prior information can be used.In lack of prior information, the estimated values of the parameters based on sample values can be used.For the support of the problem, an empirical study as well as a Monte Carlo simulation study has been made.The results obtained from the Monte Carlo simulation study are found to be similar to the results based on the empirical study.On the basis of the empirical study, we observe that use of known coefficient of variation of the study variable in the proposed estimators for population mean is found to be more useful in increasing the precision of the proposed estimators with respect to the relevant estimators for the fixed cost .The total cost for the proposed estimators is also less than the relevant estimators for the specified variance

Figure2:
Figure2: MSE of different estimators for different value of .

Determination of the Sample Size for a Fixed Cost One
aspect for choosing the sample size so that the available resources are used in an effective way is to maximize the precision of the estimator for a fixed cost.Let be the total coat (fixed) of the survey apart from the overhead cost.The cost function can be written as , (44) where, is the cost per unit in the sample.Since will vary from sample to sample, so the expected cost can be written

Table 1 :
Relative efficiencies of different estimators ̅ (special case no.xii) 488.99 (161.67)Figure1: MSE of different estimators for different value of and Mean square error

Table 2 :
Relative efficiencies of different estimators ̅

Table 4 :
Simulated percent relative efficiencies (with respect to