On the Estimation Of Population Mean Under Systematic Sampling Using Auxiliary Attributes

Systematic random sampling is the simplest type of sampling scheme, requires only one random start. It provides good results in some situations like; forest regions for assessing the volume of the timber etc. For details see Murthy (1967) and Cochran (1977). When supplementary information is available Swain (1964), Shukla (1971), Singh and Solanki (2012), and Singh et al. (2012) have developed some estimators for using available supplementary information. But none of these have paid their attention towards auxiliary attributes. So in our work, we utilize available auxiliary attributes.


INTRODUCTION
Systematic random sampling is the simplest type of sampling scheme, requires only one random start.It provides good results in some situations like; forest regions for assessing the volume of the timber etc.For details see Murthy (1967) and Cochran (1977).When supplementary information is available Swain (1964), Shukla (1971), Singh and Solanki (2012), and Singh et al. (2012) have developed some estimators for using available supplementary information.But none of these have paid their attention towards auxiliary attributes.So in our work, we utilize available auxiliary attributes.
In the theor y of sur vey sampling, supplementary information plays a vital role for increasing the efficiency of population parameters.A number of authors have developed estimators based on auxiliary information.Another way to enhance the efficiency of an estimator is to utilize auxiliary attributes.Naik and Gupta (1996)

Preliminaries and Adapted Estimators
Let l be the finite population having units 1 to N. Further, we consider N=nk, where n and k are positive whole numbers.Hence there will be k samples of size n.Let M be random variable having range 1 to k.The systematic random sample is then selected by the following random sequence as , .where a and b be any known population characteristics or 1.
Let we express in terms of and as follows where Now by simplifying, we get Let we take expectation on both sides and get the bias of as Now squaring both sides of as

Efficiency Comparison
In current section, we find the efficiency conditions for the proposed estimators by looking at the mean square error of the existing estimators as given below From the above mentioned conditions, we can say that proposed estimators are more efficient as compare to adapted estimators.

Numerical Illustration
The performance of proposed and existing estimators examined through two real data sets.

CONClUSION
We have proposed a class of estimators for using available auxiliary attributes under systematic random sampling scheme and obtained its bias and minimum MSE equations.All the adapted estimators are compared with proposed estimators using MSE.With the help of these comparisons, efficiency condition has been found where proposed estimators perform much better.The theoretical conditions and numerical illustrations show that proposed estimators are much better.Hence, it is advisable to use the proposed class of estimators.
, Singh et al. (2007), Abd-Elfattah et al. (2010), Solanki and Singh (2012) and Koyuncu (2012) introduced various estimators utilizing available auxiliary attributes under simple random sampling.Taking motivation from these we are going to propose a family of estimators under systematic random sampling scheme using available auxiliary attributes.
of the study variable and auxiliary attribute for (i=1,2,…,k) and (j=1,2,…,n).Note thatis the binary character so it can take only two possible values i.e., =1, if the ith unit of the population possesses attribute F, =0, otherwise.Let A= and a= denote the total number of units in the population and sample respectively, possessing an auxiliary attribute F. Hence, the corresponding sample and population proportions are and .S i m i l a r l y, intra-class correlation of P, is the intra-class correlation of Y and is the correlation between P and Y.The variance of the traditional sample mean is V FollowingNaik and Gupta (1996), we propose the usual ratio and product estimators utilizing available auxiliary attributes under systematic random sampling scheme, Table 1: Some members of proposed class et al. (2007), we propose the exponential ratio and product estimators utilizing available auxiliary attributes under systematic random sampling scheme, of Abd-Elfattah et al. (2010), we propose the family of estimators utilizing available auxiliary attributes under systematic random sampling scheme, The minimum MSE of these estimators ( ) is equal to the MSE of regression estimator i.e.MSE = Where Solanki and Singh (2013) developed the generalized estimator, given below , Where by putting (α=0, 1, -1) we get and respectively.The Proposed Family of Estimators Taking motivation from Abd-Elfattah et al. (2010), we propose the following family of mean square error of , i.e

Population 1 2
Data is taken from Mur thy(1967)    where Y=Volume of the timber and F=length .Descriptives of the population are N=176, Data is taken fromMurthy (1967) where Y=Volume of the timber and Y= Volume .Descriptives of the population are N=176,