Two Phase Sampling Exponential Type Estimators for Ratio and Product of two Population Means in the Presence of Non-response

— In this paper, two phase sampling exponential type estimators for ratio and product of two population means in the presence of non-response have been proposed and the expressions for the mean suare error of the proposed estimators for ratio and product of two population means in case of fixed sample sizes and also in case of fixed cost, are obtained. The expressions for optimum values of sample sizes are obtained in case of fixed cost and also in case of specified variance. The proposed estimators have been found to be more efficient than the relevant estimators for the fixed values of sample sizes, under the specified conditions. The proposed estimators are also more efficient than the relevant estimators in case of the fixed cost and have less total cost in comparison to the cost incurred in case of relevant estimators for the specified variance. Empirical studies have been given in support of the problem under investigation.


I. INTRODUCTION
Estimation of ratio and product of two population means is widely used in the field of agriculture, socio-economics and medical sciences. A lot of research works for estimating the ratio and product of two population means using one auxiliary character, have been done by Singh [1,2,3], Shah and Shah [4], Tripathi [5], Singh [6], Singh [7], Birader and Singh [8] and Upadhayay et al. [9]. For estimating the population mean by using sample values sometimes the information on all units selected in the sample is not collected due to the problem of non-response which occurs due to lack of interest, not at home, refusal and lack of the knowledge regarding to the surveys. Hansen and Hurwitz [10] first suggested a technique of sub sampling from nonrespondent to deal with the problem of non-response. Further by using the known population mean of auxiliary character, various ratio, product and regression type estimators for population mean in the presence of non-response have been proposed by Rao [11,12], Khare and Srivastava [13,14], Singh et al. [15], Kumar and Bhougal [16] and Kumar and Kumar [17]. Sometimes the population mean of auxiliary character is not known. In this situation, two phase sampling ratio, product and regression type estimators have been proposed by Khare and Srivastava [18,19], Singh and Kumar [20] and Khare et al. [21]. The research works for estimating the ratio of two population means using auxiliary characters in the presence of non-response have also been done by Khare and Pandey [22] and Khare and Sinha [23,24]. In this paper, we have proposed two phase sampling exponential type estimators for ratio and product of two population means in the presence of non-response. The expressions for the mean suare error of the proposed estimators for ratio and product of two population means in case of fixed sample sizes are obtained. The conditions in which proposed estimators are more efficient in comprision to the relevant estimators are obtained. The expressions for the optimum values of sample sizes are also obtained in case of fixed cost and also in case of specified variance. Empirical studies are also given to show the performance of the proposed estimators in comprision to the relevant estimators. x for the entire population and for the non responding part of the population. The classical ratio and product estimators are useful only when the linear relationship between study character and auxiliary character is very strong. Sometimes the linear relationship between study character and auxiliary character is not very strong. In this situation, Singh and Vishwakarma [25] proposed the two phase sampling exponential ratio ) ( R t and exponential product ) ( P t estimators for the population mean which are given as follows:  [26] and Lakhre [27] studied Singh and Vishwakarma [25] ratio and product type exponential estimators in double sampling for stratification. Now using two phase sampling exponential product and exponential ratio estimators, we propose conventional two phase sampling exponential type estimators for ratio and product of two population means in the presence of nonresponse which are given as follows: Using two phase sampling exponential product and exponential ratio estimators, we propose the alternate two phase sampling exponential type estimators for ratio and product of two population means in the presence of nonresponse which are given as follows:

III. THE EXPRESSIONS FOR THE MEAN SQUARE ERROR (MSE) OF THE PROPOSED ESTIMATORS
In order to derive the expressions for the mean square error of the proposed estimators.
By using simple random sampling without replacement method of sampling, we have Using the large sample approximations, the expressions for the mean square error of the proposed estimators 1 , , (

IV. COMPARISONS OF THE PROPOSED ESTIMATORS WITH RELEVANT ESTIMATORS
The expected cost C to be incurred on the survey apart from overhead expenses, is given by

VII.1 Data Set I [Khare and Sinha [24]]
The present data belong to the data on growth of upper socio-economic group of 95 school going children of Varanasi under an ICMR study, Department of Pediatrics, y (the height of the children in cm) and 2 y (weight of the children in kg) and auxiliary character x (chest circumference of the children in cm) have been given as follows: The problem considered is to estimate the ratio between height and weight of the male children aged 6-7 years using chest circumference as the auxiliary character.

VIII. CONCLUSION
From tables 1, 2, 3, we conclude that when linear relationship between study character and auxiliary character is not very strong, the proposed estimators for ratio of two population means in the presence of non-response are more efficient in comprision to the usual estimator in case of the fixed sample sizes and also in case of the fixed cost. The proposed estimators have less total cost in comprision to the usual estimator in case of the specified variance. From tables 4, 5, 6, we conclude that when linear relationship between study character and auxiliary character is not very strong, the proposed estimators for product of two population means in the presence of non-response are more efficient in comprision to the usual estimator in case of the fixed sample sizes and also in case of the fixed cost. The proposed estimators have less total cost in comprision to the usual estimator in case of the specified variance.