Improvement on the Non-response in the Population Ratio of Mean for Current Occasion in Sampling on Two Occasions

In this article, we attempt the problem of estimation of the population ratio of mean in mail surveys. This problem is conducted for current occasion in the context of sampling on two occasions when there is nonresponse (i) on both occasions, (ii) only on the first occasion and (iii) only on the second occasion. We obtain the gain in efficiency of all the estimators over the direct estimate using no information gathered on the first occasion. We derive the sample sizes and the saving in cost for all the estimators, which have the same precision than the direct estimate using no information gathered on the first occasion. An empirical study that allows us to investigate the performance of the proposed strategy is carried out.


Introduction
, Jessen (1942), Patterson (1950), Raj (1968), Tikkiwal (1951) and Yates (1949) contributed towards the development of the theory of Unbiased estimation of mean of characteristics in Successive sampling. In many practical situations the Estimate of the population Ratio and Product of two characters for the most recent occasion may be of considerable interest. The theory of estimation of the population Ratio of two characters over two occasions has been considered by Artes and Garcia (2001), Garcia and Artes (2002), Okafor (1992), Okafor and Arnab (1987), Rao (1957) and Rao and Pereira (1968) among others. _____________________________________ Further, Garcia (2008) presented some sampling strategies for estimating, by a Linear Estimate, the population Product of two characters over two occasions. Hansen and Hurwitz (1946) suggested a Technique for handling the Non-response in mail surveys. These surveys have the advantage that the data can be collected in a relatively inexpensive way. Okafor (2001)  Also, Cochran (1977) and Okafor and Lee (2000) extended the HH Technique to the case when the information on the characteristic under study is also available on auxiliary characteristic.
In this article, we develop the HH Technique to Estimate the population Ratio of mean for current occasion in the context of sampling on two occasions when there is Non-response (i) on both occasions, (ii) only on the first occasion and (iii) only on the second occasion. An empirical study that allows us to investigate the performance of the proposed strategy is carried out.

The Technique
Consider a finite population of N identifiable units. Let (xi; yi) be, for I = 1, 2, …, N, the values of the characteristic on the first and second occasions, respectively. We assume that the population can be divided into two classes, those who respond at the first attempt and those who not. Let, the sizes of these two classes be N 1 and N 2 , respectively. Let on the first occasion, schedules through mail are sent to 'n' units selected by simple random sampling. On the second occasion, a simple random sample of m = np units, for 0 < p < 1, is retained while an independent sample of u = nq = n-m units, for q = 1-p, is selected (unmatched with the first

The Non-Response in the Population Ratio of Mean for Current
Occasion in Sampling on Two Occasions _______________________________________________________________________________ 177 occasion). We assume that in the unmatched portion of the sample on two occasions, 'u 1 ' units respond and 'u 2 ' units do not. Similarly, in the matched portion 'm 1 ' units respond and 'm 2 ' units do not.
Let denotes the size of the subsample drawn from the Non-response class from the matched portion of the sample on the two occasions for collecting information through personal interview. Similarly, denote by the size of the sub-sample drawn from the Non-response class in the unmatched portion of the sample on the two occasions. Also, let , ; j=1, 2 and j = 1, 2 denote the population variance and population variance pertaining to the Nonresponse class, respectively. In addition, let ̅ , ̅ , ̅ and ̅ denote the Estimator for matched and unmatched portions of the sample on the first occasion, respectively. Let the corresponding Estimator for the second occasion be denoted by ̅ , ̅ , ̅ and ̅ Thus, have the following setup: x i (y i ), the variable x (y) on ith occasion, i = 1, 2.

Estimation of the Population Ratio of Mean for Current Occasion in the Presence of Non-response on both
where ̂ is the usual Estimator of the Ratio of mean for the current occasion in the context of sampling on two occasions when there is complete response, that is ̂ ̂ ̂ ̂ ̂ Similarly an Estimate of the first occasion is, Its variance is,

̂ ̅
Equating to zero the derivative of ̂ with respect to , we find that the variance ̂ will have its minimum value if we choose: However, if only the Estimate using information gathered on the second occasion is considered, the Estimator of the population Ratio is, and its variance is,

Estimation of the Population Ratio of Mean for the Current Occasion in the Presence of Non-response on the First Occasion:
When there is Non-response only on the first occasion, the Minimum Variance Linear Unbiased Estimator for the population Ratio on current occasion can be obtained as follows: Thus the Estimator ̂ turns out to be

The Non-Response in the Population Ratio of Mean for Current Occasion in Sampling on Two Occasions
where ̂ is the usual Estimator of the Ratio of mean for the current occasion in the context of sampling on two occasions when there is complete Response, that is, The optimum fraction to be unmatched is given by and thus the minimum variance of ̂ is,

Estimation of the Population Ratio of Mean for the Current Occasion in the Presence of Non-response on the Second Occasion:
When there is Non-response only on the second occasion, the Minimum Variance Linear Unbiased Estimator for the population Ratio on current occasion can be obtained as follows:

Comparing Estimators in terms of Survey Cost
We give some ideas about how saving in cost through mail surveys in the context of Successive sampling on two occasions for different assumed values of , , , , , , , and . Let , , , , and (see Choudhary et al., 2004) where , , and denote the cost per unit for mailing a questionnaire, processing the results from the first attempt respondents, and collecting data through personal interview, respectively. In addition, is the total cost incurred for collecting the data by personal interview from the whole sample, i.e., when there is no Non-Response. The Cost Function in this case is given by (assuming the cost incurred on data collection for the matched and unmatched portion of the sample are same and cost incurred on the data collection on both occasions is same). (4.1) Substituting the values of and in equation (4.1), the total cost work out to be 4500. By equating the variances ̂ , ̂ , and ̂ , respectively, to ̂ and using the assumed values of different parameters, the values of the sample size for the three cases and the corresponding expected cost of survey were determined with respect of ̂ , ̂ , and ̂ . The sample sizes associated with the three Estimators which provide equal precision to the Estimator ̂ are denoted by , and . The results of this exercise are presented in Tables 3-4. From these tables, we obtain the following conclusions:

Conclusions
In this paper, we have used the HH Technique for estimating the population Ratio of mean in mail surveys. This problem is conducted for current occasion in the context of sampling on two occasions when there is Non-Response (i) on both occasions, (ii) only on the first occasion and (iii) only on the second occasion. The results obtained reveals that the loss in precision is maximum for the estimation of the Ratio of mean when there is Non-Response only on the second occasion, whereas it is least for the estimation of the Ratio of mean when there is Non-Response on both occasions and when there is Non-Response only on the first occasion. Also, we derive the sample sizes and the saving in cost for all the Estimators, which have the same precision than the Estimator of the population Ratio of mean when there is no non-Response. In the majority of the cases the sample sizes and the saving in cost is maximum for the estimation of the Ratio of mean when there is Non-Response on both occasions, whereas it is least for the estimation of the Ratio of mean when there is Non-Response only on the first occasion and when there is Non-Response only on the second occasion.