Certain Monte Carlo Studies in Sample Surveys

Abstract

Sample survey statistician is generally faced with the problem as how he should make the best use of available information on some auxiliary variable which is highly correlated with the variable under study. Information on auxiliary variable could be made use of at three stages, viz, (I) for stratifying the sampling units (II) for selecting the sampling units and (III) at the estimation stage i.e., on using the information for ratio or regression method of estimation. The types of population which we generally come across in surveys for estimating the production or extent of cultivation of agricultural commodities generally follow a linear regression relationship on auxiliary character X which is such that the condition variance for a given value of X depends on X itself. It has been suggested (Rao, Hartley and Cochran) that the functional relationship between the conditional variance and X. Where X lies between 1and 2. In this thesis three sets of data horticulture crops have been examined with a view to finding out the goodness of fit of the above relationship. The examination of these data have revealed that the value of X lies between 1 to 2, so long as the sampling unit is fairly small like an orchard, However, when the sampling unit is taken as big as a village, the value is found to lie somewhere about 3. The main object of the present study was to find out that in the populations which follow the pattern as discussed above, whether the auxiliary information could be better made use of for stratifying the units in the population or for estimating the character under study through ratio or regression method of estimation. The study is carried out through Monte Carlo methods which envisage the simulation of populations which have the given pattern of above variance relationship for some reasonably fixed values of regression parameters and ten values of varying from 1.50 to 4.00. from each of the generated hypothetical populations consisting of 200 units, corresponding to a given value, 25 independent samples each of size 12units were selected. Each of these samples provided estimates of population total by using different methods of estimation along with the corresponding estimates of variances and gain in precision of the given method of estimation over simple mean estimator. A part from this, each of the generated populations also stratified into four equal strata and 25 samples each size 12 allocated equally to different strata were selected, with a view to comparing the stratified sample estimators with those based in ratio and regression methods of estimation. One estimate of the amount of bias in different parameters estimated through ratio and regression methods of estimation is provided by taking difference between the average value of estimates taken over 25 samples and the true value of the corresponding parameter. Similarly, the mean square of 25 samples values of the estimates of different parameters provides estimate of the true value of the variance of an estimate of a given parameter. The present study has revealed the following results. i. While comparing stratified sample estimator with ratio as well as regression estimator it has been found that for value of lamda between 1.50 to 3.00, use of auxiliary information for ratio as well as regression method of estimation will be more efficient as compared to its use for stratifying the units while for value of lamda between 3.00 to 4.00, the efficiency of stratification is more or less of the same order as that of ratio or regression method of estimation. ii. So far as the amount of bias in estimating the population total through ratio and regression estimators is concerned, the study has revealed that its value is appreciably small being less than 4 percent of the total for all values. However, it has been observed that the amount of bias in regression estimator is always less than that in ratio estimator for all values. iii. Comparing the performance of ratio and regression methods for estimating the population total, it has been found that the true variance as estimated by the mean square between the estimates of the total provided by 25 independent samples is consistently smaller in the case of ratio then that in the case of regression method of estimation. This in addition to the simplicity in the calculation suggests that ratio is superior to regression method of estimation for such populations as discussed in this thesis. iv. So far as estimate of variance of ratio and regression estimators is concerned, the study has revealed that the coefficient of variation in the estimate of variance varies from 23 percent to as much as 83 percent for values between 1.50 to 4.00. This fact suggests that precision of the existing variance estimator both in the case of ratio and regression methods is very poor. The similar conclusions have also been drawn for estimating the gain in precision of either ratio or regression estimator over simple mean estimator. v. While examining the difference between the variance obtained by using approximate formulas for evaluating the variance of population total through ratio and regression method of estimation and the corresponding true variance, it has been found that approximate formulas used for evaluating the variance of ratio estimator over evaluates the true variance for most of the values. The reverse is found to be true in the case of regression method of estimation. vi. For the population which have been generated in this thesis by taking the value of regression coefficient between Y and X equal to 30.00 and fixing the value of y in (7.1.1.) equal to 0.60, it has been found that the variance of ratio and regression estimators of the total is more or less doubled when the value of X is increased by 0.85. This fact suggests that although the present investigations have been carried out only for 10 values of X, yet one can infer the behavior of the variances of ratio and regression estimators for intermediate values of X also. vii. Efforts was also made to study the type of frequency distributions followed by estimates of parameters of population total and its variance based on stratified sample and ratio and regression methods of estimation. The investigation has revealed that in majority of cases, the frequency curves of various estimators were of Pearson Type I. However, in few cases the frequency curves were of Type IV and VI. On the basis of 25 samples, effort has also been made to estimate the parameters of these frequency distributions. It will be mentioned that the precision of these estimates is not very high, since the total sample size for estimating the parameters is very small i.e., 25 samples. However, these estimates do provide the basis for comparing various methods of using the auxiliary information.