On a new estimator for the variance of the ratio estimator with small sample corrections
Section 4. Conclusions

Table of contents

In this paper we have derived a new approximation formula for $MSE ({\hat{\bar{Y}}}_{R})$ of order $1 / n^{2}$ and a new formula for the bias of $s_{\hat{e}}^{2}$ of order $1 / n .$ The new estimator ${\hat{MSE}}_{2} ({\hat{\bar{Y}}}_{R})$ which takes into account the bias of $s_{\hat{e}}^{2}$ appears to be less biased than ${\hat{MSE}}_{0} ({\hat{\bar{Y}}}_{R}) = \hat{Var} ({\hat{\bar{Y}}}_{R})$ and ${\hat{MSE}}_{1} ({\hat{\bar{Y}}}_{R}) .$ For $n \geq 8,$ the bias of ${\hat{MSE}}_{2} ({\hat{\bar{Y}}}_{R})$ was in all cases of the simulation study less than 7% which is much better than the standard variance estimator; in most cases, this result even holds for $n \geq 4.$ For very small $n,$ ${\hat{MSE}}_{2} ({\hat{\bar{Y}}}_{R})$ may have a large negative bias if the population has a large coefficient of variation $C_{x} .$ From our simulation study this issue appears to be unlikely to occur as long as $C_{x} < 0 .8 .$

Finally, recall that for the populations in this simulation study, the bias of the ratio estimator itself was consistently small, even for $n = 4.$ In general, for other populations this bias may not be negligible. Cochran (1977, pages 174-175) discusses several alternative ratio estimators that are unbiased.

References

Cochran, W.G. (1977). Sampling Techniques. New York: John Wiley & Sons, Inc.

David, I.P., and Sukhatme, B.V. (1974). On the bias and mean square error of the ratio estimator. Journal of the American Statistical Association, 69, 464-466.

Kendall, M.G., and Stuart, A. (1958). The Advanced Theory of Statistics, Volume I. London: Charles Griffin and Company.

Kish, L. (1995). Survey Sampling. New York: John Wiley & Sons, Inc.

Koop, J.C. (1968). An exercise in ratio estimation. The American Statistician, 22, 29-30.

Nath, S.N. (1968). On product moments from a finite universe. Journal of the American Statistical Association, 63, 535-541.

Rao, J.N.K. (1969). Ratio and regression estimators. In New Developments in Survey Sampling, (Eds., N.L. Johnson and H. Smith), New York: John Wiley & Sons, Inc., 213-234.

Sukhatme, P.V. (1954). Sampling Theory of Surveys with Applications, Iowa State College Press, Ames, IA.

Tin, M. (1965). Comparison of some ratio estimators. Journal of the American Statistical Association, 60, 294-307.

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On a new estimator for the variance of the ratio estimator with small sample corrections
Section 4. Conclusions

References

On a new estimator for the variance of the ratio estimator with small sample corrections Section 4. Conclusions

References

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

On a new estimator for the variance of the ratio estimator with small sample corrections
Section 4. Conclusions