The jackknife is a popular method in survey sampling which is widely used for standard error estimation (e.g. Shao & Tu (1995) and Wolter (2007)). The applicability and theoretical properties of the jackknife variance estimation under unequal probability without-replacement sampling have been studied to a limited extent in the survey sampling literature. Some examples are given by Campbell (1980), Berger & Skinner (2005), Berger & Rao (2006) and Berger (2007). In these works the jackknife variance estimation is defined for functions of Hajek (1971) point estimators. In practice, however, it is well known that the performance of Hajek point estimation may be restricted by common situations found in multipurpose surveys, e.g. various weight-adjustments, small sample sizes that are result of high stratification, and in general, the use of survey weights which are poorly correlated with some variables of interest contained in the survey.
We propose a set of generalised jackknife variance estimators suitable for functions of Horvitz-Thompson (1952) point estimators. The proposed variance estimators naturally include finite population corrections and are defined for any without-replacement unequal probability sampling design. We explore briefly some of their limit properties. Also, a simulation study shows that the proposed methods may improve customary jackknifes which handle unequal probabilities (e.g. Campbell (1980)).
Berger, Y.G. (2007). A jackknife variance estimator for unistage stratified samples with unequal probabilities. Biometrika. 94, 4, 953-964.
Berger, Y.G. and Rao, J.N.K. (2006). Adjusted jackknife for imputation under unequal probability sampling without replacement. J. R. Statist. Soc. B. 68, 3, 531-547.
Berger, Y.G. and Skinner, C.J. (2005). A jackknife variance estimator for unequal probability sampling. J. R. Statist. Soc. B. 67, 1, 79-89.
Campbell, C. (1980). A different view of finite population estimation. Proc. Surv. Res. Meth. Sect. Am. Statist. Assoc. 319-324.
Hajek, J. (1971). Comment on a paper by Basu, D. in Foundations of Statistical Inference (Godambe, V.P. and Sprott, D.A. eds.). p. 236. Toronto: Holt, Rinehart and Winston.
Horvitz, D.G. and Thompson, D.J. (1952). A generalization of sampling without replacement from a finite universe. J. Am. Statist. Assoc. 47, 663-685.
Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. New York: Springer.
Wolter, K.M. (2007). Introduction to Variance Estimation. 2nd Ed. New York: Springer.
Keywords: pseudovalues; smooth function of totals; stratification; inclusion probabilities
Biography: He has a MSc in mathematics and he is doing a PhD in social statistics in the United Kingdom since 2008. His practical experience in survey sampling includes national-scale household and telephone surveys, polling and electoral rapid counts in Mexico. His professional background includes working for the Mexico's national elections office, the Mexico's presidential office and the Mexico's National Supreme Court of Justice.