| Abstrakt: |
In estimation problems, researchers generally use estimators, which are symmetric in observations. Given any function f (x1, x2, ... xn) in n variables with values in an abelian group, a symmetric function can be constructed by summing values of f (x1, x2, ... xn) over all permutations of the arguments. Similarly, an antisymmetric function can be constructed by summing over even permutations and subtracting the sum over odd permutations. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions f (x1, x2, ... xn). The only general case where f (x1, x2, ... xn) can be recovered if both its symmetrization and anti symmetrization are known, is when n = 2 and the abelian group admits a division by 2 (inverse of doubling); then f (x1, x2, ... xn) is equal to half the sum of its symmetrization and its anti symmetrization [Joseph et al. (2009)]. To study properties of some of these estimators, a special class of symmetric functions, called U-statistics, has been found useful, particularly in non-parametric statistical theory. Fraser (1957) has given a comprehensive discussion of U-Statistics. In this paper, we have reviewed some of the properties of U-statistics and have proved necessary and sufficient condition that a jackknife estimator reduces to the original estimator, if it is in the form of a U-statistic. It has been illustrated by taking an estimator of P(X < Y) when data are censored and if U is any censoring point, then by taking the degree of kernel as one, the jackknife estimator of a U-statistic based on a sample of size n reduces to the same estimator. These results will be very useful in non-parametric statistical theory. [ABSTRACT FROM AUTHOR] |
