On the Rate of Convergence in Normal Approximation and Large Deviation Probabilities for a Class of Statistics

Abstract

A new class of statistics is introduced to include, as special cases, unsigned linear rank statistics, signed linear rank statistics, linear combination of functions of order statistics, linear functions of concomitants of order statistics, and a rank combinatorial statistic. For this class, the rate of convergence to normality and Cramér’s type large deviation probabilities are investigated. Under the assumption that underlying observations are only independent, it is shown that this rate is $O(N^{ - {\delta / 2}} \log N)$ if the first derivative of the score generating function $\varphi $ satisfies Lipschitz’s condition of order $\delta $, $0 < \delta \leqq 1$, and it is $O(N^{ - {1 / 2}} )$ if $\varphi ''$ satisfies Lipschitz’s condition of order $\delta \geqq \frac{1}{2}$; and that Cramér’s large deviation theorem holds in the optimal range $0 < x \leqq \rho _N N^{{1 / 6}} $, $\rho _N = o(1)$. The results obtained provide new results and extend as well as generalize a number of known results obtained in this direction.