Laws of the iterated logarithm for sample moments and applications

Many important statistics are functions of sample moments, for instance, sample skewness, sample kurtosis, sample odds ratio, sample correlation coefficient, sample quantiles, sample process capability indices [Kotz, S. and Johnson, N. L. (1993). Process Capability Indices. Chapman and Hall; Kotz, S. and Lovelace, C. R. (1998). Process Capability Indices in Theory and Practice. Arnold.], student t-type statistics, etc. In this article, we first derive the laws of the iterated logarithm for sample moments and then the laws of the iterated logarithm for sample skewness, sample kurtosis, sample odds ratio, and sample correlation coefficient. The other functions of sample moments can be dealt with without difficulty. The results provide the basis for concepts of 100% confidence intervals and tests of power 1 in statistical inferences [Robbins, H. (1970). Statistical methods related to the law of the iterated logarithm. Ann. Math. Stat., 41, 1397-1409; Robbins, H. and Siegmund, D. (1973). Statistical tests of power one and the integral representation of solutions of certain partial differential equations. Bull. Inst. Math. Acad. Sinica, 1, 93-120; Robbins, H. and Siegmund, D. (1974). The expected sample size of some test of power one. Ann. Stat., 2, 415-436; Lai, T. L. (1977). Power one tests based on sample sums. Ann. Stat., 5, 866-880.].