Laws of the iterated logarithm for sample moments and applications. (English) Zbl 1130.62019
Summary: 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, Student \(t\)-type statistics, etc. 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 coefficients. 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.
References:
| [1] | Agresti A, Categorical Data Analysis, Wiley (1990) |
| [2] | Chung KL, A Course in Probability Theory (1974) |
| [3] | DOI: 10.2307/2371287 · Zbl 0024.15802 · doi:10.2307/2371287 |
| [4] | Hoffmann-Jorgensen J, Probability with a View Toward Statistics 2 (1994) |
| [5] | Kotz S, Process Capability Indices, Chapman and Hall (1993) |
| [6] | Kotz S, Process Capability Indices in Theory and Practice, Arnold (1998) |
| [7] | Lai TL, Ann. Stat. 5 pp 866– (1977) · Zbl 0368.62011 · doi:10.1214/aos/1176343943 |
| [8] | DOI: 10.1214/aoms/1177696786 · Zbl 0239.62025 · doi:10.1214/aoms/1177696786 |
| [9] | Robbins H, Bull. Inst. Math. Acad. Sinica 1 pp 93– (1973) |
| [10] | DOI: 10.1214/aos/1176342704 · Zbl 0318.62069 · doi:10.1214/aos/1176342704 |
| [11] | DOI: 10.2307/2685263 · doi:10.2307/2685263 |
| [12] | DOI: 10.2307/2684692 · doi:10.2307/2684692 |
| [13] | Serfling RJ, Approximation Theorems of Mathematical Statistics, John Wiley and Sons (1980) · Zbl 0538.62002 · doi:10.1002/9780470316481 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
