Tail bounds for empirically standardized sums

Exponential tail bounds for sums play an important role in statistics, but the example of t-statistic shows that exponential decay may be lost when population parameters need to estimated from data. However, it turns out if Studentizing is accompanied by estimating location parameter a suitable way, then regains behavior. Motivated this example, paper analyzes other ways empirically standardizing and establishes are sub-Gaussian or even closer normal following settings: Standardization with Studentized contrasts observations, standardization log likelihood ratio statistic observations family, via self-normalization symmetric distribution unknown center symmetry. The latter gives rise novel scan heteroscedastic data whose asymptotic power analyzed case where have log-concave distribution.