  Massive data analysis becomes increasingly prevalent, subsampling methods
like BLB (Bag of Little Bootstraps) serves as powerful tools for assessing the
quality of estimators for massive data. However, the performance of the
subsampling methods are highly influenced by the selection of tuning parameters
( e.g., the subset size, number of resamples per subset ). In this article we
develop a hyperparameter selection methodology, which can be used to select
tuning parameters for subsampling methods. Specifically, by a careful
theoretical analysis, we find an analytically simple and elegant relationship
between the asymptotic efficiency of various subsampling estimators and their
hyperparameters. This leads to an optimal choice of the hyperparameters. More
specifically, for an arbitrarily specified hyperparameter set, we can improve
it to be a new set of hyperparameters with no extra CPU time cost, but the
resulting estimator's statistical efficiency can be much improved. Both
simulation studies and real data analysis demonstrate the superior advantage of
our method.
