  A significant hurdle for analyzing large sample data is the lack of effective
statistical computing and inference methods. An emerging powerful approach for
analyzing large sample data is subsampling, by which one takes a random
subsample from the original full sample and uses it as a surrogate for
subsequent computation and estimation. In this paper, we study subsampling
methods under two scenarios: approximating the full sample ordinary
least-square (OLS) estimator and estimating the coefficients in linear
regression. We present two algorithms, weighted estimation algorithm and
unweighted estimation algorithm, and analyze asymptotic behaviors of their
resulting subsample estimators under general conditions. For the weighted
estimation algorithm, we propose a criterion for selecting the optimal sampling
probability by making use of the asymptotic results. On the basis of the
criterion, we provide two novel subsampling methods, the optimal subsampling
and the predictor- length subsampling methods. The predictor-length subsampling
method is based on the L2 norm of predictors rather than leverage scores. Its
computational cost is scalable. For unweighted estimation algorithm, we show
that its resulting subsample estimator is not consistent to the full sample OLS
estimator. However, it has better performance than the weighted estimation
algorithm for estimating the coefficients. Simulation studies and a real data
example are used to demonstrate the effectiveness of our proposed subsampling
methods.
