Maximum sampled conditional likelihood estimation for informative  subsample

Subsampling is an effective approach to extract useful information from massive data sets when computing resources are limited. Existing investigations focus on developing better sampling procedures and deriving probabilities with higher estimation efficiency. After a subsample is taken from the full data, most available methods use an inverse probability weighted target function to define the estimator. This type of weighted estimator reduces the contributions of more informative data points, and thus it does not fully utilize information in the selected subsample. This paper focuses on parameter estimation with selected subsample, and proposes to use the maximum sampled conditional likelihood estimator (MSCLE) based on the sampled data. We established the asymptotic normality of the MSCLE, and prove that its variance covariance matrix reaches the lower bound of asymptotically unbiased estimators. Specifically, the MSCLE has a higher estimation efficiency than the weighted estimator. We further discuss the asymptotic results with the L-optimal subsampling probabilities, and illustrate the estimation procedure with generalized linear models. Numerical experiments are provided to evaluate the practical performance of the proposed method.