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Register a new userOptimal Distributed Subsampling for Maximum Quasi-Likelihood Estimators with Massive Data
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Nonuniform subsampling methods are effective to reduce computational burden and maintain estimation efficiency for massive data. Existing methods mostly focus on subsampling with replacement due to its high computational efficiency. If the data volume is so large that nonuniform subsampling probabilities cannot be calculated all at once, then subsampling with replacement is infeasible to implement. This paper solves this problem using Poisson subsampling. We first derive optimal Poisson subsampling probabilities in the context of quasi-likelihood estimation under the A- and L-optimality criteria. For a practically implementable algorithm with approximated optimal subsampling probabilities, we establish the consistency and asymptotic normality of the resultant estimators. To deal with the situation that the full data are stored in different blocks or at multiple locations, we develop a distributed subsampling framework, in which statistics are computed simultaneously on smaller partitions of the full data. Asymptotic properties of the resultant aggregated estimator are investigated. We illustrate and evaluate the proposed strategies through numerical experiments on simulated and real data sets.
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To fast approximate maximum likelihood estimators with massive data, this paper studies the Optimal Subsampling Method under the A-optimality Criterion (OSMAC) for generalized linear models. The consistency and asymptotic normality of the estimator from a general subsampling algorithm are established, and optimal subsampling probabilities under the A- and L-optimality criteria are derived. Furthermore, using Frobenius norm matrix concentration inequalities, finite sample properties of the subsample estimator based on optimal subsampling probabilities are also derived. Since the optimal subsampling probabilities depend on the full data estimate, an adaptive two-step algorithm is developed. Asymptotic normality and optimality of the estimator from this adaptive algorithm are established. The proposed methods are illustrated and evaluated through numerical experiments on simulated and real datasets.
To analyse a very large data set containing lengthy variables, we adopt a sequential estimation idea and propose a parallel divide-and-conquer method. We conduct several conventional sequential estimation procedures separately, and properly integrate their results while maintaining the desired statistical properties. Additionally, using a criterion from the statistical experiment design, we adopt an adaptive sample selection, together with an adaptive shrinkage estimation method, to simultaneously accelerate the estimation procedure and identify the effective variables. We confirm the cogency of our methods through theoretical justifications and numerical results derived from synthesized data sets. We then apply the proposed method to three real data sets, including those pertaining to appliance energy use and particulate matter concentration.
Massive data bring the big challenges of memory and computation for analysis. These challenges can be tackled by taking subsamples from the full data as a surrogate. For functional data, it is common to collect multiple measurements over their domains, which require even more memory and computation time when the sample size is large. The computation would be much more intensive when statistical inference is required through bootstrap samples. To the best of our knowledge, this article is the first attempt to study the subsampling method for the functional linear model. We propose an optimal subsampling method based on the functional L-optimality criterion. When the response is a discrete or categorical variable, we further extend our proposed functional L-optimality subsampling (FLoS) method to the functional generalized linear model. We establish the asymptotic properties of the estimators by the FLoS method. The finite sample performance of our proposed FLoS method is investigated by extensive simulation studies. The FLoS method is further demonstrated by analyzing two large-scale datasets: the global climate data and the kidney transplant data. The analysis results on these data show that the FLoS method is much better than the uniform subsampling approach and can well approximate the results based on the full data while dramatically reducing the computation time and memory.
Subsampling is a computationally effective approach to extract information from massive data sets when computing resources are limited. After a subsample is taken from the full data, most available methods use an inverse probability weighted objective function to estimate the model parameters. This type of weighted estimator does not fully utilize information in the selected subsample. In this paper, we propose 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 asymptotic variance covariance matrix is the smallest among a class of asymptotically unbiased estimators, including the inverse probability 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.
This paper develops a bias correction scheme for a multivariate heteroskedastic errors-in-variables model. The applicability of this model is justified in areas such as astrophysics, epidemiology and analytical chemistry, where the variables are subject to measurement errors and the variances vary with the observations. We conduct Monte Carlo simulations to investigate the performance of the corrected estimators. The numerical results show that the bias correction scheme yields nearly unbiased estimates. We also give an application to a real data set.
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