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In this paper, we propose improved estimation method for logistic regression based on subsamples taken according the optimal subsampling probabilities developed in Wang et al. (2018). Both asymptotic results and numerical results show that the new estimator has a higher estimation efficiency. We also develop a new algorithm based on Poisson subsampling, which does not require to approximate the optimal subsampling probabilities all at once. This is computationally advantageous when available random-access memory is not enough to hold the full data. Interestingly, asymptotic distributions also show that Poisson subsampling produces a more efficient estimator if the sampling ratio, the ratio of the subsample size to the full data sample size, does not converge to zero. We also obtain the unconditional asymptotic distribution for the estimator based on Poisson subsampling. Pilot estimators are required to calculate subsampling probabilities and to correct biases in un-weighted estimators; interestingly, even if pilot estimators are inconsistent, the proposed method still produce consistent and asymptotically normal estimators.
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Drift Estimation of Multiscale Diffusions Based on Filtered Data
Abstract We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.
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- Award ID(s):
- 1818977
- PAR ID:
- 10338468
- Date Published:
- Journal Name:
- Foundations of Computational Mathematics
- ISSN:
- 1615-3375
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10208-021-09541-9
