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1. Optimal subsampling for quantile regression in big data

   https://doi.org/10.1093/biomet/asaa043  

   Wang, Haiying; Ma, Yanyuan (July 2020, Biometrika)

   null (Ed.)

   Summary We investigate optimal subsampling for quantile regression. We derive the asymptotic distribution of a general subsampling estimator and then derive two versions of optimal subsampling probabilities. One version minimizes the trace of the asymptotic variance-covariance matrix for a linearly transformed parameter estimator and the other minimizes that of the original parameter estimator. The former does not depend on the densities of the responses given covariates and is easy to implement. Algorithms based on optimal subsampling probabilities are proposed and asymptotic distributions, and the asymptotic optimality of the resulting estimators are established. Furthermore, we propose an iterative subsampling procedure based on the optimal subsampling probabilities in the linearly transformed parameter estimation which has great scalability to utilize available computational resources. In addition, this procedure yields standard errors for parameter estimators without estimating the densities of the responses given the covariates. We provide numerical examples based on both simulated and real data to illustrate the proposed method. 

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2. Sampling-based Gaussian Mixture Regression for Big Data

   https://doi.org/10.6339/22-JDS1057  

   Lee, JooChul; Schifano, Elizabeth D.; Wang, HaiYing (August 2022, Journal of Data Science)

   This paper proposes a nonuniform subsampling method for finite mixtures of regression models to reduce large data computational tasks. A general estimator based on a subsample is investigated, and its asymptotic normality is established. We assign optimal subsampling probabilities to data points that minimize the asymptotic mean squared errors of the general estimator and linearly transformed estimators. Since the proposed probabilities depend on unknown parameters, an implementable algorithm is developed. We first approximate the optimal subsampling probabilities using a pilot sample. After that, we select a subsample using the approximated subsampling probabilities and compute estimates using the subsample. We evaluate the proposed method in a simulation study and present a real data example using appliance energy data. 

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3. Optimal subsampling for semi‐parametric accelerated failure time models with massive survival data using a rank‐based approach

   https://doi.org/10.1002/sim.10200  

   Yang, Zehan; Wang, HaiYing; Yan, Jun (October 2024, Statistics in Medicine)

   Subsampling is a practical strategy for analyzing vast survival data, which are progressively encountered across diverse research domains. While the optimal subsampling method has been applied to inferences for Cox models and parametric accelerated failure time (AFT) models, its application to semi‐parametric AFT models with rank‐based estimation have received limited attention. The challenges arise from the non‐smooth estimating function for regression coefficients and the seemingly zero contribution from censored observations in estimating functions in the commonly seen form. To address these challenges, we develop optimal subsampling probabilities for both event and censored observations by expressing the estimating functions through a well‐defined stochastic process. Meanwhile, we apply an induced smoothing procedure to the non‐smooth estimating functions. As the optimal subsampling probabilities depend on the unknown regression coefficients, we employ a two‐step procedure to obtain a feasible estimation method. An additional benefit of the method is its ability to resolve the issue of underestimation of the variance when the subsample size approaches the full sample size. We validate the performance of our estimators through a simulation study and apply the methods to analyze the survival time of lymphoma patients in the surveillance, epidemiology, and end results program. 

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4. Optimal Nonparametric Inference with Two-Scale Distributional Nearest Neighbors

   https://doi.org/10.1080/01621459.2022.2115375  

   Demirkaya, Emre; Fan, Yingying; Gao, Lan; Lv, Jinchi; Vossler, Patrick; Wang, Jingbo (October 2022, Journal of the American Statistical Association)

   The weighted nearest neighbors (WNN) estimator has been popularly used as a flexible and easy-to-implement nonparametric tool for mean regression estimation. The bagging technique is an elegant way to form WNN estimators with weights automatically generated to the nearest neighbors (Steele, 2009; Biau et al., 2010); we name the resulting estimator as the distributional nearest neighbors (DNN) for easy reference. Yet, there is a lack of distributional results for such estimator, limiting its application to statistical inference. Moreover, when the mean regression function has higher-order smoothness, DNN does not achieve the optimal nonparametric convergence rate, mainly because of the bias issue. In this work, we provide an in-depth technical analysis of the DNN, based on which we suggest a bias reduction approach for the DNN estimator by linearly combining two DNN estimators with different subsampling scales, resulting in the novel two-scale DNN (TDNN) estimator. The two-scale DNN estimator has an equivalent representation of WNN with weights admitting explicit forms and some being negative. We prove that, thanks to the use of negative weights, the two-scale DNN estimator enjoys the optimal nonparametric rate of convergence in estimating the regression function under the fourth order smoothness condition. We further go beyond estimation and establish that the DNN and two-scale DNN are both asymptotically normal as the subsampling scales and sample size diverge to infinity. For the practical implementation, we also provide variance estimators and a distribution estimator using the jackknife and bootstrap techniques for the two-scale DNN. These estimators can be exploited for constructing valid confidence intervals for nonparametric inference of the regression function. The theoretical results and appealing nite-sample performance of the suggested two-scale DNN method are illustrated with several simulation examples and a real data application. 

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5. Minimax estimation of low-rank quantum states and their linear functionals

   https://doi.org/10.3150/23-BEJ1610  

   Lahiry, Samriddha; Nussbaum, Michael (February 2024, Bernoulli)

   In classical statistics, a well known paradigm consists in establishing asymptotic equivalence between an experiment of i.i.d. observations and a Gaussian shift experiment, with the aim of obtaining optimal estimators in the former complicated model from the latter simpler model. In particular, a statistical experiment consisting of n i.i.d. observations from d-dimensional multinomial distributions can be well approximated by an experiment consisting of d − 1 dimensional Gaussian distributions. In a quantum version of the result, it has been shown that a collection of n qudits (d-dimensional quantum states) of full rank can be well approximated by a quantum system containing a classical part, which is a d − 1 dimensional Gaussian distribution, and a quantum part containing an ensemble of d(d − 1)/2 shifted thermal states. In this paper, we obtain a generalization of this result when the qudits are not of full rank. We show that when the rank of the qudits is r, then the limiting experiment consists of an r − 1 dimensional Gaussian distribution and an ensemble of both shifted pure and shifted thermal states. For estimation purposes, we establish an asymptotic minimax result in the limiting Gaussian model. Analogous results are then obtained for estimation of a low rank qudit from an ensemble of identically prepared, independent quantum systems, using the local asymptotic equivalence result. We also consider the problem of estimation of a linear functional of the quantum state. We construct an estimator for the functional, analyze the risk and use quantum local asymptotic equivalence to show that our estimator is also optimal in the minimax sense. 

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