Subsampling tests of parameter hypotheses and overidentifying restrictions with possible failure of identification

We introduce a general testing procedure in models with possible identification failure that has exact asymptotic rejection probability under the null hypothesis. The procedure is widely applicable and in this paper we apply it to tests of arbitrary linear parameter hypotheses as well as to tests of overidentification in time series models given by unconditional moment conditions. The main idea is to subsample classical tests, like for example the Wald or the J test. More precisely, instead of using critical values based on asymptotic theory, we compute data-dependent critical values based on the subsampling technique.We show that under full identification the resulting tests are consistent against fixed alternatives and that they have exact asymptotic rejection probabilities under the null hypothesis independent of identification failure. Furthermore, the subsampling tests of parameter hypotheses are shown to have the same local power as the original tests under full identification.An algorithm is provided that automates the block size choice needed to implement the subsampling testing procedure. A Monte Carlo study shows that the tests have reasonable size properties and often outperform other robust tests in terms of power.     

A Subsampling Method for the Computation of Multivariate Estimators with High Breakdown Point

Abstract

All known robust location and scale estimators with high breakdown point for multivariate samples are very expensive to compute. In practice, this computation has to be carried out using an approximate subsampling procedure. In this article we describe an alternative subsampling scheme, applicable to both the Stahel-Donoho estimator and the minimum volume ellipsoid estimator, with the property that the number of subsamples required can be substantially reduced with respect to the standard subsampling procedures used in both cases. We also discuss some bias and variability properties of the estimator obtained from the proposed subsampling process.     

Optimal subsampling for large-scale quantile regression

To deal with massive data sets, subsampling is known as an effective method which can significantly reduce computational costs in estimating model parameters. In this article, an efficient subsampling method is developed for large-scale quantile regression via Poisson sampling framework, which can solve the memory constraint problem imposed by big data. Under some mild conditions, large sample properties for the estimator involving the weak and strong consistencies, and asymptotic normality are established. Furthermore, the optimal subsampling probabilities are derived according to the A-optimality criterion. It is shown that the estimator based on the optimal subsampling asymptotically achieves a smaller variance than that by the uniform random subsampling. The proposed method is illustrated and evaluated through numerical analyses on both simulated and real data sets.     

Drift Estimation of Multiscale Diffusions Based on Filtered Data

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.

     

Moderate deviations in subsampling distribution estimation

In Politis and Romano (1994) the subsampling methodology was put forth for approximating the sampling distribution (and the corresponding quantiles) of general statistics from i.i.d. and stationary data. In this note, we address the question of how well the subsampling distribution approximates the tail of the target distribution. In the regular setting of the sample mean of an -dependent sequence we show a moderate deviation property of the subsampling distribution.

     

Optimal subsampling for least absolute relative error estimators with massive data

Due to the limitation of computational resources, traditional statistical methods are no longer applicable to large data sets. Subsampling is a popular method which can significantly reduce computational burden. This paper considers a subsampling strategy based on the least absolute relative error in the multiplicative model for massive data. In addition, we employ the random weighting and the least squares methods to handle the problem that the asymptotic covariance of the estimator is difficult to be estimated directly. Moreover, the comparison among the least absolute relative error, least absolute deviation and least squares under the optimal subsampling strategy are given in simulation studies and real examples.     

Subsampling bias and the best-discrepancy systematic cross validation

Statistical machine learning models should be evaluated and validated before putting to work.Conventional k-fold Monte Carlo cross-validation(MCCV)procedure uses a pseudo-random sequence to partition instances into k subsets,which usually causes subsampling bias,inflates generalization errors and jeopardizes the reliability and effectiveness of cross-validation.Based on ordered systematic sampling theory in statistics and low-discrepancy sequence theory in number theory,we propose a new k-fold cross-validation procedure by replacing a pseudo-random sequence with a best-discrepancy sequence,which ensures low subsampling bias and leads to more precise expected-prediction-error(EPE)estimates.Experiments with 156 benchmark datasets and three classifiers(logistic regression,decision tree and na?ve bayes)show that in general,our cross-validation procedure can extrude subsampling bias in the MCCV by lowering the EPE around 7.18%and the variances around 26.73%.In comparison,the stratified MCCV can reduce the EPE and variances of the MCCV around 1.58%and 11.85%,respectively.The leave-one-out(LOO)can lower the EPE around 2.50%but its variances are much higher than the any other cross-validation(CV)procedure.The computational time of our cross-validation procedure is just 8.64%of the MCCV,8.67%of the stratified MCCV and 16.72%of the LOO.Experiments also show that our approach is more beneficial for datasets characterized by relatively small size and large aspect ratio.This makes our approach particularly pertinent when solving bioscience classification problems.Our proposed systematic subsampling technique could be generalized to other machine learning algorithms that involve random subsampling mechanism.     

Nonsingular subsampling for regression S estimators with categorical predictors

Simple random subsampling is an integral part of S estimation algorithms for linear regression. Subsamples are required to be nonsingular. Usually, discarding a singular subsample and drawing a new one leads to a sufficient number of nonsingular subsamples with a reasonable computational effort. However, this procedure can require so many subsamples that it becomes infeasible, especially if levels of categorical variables have low frequency. A subsampling algorithm called nonsingular subsampling is presented, which generates only nonsingular subsamples. When no singular subsamples occur, nonsingular subsampling is as fast as the simple algorithm, and if singular subsamples do occur, it maintains the same computational order. The algorithm works consistently, unless the full design matrix is singular. The method is based on a modified LU decomposition algorithm that combines sample generation with solving the least squares problem. The algorithm may also be useful for ordinary bootstrapping. Since the method allows for S estimation in designs with factors and interactions between factors and continuous regressors, we study properties of the resulting estimators, both in the sense of their dependence on the randomness of the sampling and of their statistical performance.     

K-sample subsampling in general spaces: The case of independent time series

The problem of subsampling in two-sample and K-sample settings is addressed where both the data and the statistics of interest take values in general spaces. We focus on the case where each sample is a stationary time series, and construct subsampling confidence intervals and hypothesis tests with asymptotic validity. Some examples are also given, and the problem of optimal block size choice is discussed.     

A Markov Chain Monte Carlo Convergence Diagnostic Using Subsampling

Abstract

A new diagnostic procedure for assessing convergence of a Markov chain Monte Carlo (MCMC) simulation is proposed. The method is based on the use of subsampling for the construction of confidence regions from asymptotically stationary time series as developed in Politis, Romano, and Wolf. The MCMC subsampling diagnostic is capable of gauging at what point the chain has “forgotten” its starting points, as well as to indicate how many points are needed to estimate the parameters of interest according to the desired accuracy. Simulation examples are also presented showing that the diagnostic performs favorably in interesting cases.     

改进的自回归过程结构变点检验

In this paper, we consider the problem of detecting for structural changes in the autoregressive processes including AR（p） process. In performing a test, we employ the conventional residual CUSUM of squares test （RCUSQ） statistic. The RCUSQ test is based on the subsampling method introduced by Jach and Kokoszka [J. Methodology and Computing in Applied Probability 25（2004）]. It is shown that under regularity conditions, the asymptotic distribution of the test statistic is the function of a standard Brownian bridge. Simulation results as to AR（1） process and an example of real data analysis are provided for illustration.     

Compressive Sampling via Random Convolution

Several recent results in compressive sampling show that a sparse signal (i.e. a signal which can be compressed in a known orthobasis) can be efficiently acquired by taking linear measurements against random test functions. In this paper, we show that these results can be extended to measurements taken by convolving with a random pulse and then subsampling. The measurement scheme is universal in that it complements (with high probability) any fixed orthobasis we use to represent the signal. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Null controllability of a class of Newtonian filtration equations

In this paper, we study the null controllability of a class of Newtonian filtration equations. Using the properties of finite extinction time and finite speed of propagation, we construct a control function in feedback form, such that system is exactly null controllable at any time T>0. If we put a nonnegative constraint on the control function, this system is not exactly null controllable any more, but approximately null controllable for a long time.     

Subsampling tests for variance changes in the presence of autoregressive parameter shifts

In this paper, we consider the problem of testing for variance changes in the linear autoregressive processes including AR(p) processes when there are autoregressive parameter shifts. In performing a test, we employ the conventional residual CUSUM of squares test (RCUSQ) statistic. The RCUSQ test is based on the subsampling method introduced by Jach and Kokoszka (2004) [16] to eliminate the influence caused by autoregressive parameter shifts. It is shown that under regularity conditions, the test statistic behaves asymptotically the function of a standard Brownian bridge. We establish the asymptotic validity of this method and assess its performance both theoretically and numerically.     

Null Extensions and Generalized Inflations of Brandt Semigroups

Clarke and Monzo defined in [3] a construction called a generalized inflation of a semigroup. It is always the case that any inflation of a semigroup is a generalized inflation, and any generalized inflation of a semigroup is a null extension of the semigroup. Clarke and Monzo proved that any associative null extension of a base semigroup which is a union of groups is in fact a generalized inflation. In this paper we study null extensions and generalized inflations of Brandt semigroups. We first prove that any generalized inflation of a Brandt semigroup is actually an inflation of the semigroup. This answers a question posed by Clarke and Monzo in [3]. Then we characterize associative null extensions of Brandt semigroups, and show that there are associative null extensions of Brandt semigroups which are not generalized inflations.     

Multiscale data sampling and function extension

We introduce a multiscale scheme for sampling scattered data and extending functions defined on the sampled data points, which overcomes some limitations of the Nyström interpolation method. The multiscale extension (MSE) method is based on mutual distances between data points. It uses a coarse-to-fine hierarchy of the multiscale decomposition of a Gaussian kernel. It generates a sequence of subsamples, which we refer to as adaptive grids, and a sequence of approximations to a given empirical function on the data, as well as their extensions to any newly-arrived data point. The subsampling is done by a special decomposition of the associated Gaussian kernel matrix in each scale in the hierarchical procedure.     

双符号等长代换系统的序列熵

系统称为null的,如果对任意序列,它的序列熵为零.双符号等长代换及其对应的代换极小系统可分成三类:有限的、离散的和连续的.容易看出离散的代换极小系统是null的,Goodman证明了连续的代换极小系统不是null的.本文将完全刻画所有的双符号等长代换极小系统的序列墒.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

Null controllability of the viscous Camassa–Holm equation with moving control

In this paper, we study the null controllability of the viscous Camassa–Holm equation on the one-dimensional torus. By using a moving distributed control, we obtain that the system is null controllable for a given data with certain regularity.     

Twisted Surfaces with Null Rotation Axis in Minkowski 3-Space

We examine curvature properties of twisted surfaces with null rotation axis in Minkowski 3-space. That is, we study surfaces that arise when a planar curve is subject to two synchronized rotations, possibly at different speeds, one in its supporting plane and one of this supporting plane about an axis in the plane. Moreover, at least one of the two rotation axes is a null axis. As is clear from its construction, a twisted surface generalizes the concept of a surface of revolution. We classify flat, constant Gaussian curvature, minimal and constant mean curvature twisted surfaces with a null rotation axis. Aside from pseudospheres, pseudohyperbolic spaces and cones, we encounter B-scrolls in these classifications. The appearance of B-scrolls in these classifications is of course the result of the rotation about a null axis. As for the cones in the classification of flat twisted surfaces, introducing proper coordinates, we prove that they are determined by so-called Clelia curves. With a Clelia curve we mean a curve that has linear dependent spherical coordinates.