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Register a new userRejoinder to Equi-energy sampler with applications in statistical inference and statistical mechanics
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Added by
Wing H. Wong
Publication date
2006
fields
Mathematical Statistics
and research's language is
English
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Rejoinder to ``Equi-energy sampler with applications in statistical inference and statistical mechanics by Kou, Zhou and Wong [math.ST/0507080]
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This paper considers distributed statistical inference for general symmetric statistics %that encompasses the U-statistics and the M-estimators in the context of massive data where the data can be stored at multiple platforms in different locations. In order to facilitate effective computation and to avoid expensive communication among different platforms, we formulate distributed statistics which can be conducted over smaller data blocks. The statistical properties of the distributed statistics are investigated in terms of the mean square error of estimation and asymptotic distributions with respect to the number of data blocks. In addition, we propose two distributed bootstrap algorithms which are computationally effective and are able to capture the underlying distribution of the distributed statistics. Numerical simulation and real data applications of the proposed approaches are provided to demonstrate the empirical performance.
In this paper, we survey some recent results on statistical inference (parametric and nonparametric statistical estimation, hypotheses testing) about the spectrum of stationary models with tapered data, as well as, a question concerning robustness of inferences, carried out on a linear stationary process contaminated by a small trend. We also discuss some question concerning tapered Toeplitz matrices and operators, central limit theorems for tapered Toeplitz type quadratic functionals, and tapered Fejer-type kernels and singular integrals. These are the main tools for obtaining the corresponding results, and also are of interest in themselves. The processes considered will be discrete-time and continuous-time Gaussian, linear or Levy-driven linear processes with memory.
We extend Hoeffdings lemma to general-state-space and not necessarily reversible Markov chains. Let ${X_i}_{i ge 1}$ be a stationary Markov chain with invariant measure $pi$ and absolute spectral gap $1-lambda$, where $lambda$ is defined as the operator norm of the transition kernel acting on mean zero and square-integrable functions with respect to $pi$. Then, for any bounded functions $f_i: x mapsto [a_i,b_i]$, the sum of $f_i(X_i)$ is sub-Gaussian with variance proxy $frac{1+lambda}{1-lambda} cdot sum_i frac{(b_i-a_i)^2}{4}$. This result differs from the classical Hoeffdings lemma by a multiplicative coefficient of $(1+lambda)/(1-lambda)$, and simplifies to the latter when $lambda = 0$. The counterpart of Hoeffdings inequality for Markov chains immediately follows. Our results assume none of countable state space, reversibility and time-homogeneity of Markov chains and cover time-dependent functions with various ranges. We illustrate the utility of these results by applying them to six problems in statistics and machine learning.
In this paper we discuss the estimation of a nonparametric component $f_1$ of a nonparametric additive model $Y=f_1(X_1) + ...+ f_q(X_q) + epsilon$. We allow the number $q$ of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating $f_1$ in the oracle model $Z= f_1(X_1) + epsilon$, for which the additive components $f_2,dots,f_q$ are known. We construct a two-step presmoothing-and-resmoothing estimator of $f_1$ and state finite-sample bounds for the difference between our estimator and some smoothing estimators $hat f_1^{text{(oracle)}}$ in the oracle model. In an asymptotic setting these bounds can be used to show asymptotic equivalence of our estimator and the oracle estimators; the paper thus shows that, asymptotically, under strong enough sparsity conditions, knowledge of $f_2,dots,f_q$ has no effect on estimation accuracy. Our first step is to estimate $f_1$ with an undersmoothed estimator based on near-orthogonal projections with a group Lasso bias correction. We then construct pseudo responses $hat Y$ by evaluating a debiased modification of our undersmoothed estimator of $f_1$ at the design points. In the second step the smoothing method of the oracle estimator $hat f_1^{text{(oracle)}}$ is applied to a nonparametric regression problem with responses $hat Y$ and covariates $X_1$. Our mathematical exposition centers primarily on establishing properties of the presmoothing estimator. We present simulation results demonstrating close-to-oracle performance of our estimator in practical applications.
In this paper we develop an online statistical inference approach for high-dimensional generalized linear models with streaming data for real-time estimation and inference. We propose an online debiased lasso (ODL) method to accommodate the special structure of streaming data. ODL differs from offline debiased lasso in two important aspects. First, in computing the estimate at the current stage, it only uses summary statistics of the historical data. Second, in addition to debiasing an online lasso estimator, ODL corrects an approximation error term arising from nonlinear online updating with streaming data. We show that the proposed online debiased estimators for the GLMs are consistent and asymptotically normal. This result provides a theoretical basis for carrying out real-time interim statistical inference with streaming data. Extensive numerical experiments are conducted to evaluate the performance of the proposed ODL method. These experiments demonstrate the effectiveness of our algorithm and support the theoretical results. A streaming dataset from the National Automotive Sampling System-Crashworthiness Data System is analyzed to illustrate the application of the proposed method.
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