Bootstrap-Based Inference for Cube Root Asymptotics

A Note on Bootstrapping M-estimates from Unstable AR(2) Process with Infinite Variance Innovations

The limiting distribution for M-estimates in a non-stationary autoregressive model with heavy-tailed error is computationally intractable. To make inferences based on the M-estimates, the bootstrap procedure can be used to approximate the sampling distribution. In this paper, we show that the bootstrap scheme with $m=o(n)$ resampling sample size when $m/n \to 0$ is approximately valid in a multiple unit roots time series with innovations in the domain of attraction of a stable law with index $0<α\leq2$.

Bootstrap consistency for general semiparametric $M$-estimation

Consider $M$-estimation in a semiparametric model that is characterized by a Euclidean parameter of interest and an infinite-dimensional nuisance parameter. As a general purpose approach to statistical inferences, the bootstrap has found wide applications in semiparametric $M$-estimation and, because of its simplicity, provides an attractive alternative to the inference approach based on the asymptotic distribution theory. The purpose of this paper is to provide theoretical justifications for the use of bootstrap as a semiparametric inferential tool. We show that, under general conditions, the…

On the uniform asymptotic validity of subsampling and the bootstrap

This paper provides conditions under which subsampling and the bootstrap can be used to construct estimators of the quantiles of the distribution of a root that behave well uniformly over a large class of distributions $\mathbf{P}$. These results are then applied (i) to construct confidence regions that behave well uniformly over $\mathbf{P}$ in the sense that the coverage probability tends to at least the nominal level uniformly over $\mathbf{P}$ and (ii) to construct tests that behave well uniformly over $\mathbf{P}$ in the sense that the size tends to no greater than the nominal level unifo…

Robust inference for threshold regression models

This paper is concerned with inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump. We nest previous works that assume either continuity or discontinuity at the threshold point and develop robust inference methods on the parameters of the model, which are valid under both specifications. In particular, we found that the parameter values under the kink restriction are irregular points of the Hessian matrix of the expected Gaussian quasi-likelihood. This irregularity destroys the asymptotic normality…

Optimal third root asymptotic bounds in the statistical estimation of thresholds

This paper is concerned with estimating the intersection point of two densities, given a sample of both of the densities. This problem arises in classification theory. The main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size. As corollaries, we obtain probabilistic bounds for the prediction error in a classification problem. The key to the proof is an entropy estimate. The lower bounds are based on bounds for general estimators, which are applicable in other contexts as well. Furthermore, we…

Sieve Wald and QLR Inferences on Semi/nonparametric Conditional Moment Models

This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. These models are often ill-posed and hence it is difficult to verify whether a (possibly nonlinear) functional is root-$n$ estimable or not. We provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is root-$n$ estimable or not. We establish the following new useful results: (1) the a…

仿射过程的近单位根理论

Near-Unit-Root Theory for Affine Processes

本文针对离散时间仿射过程建立了近单位根渐近理论，揭示了其时变条件方差在近单位根下的非渐近可忽略性，并分析了局部单位根、温和爆炸和温和平稳三种框架下的估计量行为与推断方法。

Two-Step Estimation and Inference with Possibly Many Included Covariates

We study the implications of including many covariates in a first-step estimate entering a two-step estimation procedure. We find that a first order bias emerges when the number of \textit{included} covariates is "large" relative to the square-root of sample size, rendering standard inference procedures invalid. We show that the jackknife is able to estimate this "many covariates" bias consistently, thereby delivering a new automatic bias-corrected two-step point estimator. The jackknife also consistently estimates the standard error of the original two-step point estimator. For inference, we …

在测量误差下对凹分布函数的推断

Inference for concave distribution functions under measurement error

本文提出了一种在测量误差下估计凹分布函数的方法，并构造了检验凹性假设的非参数检验，通过理论分析和模拟研究验证了其有效性。

下尾Spearman's rho曲线的Bernstein平滑估计与bootstrap推断

Bernstein-smoothed estimation and bootstrap inference for the lower-tail Spearman's rho curve

本文提出下尾Spearman's rho曲线的Bernstein平滑估计量，建立一致性和弱收敛理论，验证平滑beta bootstrap推断，并通过模拟和保险数据展示其降低方差和误差的优势。

Nonparametric estimation of the incubation time distribution

We discuss nonparametric estimators of the distribution of the incubation time of a disease. The classical approach in these models is to use parametric families like Weibull, log-normal or gamma in the estimation procedure. We analyze instead the nonparametric maximum likelihood estimator (MLE) and show that, under some conditions, its rate of convergence is cube root $n$ and that its limit behavior is given by Chernoff's distribution. We also study smooth estimates, based on the MLE. The density estimates, based on the MLE, are capable of catching finer or unexpected aspects of the density, …