A change point detection procedure using the method of moment estimators is proposed. The test statistics is based on a suitable $Z$-process. The asymptotic behavior of this process is established under both the null and the alternative hypothesis and the consistency of the test is also proved. An estimator for the change point is proposed and its consistency is derived. Some examples of this method applied to a parametric family of random variables are presented.
We introduce two types of estimators of the finite-dimensional parameters in the case of observations of inhomogeneous Poisson processes. These are the estimators of the method of moments and multi-step MLE. It is shown that the estimators of the method of moments are consistent and asymptotically normal and the multi-step MLE are consistent and asymptotically efficient. The construction of multi-step MLE-process is done in two steps. First we construct a consistent estimator by the observations on some learning interval and then this estimator is used for construction of one-step and two-step…
In this paper we continue the investigation of the SRCEN estimator of the extreme value index $γ$ (or the tail index $α=1/γ$) proposed in \cite{MCE} for $γ>1/2$. We propose a new estimator based on the local maximum. This, in fact, is a modification of the SRCEN estimator to the case $γ>0$. We establish the consistency and asymptotic normality of the newly proposed estimator for i.i.d. data. Also, a short discussion on the comparison of the estimators is included.
In this paper, we consider an inference problem for the first order autoregressive process with non-zero mean driven by a long memory stationary Gaussian process. Suppose that the covariance function of the noise can be expressed as $|k|^{2H-2}$ times a positive constant when $k$ tends to infinity, and the fractional Gaussian noise and the fractional ARIMA model are special examples that satisfy this assumption. We propose moment estimators and prove the strong consistency, the asymptotic normality and joint asymptotic normality.
In this paper, we consider an inference problem for the first order autoregressive process driven by a long memory stationary Gaussian process. Suppose that the covariance function of the noise can be expressed as $\abs{k}^{2H-2}$ times a function slowly varying at infinity. The fractional Gaussian noise and the fractional ARIMA model and some others Gaussian noise are special examples that satisfy this assumption. We propose a second moment estimator and prove the strong consistency and give the asymptotic distribution. Moreover, when the limit distribution is Gaussian, we give the upper Berr…
The q-weighted CUSUM and their corresponding estimator are well known statistics for change-point detection and estimation. They have the difficulty that the performance is highly dependent on the location of the change. An adaptive estimator with data-driven weights is presented to overcome this problem, and it is shown that the corresponding adaptive change-point tests are valid.
We define a moment-based estimator that maximizes the empirical saddlepoint (ESP) approximation of the distribution of solutions to empirical moment conditions. We call it the ESP estimator. We prove its existence, consistency and asymptotic normality, and we propose novel test statistics. We also show that the ESP estimator corresponds to the MM (method of moments) estimator shrunk toward parameter values with lower estimated variance, so it reduces the documented instability of existing moment-based estimators. In the case of just-identified moment conditions, which is the case we focus on, …
This work studies the total variation regularized $\ell_2$ estimator (fused lasso) in the setting of a change point detection problem. Compared with existing works that focus on the sum of squared estimation errors, we give bound on the element-wise estimation error. Our bound is nearly optimal in the sense that the sum of squared error matches the best existing result, up to a logarithmic factor. This analysis of the element-wise estimation error allows a screening method that can approximately detect all the change points. We also generalize this method to the muitivariate setting, i.e., to …
A variety of estimators for the parameters of the Generalized Pareto distribution, the approximating distribution for excesses over a high threshold, have been proposed, always assuming the underlying data to be independent. We recently proved that the likelihood moment estimators are consistent estimators for the parameters of the Generalized Pareto distribution for the case where the underlying data arises from a (stationary) linear process with heavy-tailed innovations. In this paper we derive the bivariate asymptotic normality under some additional assumptions and give an explicit example …
Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantile…
In this paper, we consider the Whittle estimator for the parameters of a stationary solution of a continuous-time linear state space model sampled at low frequencies. In our context the driving process is a Lévy process which allows flexible margins of the underlying model. The Lévy process is supposed to have finite second moments. It is well known that then the class of stationary solutions of linear state space models and the class of multivariate CARMA processes coincides. We prove that the Whittle estimator, which is based on the periodogram, is strongly consistent and asymptotically norm…
We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requi…
The quasi-maximum likelihood estimation is a commonly-used method for estimating GARCH parameters. However, such estimators are sensitive to outliers and their asymptotic normality is proved under the finite fourth moment assumption on the underlying error distribution. In this paper, we propose a novel class of estimators of the GARCH parameters based on ranks, called R-estimators, with the property that they are asymptotic normal under the existence of a more than second moment of the errors and are highly efficient. We also consider the weighted bootstrap approximation of the finite sample …
The change-plane Cox model is a popular tool for the subgroup analysis of survival data. Despite the rich literature on this model, there has been limited investigation into the asymptotic properties of the estimators of the finite-dimensional parameter. Particularly, the convergence rate, not to mention the asymptotic distribution, has not been fully characterized for the general model where classification is based on multiple covariates. To bridge this theoretical gap, this study proposes a maximum smoothed partial likelihood estimator and establishes the following asymptotic properties. Fir…
We consider the class of all stationary Gaussian process with explicit parametric spectral density. Under some conditions on the autocovariance function, we defined a GMM estimator that satisfies consistency and asymptotic normality, using the Breuer-Major theorem and previous results on ergodicity. This result is applied to the joint estimation of the three parameters of a stationary Ornstein-Uhlenbeck (fOU) process driven by a fractional Brownian motion. The asymptotic normality of its GMM estimator applies for any H in (0,1) and under some restrictions on the remaining parameters. A numeric…
We consider two stage estimation with a non-parametric first stage and a generalized method of moments second stage, in a simpler setting than (Chernozhukov et al. 2016). We give an alternative proof of the theorem given in (Chernozhukov et al. 2016) that orthogonal second stage moments, sample splitting and $n^{1/4}$-consistency of the first stage, imply $\sqrt{n}$-consistency and asymptotic normality of second stage estimates. Our proof is for a variant of their estimator, which is based on the empirical version of the moment condition (Z-estimator), rather than a minimization of a norm of t…
We prove the consistency and asymptotic normality of the Laplacian Quasi-Maximum Likelihood Estimator (QMLE) for a general class of causal time series including ARMA, AR($\infty$), GARCH, ARCH($\infty$), ARMA-GARCH, APARCH, ARMA-APARCH,..., processes. We notably exhibit the advantages (moment order and robustness) of this estimator compared to the classical Gaussian QMLE. Numerical simulations confirms the accuracy of this estimator.
We introduce an estimation method for the scaled skewness coefficient of the sample mean of short and long memory linear processes. This method can be extended to estimate higher moments such as curtosis coefficient of the sample mean. Also a general result on computing all asymptotic moments of partial sums is obtained, allowing in particular a much easier derivation of some existing central limit theorems for linear processes. The introduced skewness estimator provides a tool to empirically examine the error of the central limit theorem for long and short memory linear processes. We also sho…
This paper proposes a Sieve Simulated Method of Moments (Sieve-SMM) estimator for the parameters and the distribution of the shocks in nonlinear dynamic models where the likelihood and the moments are not tractable. An important concern with SMM, which matches sample with simulated moments, is that a parametric distribution is required. However, economic quantities that depend on this distribution, such as welfare and asset-prices, can be sensitive to misspecification. The Sieve-SMM estimator addresses this issue by flexibly approximating the distribution of the shocks with a Gaussian and tail…
A robust approach for estimating change-points in the mean of an AR(1) process
本文考虑高斯AR（1）过程均值的多个变点点估计问题.考虑到依赖结构不允许我们使用动态规划算法，这是在独立情况下给出最优解的唯一算法。我们提出了自相关参数的一个稳健估计，它是相容的，并满足中心极限定理。然后，我们建议按照经典的推理方法，通过插入这个估计在用于变点估计的标准。我们证明了这些估计的渐近性质与独立框架下的经典估计相同。然后使用相同的插件方法来近似修改的BIC并选择段的数量。该方法在R包AR1seg中实现，可从Comprehensive R Archive Network（CRAN）获得。该软件包用于模拟部分，其中我们表明，对于有限的样本量，考虑到依赖结构，提高了变点估计和选择标准的统计性能。
This paper studies multivariate nonparametric change point localization and inference problems. The data consists of a multivariate time series with potentially short range dependence. The distribution of this data is assumed to be piecewise constant with densities in a Hölder class. The change points, or times at which the distribution changes, are unknown. We derive the limiting distributions of the change point estimators when the minimal jump size vanishes or remains constant, a first in the literature on change point settings. We are introducing two new features: a consistent estimator th…
This paper develops theory for feasible estimators of finite-dimensional parameters identified by general conditional quantile restrictions, under much weaker assumptions than previously seen in the literature. This includes instrumental variables nonlinear quantile regression as a special case. More specifically, we consider a set of unconditional moments implied by the conditional quantile restrictions, providing conditions for local identification. Since estimators based on the sample moments are generally impossible to compute numerically in practice, we study feasible estimators based on …
In this paper we investigate the performance of a variety of estimation techniques for the scale and shape parameter of the Lomax distribution. These methods include traditional methods such as the maximum likelihood estimator and the method of moments estimator. A version of the maximum likelihood estimator adjusted for bias is also included. Furthermore, alternative moment-based estimation techniques such as the $L$-moment estimator and the probability weighted moments estimator are included along with three different minimum distance estimators. The finite sample performances of each of the…
Large-scale sequential data is often exposed to some degree of inhomogeneity in the form of sudden changes in the parameters of the data-generating process. We consider the problem of detecting such structural changes in a high-dimensional regression setting. We propose a joint estimator of the number and the locations of the change points and of the parameters in the corresponding segments. The estimator can be computed using dynamic programming or, as we emphasize here, it can be approximated using a binary search algorithm with $O(n \log(n) \mathrm{Lasso}(n))$ computational operations while…
A Moment-assisted Approach for Improving Subsampling-based MLE with Large-scale data
Beijing University of Posts and Telecommunications（北京邮电大学）；Zhejiang Gongshang University（浙江工商大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Tsinghua University（清华大学）
