Abstract:
In this paper, in order to test whether changes have occurred in a nonlinear parametric regression, we propose a nonparametric method based on the empirical likelihood. Firstly, we test the null hypothesis of no-change against the alternative of one change in the regression parameters. Under null hypothesis, the consistency and the convergence rate of the regression parameter estimators are proved. The asymptotic distribution of the test statistic under the null hypothesis is obtained, which allows to find the asymptotic critical value. On the other hand, we prove that the proposed test statistic has the asymptotic power equal to 1. These theoretical results allows find a simple test statistic, very useful for applications. The epidemic model, a particular model with two change-points under the alternative hypothesis, is also studied. Numerical studies by Monte-Carlo simulations show the performance of the proposed test statistic, compared to an existing method in literature.

Abstract:
We investigate the power of the CUSUM test and the Wilcoxon change-point test for a shift in the mean of a process with long-range dependent noise. We derive analytiv formulas for the power of these tests under local alternatives. These results enable us to calculate the asymptotic relative efficiency (ARE) of the CUSUM test and the Wilcoxon change point test. We obtain the surprising result that for Gaussian data, the ARE of these two tests equals 1, in contrast to the case of i.i.d. noise when the ARE is known to be $3/\pi$.

Abstract:
In this paper we study the consistency of different bootstrap procedures for constructing confidence intervals (CIs) for the unique jump discontinuity (change-point) in an otherwise smooth regression function in a stochastic design setting. This problem exhibits nonstandard asymptotics and we argue that the standard bootstrap procedures in regression fail to provide valid confidence intervals for the change-point. We propose a version of smoothed bootstrap, illustrate its remarkable finite sample performance in our simulation study, and prove the consistency of the procedure. The $m$ out of $n$ bootstrap procedure is also considered and shown to be consistent. We also provide sufficient conditions for any bootstrap procedure to be consistent in this scenario.

Abstract:
Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac-Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the spacing test to the frame of unknown noise variance. More precisely, we investigate the power of the spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level $\alpha$. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level $\alpha$ goes to zero, we show that the probability of getting a true positive is much greater than $\alpha$; and we give a detailed description of its power in the case of two predictors. Moreover, we numerically investigate a comparison between the spacing test for LARS and the Pearson's chi-squared test (goodness of fit).

Abstract:
A number of statistical tests are proposed for the purpose of change-point detection in a general nonparametric regression model under mild conditions. New proofs are given to prove the weak convergence of the underlying processes which assume remove the stringent condition of bounded total variation of the regression function and need only second moments. Since many quantities, such as the regression function, the distribution of the covariates and the distribution of the errors, are unspecified, the results are not distribution-free. A weighted bootstrap approach is proposed to approximate the limiting distributions. Results of a simulation study for this paper show good performance for moderate samples sizes.

Abstract:
We consider a high-dimensional regression model with a possible change-point due to a covariate threshold and develop the Lasso estimator of regression coefficients as well as the threshold parameter. Our Lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non-asymptotic oracle inequalities for both the prediction risk and the $\ell_1$ estimation loss for regression coefficients. Since the Lasso estimator selects variables simultaneously, we show that oracle inequalities can be established without pretesting the existence of the threshold effect. Furthermore, we establish conditions under which the estimation error of the unknown threshold parameter can be bounded by a nearly $n^{-1}$ factor even when the number of regressors can be much larger than the sample size ($n$). We illustrate the usefulness of our proposed estimation method via Monte Carlo simulations and an application to real data.

Abstract:
The deviation test belong to core tools in point process statistics, where hypotheses are typically tested considering differences between an empirical summary function and its expectation under the null hypothesis, which depend on a distance variable r. This test is a classical device to overcome the multiple comparison problem which appears since the functional differences have to be considered for a range of distances r simultaneously. The test has three basic ingredients: (i) choice of a suitable summary function, (ii) transformation of the summary function or scaling of the differences, and (iii) calculation of a global deviation measure. We consider in detail the construction of such tests both for stationary and finite point processes and show by two toy examples and a simulation study for the case of the random labelling hypothesis that the points (i) and (ii) have great influence on the power of the tests.

Abstract:
In a variety of different settings cumulative sum (CUSUM) procedures have been applied for the sequential detection of structural breaks in the parameters of stochastic models. Yet their performance depends strongly on the time of change and is best under early-change scenarios. For later changes their finite sample behavior is rather questionable. We therefore propose modified CUSUM procedures for the detection of abrupt changes in the regression parameter of multiple time series regression models, that show a higher stability with respect to the time of change than ordinary CUSUM procedures. The asymptotic distributions of the test statistics and the consistency of the procedures are provided. In a simulation study it is shown that the proposed procedures behave well in finite samples. Finally the procedures are applied to a set of capital asset pricing data related to the Fama-French extension of the capital asset pricing model.

Abstract:
We propose a class of difference-based estimators for the autocovariance in nonparametric regression when the signal is discontinuous (change-point model), possibly highly fluctuating, and the errors form a stationary $m$-dependent Gaussian process. These estimators circumvent the explicit pre-estimation of the unknown regression function, a task that is particularly challenging in change-point regression with correlated errors. We provide finite sample expressions for their mean squared errors when the signal function is piecewise constant. Based on this, we distinguish the signal as the major source of the mean squared errors and derive biased-optimized estimates. These optimal estimates do not depend on the particular (unknown) autocovariance structure, and notably, for positive correlated errors, in addition, they minimize that part of the variance which is influenced by the unknown regression function. Further, we provide some asymptotic analysis of our estimators. We show their $\sqrt{n}$-consistency in the context of a piecewise H\"older signal with non-Gaussian stationary $m$-dependent errors and when the number of change-points tends to infinity. Finally, we combine our biased-optimized autocovariance estimates with a projection-based approach and derive covariance matrix estimates for change-point regression, a method which is of independent interest. We also provide a practical method to estimate $m$ which, along with all the methods presented in this paper, can be found in our R package dbacf. Several simulation studies as well as applications to two datasets from biophysics complement this paper.

Abstract:
Traditional change-points detection is based on exact data set which cannot reflect prior information of data. This paper introduced a regression-class mixture decomposition method for fuzzy point data. In the method, different regression classes were mined sequentially in fuzzy point data set, and then the regression change points can be determined. So the number of change points which can be gotten automatically, need not prespecifying. Experiments prove that the proposed method is very robust, and by using fuzzy point data we introduced the prior information of the analysis data into the process of mining regression classes, which made the change-point we got in fuzzy point data be more practical than we got in exact data set.