arXiv subjects

Lajos Horváth

Publications and source records attributed to Lajos Horváth.

At least 19 recordsLinked to original sources

arXiv2026

Online detection of distributional changes for time series in metric spaces

We propose an online testing framework for detecting distributional changes in serially dependent data with values in a separable metric space. Based on two-sample $U$-statistics, the framework encompasses sequential analogs of energy distance and maximum mean discrepancy (MMD) procedures while accommodating temporal dependence. We establish asymptotic theory for finite and open-ended monitoring horizons that characterizes the full asymptotic run-length distribution under $H_0$ and yields asymptotic false-alarm control. We further establish new spectral approximation results for kernel matrices formed from serially dependent observations, and use them to construct a feasible Monte Carlo calibration procedure. Our flexible window construction encompasses classical, Page-type, and full-scan historical-baseline monitoring and can achieve short detection delays for both early and late changepoints, without requiring sub-Gaussianity or high-order moments of the raw observations. Simulations show reliable false-alarm control across linear, nonlinear, high-dimensional, and functional time-series models and further demonstrate that, over a broad range of alternatives and changepoint locations, the proposed method can achieve substantially shorter delays than recent procedures designed specifically for rapid detection. Applications to foreign exchange rates, electricity-market curves, and daily air transportation networks illustrate the methodology across scalar, functional, and network-valued time series.

stat.ME↗

arXiv2026

Sequential monitoring for distributional changepoints using degenerate U-statistics

We investigate the online detection of changepoints in the distribution of a sequence of observations using a class of degenerate \textit{U}-statistic-type processes. We consider an ordinary (Chu--Stinchcombe--White-type) detector and a Page-type detector under open- and closed-ended monitoring, and introduce an expanding-baseline Page-type procedure that incorporates sufficiently old monitoring observations into the baseline sample. Under the null, we derive weak limits for all three procedures and justify a Monte Carlo approximation to their critical values. For the ordinary and Page-type detectors, we also establish consistency and limiting distributions for detection delays under both early and late changes. The theory requires only square summability of the eigenvalues associated with the degenerate kernel operator, rather than the stronger absolute-summability condition often imposed in related work. Simulations show competitive performance relative to recent mean-, covariance-, and empirical-CDF-based monitors, and an application to multivariate compressor-sensor data from a metro train illustrates the methodology.

math.ST↗

arXiv2023

Real-time monitoring with RCA models

We propose a family of weighted statistics based on the CUSUM process of the WLS residuals for the online detection of changepoints in a Random Coefficient Autoregressive model, using both the standard CUSUM and the Page-CUSUM process. We derive the asymptotics under the null of no changepoint for all possible weighing schemes, including the case of the standardised CUSUM, for which we derive a Darling-Erdos-type limit theorem; our results guarantee the procedure-wise size control under both an open-ended and a closed-ended monitoring. In addition to considering the standard RCA model with no covariates, we also extend our results to the case of exogenous regressors. Our results can be applied irrespective of (and with no prior knowledge required as to) whether the observations are stationary or not, and irrespective of whether they change into a stationary or nonstationary regime. Hence, our methodology is particularly suited to detect the onset, or the collapse, of a bubble or an epidemic. Our simulations show that our procedures, especially when standardising the CUSUM process, can ensure very good size control and short detection delays. We complement our theory by studying the online detection of breaks in epidemiological and housing prices series.

stat.ME↗

arXiv2023

Change point detection in high dimensional data with U-statistics

We consider the problem of detecting distributional changes in a sequence of high dimensional data. Our approach combines two separate statistics stemming from $L_p$ norms whose behavior is similar under $H_0$ but potentially different under $H_A$, leading to a testing procedure that that is flexible against a variety of alternatives. We establish the asymptotic distribution of our proposed test statistics separately in cases of weakly dependent and strongly dependent coordinates as $\min\{N,d\}\to\infty$, where $N$ denotes sample size and $d$ is the dimension, and establish consistency of testing and estimation procedures in high dimensions under one-change alternative settings. Computational studies in single and multiple change point scenarios demonstrate our method can outperform other nonparametric approaches in the literature for certain alternatives in high dimensions. We illustrate our approach though an application to Twitter data concerning the mentions of U.S. Governors.

math.ST↗

arXiv2023

On changepoint detection in functional data using empirical energy distance

We propose a novel family of test statistics to detect the presence of changepoints in a sequence of dependent, possibly multivariate, functional-valued observations. Our approach allows to test for a very general class of changepoints, including the "classical" case of changes in the mean, and even changes in the whole distribution. Our statistics are based on a generalisation of the empirical energy distance; we propose weighted functionals of the energy distance process, which are designed in order to enhance the ability to detect breaks occurring at sample endpoints. The limiting distribution of the maximally selected version of our statistics requires only the computation of the eigenvalues of the covariance function, thus being readily implementable in the most commonly employed packages, e.g. R. We show that, under the alternative, our statistics are able to detect changepoints occurring even very close to the beginning/end of the sample. In the presence of multiple changepoints, we propose a binary segmentation algorithm to estimate the number of breaks and the locations thereof. Simulations show that our procedures work very well in finite samples. We complement our theory with applications to financial and temperature data.

stat.ME↗

arXiv2020

Limit results for $L^p$ functionals of weighted CUSUM processes

The cumulative sum (CUSUM) process is often used in change point analysis to detect changes in the mean of sequentially observed data. We provide a full description of the asymptotic distribution of $L^p, 1\leq p <\infty$, functionals of the weighted CUSUM process for time series under general conditions.

math.PR↗

arXiv2020

Sequential Monitoring of Changes in Housing Prices

We propose a sequential monitoring scheme to find structural breaks in real estate markets. The changes in the real estate prices are modeled by a combination of linear and autoregressive terms. The monitoring scheme is based on a detector and a suitably chosen boundary function. If the detector crosses the boundary function, a structural break is detected. We provide the asymptotics for the procedure under the stability null hypothesis and the stopping time under the change point alternative. Monte Carlo simulation is used to show the size and the power of our method under several conditions. We study the real estate markets in Boston, Los Angeles and at the national U.S. level. We find structural breaks in the markets, and we segment the data into stationary segments. It is observed that the autoregressive parameter is increasing but stays below 1.

econ.EM↗

arXiv2019

A new class of change point test statistics of Rényi type

A new class of change point test statistics is proposed that utilizes a weighting and trimming scheme for the cumulative sum (CUSUM) process inspired by Rényi (1953). A thorough asymptotic analysis and simulations both demonstrate that this new class of statistics possess superior power compared to traditional change point statistics based on the CUSUM process when the change point is near the beginning or end of the sample. Generalizations of these "Rényi" statistics are also developed to test for changes in the parameters in linear and non-linear regression models, and in generalized method of moments estimation. In these contexts we applied the proposed statistics, as well as several others, to test for changes in the coefficients of Fama-French factor models. We observed that the Rényi statistic was the most effective in terms of retrospectively detecting change points that occur near the endpoints of the sample.

math.ST↗

arXiv2015

Empirical eigenvalue based testing for structural breaks in linear panel data models

Testing for stability in linear panel data models has become an important topic in both the statistics and econometrics research communities. The available methodologies address testing for changes in the mean/linear trend, or testing for breaks in the covariance structure by checking for the constancy of common factor loadings. In such cases when an external shock induces a change to the stochastic structure of panel data, it is unclear whether the change would be reflected in the mean, the covariance structure, or both. In this paper, we develop a test for structural stability of linear panel data models that is based on monitoring for changes in the largest eigenvalue of the sample covariance matrix. The asymptotic distribution of the proposed test statistic is established under the null hypothesis that the mean and covariance structure of the panel data's cross sectional units remain stable during the observation period. We show that the test is consistent assuming common breaks in the mean or factor loadings. These results are investigated by means of a Monte Carlo simulation study, and their usefulness is demonstrated with an application to U.S. treasury yield curve data, in which some interesting features of the 2007-2008 subprime crisis are illuminated.

stat.ME↗

arXiv2015

Estimation of the time of change in panel data

We consider the problem of estimating the common time of a change in the mean parameters of panel data when dependence is allowed between the panels in the form of a common factor. A CUSUM type estimator is proposed, and we establish first and second order asymptotics that can be used to derive consistent confidence intervals for the time of change. Our results improve upon existing theory in two primary directions. Firstly, the conditions we impose on the model errors only pertain to the order of their long run moments, and hence our results hold for nearly all stationary time series models of interest, including nonlinear time series like the ARCH and GARCH processes. Secondly, we study how the asymptotic distribution and norming sequences of the estimator depend on the magnitude of the changes in each panel and the common factor loadings. The performance of our results in finite samples is demonstrated with a Monte Carlo simulation study, and we consider applications to two real data sets: the exchange rates of 23 currencies with respect to the US dollar, and the GDP per capita in 113 countries.

math.ST↗

arXiv2015

On the asymptotic normality of kernel estimators of the long run covariance of functional time series

We consider the asymptotic normality in $L^2$ of kernel estimators of the long run covariance kernel of stationary functional time series. Our results are established assuming a weakly dependent Bernoulli shift structure for the underlying observations, which contains most stationary functional time series models, under mild conditions. As a corollary, we obtain joint asymptotics for functional principal components computed from empirical long run covariance operators, showing that they have the favorable property of being asymptotically independent.

math.ST↗

arXiv2013

A test of significance in functional quadratic regression

We consider a quadratic functional regression model in which a scalar response depends on a functional predictor; the common functional linear model is a special case. We wish to test the significance of the nonlinear term in the model. We develop a testing method which is based on projecting the observations onto a suitably chosen finite dimensional space using functional principal component analysis. The asymptotic behavior of our testing procedure is established. A simulation study shows that the testing procedure has good size and power with finite sample sizes. We then apply our test to a data set provided by Tecator, which consists of near-infrared absorbance spectra and fat content of meat.

math.ST↗

arXiv2013

Limit Laws in Transaction-Level Asset Price Models

We consider pure-jump transaction-level models for asset prices in continuous time, driven by point processes. In a bivariate model that admits cointegration, we allow for time deformations to account for such effects as intraday seasonal patterns in volatility, and non-trading periods that may be different for the two assets. We also allow for asymmetries (leverage effects). We obtain the asymptotic distribution of the log-price process. We also obtain the asymptotic distribution of the ordinary least-squares estimator of the cointegrating parameter based on data sampled from an equally-spaced discretization of calendar time, in the case of weak fractional cointegration. For this same case, we obtain the asymptotic distribution for a tapered estimator under more

math.ST↗

arXiv2013

Functional Data Analysis with Increasing Number of Projections

Functional principal components (FPC's) provide the most important and most extensively used tool for dimension reduction and inference for functional data. The selection of the number, d, of the FPC's to be used in a specific procedure has attracted a fair amount of attention, and a number of reasonably effective approaches exist. Intuitively, they assume that the functional data can be sufficiently well approximated by a projection onto a finite-dimensional subspace, and the error resulting from such an approximation does not impact the conclusions. This has been shown to be a very effective approach, but it is desirable to understand the behavior of many inferential procedures by considering the projections on subspaces spanned by an increasing number of the FPC's. Such an approach reflects more fully the infinite-dimensional nature of functional data, and allows to derive procedures which are fairly insensitive to the selection of d. This is accomplished by considering limits as d tends to infinity with the sample size. We propose a specific framework in which we let d tend to infinity by deriving a normal approximation for the two-parameter partial sum process of the scores ξ_{i,j} of the i-th function with respect to the j-th FPC. Our approximation can be used to derive statistics that use segments of observations and segments of the FPC's. We apply our general results to derive two inferential procedures for the mean function: a change-point test and a two-sample test. In addition to the asymptotic theory, the tests are assessed through a small simulation study and a data example.

math.ST↗

arXiv2012

Asymptotics of trimmed CUSUM statistics

There is a wide literature on change point tests, but the case of variables with infinite variances is essentially unexplored. In this paper we address this problem by studying the asymptotic behavior of trimmed CUSUM statistics. We show that in a location model with i.i.d. errors in the domain of attraction of a stable law of parameter $0<α<2$, the appropriately trimmed CUSUM process converges weakly to a Brownian bridge. Thus, after moderate trimming, the classical method for detecting change points remains valid also for populations with infinite variance. We note that according to the classical theory, the partial sums of trimmed variables are generally not asymptotically normal and using random centering in the test statistics is crucial in the infinite variance case. We also show that the partial sums of truncated and trimmed random variables have different asymptotic behavior. Finally, we discuss resampling procedures which enable one to determine critical values in the case of small and moderate sample sizes.

math.ST↗

arXiv2011

Testing the Equality of Covariance Operators in Functional Samples

We propose a robust test for the equality of the covariance structures in two functional samples. The test statistic has a chi-square asymptotic distribution with a known number of degrees of freedom, which depends on the level of dimension reduction needed to represent the data. Detailed analysis of the asymptotic properties is developed. Finite sample performance is examined by a simulation study and an application to egg-laying curves of fruit flies.

stat.ME↗

arXiv2009

Break detection in the covariance structure of multivariate time series models

In this paper, we introduce an asymptotic test procedure to assess the stability of volatilities and cross-volatilites of linear and nonlinear multivariate time series models. The test is very flexible as it can be applied, for example, to many of the multivariate GARCH models established in the literature, and also works well in the case of high dimensionality of the underlying data. Since it is nonparametric, the procedure avoids the difficulties associated with parametric model selection, model fitting and parameter estimation. We provide the theoretical foundation for the test and demonstrate its applicability via a simulation study and an analysis of financial data. Extensions to multiple changes and the case of infinite fourth moments are also discussed.

math.ST↗

arXiv2008

Testing for changes in polynomial regression

We consider a nonlinear polynomial regression model in which we wish to test the null hypothesis of structural stability in the regression parameters against the alternative of a break at an unknown time. We derive the extreme value distribution of a maximum-type test statistic which is asymptotically equivalent to the maximally selected likelihood ratio. The resulting test is easy to apply and has good size and power, even in small samples.

math.ST↗