arXiv subjects

Josua Gösmann

Publications and source records attributed to Josua Gösmann.

5 recordsLinked to original sources

arXiv2021

Relevant change points in high dimensional time series

This paper investigates the problem of detecting relevant change points in the mean vector, say $μ_t =(μ_{1,t},\ldots ,μ_{d,t})^T$ of a high dimensional time series $(Z_t)_{t\in \mathbb{Z}}$. While the recent literature on testing for change points in this context considers hypotheses for the equality of the means $μ_h^{(1)}$ and $μ_h^{(2)}$ before and after the change points in the different components, we are interested in a null hypothesis of the form $$ H_0: |μ^{(1)}_{h} - μ^{(2)}_{h} | \leq Δ_h ~~~\mbox{ for all } ~~h=1,\ldots ,d $$ where $Δ_1, \ldots , Δ_d$ are given thresholds for which a smaller difference of the means in the $h$-th component is considered to be non-relevant. We propose a new test for this problem based on the maximum of squared and integrated CUSUM statistics and investigate its properties as the sample size $n$ and the dimension $d$ both converge to infinity. In particular, using Gaussian approximations for the maximum of a large number of dependent random variables, we show that on certain points of the boundary of the null hypothesis a standardised version of the maximum converges weakly to a Gumbel distribution.

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arXiv2020

Sequential change point detection in high dimensional time series

Change point detection in high dimensional data has found considerable interest in recent years. Most of the literature either designs methodology for a retrospective analysis, where the whole sample is already available when the statistical inference begins, or considers online detection schemes controlling the average time until a false alarm. This paper takes a different point of view and develops monitoring schemes for the online scenario, where high dimensional data arrives successively and the goal is to detect changes as fast as possible controlling at the same time the probability of a type I error of a false alarm. We develop a sequential procedure capable of detecting changes in the mean vector of a successively observed high dimensional time series with spatial and temporal dependence. The statistical properties of the method are analyzed in the case where both, thesample size and dimension tend to infinity. In this scenario, it is shown that the new monitoring scheme has asymptotic level alpha under the null hypothesis of no change and is consistent under the alternative of a change in at least one component of the high dimensional mean vector. The approach is based on a new type of monitoring scheme for one-dimensional data which turns out to be often more powerful than the usually used CUSUM and Page-CUSUM methods, and the component-wise statistics are aggregated by the maximum statistic. For the analysis of the asymptotic properties of our monitoring scheme we prove that the range of a Brownian motion on a given interval is in the domain of attraction of the Gumbel distribution, which is a result of independent interest in extreme value theory. The finite sample properties of the new methodology are illustrated by means of a simulation study and in the analysis of a data example.

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arXiv2020

A new approach for open-end sequential change point monitoring

We propose a new sequential monitoring scheme for changes in the parameters of a multivariate time series. In contrast to procedures proposed in the literature which compare an estimator from the training sample with an estimator calculated from the remaining data, we suggest to divide the sample at each time point after the training sample. Estimators from the sample before and after all separation points are then continuously compared calculating a maximum of norms of their differences. For open-end scenarios our approach yields an asymptotic level $α$ procedure, which is consistent under the alternative of a change in the parameter. By means of a simulation study it is demonstrated that the new method outperforms the commonly used procedures with respect to power and the feasibility of our approach is illustrated by analyzing two data examples.

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arXiv2018

A likelihood ratio approach to sequential change point detection for a general class of parameters

In this paper we propose a new approach for sequential monitoring of a parameter of a $d$-dimensional time series, which can be estimated by approximately linear functionals of the empirical distribution function. We consider a closed-end-method, which is motivated by the likelihood ratio test principle and compare the new method with two alternative procedures. We also incorporate self-normalization such that estimation of the long-run variance is not necessary. We prove that for a large class of testing problems the new detection scheme has asymptotic level $α$ and is consistent. The asymptotic theory is illustrated for the important cases of monitoring a change in the mean, variance and correlation. By means of a simulation study it is demonstrated that the new test performs better than the currently available procedures for these problems.Finally the methodology is illustrated by a small data example investigating index prices from the dot-com bubble.

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arXiv2017

Optimal designs for regression with spherical data

In this paper optimal designs for regression problems with spherical predictors of arbitrary dimension are considered. Our work is motivated by applications in material sciences, where crystallographic textures such as the missorientation distribution or the grain boundary distribution (depending on a four dimensional spherical predictor) are represented by series of hyperspherical harmonics, which are estimated from experimental or simulated data. For this type of estimation problems we explicitly determine optimal designs with respect to Kiefers $Φ_p$-criteria and a class of orthogonally invariant information criteria recently introduced in the literature. In particular, we show that the uniform distribution on the $m$-dimensional sphere is optimal and construct discrete and implementable designs with the same information matrices as the continuous optimal designs. Finally, we illustrate the advantages of the new designs for series estimation by hyperspherical harmonics, which are symmetric with respect to the first and second crystallographic point group.

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