arXiv subjects

Carsten Jentsch

Publications and source records attributed to Carsten Jentsch.

25 records · Page 2Linked to original sources

arXiv2023

Bootstrap Consistency for the Mack Bootstrap

Mack's distribution-free chain ladder reserving model belongs to the most popular approaches in non-life insurance mathematics. Proposed to determine the first two moments of the reserve, it does not allow to identify the whole distribution of the reserve. For this purpose, Mack's model is usually equipped with a tailor-made bootstrap procedure. Although widely used in practice to estimate the reserve risk, no theoretical bootstrap consistency results exist that justify this approach. To fill this gap in the literature, we adopt the framework proposed by Steinmetz and Jentsch (2022) to derive asymptotic theory in Mack's model. By splitting the reserve into two parts corresponding to process and estimation uncertainty, this enables - for the first time - a rigorous investigation also of the validity of the Mack bootstrap. We prove that the (conditional) distribution of the asymptotically dominating process uncertainty part is correctly mimicked by Mack's bootstrap if the parametric family of distributions of the individual development factors is correctly specified. Otherwise, this is not the case. In contrast, the (conditional) distribution of the estimation uncertainty part is generally not correctly captured by Mack's bootstrap. To tackle this, we propose an alternative Mack-type bootstrap, which is designed to capture also the distribution of the estimation uncertainty part. We illustrate our findings by simulations and show that the newly proposed alternative Mack bootstrap performs superior to the Mack bootstrap.

math.ST↗

arXiv2022

Testing exogeneity in the functional linear regression model

We propose a novel test statistic for testing exogeneity in the functional linear regression model. In contrast to Hausman-type tests in finite dimensional linear regression setups, a direct extension to the functional linear regression model is not possible. Instead, we propose a test statistic based on the sum of the squared difference of projections of the two estimators for testing the null hypothesis of exogeneity in the functional linear regression model. We derive asymptotic normality under the null and consistency under general alternatives. Moreover, we prove bootstrap consistency results for residual-based bootstraps. In simulations, we investigate the finite sample performance of the proposed testing approach and illustrate the superiority of bootstrap-based approaches. In particular, the bootstrap approaches turn out to be much more robust with respect to the choice of the regularization parameter.

math.ST↗

arXiv2022

A Bootstrap-Assisted Self-Normalization Approach to Inference in Cointegrating Regressions

Traditional inference in cointegrating regressions requires tuning parameter choices to estimate a long-run variance parameter. Even in case these choices are "optimal", the tests are severely size distorted. We propose a novel self-normalization approach, which leads to a nuisance parameter free limiting distribution without estimating the long-run variance parameter directly. This makes our self-normalized test tuning parameter free and considerably less prone to size distortions at the cost of only small power losses. In combination with an asymptotically justified vector autoregressive sieve bootstrap to construct critical values, the self-normalization approach shows further improvement in small to medium samples when the level of error serial correlation or regressor endogeneity is large. We illustrate the usefulness of the bootstrap-assisted self-normalized test in empirical applications by analyzing the validity of the Fisher effect in Germany and the United States.

econ.EM↗

arXiv2020

Random boosting and random^2 forests -- A random tree depth injection approach

The induction of additional randomness in parallel and sequential ensemble methods has proven to be worthwhile in many aspects. In this manuscript, we propose and examine a novel random tree depth injection approach suitable for sequential and parallel tree-based approaches including Boosting and Random Forests. The resulting methods are called \emph{Random Boost} and \emph{Random$^2$ Forest}. Both approaches serve as valuable extensions to the existing literature on the gradient boosting framework and random forests. A Monte Carlo simulation, in which tree-shaped data sets with different numbers of final partitions are built, suggests that there are several scenarios where \emph{Random Boost} and \emph{Random$^2$ Forest} can improve the prediction performance of conventional hierarchical boosting and random forest approaches. The new algorithms appear to be especially successful in cases where there are merely a few high-order interactions in the generated data. In addition, our simulations suggest that our random tree depth injection approach can improve computation time by up to 40%, while at the same time the performance losses in terms of prediction accuracy turn out to be minor or even negligible in most cases.

stat.ML↗

arXiv2020

Improving Reliability of Latent Dirichlet Allocation by Assessing Its Stability Using Clustering Techniques on Replicated Runs

For organizing large text corpora topic modeling provides useful tools. A widely used method is Latent Dirichlet Allocation (LDA), a generative probabilistic model which models single texts in a collection of texts as mixtures of latent topics. The assignments of words to topics rely on initial values such that generally the outcome of LDA is not fully reproducible. In addition, the reassignment via Gibbs Sampling is based on conditional distributions, leading to different results in replicated runs on the same text data. This fact is often neglected in everyday practice. We aim to improve the reliability of LDA results. Therefore, we study the stability of LDA by comparing assignments from replicated runs. We propose to quantify the similarity of two generated topics by a modified Jaccard coefficient. Using such similarities, topics can be clustered. A new pruning algorithm for hierarchical clustering results based on the idea that two LDA runs create pairs of similar topics is proposed. This approach leads to the new measure S-CLOP ({\bf S}imilarity of multiple sets by {\bf C}lustering with {\bf LO}cal {\bf P}runing) for quantifying the stability of LDA models. We discuss some characteristics of this measure and illustrate it with an application to real data consisting of newspaper articles from \textit{USA Today}. Our results show that the measure S-CLOP is useful for assessing the stability of LDA models or any other topic modeling procedure that characterize its topics by word distributions. Based on the newly proposed measure for LDA stability, we propose a method to increase the reliability and hence to improve the reproducibility of empirical findings based on topic modeling. This increase in reliability is obtained by running the LDA several times and taking as prototype the most representative run, that is the LDA run with highest average similarity to all other runs.

cs.CL↗

arXiv2015

Covariance matrix estimation and linear process bootstrap for multivariate time series of possibly increasing dimension

Multivariate time series present many challenges, especially when they are high dimensional. The paper's focus is twofold. First, we address the subject of consistently estimating the autocovariance sequence; this is a sequence of matrices that we conveniently stack into one huge matrix. We are then able to show consistency of an estimator based on the so-called flat-top tapers; most importantly, the consistency holds true even when the time series dimension is allowed to increase with the sample size. Second, we revisit the linear process bootstrap (LPB) procedure proposed by McMurry and Politis [J. Time Series Anal. 31 (2010) 471-482] for univariate time series. Based on the aforementioned stacked autocovariance matrix estimator, we are able to define a version of the LPB that is valid for multivariate time series. Under rather general assumptions, we show that our multivariate linear process bootstrap (MLPB) has asymptotic validity for the sample mean in two important cases: (a) when the time series dimension is fixed and (b) when it is allowed to increase with sample size. As an aside, in case (a) we show that the MLPB works also for spectral density estimators which is a novel result even in the univariate case. We conclude with a simulation study that demonstrates the superiority of the MLPB in some important cases.

math.ST↗

arXiv2015

Testing equality of spectral densities using randomization techniques

In this paper, we investigate the testing problem that the spectral density matrices of several, not necessarily independent, stationary processes are equal. Based on an $L_2$-type test statistic, we propose a new nonparametric approach, where the critical values of the tests are calculated with the help of randomization methods. We analyze asymptotic exactness and consistency of these randomization tests and show in simulation studies that the new procedures posses very good size and power characteristics.

math.ST↗