Abstract
This paper rigourously introduces the asymptotic concept of high dimensional efficiency which quantifies the detection power of different statistics in high dimensional multivariate settings. It allows for comparisons of different high dimensional methods with different null asymptotics and even different asymptotic behavior such as extremal-type asymptotics. The concept will be used to understand the power behavior of different test statistics as the performance will greatly depend on the assumptions made, such as sparseness or denseness of the signal. The effect of misspecification of the covariance on the power of the tests is also investigated, because in many high dimensional situations estimation of the full dependency (covariance) between the multivariate observations in the panel is often either computationally or even theoretically infeasible. The theoretic quantification by the theory is accompanied by simulation results which confirm the theoretic (asymptotic) findings for surprisingly small samples. The development of this concept was motivated by, but is by no means limited to, high-dimensional change point tests. It is shown that the concept of high dimensional efficiency is indeed suitable to describe small sample power.
Highlights
There has recently been a renaissance in research for statistical methods for change point problems (Horvath and Rice, 2014)
Kirch points in a wide variety of settings, such as multivariate (Horvath et al, 1999; Ombao et al, 2005; Aue et al, 2009b; Kirch et al, 2015) functional (Berkes et al, 2009; Aue et al, 2009a; Hormann and Kokoszka, 2010; Aston and Kirch, 2012a; Torgovitski, 2015) and high dimensional settings (Bai, 2010; Horvath and Huskova, 2012; Chan et al, 2012; Enikeeva and Harchaoui, 2013; Cho and Fryzlewicz, 2015) have recently been proposed. These include methods based on taking maxima statistics across panels coordinate-wise (Jirak, 2015), use of scan statistic approaches (Enikeeva and Harchaoui, 2013), using sparsified binary segmentation for multiple change point detection (Cho and Fryzlewicz, 2015), uses of double CUSUM procedures (Cho, 2015), as well as those based on structural assumptions such as sparsity (Wang and Samworth, 2016)
The new concept of high dimensional efficiency allows a comparison of the magnitude of changes that can be detected asymptotically as the number of dimensions increases
Summary
There has recently been a renaissance in research for statistical methods for change point problems (Horvath and Rice, 2014). The panel data ( known as “small n large p” or “high dimensional low sample size”) setting is able to capture the small sample properties very well in situations where d is comparable or even larger than T using asymptotic considerations In this asymptotic framework the detection ability or efficiency of various tests can be defined by the rates at which vanishing alternatives can still be detected. Rather than a separate simulation section, simulations will be interspersed throughout the theory They complement the theoretic results, confirming that the conclusion are already valid for small samples, verifying that the concept of high-dimensional efficiency is suitable to understand the power behavior of different test statistics.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
