Agnieszka Jach

Subsampling inference for the mean of heavy-tailed long-memory time series

In this article, we revisit a time series model introduced by MCElroy and Politis (2007a) and generalize it in several ways to encompass a wider class of stationary, nonlinear, heavy‐tailed time series with long memory. The joint asymptotic distribution for the sample mean and sample variance under the extended model is derived; the associated convergence rates are found to depend crucially on the tail thickness and long memory parameter. A self‐normalized sample mean that concurrently captures the tail and memory behaviour, is defined. Its asymptotic distribution is approximated by subsampling without the knowledge of tail or/and memory parameters; a result of independent interest regarding subsampling consistency for certain long‐range dependent processes is provided. The subsampling‐based confidence intervals for the process mean are shown to have good empirical coverage rates in a simulation study. The influence of block size on the coverage and the performance of a data‐driven rule for block size selection are assessed. The methodology is further applied to the series of packet‐counts from ethernet traffic traces.

Wavelet-Based Index of Magnetic Storm Activity

[1] A wavelet-based method of computing an index of storm activity is put forward. The new index can be computed automatically using statistical procedures and does not require selecting quiet days and removing the secular component by polynomial fitting. This 1-min index is designed to facilitate the study of the fine structure of geomagnetic storm events and requires only the most recent magnetogram records, e.g., the 2 months including the storm event of interest. It can thus be computed over a moving window as soon as new magnetogram records become available. Averaged over 1-hour periods, it is practically indistinguishable from the standard Dst index.

Wavelet-domain test for long-range dependence in the presence of a trend

We propose a test to distinguish a weakly-dependent time series with a trend component, from a long-memory process, possibly with a trend. The test uses a generalized likelihood ratio statistic based on wavelet domain likelihoods. The trend is assumed to be a polynomial whose order does not exceed a known value. The test is robust to trends which are piecewise polynomials. We study the empirical size and power by means of simulations and find that they are good and do not depend on specific choices of wavelet functions and models for the wavelet coefficients. The test is applied to annual minima of the Nile River and confirms the presence of long-range dependence in this time series.

Subsampling inference for the autocovariances and autocorrelations of long‐memory heavy‐ tailed linear time series

We provide a self‐normalization for the sample autocovariances and autocorrelations of a linear, long‐memory time series with innovations that have either finite fourth moment or are heavy‐tailed with tail index 2 d and α, and which consequently lead to three different limit distributions; for the sample autocorrelation the limit distribution only depends on d. We introduce a self‐normalized sample autocovariance statistic, which is computable without knowledge of α or d (or their relationship), and which converges to a non‐degenerate distribution. We also treat self‐normalization of the autocorrelations. The sampling distributions can then be approximated non‐parametrically by subsampling, as the corresponding asymptotic distribution is still parameter‐dependent. The subsampling‐based confidence intervals for the process autocovariances and autocorrelations are shown to have satisfactory empirical coverage rates in a simulation study. The impact of subsampling block size on the coverage is assessed. The methodology is further applied to the log‐squared returns of Merck stock.

Tail index estimation in the presence of long-memory dynamics

Most tail index estimators are formulated under assumptions of weak serial dependence, but nevertheless are applied in practice to long-range dependent time series data. This issue arises because for many time series found in teletraffic and financial econometric applications, both heavy tails and long memory are prevalent features. For a certain class of Heavy-Tail Long-Memory (HTLM) processes, McElroy and Politis (2007a) and Jach et al. (2011) found that the probabilistic behavior of the sample mean depends delicately on the interplay of the tail index and the long memory parameter. In contrast, results in Kulik and Soulier (2011) indicate that the sample quantiles for a related HTLM process are unaffected by long-range dependence. Motivated by these results, we undertake an extensive numerical study to compare the finite-sample performance of several tail index estimators-both those based on sample quantiles, such as the Hill and DEdH (Hill (1975) and Dekkers et al. (1989)) as well as those based on moments, e.g. Meerschaert and Scheffler (1998)-in the HTLM context. Our results largely confirm and expand those of Kulik and Soulier (2011), in that the Hill and DEdH estimators perform well despite the presence of long memory.

Wavelet-based confidence intervals for the self-similarity parameter

We propose and compare several methods of constructing wavelet-based confidence intervals for the self-similarity parameter in heavy-tailed observations. We use empirical coverage probabilities to assess the procedures by applying them to Linear Fractional Stable Motion with many choices of parameters. We find that the asymptotic confidence intervals provide empirical coverage often much lower than nominal. We recommend the use of resampling confidence intervals. We also propose a procedure for monitoring the constancy of the self-similarity parameter and apply it to Ethernet data sets.

Assessing scale-wise similarity of curves with a thick pen: As illustrated through comparisons of spectral irradiance

Abstract Forest canopies create dynamic light environments in their understorey, where spectral composition changes among patterns of shade and sunflecks, and through the seasons with canopy phenology and sun angle. Plants use spectral composition as a cue to adjust their growth strategy for optimal resource use. Quantifying the ever‐changing nature of the understorey light environment is technically challenging with respect to data collection. Thus, to capture the simultaneous variation occurring in multiple regions of the solar spectrum, we recorded spectral irradiance from forest understoreys over the wavelength range 300–800 nm using an array spectroradiometer. It is also methodologically challenging to analyze solar spectra because of their multi‐scale nature and multivariate lay‐out. To compare spectra, we therefore used a novel method termed thick pen transform (TPT), which is simple and visually interpretable. This enabled us to show that sunlight position in the forest understorey (i.e., shade, semi‐shade, or sunfleck) was the most important factor in determining shape similarity of spectral irradiance. Likewise, the contributions of stand identity and time of year could be distinguished. Spectra from sunflecks were consistently the most similar, irrespective of differences in global irradiance. On average, the degree of cross‐dependence increased with increasing scale, sometimes shifting from negative (dissimilar) to positive (similar) values. We conclude that the interplay of sunlight position, stand identity, and date cannot be ignored when quantifying and comparing spectral composition in forest understoreys. Technological advances mean that array spectroradiometers, which can record spectra contiguously over very short time intervals, are being widely adopted, not only to measure irradiance under pollution, clouds, atmospheric changes, and in biological systems, but also spectral changes at small scales in the photonics industry. We consider that TPT is an applicable method for spectral analysis in any field and can be a useful tool to analyze large datasets in general.

Change Point Detection Based on Empirical Quantiles

The paper studies the problem of testing the null hypothesis that the observations X 1,…, X n have the same distribution or follow the same stochastic model versus the alternative that the distribution or the model change at some point in the sample. Using L 1, L 2 and max type statistics based on empirical quantiles, the validity of the subsampling method for independent observations and GARCH processes is verified. A simulation study assesses the performance of the tests in finite samples.

Wavelet semi-parametric inference for long memory in volatility in the presence of a trend

ABSTRACT Risk of investing in a financial asset is quantified by functionals of squared returns. Discrete time stochastic volatility (SV) models impose a convenient and practically relevant time series dependence structure on the log-squared returns. Different long-term risk characteristics are postulated by short-memory SV and long-memory SV models. It is therefore important to test which of these two alternatives is suitable for a specific asset. Most standard tests are confounded by deterministic trends. This paper introduces a new, wavelet-based, test of the null hypothesis of short versus long memory in volatility which is robust to deterministic trends. In finite samples, the test performs better than currently available tests which are based on the Fourier transform.