Research

2025

1. JOE

   On changepoint detection in functional data using empirical energy distance

   Boniece, B.C., Horváth, L., and Trapani, L.

   Journal of Econometrics (2025) [arXiv] [doi]

   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.

2. BEJ

   Data-driven fixed-point tuning for truncated realized variations

   Boniece, B.C., Figueroa-López, J. E., and Han, Y.

   To appear in Bernoulli (2025) [arXiv]

2024

1. AoAP

   On high-dimensional wavelet eigenanalysis

   Abry, P., Boniece, B. C., Didier, G., and Wendt, H.

   Annals of Applied Probability (2024) [arXiv] [doi]

   We study the scaling behavior of eigenvalues of large $p$-dimensional wavelet (sample covariance) random matrices, with the goal of statistically characterizing a low-dimensional ($r$-variate, $r\ll p$) scaling structure in the presence of additive high-dimensional noise. The asymptotic setting considered is where the sample size, $n$, dimension, $p(n)$, and scale, $a(n)$, tend to infinity such that $$\lim_{n\to\infty}\frac{p(n)a(n)}{n} = c\in[0,\infty).$$ Starting from possibly non-Gaussian, finite-variance measurements, correlated in time and between rows, we show that the $r$ largest eigenvalues of wavelet random matrices display asymptotically scale invariant behavior, and also exhibit asymptotically Gaussian distributions up to a log transformation.

2. SPA

   Efficient integrated volatility estimation in the presence of infinite variation jumps via debiased truncated realized variations

   Boniece, B.C., Figueroa-López, J. E., and Han, Y.

   Stochastic Processes and their Applications (2024) [arXiv] [doi]

   The integrated variance of an Itô semimartingale plays a special role in financial econometrics, and its estimation has attracted considerable research attention over the past two decades. In the high-frequency sampling limit, the integrated variance can be estimated efficiently in a variety of ways when the jump activity of the underlying process is of finite variation. However, many well-known estimation methods either completely fail or display slow convergence rates when jumps are of unbounded variation, rendering them impractical for many applications. In this work, we propose an estimation method for a broad class of Itô semimartingales with locally stable jumps that achieves optimal asymptotic behavior even when jumps are of unbounded variation. Our approach is based on modifying an existing estimator known as the truncated realized variance (TRV), defined as $$ \widehat{C}_{n}(\varepsilon)=\sum_{i=1}^{n}(\Delta_i^n X )^2{\bf 1}_{\{|\Delta_i^n X|\leq\varepsilon\}}, $$ where $X$ is the process observed at times $t_0<\ldots<t_n$, $\Delta_i^n X =X_{t_i}-X_{t_{i-1}}$, and $\varepsilon \to 0$ at a suitable rate as $n\to\infty$. Specifically, we augment the TRV $\widehat{C}_{n}(\varepsilon)$ using a two-step procedure that estimates bias from the available data and removes it, thereby allowing for efficient estimation provided the underlying jump index is less than $8/5$. Through Monte Carlo studies, we show our method can outperform the only other existing efficient method in the literature.

2023

1. TEST

   Change-point detection in high-dimensional data with U-statistics

   Boniece, B.C., Horváth, L., and Jacobs, P.

   TEST (2023) [arXiv] [doi]

   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.

2022

1. SISP

   Wavelet eigenvalue regression in high dimensions

   Abry, P., Boniece, B. C., Didier, G., and Wendt, H.

   Statistical Inference for Stochastic Processes (2022) [arXiv] [doi]

   We propose a wavelet eigenvalue regression methodology for a high dimensional setting. Starting from the $p$-dimensional high-dimensional fractional signal-plus-noise model considered in the companion work (Abry, Boniece, Didier and Wendt (2021)), we propose a multiscale wavelet eigenvalue regression estimator $$ \widehat{\ell}_{i} := \frac{1}{2}\Big(\sum_{j=j_1}^{j_2} w_j \log_2 \lambda_{i}\big({\mathbf W}(a(n)2^j)\big) -1\Big),\quad i=1,\ldots,p, $$ for fixed octaves $j_1$, $j_2$ and regression weights $w_j$, where ${\mathbf W}(a(n)2^j)$ denotes the wavelet (sample covariance) random matrix at scale $a(n)2^j$, and $\lambda_i(\cdot)$ its $i^{th}$ eigenvalue in nondecreasing order. In a high-dimensional large-scale limit, we show that the fixed-dimensional subvector $$ \{\widehat{h}_{q}\}_{q=1,\ldots,r} := \{ \widehat{\ell}_{i} \}_{i = p-r+1,\ldots,p} $$ is a consistent and, under additional assumptions, asymptotically Gaussian estimator of the scaling eigenvalue structure $\{h_q\}_{q=1,\ldots,r}$ of the model, where $r$ denotes the effective dimension of the underlying process. We further construct a consistent estimator of the effective dimension $r$ that significantly increases the robustness of the methodology. The finite-sample performance of the proposed estimators are studied through simulations.

2. AAP

   Operator fractional Lévy motion: integral representations and time reversibility

   Boniece, B. C. and Didier, G.

   Advances in Applied Probability (2022) [arXiv] [doi]

   A stochastic process $X=\{X(t)\}_{t\in\mathbb R}$ is called covariance operator self-similar (cov.-o.s.s.) if $$ \text{Cov}(X(cs),X(ct)) = c^{H} \text{Cov}(X(s),X(t))c^{H^*}, \quad s,t \in \mathbb R, \quad c > 0, $$ for some matrix $H$ satisfying $\Re(\text{eig}(H)) \subseteq (0,1]$. In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are cov-o.s.s. and have wide-sense stationary increments. The ofLm can be viewed as a Lévy-driven extension of operator fractional Brownian motion, or also as a cov-o.s.s. extension of the fractional Lévy motion, and consists of two distinct subclasses: the harmonizable and the moving-average types. We show that both types of ofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We characterize time reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that the parametric conditions for time reversibility are generally more restrictive than those for the Gaussian case (ofBm).

2021

1. ACHA

   Tempered fractional Brownian motion: wavelet estimation, modeling and testing

   Boniece, B. C., Didier, G., and Sabzikar, F.

   Applied and Computational Harmonic Analysis (2021) [arXiv] [doi]

   Tempered fractional Brownian motion (tfBm) is a modification of fractional Brownian motion (fBm) that displays power-law scaling at moderate frequencies but remains bounded at low frequencies. More precisely, for a tfBm $X$ with Hurst and tempering parameters $H$ and $\lambda>0$, respectively, its autocovariance $\gamma_X(h) = \mathbb E X(t)X(t+h)$, $h \in \mathbb Z$, has decay $$ \gamma_X(h) \sim C\frac{h^{H-1/2}}{ e^{\lambda h}}, \quad |h| \rightarrow \infty, $$ for some constant $C>0$ depending on $X$. We use wavelets to construct the first estimation method for tfBm and a simple and computationally efficient test for fBm vs tfBm alternatives. The properties of the wavelet estimator and test are mathematically and computationally established. An application of the methodology shows that tfBm is a better model than fBm for a geophysical flow data set.

2020

1. JSP

   On fractional Lévy processes: tempering, sample path properties and stochastic integration

   Boniece, B. C., Didier, G., and Sabzikar, F.

   Journal of Statistical Physics (2020) [arXiv] [doi]

   We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP II, respectively). TFLP and TFLP II make up very broad finite-variance, generally non-Gaussian families of models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Lévy process. We establish the sample path properties of TFLP and TFLP II. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP II, which may not be semi-martingales depending on the value of the memory parameter and choice of marginal distribution.

2019

1. CAMSAP

   Wavelet-based detection and estimation of fractional Lévy signals in high dimensions

   Boniece, B. C., Wendt, H., Didier, G., and Abry, P.

   8th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP) (2019) [doi] [pdf]

   We put forward a statistical methodology for identifying the number of multivariate self-similar, Lévy-driven components immersed in high-dimensional noise, as well as for estimating the underlying scaling exponents. It relies on the analysis of the evolution over scales of the eigenvalues of random wavelet matrices. The mathematical framework further allows us to analyze the impact of the tails of the Lévy noise marginal distribution on the estimation performance.

2. EUSIPCO

   On multivariate non-Gaussian scale invariance: fractional Lévy processes and wavelet estimation

   Boniece, B. C., Wendt, H., Didier, G., and Abry, P.

   European Signal Processing Conference (EUSIPCO) (2019) [doi] [pdf]

   We examine the performance of joint wavelet eigenanalysis estimation for the Hurst parameters (scaling exponents) of non-Gaussian multivariate processes. We propose a new process called operator fractional Lévy motion (ofLm) as a Lévy-type model or non-Gaussian multivariate scale invariance.

2018

1. SSP

   Tempered fractional Brownian motion: wavelet estimation and modeling of turbulence in geophysical flows

   Boniece, B. C., Sabzikar, F., and Didier, G.

   IEEE Statistical Signal Processing Workshop (SSP) (2018) [doi] [pdf]

   This paper introduces a wavelet framework to construct the first estimation method for tempered fractional Brownian motion (tfBm). The properties of the wavelet coefficients and spectrum of tfBm are studied, and estimation performance is assessed by means of Monte Carlo experiments.