Sieve-based confidence intervals and bands for L\'{e}vy densities

The estimation of the L\'{e}vy density, the infinite-dimensional parameter controlling the jump dynamics of a L\'{e}vy process, is considered here under a discrete-sampling scheme. In this setting, the jumps are latent variables, the statistical properties of which can be assessed when the frequency and time horizon of observations increase to infinity at suitable rates. Nonparametric estimators for the L\'{e}vy density based on Grenander's method of sieves was proposed in Figueroa-L\'{o}pez [IMS Lecture Notes 57 (2009) 117--146]. In this paper, central limit theorems for these sieve estimators, both pointwise and uniform on an interval away from the origin, are obtained, leading to pointwise confidence intervals and bands for the L\'{e}vy density. In the pointwise case, our estimators converge to the L\'{e}vy density at a rate that is arbitrarily close to the rate of the minimax risk of estimation on smooth L\'{e}vy densities. In the case of uniform bands and discrete regular sampling, our results are consistent with the case of density estimation, achieving a rate of order arbitrarily close to $\log^{-1/2}(n)\cdot n^{-1/3}$, where $n$ is the number of observations. The convergence rates are valid, provided that $s$ is smooth enough and that the time horizon $T_n$ and the dimension of the sieve are appropriately chosen in terms of $n$.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ286 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Multiscale change-point segmentation: beyond step functions.

Modern multiscale type segmentation methods are known to detect multiple change-points with high statistical accuracy, while allowing for fast computation. Underpinning (minimax) estimation theory has been developed mainly for models that assume the signal as a piecewise constant function. In this paper, for a large collection of multiscale segmentation methods (including various existing procedures), such theory will be extended to certain function classes beyond step functions in a nonparametric regression setting. This extends the interpretation of such methods on the one hand and on the other hand reveals these methods as robust to deviation from piecewise constant functions. Our main finding is the adaptation over nonlinear approximation classes for a universal thresholding, which includes bounded variation functions, and (piecewise) Holder functions of smoothness order 0 < alpha <= 1 as special cases. From this we derive statistical guarantees on feature detection in terms of jumps and modes. Another key finding is that these multiscale segmentation methods perform nearly (up to a log-factor) as well as the oracle piecewise constant segmentation estimator (with known jump locations), and the best piecewise constant approximants of the (unknown) true signal. Theoretical findings are examined by various numerical simulations

Consistencies and rates of convergence of jump-penalized least squares estimators

We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in $L^2([0,1))$ our results cover other metrics like Skorokhod metric on the space of c\`{a}dl\`{a}g functions and uniform metrics on $C([0,1])$. We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) H\"{o}lder continuous functions of order $0<\alpha\le1$ and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.Comment: Published in at http://dx.doi.org/10.1214/07-AOS558 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Statistical Skorohod embedding problem and its generalizations

Given a L\'evy process $L$, we consider the so-called statistical Skorohod embedding problem of recovering the distribution of an independent random time $T$ based on i.i.d. sample from $L_{T}.$ Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose a consistent estimator for the density of $T,$ derive its convergence rates and prove their optimality. It turns out that the convergence rates heavily depend on the decay of the Mellin transform of $T.$ We also consider the application of our results to the problem of statistical inference for variance-mean mixture models and for time-changed L\'evy processes

Symmetry properties and spectra of the two-dimensional quantum compass model

We use exact symmetry properties of the two-dimensional quantum compass model to derive nonequivalent invariant subspaces in the energy spectra of $L\times L$ clusters up to L=6. The symmetry allows one to reduce the original $L\times L$ compass cluster to the $(L-1)\times (L-1)$ one with modified interactions. This step is crucial and enables: (i) exact diagonalization of the $6\times 6$ quantum compass cluster, and (ii) finding the specific heat for clusters up to L=6, with two characteristic energy scales. We investigate the properties of the ground state and the first excited states and present extrapolation of the excitation energy with increasing system size. Our analysis provides physical insights into the nature of nematic order realized in the quantum compass model at finite temperature. We suggest that the quantum phase transition at the isotropic interaction point is second order with some admixture of the discontinuous transition, as indicated by the entropy, the overlap between two types of nematic order (on horizontal and vertical bonds) and the existence of the critical exponent. Extrapolation of the specific heat to the $L\to\infty$ limit suggests the classical nature of the quantum compass model and high degeneracy of the ground state with nematic order.Comment: 15 pages, 12 figures; accepted for publication in Physical Review