TY  - JOUR
AB  - Consider the sum Y = B + B(H) of a Brownian motion B and an independent fractional Brownian motion B(H) with Hurst parameter H ∈ (0, 1). Even though B(H) is not a semimartingale, it was shown by Cheridito (Bernoulli 7 (2001) 913–934) that Y is a semimartingale if H > 3/4. Moreover, Y is locally equivalent to B in this case, so H cannot be consistently estimated from local observations of Y. This paper pivots on another unexpected feature in this model: if B and B(H) become correlated, then Y will never be a semimartingale, and H can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called mixed semimartingales, which are semiparametric extensions of Y with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense.
AU  - Chong, C.H.*
AU  - Delerue, T.
AU  - Mies, F.*
C1  - 73624
C2  - 57145
SP  - 219-244
TI  - Rate-optimal estimation of mixed semimartingales.
JO  - Ann. Stat.
VL  - 53
IS  - 1
PY  - 2025
SN  - 0090-5364
ER  - 

TY  - JOUR
AB  - 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 cadlag 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) Holder continuous functions of order 0 < alpha <= 1 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.
AU  - Boysen, L.*
AU  - Kempe, A.
AU  - Liebscher, V.*
AU  - Munk, A.*
AU  - Wittich, O.*
C1  - 433
C2  - 26416
SP  - 157-183
TI  - Consistencies and rates of convergence of jump-penalized least squares estimators.
JO  - Ann. Stat.
VL  - 37
IS  - 1
PB  - Inst Mathematical Statistics
PY  - 2009
SN  - 0090-5364
ER  -