TY - JOUR
T1 - Recovery of periodicities hidden in heavy-tailed noise
AU - Karabash, Illya M.
AU - Prestin, Jürgen
PY - 2018/1/1
Y1 - 2018/1/1
N2 - We address a parametric joint detection-estimation problem for discrete signals of the form x(t) = ΣN
n=1αne-iλnt + ϵt, t ∈ N, with an additive noise represented by independent centered complex random variables ϵt. The distributions of ϵt are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy-tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., frequencies λn, their number N, and complex coefficients αn. For example, one of considered classes of noise is the following: ϵt are independent identically distributed random variables with E(ϵt)=0 and E피(|ϵt|ln|ϵt|) < ∞. The construction of estimators is based on detection of singularities of anti-derivatives for Z-transforms and on a two-level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series.
AB - We address a parametric joint detection-estimation problem for discrete signals of the form x(t) = ΣN
n=1αne-iλnt + ϵt, t ∈ N, with an additive noise represented by independent centered complex random variables ϵt. The distributions of ϵt are assumed to be unknown, but satisfying various sets of conditions. We prove that in the case of a heavy-tailed noise it is possible to construct asymptotically strongly consistent estimators for the unknown parameters of the signal, i.e., frequencies λn, their number N, and complex coefficients αn. For example, one of considered classes of noise is the following: ϵt are independent identically distributed random variables with E(ϵt)=0 and E피(|ϵt|ln|ϵt|) < ∞. The construction of estimators is based on detection of singularities of anti-derivatives for Z-transforms and on a two-level selection procedure for special discretized versions of superlevel sets. The consistency proof relies on the convergence theory for random Fourier series.
UR - http://www.scopus.com/inward/record.url?scp=85026442380&partnerID=8YFLogxK
U2 - 10.1002/mana.201600361
DO - 10.1002/mana.201600361
M3 - Journal articles
AN - SCOPUS:85026442380
SN - 0025-584X
VL - 291
SP - 86
EP - 102
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
IS - 1
ER -