Statistics,2018, 52(5): 1156-1192.

Liu G, Zhang L, Wang M. 

DOI: https://doi.org/10.1080/02331888.2018.1500578

Abstract 

In this paper, we investigate the asymptotic properties of the threshold multipower variation, based on the generalized increments, for a fractional process with jumps and noise ∫0tbsds+∫0tσsdBsH+Jt+ϵt, where b={bt,t≥0} is a drift process, BH={BtH,t≥0} is a fractional Brownian motion with the Hurst parameter H∈(0,1), σ={σt,t≥0} is a stochastic process with paths of finite p-variation for p<1/(1−H), J={Jt,t≥0} is a jump process, and ϵ={ϵt,t≥0} is a noise process independent of the signal process ∫0tbsds+∫0tσsdBsH+Jt. We obtain the large number laws and the corresponding central limit theorems for the generalized threshold multipower variation. We apply these theorems to estimate H in the presence of jumps and noise and obtain the large number laws and the central limit theorems of the estimator. We observe that all of the central limit theorems have the same convergence rate for all domain H∈(0,1). Simulations are conducted to evaluate the performance of the proposed estimator. Finally, real data applications are implemented for illustrative purposes.