ADAPTIVE DENSITY ESTIMATION FOR GENERAL ARCH MODELS
研究在一般ARCH模型中，当波动率与噪声不独立时，如何基于观测数据自适应估计波动率对数平方的密度，并证明估计量达到独立同分布情形下的最优收敛速度。
We consider a model Y t = σ t η t in which (σ t ) is not independent of the noise process (η t ) but σ t is independent of η t for each t . We assume that (σ t ) is stationary, and we propose an adaptive estimator of the density of ln(σ t 2 ) based on the observations Y t . Under a new dependence structure, the τ-dependency defined by Dedecker and Prieur (2005, Probability Theory and Related Fields 132, 203–236), we prove that the rates of this nonparametric estimator coincide with the rates obtained in the independent and identically distributed (i.i.d.) case when (σ t ) and (η t ) are independent. The results apply to various linear and nonlinear general autoregressive conditionally heteroskedastic (ARCH) processes. They are illustrated by simulations applying the deconvolution algorithm of Comte, Rozenholc, and Taupin (2006, Canadian Journal of Statistics 34, 431–452) to a new noise density.