NONPARAMETRIC ESTIMATION OF THE DERIVATIVES OF THE STATIONARY DENSITY FOR STATIONARY PROCESSES

作者：Schmisser Emeline^*

来源：ESAIM: Probability and Statistics , 2013, 17: 33-69.

DOI：10.1051/ps/2011102

• Derivatives of the stationary density
• diffusion processes
• mixing processes
• nonparametric estimation
• stationary processes

In this article, our aim is to estimate the successive derivatives of the stationary density f of a strictly stationary and beta-mixing process (Xt)(t %26gt;= 0). This process is observed at discrete times t = 0, Delta, ..., n Delta. The sampling interval Delta can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative f((j)) belongs to the Besov space B-2,infinity(alpha), then our estimator converges at rate (n Delta)(-alpha/(2 alpha+2j+1)). Then we consider a diffusion with known diffusion coefficient. We use the particular form of the stationary density to compute an adaptive estimator of its first derivative f%26apos;. When the sampling interval Delta tends to 0, and when the diffusion coefficient is known, the convergence rate of our estimator is (n Delta)(-alpha/(2 alpha+1)). When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data.

• 出版日期2013-1