Optimal sampling for density estimation in continuous time

Abstract

For nonparametric density estimation, first we give optimal sampling schemes of continuous time processes. Next we study effects of known or small errors-in-variables on such samplings. Throughout the paper various simulations are also presented.MSC 2000: Primary 62G07 Secondary 62D05, 62M09, 60H35.

Estimation of stochastic volatility models by nonparametric filtering

“…The estimation problem in the second step can be seen as a kernel estimation with errors-in-variables. The implications of this for kernel regression was analyzed in Rilstone (1996) and Sperlich (2007) in a cross-sectional framework while kernel density estimation of stochastic processes with errors-in-variables was considered in Blanke and Pumo (2003). We follow a similar strategy to Rilstone (1996) and Sperlich (2007) when analyzing the impact of the …rst-step estimation of f 2 t g on the nonparametric estimators of the SV model in the second step: We split up the total estimation error into two components: One component due to the estimation of f 2 t g in the …rst step, and a second component due to the sampling error of the estimator based on the actual process.…”

Piecewise linear density estimation for sampled data

“…First, if the total time of observation is a given and large enough T n , the value of a minimal δ * n allows to select a maximal number n * of points in [0, T n ] to estimate f . On the other hand, consider that a maximal and large enough sample size n is available, then we can deduce from δ * n a minimal sufficient time T * n = nδ * n of observation (see Blanke and Pumo [5]). Furthermore, we will emphasize the convenience of such a framework to sample a continuous-time process.…”

“…Assumptions A ′ 1 are in the spirit of those made (and widely discussed) by Blanke and Pumo [5]. Here A ′ 1 (i) is used with high frequency sampling to obtain optimal rates together with a short sampling step δ n depending on a positive known coefficient γ 0 .…”

“…Hence it seems more natural to plan an estimation approach based upon n discrete values of the process collected with a suitable time sampling procedure. In the context of nonparametric density estimation, the three sampling procedures considered in the present work have been investigated by Masry [24], Prakasa Rao [25], Wu [30] (random sampling), Bosq [6,7] and Blanke and Pumo [5] ("high frequency" periodic sampling), among others. As far as we know, most of existing papers -including those cited above -only deal with kernel estimation, and none have yet focused a special attention on piecewise linear estimators.…”

A note on wavelet density deconvolution for weakly dependent data

“…The optimality of the pointwise rates of such estimators was recently proved by Butucea (2004). It is also worth mentioning that Blanke and Pumo (2003) consider the optimal choice of the data collecting scheme depending on the features of the bivariate density f (Y i ,Y j ) . See also the papers Fan (2002) and Walter (1999) and the references therein for related work on wavelet deconvolution.…”