Asymptotically Sufficient Statistics in Nonparametric Regression Experiments with Correlated Noise

We find asymptotically sufficient statistics that could help simplify inference in nonparametric regression problems with correlated errors. These statistics are derived from a wavelet decomposition that is used to whiten the noise process and to effectively separate high-resolution and low-resolution components. The lower-resolution components contain nearly all the available information about the mean function, and the higher-resolution components can be used to estimate the error covariances. The strength of the correlation among the errors is related to the speed at which the variance of the higher-resolution components shrinks, and this is considered an additional nuisance parameter in the model. We show that the NPR experiment with correlated noise is asymptotically equivalent to an experiment that observes the mean function in the presence of a continuous Gaussian process that is similar to a fractional Brownian motion. These results provide a theoretical motivation for some commonly proposed wavelet estimation techniques.

Weak approximation for a class of Gaussian processes

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Weak approximation for a class of Gaussian processes

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

The first-passage density of a continuous gaussian process to a general boundary

Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.

The first-passage density of a continuous gaussian process to a general boundary

Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.