On boundary Crossing Probabilities of Brownian Bridge/Motion with Trend : Applications and Generalizations

Kurzfassung/Abstract

We consider a signal--plus--noise model $B_0+h$ with Brownian bridge $B_0$ as noise and $h$ as signal. We show a lower and an upper bound for the boundary crossing probability $\bf{P}\exists z\in [0,1]:h(z)+B_0(z)<\ell(z)$ or $u(z)<h(z)+B_0(z)$ where $\ell, u$ are boundary functions. The probability considered above corresponds with the power of tests of Kolmogorov--Smirnov type for testing
$$
H_0: h\equiv 0 \mbox{ against } K: h\neq 0.
$$
Such tests can be used to check for regression by using a residual partial sums limit approach. Especially, in case an unknown constant is assumed as regression function in the linear model we consider the least squares residuals of this model. Then under the hypothesis $H_0$ that the linear model is true the residual partial sums limit process is a Brownian bridge. In case the hypothesis is not true then the residual partial sums limit process is a Brownian bridge with some trend $h\neq 0$. For more complicated linear regression models one gets more complicated
Gaussian processes with mean zero if the assumed regression model is true and with trend $h\neq 0$ if the assumed regression model is not true.
Our bounds can be easily transformed to bounds for a boundary crossing probability of Brownian motion with trend. This result is also useful to check for a certain regression in an analogous way as described above.