Testing for the Mean of Random Curves: A Penalization Approach

Let X ₁,...,X _n be an i.i.d. sample of random curves, viewed as Hilbert space valued random elements, with mean curve m. An asymptotic test of m = m ₀ vs m ≠ m ₀ is proposed, when m ₀ is a fixed known function. The test statistics converges under very mild assumptions and relies on the pseudo-inversion of the covariance operator (leading to a non standard inverse problem). The power against local alternatives is investigated. In final form November 2004