Comparison of a large number of regression curves

We revisit the classical statistical inference problem of comparing regression curves. Traditional methods assume that the number of curves is small and fixed, while the sample size on which each curve is based tends to infinity. In contrast, we consider the case where the number of curves tends to infinity and the sample sizes are bounded by a common value. Our test is motivated by the fact that two Borel measurable functions are equivalent if and only if their Fourier transforms are identical (Bierens, 1994). An unbiased statistic is then proposed to avoid noise accumulation in a high-dimensional context. The asymptotic null distribution of the test statistic is derived and its power is studied via simulation. An illustration involving cholesterol data is provided.