In the modern world, massive data are often so easily collected that the data are not just univariate nor multivariate, they are sampled from continuous functions and the observed data are a set of curves. More precisely, we have observations of form { X_{i}(t_{j}) \}: the observation of the i^{th} subject at time t_{j}. Rice and Silverman (1991) gave a gait cycle analysis. Ramsay and Dalzell (1991) analyzed temperature precipitation patterns.

Our goal is to provide a powerful method to test the significance between two or more sets of curves. The Kolmogorov-Smirnov test and the Cramer-Von Mises test are two traditional methods. We will show that these two methods test only a few dimensions and have low power in detecting features such as local bumps and high frequency components.

We propose two orthogonal transforms to compress the information. One is a discrete Fourier transform and the other is a discrete wavelet transform. The discrete Fourier transform is applied when curves are smooth; then we apply adaptive Neyman procedure to choose a statistic which has the maximum power. The discrete wavelet transform is applied when curves have local properties; then we apply a thresholding procedure to select some signals into a statistic. Simulation results and an application to business commercial data will be given in this talk.