Novel Test for the Equality of Continuous Curves with Homoscedastic or Heteroscedastic Measurement Errors

Testing equality of two curves occurs often in functional data analysis. In this paper, we develop procedures for testing if two curves measured with either homoscedastic or heteroscedastic errors are equal. The method is applicable to a general class of curves. Compared with existing tests, ours does not require repeated measurements to obtain the variances at each of the explanatory values. Instead, our test calculates the overall variances by pooling all of the data points. The null distribution of the test statistic is derived and an approximation formula to calculate the p value is developed when the heteroscedastic variances are either known or unknown. Simulations are conducted to show that this procedure works well in the finite sample situation. Comparisons with other test procedures are made based on simulated data sets. Applications to our motivating example from an environmental study will be illustrated. An R package was created for ease of general applications.