A test for independence and identical distribution of functional observations is proposed in this thesis. To reduce dimension, curves are projected on the most important functional principal components. Then a test statistic based on lagged cross--covariances of the resulting vectors is constructed. We show that this dimension reduction step introduces asymptotically negligible terms, i.e. the projections behave asymptotically as iid vector--valued observations. A complete asymptotic theory based on correlations of random matrices, functional principal component expansions, and Hilbert space techniques is developed. The test statistic has chi-square asymptotic null distribution. Two inferential tests for error correlation in the functional linear model are put forward. To construct them, finite dimensional residuals are computed in two different ways, and then their autocorrelations are suitably defined. From these autocorrelation matrices, two quadratic forms are constructed whose limiting distributions are chi--squared with known numbers of degrees of freedom (different for the two forms). A test for detecting a change point in the mean of functional observations is developed. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. A comprehensive asymptotic theory for the estimation of a change--point in the mean function of functional observations is developed. The procedures developed in this thesis can be readily computed using the R package fda. All theoretical insights obtained in this thesis are confirmed by simulations and illustrated by real life-data examples.

Correlation tests were run on yearly snowfall and tornado activity data collected between water-years 1965/66 and 2010/11. Snowfall activity was evaluated using two separate measurements, which included snowfall days and daily snow depth. Tornado activity was measured through tornado days and total yearly tornado occurrences. Tornado days were defined as a 24-h period (0600–0600 UTC) during which either one or more tornadoes occurred within a chosen southern region boundary. Correlation tests revealed an absence of any mutual relationship between the snowfall and tornado activity. Three prominent teleconnections (ENSO, NAO, & AO), 6-month (Oct-Dec and Feb-Apr) and 3-month means (Oct-May) were also analyzed to reveal possible correlations with the tornado and snowfall activity. Significant negative correlations were found between ONI × tornado days; ONI × tornado totals; NAOI × snowfall days; NAOI × snow depth; AOI × snowfall days; and AOI × snow depth