10. Same α ( t ) and β ( t ) as in Model 9, with two-knot warping as in Model 5. For each sample we computed the common (un-warped) ANOVA estimator, the warped ANOVA estimator proposed in this paper, and a naive two-step warped ANOVA estimator. The latter is computed as follows: first the curves are aligne d by least-squares registration (i.e. minimizing ∑ n i =1 ‖ x$_{i}$ ◦ w$_{i}$ − µ ‖ 2 over w$_{i}$ s in W τ $_{0}$) and then the common ANOVA estimators are computed on the warped data. We used cubic B-splines with 10 equis- paced knots as basis functions for the functional parameters. As warping families we used interpolating Hermite splines with τ 0 = . 3 for models 1–4 and 7–9, and τ 0 = ( . 3 , . 6) for models 5–6 and 10. As error measures we used the bias, the s tandard deviation and the root mean squared error, defined as follows: if f 0 ∈ L $^{2}$( I ) and ˆ f is the estima- tor, then bias( ˆ f ) = [ ∫ { E ˆ f ( t ) − f 0 ( t ) } $^{2}$dt ] 1 / $^{2}$, sd( ˆ f ) = [ ∫ E { ˆ f ( t ) − E ˆ f ( t ) } $^{2}$dt ] 1 / 2 and rmse( ˆ f ) = { bias $^{2}$( ˆ f ) + sd $^{2}$( ˆ f ) } 1 / $^{2}$. Some care must be taken with the principal compo- nents, because their sign is undefined: to determine the “rig ht” sign, we multiplied ˆ φ$_{1}$ and ˆ ψ$_{1}$ by 〈 ˆ φ$_{1}$, φ$_{1}$ 〉 and 〈 ˆ ψ$_{1}$, ψ$_{1}$ 〉 , respectively. The estimation errors based on 200 Monte Carlo replications for each model are shown in Table 1. The effect of warping is more clearly seen in the estimators of the principal components φ and ψ . The common ANOVA estimators, as expected, have the largest bi- ases; lacking a specific mechanism to handle time variabilit y, common ANOVA estimators ˆ φ and ˆ ψ attempt to fit amplitude and phase variability at the same tim e and get severely distorted compared to the true φ and ψ . The two warped estimators, on the other hand, can handle phase variability well. The maximum-likelihood estimator proposed in this paper always has smaller bias than the naive two-step approach; this is to be expected, since the warping step of the two-step estimator minimizes variation about the mean µ without taking into account amplitude variability or the dependence structure in the data, whereas the maximum likelihood estimator explicitly models φ and ψ . The down side of the new estimators is that, as always, the bias reduction provided by the more complex model is 17