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**Additional info for Analysis of variance for functional data**

**Sample text**

The measurements are shown as circles, and the standard regression spline ﬁt, the ﬁrst-, second-, and third-iteration regression spline ﬁts are shown as the dotdashed, dashed, dotted, and solid curves, respectively. It is seen that the usual regression spline ﬁt is indeed aﬀected by the outlier, the ﬁrst-iteration regression spline ﬁt improves the usual regression spline ﬁt and it is improved by the second-iteration regression spline ﬁt. The second- and third-iteration regression spline ﬁts are close to each other.

For convenience, it may be called a regression spline of degree k with knots τ1 , τ2 , · · · , τK . 15) associated with k = 1, 2, and 3 are usually called linear, quadratic, and cubic regression splines, respectively. We can see that within any subinterval or local neighborhood [τr , τr+1 ), we have k r βk+l (t − τl )k , βs ts + f (t) = s=0 l=1 which is a kth degree polynomial. (βk + r βk+l ). βk+r . βk+r for r = 1, 2, · · · , K. ). 16) where p = K + k + 1 denotes the number of the basis functions involved.

Let ⎤ ⎡ 1 (t1 − t0 ) · · · (t1 − t0 )p ⎥ ⎢ .. .. X = ⎣ ... ⎦, . . 1 (tn − t0 ) · · · (tn − t0 )p and W = diag(Kh (t1 − t0 ), · · · , Kh (tn − t0 )), be the design matrix and the weight matrix for the LPK ﬁt around t0 . 5) where y = (y1 , · · · , yn )T and β = (β0 , β1 , · · · , βp )T . eTr+1,p+1 S−1 n Tn y, where er+1,p+1 denotes a (p + 1)-dimensional unit vector whose (r + 1)st entry is 1 and the other entries are 0, and Sn = XT WX, Tn = XT W. 22 NONPARAMETRIC SMOOTHERS FOR A SINGLE CURVE When t0 runs over the whole support T of the design time points, a whole (r) range estimation of f (r) (t) is obtained.