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CPS1891 Tzee-Ming Huang
In summary, for a smooth , a simple knot selection algorithm using equally
̂
spaced knots like Method 3 is sufficient for giving a satisfactory estimator .
For with a sharp change, a

Algorithm 1 Method 2 Method 3
1 2.76 × 10 1.75 × 10 8.45 × 10
−3
−4
−5
2 0.00149 0.00113 0.000289
3 2.51 × 10 7.69 × 10 8.53 × 10
−5
−4
−4
4 1.70 × 10 1.86 × 10 2.44 × 10
−4
−3
−4
Table 3: Standard deviations of the mean squared errors

more flexible knot selection algorithm such as Algorithm 1 or Method 2, is
needed. Algorithm 1 is more stable than Method 2.

4. Discussion and conclusion
In this article, a knot selection algorithm (Algorithm 1) based on a statistical
test is proposed. Based on the simulation results in Section 3, overall, the
performance of Algorithm 1 is satisfactory. When has a sharp change,
Algorithm 1 outperforms Method 3, which uses equally spaced knots only, and
Algorithm 1 sometimes even performs better than Method 2, which allows for
more flexible knot selection. This satisfactory result is probably due to the fact
that Algorithm 1 is flexible enough to use unequally spaced knots yet not too
flexible to allow for arbitrary knot locations, which makes it more stable than
a knot estimation method such as Method 2.
For future work, a possible direction is to extend Algorithm 1 to the case
where is multi-variate and tensor basis functions are used. Given that
Algorithm 1 does not require searching for best knot locations, it is expected
that the extension to the multivariate case is feasible.

References
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Geometrically designed, variable knot regression splines. Computational
Statistics, 31(3):1079{1105.
2. Lindstrom, M. J. (1999). Penalized estimation of free-knot splines. Journal
of Computational and Graphical Statistics, 8(2):333{352.
3. Stone, C. J. (1982). Optimal global rates of convergence for
nonparametric regression. Ann. Statist., 10(4):1040{1053.
4. Stone, C. J. (1994). The use of polynomial splines and their tensor
products in multivariate function estimation. Ann. Statist., 22(1):118{171.
5. Yuan, Y., Chen, N., and Zhou, S. (2013). Adaptive B-spline knot selection
using multi-resolution basis set. IIE Transactions, 45(12):1263{1277.
6. Zhou, S. and Shen, X. (2001). Spatially adaptive regression splines and
accurate knot selection schemes. Journal of the American Statistical
Association, 96:247{259.

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