Page 384 - Contributed Paper Session (CPS) - Volume 2

CPS1891 Tzee-Ming Huang
number of knots. For this approach, Stone (1994) showed that, under certain
conditions, if the number of knots grows at a proper rate as the sample size
grows, then then attains the optimal convergence rate in the sense of Stone
̂
(1982).
It is also possible to use unequally spaced knots for and there are two
̂
types of approaches for doing so. In the first type of approach, the knot
locations are estimated as parameters, such as in Lindstrom (1999) and Zhou
and Shen (2001). In the other type of approach, a set of knots is considered
and knot selection is performed, such as in Yuan et al. (2013) and Kaishev et
al. (2016). While allowing the knot locations to be estimated gives more
flexiblity, the knot selection approach can be less time-consuming.
In this study, the order is fixed and the focus is on knot selection. A knot
selection algorithm is proposed. In Section 2, the proposed algorithm is
introduced. In Section 3, results for a simulation experiment are reported to
demonstrate the performance of the proposed algorithm. Concluding remarks
are given in Section 4.

2. Methodology
The proposed algorithm for knot selection is to first consider a large set of
candidate knots and then perform knot screening to remove knots that are
not needed. The knot screening is based on a statistical test. In this section, I
will first describe the idea and details of such a test, and then gives the
proposed algorithm.
Suppose that the regression in (1) can be approximated well using a
spline of order and knot , … , . Then an approximate model for (1) is
1

+
= ∑ −1 + ∑ ( − ) −1 + , = 1, … , , (2)

+
=1 =+1

where m is given and , … , are unknown knots. Note that for ℓ ∈

1
{1, … , }, we have

= ∑ −1 + (3)

1,
=1

for ( , )s such that < and > ℓ−1 (if ℓ > 1), and

ℓ

= ∑ −1 + (4)

2,
=1

for ( , )s such that > and < ℓ+1 (if ℓ < ). If the coefficient vectors

ℓ
( 1,1 , … , 1, ) and ( 2,1 , … , 2, ) are the same, then (2 remains the same with
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