# Appropriateness of Constrained Segmented Univariate Polynomial Regression Model

1. Dec 4, 2011

### midnite131

Hi all,

I've learned that in unconstrained polynomial regression, the optimal order can be determined using two F tests : one to test for the significance of the overall regression, the other to test for the significance of the higher coefficients (assuming the first test passed of course).

However in my application, I'm interested in testing for the appropriateness of a segmented polynomial fit subjected to be first order continuous, with boundary conditions placed at the end of the entire domain as well. The polynomials do not have to have the same order between segments. I should also mention that the join points are known, so I don't have to estimate them.

The closest paper I could find that broaches this topic is Gallant and Fuller's work [1]. Here, they also have a segmented polynomial fit with C1 continuity, but with no constraints. Frustratingly, they make up a test statistic for the appropriateness of their fit "by analogy to linear models theory", yet they provide no references.

I've tried to search for other papers on this topic but to no avail. This leads me to question - is this test statistic trivial to derive for the constrained case, and if so, could you please point me to resources that could help me understand how to do it?

References

[1] Gallant, A.R. and Fuller, W.A. (1973). Fitting segmented polynomial regression models whose join points have to be estimated. J. Amer. Statist. Assoc., 68, 144-147

p.s. If it helps, I'm an engineer who regrettably didn't pay too much attention in his stats course, so any material no matter how trivial it may seem would be appreciated!

2. Dec 8, 2011

### Stephen Tashi

That's probably true under a certain set of assumptions that are commonly made. Without stating the assumptions, it sounds like too grand a claim.

Speaking only as an operator of Google, there is this paper online:

ANALYSIS OF VARIANCE, COEFFICIENT OF DETERMINATION AND F -TEST FOR LOCAL POLYNOMIAL REGRESSION By Li-Shan Huang and Jianwei Chen

It has other references.