Local Asymptotics for Regression Splines and Confidence Regions

Penalized spline estimation in the partially linear model

“…Newey (1997), Zhou et al (1998), andClaeskens et al (2009);and Huang (2003a) claims this assumption is minimal. We use it to bound the eigenvalues of the design matrix away from zero and to obtain an asymptotic expression for the eigenvalues used in the decomposition of the penalization matrix.…”

“…h j,k , and M j > 0 is a constant, in order to guarantee (Zhou, Shen & Wolfe (1998)). A weaker alternative is…”

“…(see for example Zhou et al (1998), De Boor (2001, Schumaker (1981)). Normalized Bsplines have the useful property…”

“…For asymptotic results, a rate condition such as K 2 n /n → 0 (Newey (1997), Zhou et al (1998), Zhou & Wolfe (2000), Claeskens et al (2009)) or K n log n/n → 0 (Huang (2003b), Huang (2003a) is assumed. In contrast, smoothing spline estimation has K n = n and includes the penalization term to compensate for the resulting large variance (Cox (1984), Utreras (1988), Utreras (1979)).…”

Change-point estimation using shape-restricted regression splines

“…References include Stone (1980), Stone (1982), Stone, Hansen, Kooperberg, and Truong (1997), Zhou, Shen, and Wolfe (1998), Huang (1998), Zhou and Wolfe (2000), and Huang (2001). If p is the order of the spline, let the number of knots increase as n 1/(2p+1) ; that is, as n 1/7 for quadratic splines and as n 1/9 for cubic splines.…”