Inequality-Constrained Multivariate Smoothing Splines with Application to the Estimation of Posterior Probabilities

Semi-nonparametric estimation of the call-option price surface under strike and time-to-expiry no-arbitrage constraints

“…Standard cross-validation, however, only applies to unconstrained estimation problems, since the degrees of freedom in constrained estimation depend on the number of active constraints. Villalobos and Wahba (1987) and Wahba (1990, Chap. 9.4) suggest cross-validation schemes that correct for this fact, yet their approaches add an additional computational burden and do not immediately translate to our estimator, which differs from theirs in terms of constraints implementation.…”

“…The first exploits the fact that in the considered spaces the shape constraints can be represented by a finite number of linear inequality constraints to achieve the desired shape constraint globally; see, inter alia, Hildreth (1958), Brunk (1970), Dierckx (1980), Ramsay (1988), He and Shi (1998), and Meyer (2008). Alternatively, one seeks only approximately to satisfy the constraints on a finite subset of the domain of the function; see, e.g., Villalobos and Wahba (1987) and Mammen and Thomas-Agnan (1999). Both strands exploit specific properties of the spline spaces under consideration and impose the conditions directly on the unknown regression function.…”

A Nonparametric Approach for Determining NMR Relaxation Distributions

“…45). We solve a sequence of subproblems to determine the pertinent active inequality constraints (Villalobos and Wahba 1987). Each subproblem is characterized with a candidate set of active constraints.…”

Near-interpolation

“…For the special case that d = 1 (i.e., the f are scalar-valued functions) and with t i fixed as in (1.1), near-interpolation is equivalent to the problem of "best interpolation subject to inequality constraints," as studied in [2], [3], [6], [12] and [13], and similar to [23] in a multivariate setting. In a second direction, when ε ij = 0 and with interpolation conditions only of the form f (t i ) = z i (i.e., not Hermite conditions) problem (1.2) reduces to the problem of "best interpolation by curves" as studied in [14], [17], [18], [19] and [22].…”