David Bickel, statistician

This chapter provides both asymptotic and finite-sample properties of a fiducial solution to the problem of free-knot splines with four or more degrees, assuming a known number of knot points. The authors lay a foundation for the solution by proving the asymptotic normality of certain multivariate fiducial estimators. After demonstrating that the fiducial solution meets the sufficient conditions for asymptotic normality, they quantify small-sample performance on the basis of simulations. The authors conclude that fiducial inference provides a promising alternative to Bayesian inference for the free-knot spline problem addressed.
The research reported reflects the recent surge in developments of Fisher’s fiducial argument [S. Nadarajah, S. Bityukov and N. Krasnikov, Stat. Methodol. 22 (2015), 23–46; MR3261595]. The work of this chapter is carried out within the framework of generalized fiducial inference [J. Hannig, Statist. Sinica 19 (2009), no. 2, 491–544; MR2514173 (2010h:62071)], which is built on the functional-model formulation of fiducial statistics [A. P. Dawid, M. Stone and M. Stone, Ann. Statist. 10 (1982), no. 4, 1054–1074; MR0673643 (83m:62008)] rather than on the broadly equivalent confidence-based tradition beginning with [G. N. Wilkinson, J. Roy. Statist. Soc. Ser. B 39 (1977), no. 2, 119–171; MR0652326 (58 #31491)] and generalized by [E. E. M. van Berkum, H. N. Linssen and D. Overdijk, J. Statist. Plann. Inference 49 (1996), no. 3, 305–317; MR1381161 (97k:62007)].