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Methods, selection and inferential scope

Source: vignettes/methods-and-scope.Rmd
methods-and-scope.Rmd

Which function should I use?

Function Role Selection
arfc() One ASR/AMR and MCP/SCAD specification Training BIC on supplied grids
arfc_select() Choose one of the four adaptive specifications Supplied independent validation
fMCP() Fixed-resolution nonadaptive MCP comparator Training BIC
fSCAD() Fixed-resolution nonadaptive SCAD comparator Training BIC

ARFC is the adaptive regional folded-concave family. ASR uses one finest partition; AMR combines multiple partitions of the same domain, sharing one spline coefficient vector. ARFC-Select chooses a penalty family and resolution using response prediction error, not truth-aware support metrics.

What is normalized?

Let the functional domain be [0,T], let I be an interval of positive length, and let the coefficient curve be beta. A regional magnitude is its RMS value on I, and its penalty multiplier is the interval’s fraction of the domain. Level weights sum to one. Consequently adding a resolution level does not multiply the accumulated regional curvature merely by adding regions.

Pilot weights adjust regional shrinkage. The public implementation clips them for numerical stability. MCP shape 2.7 and SCAD shape 3.7 have equal maximum scalar concavity; this does not make their estimates identical or establish the superiority of either penalty.

The paper proves a mass identity, a matrix weak-convexity envelope, idealized majorization descent, conditional regional-oracle results, and a finite-library validation-risk bound. Oracle recovery requires signal/score separation and design conditions. These results do not certify every numerical initialization, post-selection confidence intervals, or support-optimal validation selection.

The software uses a regional-norm floor, coefficient cutoff and active-set refit. These conventions are exposed and retained in diagnostics. Failed paths are not silently discarded from diagnostic tables. Inspect the chosen fit’s convergence before reporting its predictions or structural zeros.

Sources and contribution

SCAD: Fan and Li (2001), doi:10.1198/016214501753382273. MCP: Zhang (2010), doi:10.1214/09-AOS729. Functional local sparsity: Lin et al. (2017), doi:10.1080/10618600.2016.1195273.

The companion study concerns scale-normalized adaptive regional geometry and joint penalty-resolution selection, not the invention of functional MCP. Use citation("AdaReFun") to cite the software version used.