Optimal design measures with singular information matrices

Design-side margins: weight bounds, singular $D_s$ designs, extreme-point selection (29 Aug 2026)

Use

prior-art frame for DS15: determinant criteria reward extremes and tolerate singular information at optima, in contrast to distortion criteria (topic 4), which penalize both. The binning problem sits in the intersection: mass dilution ($W_b\mu_b\mu_b^\top/W_b$-structure) restores the distortion side's (M2) while keeping the design side's (M3) failure.

Does not transfer

design measures are free of the conditional-mean coupling; exact-design theory has no equivalence theorem; nothing controls the partition-restricted optimum.

Papers

Sibson & Kenny (1975, JRSS-B 37:288–292): in a D-optimal design, either the support is saturated with equal weights $1/k$, or every weight is strictly below $1/k$ (Atwood's inequality) — the design-side mass bound runs upward; no published lower bound on D/$D_s$-optimal support weights was found (an earlier, since-closed question audit). Silvey (1978, Biometrika 65:553–559, secondary-verified): $D_s$-optimal design measures can carry singular full information matrices, handled by generalized-inverse equivalence theory — the design-side analogue of DS15's (M3) failure at $D_s$-optimal partitions, and independent support for the project-don't-ridge invariant. Wang, Yang & Stufken (2019, JASA 114): the D-determinant over an i.i.d. sample is maximized by extreme order statistics, with per-point information scale $2\log n$ for normal covariates (Thm 2, 6) — the undiluted-mass side of the extreme-cell heuristic.