Breakdown points for maximum likelihood estimators of location-scale mixtures

Outlier and singleton cells at exact clustering optima

Use

the published record of the finite-sample singleton phenomenon at exact optima — the qualitative shape of the $N\le18$ evidence (N-DS-BRIDGE-TREND). Hennig also supplies the normative point that one-point clusters are a modeling choice, not an invalidity.

Does not transfer

all statements are adversarial-contamination ("there exists a modification"), not i.i.d. extreme-value asymptotics; none quantifies when the sample's own maximum earns a singleton, and for distortion criteria the singleton is a failure mode where for $D_s$ it is objective-rewarded. The i.i.d. extreme-cell question itself is a confirmed search gap (29 Aug 2026, an earlier, since-closed question audit): no published result addresses whether exact optimal partitions of an i.i.d. sample isolate extreme points, under any criterion. Resolved for $D_s$ by DS15: they do not, asymptotically — min cell mass at exact scalar optima converges to the positive population masses (N-DS-SCALAR-MASS).

Sources

García-Escudero & Gordaliza (1999, JASA 94:956–969): generalized $k$-means are non-robust — contamination drives a cluster to chase an outlier; trimming repairs it. Hennig (2004, Ann. Statist. 32:1313–1340; restated verbatim in arXiv:2308.14478): adding a single far-enough point to a fixed-$K$ ML clustering makes the exact optimum give the outlier a one-point cluster and merge two original clusters. Georgogiannis (NeurIPS 2016): universal breakdown points 1/n–2/n for robust $k$-means variants.