bibkey: “kotekal2022sparse” authors: “Subhodh Kotekal” year: 2022 title: “Statistical limits of sparse mixture detection” doi: “10.1214/22-EJS2053” url: “https://arxiv.org/abs/2104.02507v3” claim: “Corollaries 1–2 express the independent sparse-mixture detection boundary through the null large-deviation rate of the normalized log-likelihood ratio, subject to a likelihood-moment condition and stated regularity assumptions.” strata_touched: [] license: “citation-only” triage: “anchor”

Statistical limits of sparse mixture detection

Electronic Journal of Statistics 16(2) (2022), DOI 10.1214/22-EJS2053. Crossref confirms the author, title, year, journal, volume, issue and DOI. The inspected primary manuscript is the 70-page arXiv:2104.02507v3, revised 26 May 2021; its first page and arXiv metadata agree on that version. The primary-paper claims below refer to this pinned manuscript.

Equations (1)–(3), PDF pages 1–2, compare independent observations from P_n against independent observations from (1-n^(-beta)) P_n + n^(-beta) Q_n. The support count in this model is random. Assumption 1, page 5, allows a separable metric observation space with a common dominating measure and Q_n << P_n; it is not restricted to continuous observations.

Theorems 1–2 and Corollaries 1–2, page 7, assume a good large-deviation principle at speed log n for the null normalized log-likelihood ratio. Condition (6) also requires, for some fixed gamma > 1, a finite limsup of log E_P[(dQ_n/dP_n)^gamma] / log n. Corollary 1 gives equality of the upper and lower boundaries when its supremum condition (9) holds. Corollary 2 supplies sufficient conditions: the rate function is right continuous at zero and the variational integrand is nonnegative somewhere on the nonnegative half-line. The resulting formula, equation (10), is

\beta^*=\frac12+\max\left\{0,\sup_{x\ge0}
 \left[x-I(x)+\frac{1\wedge I(x)}2\right]\right\}.

For the parity model’s independent compound-Poisson comparison coordinate, log E_0 L^theta = lambda g_r(theta) with lambda/log M -> alpha. The limiting cumulant is finite, smooth and steep; its Legendre transform is a finite continuous good rate function. The moment condition holds for every fixed gamma > 1. At x = alpha phi(r), one has I(x) = x > 0, so the sufficient nonnegativity condition also holds. Substitution in the published variational formula yields the scalar three-branch curve used in Chapter 17. The general formula is literature-attested; this scalar specialization is not claimed as a new general detection principle.

Corollaries 1–2 do not assert the same boundary for a fixed-cardinality positive support in a compensated kernel or for stationary Markov paths. The actual-direction relative row-count estimates, support-uniform covariances and growing-overlap likelihood tilt require the separate repo-derived proof. No full-experiment equivalence or worldwide originality claim follows from these scope differences.