bibkey: “caiwu2014sparse” authors: “T. Tony Cai; Yihong Wu” year: 2014 title: “Optimal Detection of Sparse Mixtures Against a Given Null Distribution” doi: “10.1109/TIT.2014.2304295” url: “https://arxiv.org/abs/1211.2265v1” claim: “The general independent sparse-mixture framework relates detection boundaries to likelihood-ratio behavior at null-tail scales; it supplies background, not a fixed-cardinality compensated Markov-path theorem.” strata_touched: [] license: “citation-only” triage: “anchor”

Optimal Detection of Sparse Mixtures Against a Given Null Distribution

IEEE Transactions on Information Theory 60(4), 2217–2232 (April 2014), DOI 10.1109/TIT.2014.2304295. Crossref corroborates the journal title, authors, date, volume, issue, pagination and DOI. The journal full text was unavailable through the checked routes; its proof text has not been inspected, and no assertion below relies on a journal correction.

The inspected primary preprint is the 32-page arXiv:1211.2265v1, dated 9 November 2012, titled Optimal Detection For Sparse Mixtures. Its first page identifies that arXiv version and date. The rendered manuscript also displays 3 April 2022; the corresponding source file Detection-110812-arxiv.tex contains the literal \date{\today}. That displayed compilation date is not the preprint’s version date.

The abstract, introduction and Section 3 relate sparse-mixture detection to the behavior of a single-coordinate log-likelihood ratio at null-tail scales. The main formulation, equation (10), compares independent samples from Q_n against independent samples from (1 - epsilon_n) Q_n + epsilon_n G_n. The paper develops general non-Gaussian detection boundaries and Higher Criticism results. This likelihood-ratio and tail-scale framework is literature-attested.

There is a source ambiguity in the inspected preprint’s Theorem 3 (PDF page 11, equations (40)–(43)): it defines F_n and z_n as the CDF and quantile function of G_n, whereas the following explanation calls for null quantiles, and its proof (pages 25–26, equation (98)) represents W_n ~ Q_n using z_n(U). The source TeX contains the same G_n definition. This note neither silently replaces that symbol nor invokes the ambiguous displayed boundary formula as an exact theorem.

The parity-kernel argument cites the general framework as background and derives its own compound-Poisson tail rates and scalar optimization. Its positive support has a fixed cardinality and is held constant over all observations. Its actual forward and reverse path laws are stationary Markov laws with a compensating background. The independent contamination prior above supplies neither their joint row-count approximation nor their truncated overlap moments. Those connections require the separate repo-derived finite-rank transfer and compensated-likelihood calculations. No claim of worldwide originality, parameter adaptation, or equivalence of the full pair and path experiments follows from this distinction.