bibkey: “ditzhaus2018detectability” authors: “Marc Ditzhaus; Arnold Janssen” year: 2018 title: “Detectability of nonparametric signals: higher criticism versus likelihood ratio” doi: “10.1214/18-EJS1502” url: “https://arxiv.org/abs/1709.07264v2” claim: “Theorem 4.10 gives beta=1 log-likelihood-ratio limits: the null limit is the constant -1/2 at r=1 and -1 for r>1, while the alternative limit has a finite atom and mass at positive infinity; independent Bernoulli indicators give a random signal count.” strata_touched: [] license: “citation-only” triage: “anchor”

Detectability of nonparametric signals: higher criticism versus likelihood ratio

Electronic Journal of Statistics 12(2) (2018). The inspected source is arXiv:1709.07264v2, revised 7 August 2018, a 43-page PDF; v1 was submitted 21 September 2017. The journal title, authors, year, volume, issue and DOI are corroborated by Crossref.

Section 1.1, PDF page 3, defines independent Bernoulli indicators for signal occurrence. Their sum is a random count. In the extreme sparse Gaussian model of §4.2, with occurrence probability 1/n, that count converges to Poisson(1). It is not fixed to one or to another prescribed positive integer.

Theorem 4.10, PDF page 19, extends the Gaussian detection boundary to beta = 1. At its signal parameter r = 1, the null log-likelihood ratio converges to -1/2, so the likelihood ratio converges to exp(-1/2); for r > 1 the corresponding limits are -1 and exp(-1). The theorem also gives the alternative log-likelihood-ratio limit, including mass at positive infinity. Its parameter r is the Gaussian logarithmic signal-strength parameter, not the bounded parity-kernel amplitude.

Nontrivial alternative critical limits with finitely many expected signals and loss of likelihood-ratio mean are therefore literature-attested. The theorem does not supply an exact fixed-cardinality support law, a compensated Markov kernel, or a full stationary path observation. For the parity construction, the uniform comparison between a genuine multi-spike likelihood and the product of single-spike likelihoods, followed by the fixed-cardinality mixture calculation, is a separate model-specific repo-derived argument. No global originality conclusion follows from this distinction.