{"paper":{"title":"Bivariate Binomial Moments and Bonferroni-type Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Eugene Seneta, Qin Ding","submitted_at":"2015-11-20T15:32:08Z","abstract_excerpt":"We obtain bivariate forms of Gumbel's, Fr\\'echet's and Chung's linear inequalities for $P(S\\ge u, T\\ge v)$ in terms of the bivariate binomial moments $\\{S_{i,j}\\}$, $1\\le i\\le k, 1\\le j\\le l$ of the joint distribution of $(S,T)$. At $u=v=1$, the Gumbel and Fr\\'echet bounds improve monotonically with non-decreasing $(k,l)$. The method of proof uses combinatorial identities, and reveals a multiplicative structure before taking expectation over sample points."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.06640","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}