[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83920-en":3,"doc-seo-83920-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83920,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Sharp Lower Bound on the Minimax Risk for Multinomial Uniformity Testing via a Conditional Central Limit Theorem","Sharp minimax lower bounds are derived for goodness-of-fit testing of distributional uniformity from n multinomial observations across N categories, targeting ℓp alternatives whose deviation from the uniform distribution has size εn. The analysis focuses on the intermediate regime N=o(n^2) where the signal-to-noise ratio un converges to a finite positive limit u*. In the Poissonized setting, prior work gives the limiting constant 2Φ(−u*/2); this note proves the matching lower bound. The key tool is a conditional central limit theorem for weighted sums under a Poisson mixture prior, conditioned on total count.","arXiv :2607 .05223v1 [math . ST] 6 Jul 2026  \nSharp Lower Bound on the Minimax Risk for Multinomial Uniformity Testing via a Conditional Central Limit Theorem  \nAlon Kipnis  \nSchool of Computer Science, Reichman University  \n[alon.kipnis@runi.ac.il](alon.kipnis@runi.ac.il)  \nAbstract  \nWe study minimax goodness-of-fit testing for uniformity from n multinomial observations over N categories against ℓp departures of size ϵn. Writing un := ϵ2nnN3/2−2/p/ √2 for the associated signal-to-noise ratio, we focus on the intermediate regime N = o (n2 ) with un → u∗ ∈ (0 , ∞ ), in which the minimax risk converges to a nontrivial constant. In the Poissonized version of the problem this constant equals 2Φ(−u∗ /2) Kipnis [2025], yielding an upper bound on the multinomial minimax risk. Here we prove the matching lower bound. The key step is a conditional central limit theorem for weighted sums under a Poisson mixture prior, conditioned on the total count. Together with the upper bound in Kipnis [2025], this gives an exact sharpconstant characterization of the multinomial minimax risk in the intermediate regime.  \nKeywords: uniformity testing; multinomial; minimax risk; de-Poissonization; conditional central limit theorem.  \nMSC 2020: 62G10 (Primary), 60F05 (Secondary) .  \n1 Problem setup  \nDenote by Oi the count of category i. For a given N-dimensional multinomial distribution q :=(q1 , ... , qN ) with qi ≥ 0 and P qi = 1, let H (q) denote the model  \nH (q) : (O1 ,..., ON ) ∼ Mult(n, q) . (1.1)  \nNamely, n independent samples from q. The problem of testing the uniformity of q corresponds to testing the null hypothesis  \nH0 = H (qunif), where qunif := (1/N,..., 1/N), (1.2)  \nagainst alternatives in which the distribution q deviates from uniformity in the ℓp sense (cf. Ingsterand Suslina [2003], Chhor and Carpentier [2022]):  \nH1 = H (q), for some q ∈ AN(ϵ,p),  \nwhere  \nAN(ϵ,p) := nq ∈ [0 , 1]N : ∥q − qunif∥p≥ ϵ, ∥q∥1 = 1 o . (1.3)  \nNotice that AN(ϵ,p) is not empty provided ϵ ≤ N 1/p−1 .  \nA test ψ maps a random sample (O1 ,..., ON ) to a decision {0, 1} . The risk of ψ for testing H1 against H0 is defined as  \nR (ψ;H1 , H0 ) := Pr[ψ = 1 | H (qunif)] + sup Pr[ψ = 0 | H (q)] .  \nq∈AN (ϵ,p)  \nThe minimax risk is defined as  \nR∗ := inf R(ψ;H1 , H0 ) . (1.4)ψ  \n2 Main results  \nAssume that n tends to infinity, with N and ϵ depending on n. Define  \nun := ϵ2~~n~~n/22−2/p . (2.1)  \nWe call un the signal-to-noise ratio (SNR) .  \nIt follows from previous works that R ∗ → 0 if and only if un → ∞ , and R∗ → 1 if and only if un → 0, provided AN(ϵ,p) is non-empty Balakrishnan and Wasserman [2019], Chhor and Carpentier [2022], with the case p = 1 famously characterized in Paninski [2008] . Furthermore, by the Poisson setting studied in Kipnis [2025], we have  \nlim sup R∗ ≤ 2Φ(−u∗ /2), (2.2) n→∞  \nwhenever un → u∗ with N = o (n2 ) for some u∗ ∈ (0 , ∞ ) . The main result of this note establishes the matching lower bound.  \nTheorem 2.1 . Consider the hypothesis testing problem under the minimax setting of (1.4) . Suppose that Nn goes to infinity with Nn = o (n2 ) , n = O (N) and limn→∞ un = u∗ , for some u∗ ∈ (0 , ∞ ) . Then  \nliminf R∗ ≥ 2Φ(−u∗ /2) .  \nn→∞  \nBy the upper bound (2.2), we get:  \nCorollary 2.2 . Under the assumptions of Theorem 2.1 ,  \nlim R∗ = 2Φ(−u∗ /2) . (2.3) n→∞  \nFor the proof of Theorem 2.1, we analyze the likelihood ratio test associated with a leastfavorable prior. For a real sequence w := (w0 , w 1 ,...), define the statistic  \nN  \nT (w) :=XwOi , (2.4)  \ni=1  \nand the weights  \nw∗m ∝ (m − n/N)2 − m, m = 0 , 1 , ... , (2 .5)  \nwhich arise as the large-N approximation to the likelihood ratio weights (Lemma 3.5 and Lemma 3.6) . A key component in the proof of Theorem 2.1 is the following conditional central limit theorem  \nfor the statistic T(w∗ ) under a Poisson mixture sampling model.  \nTheorem 2 .3. For each n, let N = Nn and let ˜O1 ,   , ˜ON be independent and identically distributed according to a Poisson mixture law  \nQi ∼ π 1 , ˜","cbCaigBwgUw0SBMb","https://ap.wps.com/l/cbCaigBwgUw0SBMb","pdf",431354,1,19,"English","en",105,"# Problem setup\n# Main results\n## Conditional central limit theorem\n# Preliminaries\n## Poissonized and Poisson mixture models","[{\"question\":\"What testing problem does the document study?\",\"answer\":\"It studies minimax goodness-of-fit testing for uniformity using n multinomial observations over N categories, comparing the null uniform distribution against ℓp-perturbed alternatives of size εn.\"},{\"question\":\"In which regime is the minimax risk characterized?\",\"answer\":\"The results focus on the intermediate regime where N=o(n^2) and the signal-to-noise ratio un converges to a finite limit u* in (0,∞), yielding a nontrivial limiting risk constant.\"},{\"question\":\"What is the main mathematical contribution?\",\"answer\":\"The note proves a matching minimax lower bound to the previously established Poissonized upper bound, using a conditional central limit theorem for weighted sums under a Poisson mixture model conditioned on the total 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testing problem does the document study?","Question",{"text":74,"@type":75},"It studies minimax goodness-of-fit testing for uniformity using n multinomial observations over N categories, comparing the null uniform distribution against ℓp-perturbed alternatives of size εn.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"In which regime is the minimax risk characterized?",{"text":79,"@type":75},"The results focus on the intermediate regime where N=o(n^2) and the signal-to-noise ratio un converges to a finite limit u* in (0,∞), yielding a nontrivial limiting risk constant.",{"name":81,"@type":72,"acceptedAnswer":82},"What is the main mathematical contribution?",{"text":83,"@type":75},"The note proves a matching minimax lower bound to the previously established Poissonized upper bound, using a conditional central limit theorem for weighted sums under a Poisson mixture model conditioned on the total 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