[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84361-en":3,"doc-seo-84361-105":30,"detail-sidebar-cat-0-en-105":83},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84361,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Why Constants Matter in Distribution Testing: From Uniformity to Calibration","Distribution goodness-of-fit testing has matured into a rate-level theory describing how sample size depends on alphabet size, separation from the null, and target error probability, with uniformity testing as the canonical example. The note addresses the unresolved constant-level question of which rate-optimal tests yield the best risk or power. Sharp constants are connected to Fisher information and Pinsker’s constant, revealing an effective signal-to-noise ratio and informing tuning choices in downstream uniformity and calibration testing.","arXiv :2607 .08378v 1 [ cs .IT] 9 Jul 2026  \nWhy Constants Matter in Distribution Testing: From Uniformity to Calibration  \nAlon Kipnis  \nJuly 10, 2026  \nAbstract  \nDistribution goodness-of-fit testing has developed a powerful rate-level theory: we often know how the required sample size scales with the alphabet size, the separation from the null, and the target error probability. Uniformity testing is the canonical example. One can distinguish the uniform distribution on N categories from alternatives at total-variation distance at least ϵ with far fewer than N samples, and the optimal scaling is now well understood.  \nBut rate-level theory leaves an important question unresolved: among several tests with the same sample-complexity order, which one actually gives the best risk or power? This is a constant-level question. It is especially relevant in modern applications where distribution testing is used not merely as an asymptotic abstraction, but as a practical design tool.  \nThis note argues that sharp constants in distribution testing play a role analogous to Fisher information in parametric estimation and Pinsker’s constant in nonparametric estimation. First, they distinguish between tests that are all rate-optimal but not equally powerful. Second, they reveal the effective signal-to-noise ratio governing the testing problem. Third, they can guide tuning-parameter choices in downstream applications. We illustrate this perspective through large-alphabet uniformity testing and then explain why the same logic matters for choosing the number of bins in calibration testing.  \n1 Distribution testing beyond rates  \nA central lesson from property testing and modern nonparametric statistics is that testing can be much easier than estimation. If p is an unknown distribution on N categories, estimating p in total variation requires order N samples. By contrast, testing whether p is uniform can often be done with order √N samples, up to the dependence on the separation parameter ϵ .  \nThe emphasis on constants is familiar from other parts of statistics. In regular parametric estimation, the n−1/2 rate is only the first-order message; the Fisher information determines the sharp asymptotic variance and therefore distinguishes between experiments and estimators with the same rate. Similarly, in nonparametric estimation, minimax rates describe how risk scales with sample size, but Pinsker’s constant gives the sharp asymptotic benchmark for the leading minimax risk over smoothness classes; see, for example, Nussbaum [1999] . These constants are not cosmetic refinements: they are the quantities that turn asymptotic theory into quantitative risk predictions. The same principle applies in distribution testing. Once the detection rate is known, the next question is the sharp constant governing the limiting risk.  \nThe testing-versus-estimation phenomenon was made especially clear by Paninski [2008], who showed that in the sparse regime the number of repeated observations, or collisions, contains enough information to detect nonuniformity even when most categories are unobserved.  \nThis rate-level perspective has been extremely successful. It tells us when testing is possible and when it is impossible. However, it does not fully determine the statistical performance of a concrete test. Two tests may both require  \nn ≍ √N  \nϵ2  \nsamples and yet have noticeably different error probabilities at the same n, N,ϵ . In other words, sample-complexity rates can identify the right scale while still hiding the most practically relevant comparison.  \nA sharper question is therefore:  \nWhat is the exact asymptotic risk, including the leading constant?  \nThis question is not only a refinement of theory. It changes the way we compare tests.  \n2 A simple Gaussian analogy  \nThe importance of constants is easiest to see in the Gaussian testing problem  \nH0 : X ∼ N(0, 1), H1 : X ∼ N (u,1) .  \nHere it is natural to decide between H0 and H1 based on whether the te","cbCaieHb0cJJB1cx","https://ap.wps.com/l/cbCaieHb0cJJB1cx","pdf",243886,4,1,9,"English","en",105,"# Distribution testing beyond rates\n## Uniformity testing and the constant-level risk question\n# A simple Gaussian analogy\n## Risk minimization via threshold choice\n# Uniformity testing as the model problem\n## Test statistics and constant-level differences","[{\"question\":\"How is calibration testing connected to the uniformity-testing discussion?\",\"answer\":\"After illustrating the perspective through large-alphabet uniformity testing, the note explains that the same constant-level logic applies when choosing parameters such as the number of bins in calibration testing.\"}]",1784195095,23,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"why-constants-matter-in-distribution-testing-from-uniformity-to-calibration","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/why-constants-matter-in-distribution-testing-from-uniformity-to-calibration/84361/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How is calibration testing connected to the uniformity-testing discussion?","Question",{"text":75,"@type":76},"After illustrating the perspective through large-alphabet uniformity testing, the note explains that the same constant-level logic applies when choosing parameters such as the number of bins in calibration testing.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,119,122,126],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]