{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T04:27:04Z","timestamp":1774585624181,"version":"3.50.1"},"reference-count":23,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[1956,7,1]],"date-time":"1956-07-01T00:00:00Z","timestamp":-426124800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1956,7,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We may often obtain the lower moments of a sampling distribution when the distribution itself is intractable. Tchebycheff\u2019s celebrated inequalities for the cumulative distribution function are of very little use in practice, being \u201cmuch too wide\u201d. The present work obtains generalizations of these inequalities by restricting the distribution function to be \u201csmooth\u201d in two senses: firstly, by restricting the number of changes of sign of some derivative, and secondly by also restricting the size of this derivative. These conditions lead to certain \u201cextremal\u201d distributions, and hence to the required inequalities. These have been proved correct only in the simpler cases, but are conjectured to hold in general. In Part I some of the simpler results are given explicitly.<\/jats:p>","DOI":"10.1111\/j.2517-6161.1956.tb00220.x","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T06:11:42Z","timestamp":1544163102000},"page":"139-168","source":"Crossref","is-referenced-by-count":44,"title":["Generalizations of Tchebycheff\u2019s Inequalities"],"prefix":"10.1093","volume":"18","author":[{"given":"C. L.","family":"Mallows","sequence":"first","affiliation":[{"name":"University College , London"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2018,12,5]]},"reference":[{"key":"2024090920480482100_cit1","first-page":"9","article-title":"Three papers (in German)","volume":"9","author":"Akhieser (Achieser, Achyeser)","year":"1934\u201335","journal-title":"Zap. kharkiv. mat. Tov. (Comm. Math. Soc. Kharkoff), Ser. 4"},{"key":"2024090920480482100_cit2","volume-title":"On Certain Problems in the Theory of Moments.","author":"Akhieser (Achieser, Achyeser)","year":"1938"},{"key":"2024090920480482100_cit3","first-page":"129","article-title":"Some remarks about three papers of M. S. Verblunsky","volume":"16","author":"Akhieser (Achieser, Achyeser)","year":"1940","journal-title":"Zap. nauch.\u2014issled. Inst. Mat. Mekh. Kharkov mat. Obshch. (Comm. Inst. Math. Mech. Sci. Univ. Kharkoff and Math. Soc. Kharkoff) Ser. 4"},{"key":"2024090920480482100_cit4","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1214\/aoms\/1177730293","article-title":"On random variables with comparable peakedness","volume":"19","author":"Birnbaum","year":"1948","journal-title":"Ann. Math. Statist."},{"key":"2024090920480482100_cit5","volume-title":"Mathematical Methods of Statistics.","author":"Cramer","year":"1946"},{"key":"2024090920480482100_cit6","author":"Dennis"},{"key":"2024090920480482100_cit7","first-page":"1","article-title":"G\u00e9n\u00e9ralit\u00e9s sur les Probabilit\u00e9s. Variables Al\u00e9atoires","volume-title":"Trait\u00e9 du Calcul des Probabilit\u00e9s et de ses Applications.","author":"Fr\u00e9chet","year":"1937"},{"key":"2024090920480482100_cit8","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1214\/aoms\/1177728543","article-title":"The extrema of the expected value of a function of independent random variables","volume":"26","author":"Hoeffding","year":"1955","journal-title":"Ann. Math. Statist."},{"key":"2024090920480482100_cit9","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1214\/aoms\/1177728489","article-title":"Bounds for the distribution function of a sum of independent, identically distributed random variables","volume":"26","author":"Hoeffding","year":"1955","journal-title":"Ann. Math. Statist."},{"key":"2024090920480482100_cit10","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1093\/biomet\/36.1-2.149","article-title":"Systems of frequency curves generated by methods of translation","volume":"36","author":"Johnson","year":"1949","journal-title":"Biometrika"},{"key":"2024090920480482100_cit11","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1214\/aoms\/1177729590","article-title":"The moment problem for unimodal distributions","volume":"22","author":"Johnson","year":"1951","journal-title":"Ann. Math. Statist."},{"key":"2024090920480482100_cit12","volume-title":"On Certain Applications of Algebraic Continued Fractions.","author":"Markoff","year":"1884"},{"key":"2024090920480482100_cit13","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1214\/aoms\/1177728977","article-title":"Bounds on a distribution function when its first n moments are given","volume":"24","author":"Royden","year":"1953","journal-title":"Ann. Math. Statist."},{"key":"2024090920480482100_cit14","doi-asserted-by":"crossref","DOI":"10.6028\/NBS.RPT.1744","article-title":"Probability inequalities of the Tchebycheff type","volume-title":"Rep. Nat. Bur. Stand.","author":"Savage","year":"1952"},{"key":"2024090920480482100_cit15","doi-asserted-by":"crossref","DOI":"10.1090\/surv\/001","article-title":"The Problem of Moments","volume-title":"Math. Surv.","author":"Shohat","year":"1943"},{"key":"2024090920480482100_cit16","doi-asserted-by":"crossref","first-page":"409","DOI":"10.24033\/asens.245","article-title":"Quelques recherches, sur les quadratures dites m\u00e9caniques","volume":"1","author":"Stieltjes","year":"1884","journal-title":"Ann. sci. \u00c9c. norm. sup., Paris, (3)"},{"key":"2024090920480482100_cit17","first-page":"586","article-title":"Sur certaines in\u00e9galit\u00e9s dues \u00e0 M. P. Tchebycheff. Article r\u00e9dige d'apr\u00e8s un manuscript in\u00e9dit","volume":"2","author":"Stieltjes","year":"1894","journal-title":"Oeuvres Compl\u00e8tes"},{"key":"2024090920480482100_cit18","first-page":"157","article-title":"Sur les valeurs limites des int\u00e9grates","volume":"19","author":"Tchebycheff","year":"1874","journal-title":"J. Math. Pures Appl., (2)"},{"key":"2024090920480482100_cit19","first-page":"181","volume":"10","author":"Tchebycheff","year":"1907","journal-title":"Collected Papers"},{"key":"2024090920480482100_cit20","first-page":"158","article-title":"An extremal problem in Mathematical Statistics","volume-title":"Skand. Aktuar Tidskr.","author":"Ulin","year":"1953"},{"key":"2024090920480482100_cit21","volume-title":"Introduction to Mathematical Probability.","author":"Uspensky","year":"1937"},{"key":"2024090920480482100_cit22","first-page":"30","article-title":"Une nouvelle g\u00e9n\u00e9ralization de l'in\u00e9galit\u00e9 de Bienaym\u00e9","volume":"12","author":"Dantzig","year":"1951","journal-title":"Ann. Inst. Poincar\u00e9"},{"key":"2024090920480482100_cit23","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1017\/S0305004100018831","article-title":"Solution of a moment problem for bounded functions","volume":"32","author":"Verblunsky","year":"1936","journal-title":"Proc. Camb. Phil. Soc."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fj.2517-6161.1956.tb00220.x","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/18\/2\/139\/49096510\/jrsssb_18_2_139.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/18\/2\/139\/49096510\/jrsssb_18_2_139.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T21:14:19Z","timestamp":1725916459000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/18\/2\/139\/7027355"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1956,7,1]]},"references-count":23,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,12,5]]},"published-print":{"date-parts":[[1956,7,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.2517-6161.1956.tb00220.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[1956,7]]},"published":{"date-parts":[[1956,7,1]]}}}