{"title":"累积分布函数的约西达近似值及其在生存分析中的应用","authors":"Miroslav Bačák","doi":"10.1016/j.jat.2024.106123","DOIUrl":null,"url":null,"abstract":"<div><div>The Yosida approximation method is a classic regularization technique in maximal monotone operator theory. In the present paper, however, we apply it to the cumulative distribution function (cdf) and study its properties in the context of statistics. In that case the Yosida approximation transforms a given cdf into a new cdf with better continuity properties, namely the new cdf is Lipschitz continuous, and its distance to the original cdf as well as its Lipschitz constant are both controlled by a parameter.</div><div>When applied to an empirical cdf, which is arguably the most important case in practice, the Yosida approximation yields a continuous piecewise linear cdf in a systematic way, underpinned by a versatile theoretical framework. This provides a new smoothing technique which to our knowledge has not been explored in the literature yet.</div><div>After establishing several theoretical statistical properties of Yosida approximations we show possible applications to survival analysis. Finally, we pose two open problems in order to stimulate further research along these lines.</div></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":"306 ","pages":"Article 106123"},"PeriodicalIF":0.9000,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Yosida approximations of the cumulative distribution function and applications in survival analysis\",\"authors\":\"Miroslav Bačák\",\"doi\":\"10.1016/j.jat.2024.106123\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The Yosida approximation method is a classic regularization technique in maximal monotone operator theory. In the present paper, however, we apply it to the cumulative distribution function (cdf) and study its properties in the context of statistics. In that case the Yosida approximation transforms a given cdf into a new cdf with better continuity properties, namely the new cdf is Lipschitz continuous, and its distance to the original cdf as well as its Lipschitz constant are both controlled by a parameter.</div><div>When applied to an empirical cdf, which is arguably the most important case in practice, the Yosida approximation yields a continuous piecewise linear cdf in a systematic way, underpinned by a versatile theoretical framework. This provides a new smoothing technique which to our knowledge has not been explored in the literature yet.</div><div>After establishing several theoretical statistical properties of Yosida approximations we show possible applications to survival analysis. Finally, we pose two open problems in order to stimulate further research along these lines.</div></div>\",\"PeriodicalId\":54878,\"journal\":{\"name\":\"Journal of Approximation Theory\",\"volume\":\"306 \",\"pages\":\"Article 106123\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-11-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Approximation Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021904524001114\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Approximation Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021904524001114","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Yosida approximations of the cumulative distribution function and applications in survival analysis
The Yosida approximation method is a classic regularization technique in maximal monotone operator theory. In the present paper, however, we apply it to the cumulative distribution function (cdf) and study its properties in the context of statistics. In that case the Yosida approximation transforms a given cdf into a new cdf with better continuity properties, namely the new cdf is Lipschitz continuous, and its distance to the original cdf as well as its Lipschitz constant are both controlled by a parameter.
When applied to an empirical cdf, which is arguably the most important case in practice, the Yosida approximation yields a continuous piecewise linear cdf in a systematic way, underpinned by a versatile theoretical framework. This provides a new smoothing technique which to our knowledge has not been explored in the literature yet.
After establishing several theoretical statistical properties of Yosida approximations we show possible applications to survival analysis. Finally, we pose two open problems in order to stimulate further research along these lines.
期刊介绍:
The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others:
• Classical approximation
• Abstract approximation
• Constructive approximation
• Degree of approximation
• Fourier expansions
• Interpolation of operators
• General orthogonal systems
• Interpolation and quadratures
• Multivariate approximation
• Orthogonal polynomials
• Padé approximation
• Rational approximation
• Spline functions of one and several variables
• Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds
• Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth)
• Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis
• Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth)
• Gabor (Weyl-Heisenberg) expansions and sampling theory.