Yosida approximations of the cumulative distribution function and applications in survival analysis

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory Pub Date : 2024-11-21 DOI:10.1016/j.jat.2024.106123

Miroslav Bačák

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引用次数: 0

Abstract

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.

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累积分布函数的约西达近似值及其在生存分析中的应用

Yosida 近似方法是最大单调算子理论中的一种经典正则化技术。但在本文中，我们将其应用于累积分布函数（cdf），并研究其在统计学中的特性。在这种情况下，约西达近似将给定的 cdf 转换为具有更好连续性的新 cdf，即新的 cdf 是 Lipschitz 连续的，它与原始 cdf 的距离以及 Lipschitz 常量都由一个参数控制。当应用于经验 cdf（这可以说是实践中最重要的情况）时，约西达近似以一种系统的方式产生了连续的片断线性 cdf，并以一个通用的理论框架为基础。这提供了一种新的平滑技术，据我们所知，该技术尚未在文献中得到探讨。在建立了约西达近似的几个理论统计特性后，我们展示了在生存分析中的可能应用。最后，我们提出了两个有待解决的问题，以激励沿着这些思路开展进一步的研究。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

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.