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A new confidence interval based on the theory of U-statistics for the area under the curve

Jürgen Kampf

Lukas H. Vogel

Iryna Dykun

Tienush Rassaf

Amir A. Mahabadi

https://doi.org/10.15559/25-VMSTA285

Pub. online: 9 Oct 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 13, Issue 1 (2026), pp. 57–82

Abstract

The area under the receiver operating characteristic curve (AUC) is a suitable measure for the quality of classification algorithms. Here we use the theory of U-statistics in order to derive new confidence intervals for it. The new confidence intervals take into account that only the total sample size used to calculate the AUC can be controlled, while the number of members of the case group and the number of members of the control group are random. We show that the new confidence intervals can not only be used in order to evaluate the quality of the fitted model, but also to judge the quality of the classification algorithm itself. We would like to take this opportunity to show that two popular confidence intervals for the AUC, namely DeLong’s interval and the Mann–Whitney intervals due to Sen, coincide.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/25-VMSTA124MI

Pub. online: 25 Sep 2025 Type: 2010 Mathematics Subject Classification Index

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 4 (2025), pp. 509–512

Keywords index

https://doi.org/10.15559/25-VMSTA124KI

Pub. online: 25 Sep 2025 Type: Keywords Index

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 4 (2025), pp. 507–508

Author index

https://doi.org/10.15559/25-VMSTA124AI

Pub. online: 25 Sep 2025 Type: Author Index

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 4 (2025), p. 505

Exponentially quasi-mixing limits for killed symmetric Lévy processes

Yunxi Wu

Hanjun Zhang Huasheng Li

https://doi.org/10.15559/25-VMSTA284

Pub. online: 16 Sep 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 13, Issue 1 (2026), pp. 39–56

Abstract

Quasi-mixing limits of the killed symmetric Lévy process are studied. It is proved that (intrinsic) ultracontractivity of the underlying process implies the existence of its (uniformly) exponentially quasi-mixing limits. As a by-product, this implication ensures that the process has (uniformly) exponential quasi-ergodicity and (uniformly) exponentially fractional quasi-ergodicity on ${L^{p}}$ ($p\ge 1$). It is noteworthy that precise rates of convergence and precise limiting equalities are provided, which are determined by spectral gaps and eigenfunction ratios of the underlying process. Finally, three examples are provided to demonstrate the theoretical results.

Asymptotic normality of local linear regression estimator for mixtures with varying concentrations

Daniel Horbunov

Rostyslav Maiboroda

https://doi.org/10.15559/25-VMSTA282

Pub. online: 2 Sep 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 13, Issue 1 (2026), pp. 1–17

Abstract

Finite mixtures with different regression models for different mixture components naturally arise in statistical analysis of biological and sociological data. In this paper a model of mixtures with varying concentrations is considered in which the mixing probabilities are different for different observations. The modified local linear regression estimator (mLLRE) is considered for nonparametric estimation of the unknown regression function for the given component of mixture. The asymptotic normality of the mLLRE is proved in the case when the regressor’s probability density function has jumps. Theoretically optimal bandwidth is derived. Simulations were made to estimate the accuracy of the normal approximation.

Laws of the iterated logarithm for iterated perturbed random walks

Oksana Braganets

https://doi.org/10.15559/25-VMSTA283

Pub. online: 28 Aug 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 13, Issue 1 (2026), pp. 19–38

Abstract

Let ${({\xi _{k}},{\eta _{k}})_{k\ge 1}}$ be independent identically distributed random vectors with arbitrarily dependent positive components and ${T_{k}}:={\xi _{1}}+\cdots +{\xi _{k-1}}+{\eta _{k}}$ for $k\in \mathbb{N}$. The random sequence ${({T_{k}})_{k\ge 1}}$ is called a (globally) perturbed random walk. Consider a general branching process generated by ${({T_{k}})_{k\ge 1}}$ and let ${Y_{j}}(t)$ denote the number of the jth generation individuals with birth times $\le t$. Assuming that $\mathrm{Var}\hspace{0.1667em}{\xi _{1}}\in (0,\infty )$ and allowing the distribution of ${\eta _{1}}$ to be arbitrary, a law of the iterated logarithm (LIL) is proved for ${Y_{j}}(t)$. In particular, an LIL for the counting process of ${({T_{k}})_{k\ge 1}}$ is obtained. The latter result was previously established in the article by Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that $\mathbb{E}{\eta _{1}^{a}}\lt \infty $ for some $a\gt 0$. In this paper, it is shown that the aforementioned additional assumption is not needed.

A change of measures technique for compound mixed renewal processes with applications in Risk Theory

Spyridon M. Tzaninis

Nikolaos D. Macheras

https://doi.org/10.15559/25-VMSTA281

Pub. online: 12 Aug 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 4 (2025), pp. 471–504

Abstract

Given a compound mixed renewal process S under a probability measure P, we provide a characterization of all progressively equivalent martingale probability measures Q on the domain of P, that convert S into a compound mixed Poisson process. This result extends earlier works of Delbaen and Haezendonck, Lyberopoulos and Macheras, and the authors, and enables us to find a wide class of price processes satisfying the condition of no free lunch with vanishing risk. Implications to the ruin problem and to the computation of premium calculation principles in an arbitrage-free insurance market are also discussed.

On the reducibility of affine models with dependent Lévy factors

Michał Barski

Rafał Łochowski

https://doi.org/10.15559/25-VMSTA280

Pub. online: 17 Jun 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 4 (2025), pp. 433–469

Abstract

The paper is devoted to the study of the short rate equation of the form

\[ \text{d}R(t)=F(R(t))\text{d}t+{\sum \limits_{i=1}^{d}}{G_{i}}(R(t-))\text{d}{Z_{i}}(t),\hspace{1em}R(0)={R_{0}}\ge 0,\hspace{3.33333pt}t\gt 0,\]

with deterministic functions $F,{G_{1}},\dots ,{G_{d}}$ and a multivariate Lévy process $Z=({Z_{1}},\dots ,{Z_{d}})$ with possibly dependent coordinates. This equation is assumed to have a nonnegative solution which generates an affine term structure model. Under some mild assumptions on the Lévy measure of Z it is shown that the same term structure is generated by an equation with affine drift term and noise being a one-dimensional α-stable process with index of stability $\alpha \in (1,2)$. For this case the shape of possible simple forward curves is characterized. A precise description of normal, inverse and humped profiles in terms of the equation coefficients and the stability index α is provided.

The paper generalizes the classical results on the Cox–Ingersoll–Ross (CIR) model [Econometrica 53 (1985), 385–408], as well as on its extended version where Z is a one-dimensional Lévy process [SIAM J. Financ. Math. 11(1) (2020), 131–147, Bond Markets with Lévy Factors, Cambridge University Press, 2020]. It is the starting point for the classification of affine models with dependent Lévy processes, in the spirit of [J. Finance 5 (2000), 1943–1978] and [Classification and calibration of affine models driven by independent Lévy processes, https://arxiv.org/abs/2303.08477].

On parameter estimation for

N (μ, σ^{2} I_{3})

based on projected data into

S^{2}

Jordi-Lluís Figueras

Aron Persson

Lauri Viitasaari

https://doi.org/10.15559/25-VMSTA279

Pub. online: 17 Jun 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 4 (2025), pp. 407–431

Abstract

The projected normal distribution, with isotropic variance, on the 2-sphere is considered using intrinsic statistics. It is shown that in this case, the expectation commutes with the projection, and that the covariance of the normal variable has a 1-1 correspondence with the intrinsic covariance of the projected normal distribution. This allows us to estimate, after the model identification, the parameters of the underlying normal distribution that generates the data.

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