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Eduardo Horta

Publications and source records attributed to Eduardo Horta.

9 recordsLinked to original sources
arXiv2026

Set risk measures

We introduce set risk measures (SRMs), real-valued maps defined on the family of non-empty closed bounded sets of essentially bounded random variables. SRMs extend traditional scalar risk measures by assigning a single capital requirement to an entire set of positions. We develop an axiomatic framework for SRMs, adapting classical properties such as monotonicity, translation invariance, convexity, and positive homogeneity to set arithmetic. The main technical contribution is a dual representation of convex SRMs through the \strict{} topology and regular $τ$-additive unit-mass measures. We also characterize worst-case SRMs and present examples related to systemic risk, Knightian uncertainty, and preference representations.

q-fin.MF↗
arXiv2024

Convolution Mode Regression

For highly skewed or fat-tailed distributions, mean or median-based methods often fail to capture the central tendencies in the data. Despite being a viable alternative, estimating the conditional mode given certain covariates (or mode regression) presents significant challenges. Nonparametric approaches suffer from the "curse of dimensionality", while semiparametric strategies often lead to non-convex optimization problems. In order to avoid these issues, we propose a novel mode regression estimator that relies on an intermediate step of inverting the conditional quantile density. In contrast to existing approaches, we employ a convolution-type smoothed variant of the quantile regression. Our estimator converges uniformly over the design points of the covariates and, unlike previous quantile-based mode regressions, is uniform with respect to the smoothing bandwidth. Additionally, the Convolution Mode Regression is dimension-free, carries no issues regarding optimization and preliminary simulations suggest the estimator is normally distributed in finite samples.

econ.EM↗
arXiv2021

Minkowski gauges and deviation measures

We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In doing so, we provide a new interpretation for such measures, namely, that they quantify how much one must shrink or deleverage a position for it to become acceptable. In particular, the Minkowski Deviation of a set which is convex, stable under scalar addition, and radially bounded at non-constants, is a generalized deviation measure. Furthermore, we explore the relations existing between mathematical and financial properties attributable to an acceptance set, and the corresponding properties of the induced measure. Hence, we fill the gap that is the lack of an acceptance set for deviation measures. Dual characterizations in terms of polar sets and support functionals are provided.

q-fin.RM↗
arXiv2020

A characterization of the strong law of large numbers for Bernoulli sequences

The law of large numbers is one of the most fundamental results in Probability Theory. In the case of independent sequences, there are some known characterizations; for instance, in the independent and identically distributed setting it is known that the law of large numbers is equivalent to integrability. In the case of dependent sequences, there are no known general characterizations -- to the best of our knowledge. We provide such a characterization for identically distributed Bernoulli sequences in terms of a product disintegration.

math.PR↗
arXiv2019

Smoothing quantile regressions

We propose to smooth the entire objective function, rather than only the check function, in a linear quantile regression context. Not only does the resulting smoothed quantile regression estimator yield a lower mean squared error and a more accurate Bahadur-Kiefer representation than the standard estimator, but it is also asymptotically differentiable. We exploit the latter to propose a quantile density estimator that does not suffer from the curse of dimensionality. This means estimating the conditional density function without worrying about the dimension of the covariate vector. It also allows for two-stage efficient quantile regression estimation. Our asymptotic theory holds uniformly with respect to the bandwidth and quantile level. Finally, we propose a rule of thumb for choosing the smoothing bandwidth that should approximate well the optimal bandwidth. Simulations confirm that our smoothed quantile regression estimator indeed performs very well in finite samples.

econ.EM↗
arXiv2017

Conjugate Processes: Theory and Application to Risk Forecasting

Many dynamical phenomena display a cyclic behavior, in the sense that time can be partitioned into units within which distributional aspects of a process are homogeneous. In this paper, we introduce a class of models - called conjugate processes - allowing the sequence of marginal distributions of a cyclic, continuous-time process to evolve stochastically in time. The connection between the two processes is given by a fundamental compatibility equation. Key results include Laws of Large Numbers in the presented framework. We provide a constructive example which illustrates the theory, and give a statistical implementation to risk forecasting in financial data.

math.ST↗
arXiv2016

Identifying the Spectral Representation of Hilbertian Time Series

We provide square-root n consistency results regarding estimation of the spectral representation of covariance operators of Hilbertian time series, in a setting with imperfect measurements. This is a generalization of the method developed in Bathia et al. (2010). The generalization relies on an important property of centered random elements in a separable Hilbert space, namely, that they lie almost surely in the closed linear span of the associated covariance operator. We provide a straightforward proof to this fact. This result is, to our knowledge, overlooked in the literature. It incidentally gives a rigorous formulation of PCA in Hilbert spaces.

math.ST↗