In this paper, we consider a novel methodology to estimate linear quantile regression models when the response is randomly right censored. The proposed methodology is based on approximating the error distribution by means of an extension of the Laplace distribution, that is flexible enough to approximate any continuous distribution as long as the number of parameters in this Enriched Laplace distribution grows to infinity. The extension is obtained by enriching the Laplace density by means of Laguerre polynomials in such a way that the new density has by construction the property that its quantile of interest is equal to zero. Hence, when the error term has an Enriched Laplace density, it satisfies by construction the constraints of a quantile regression model. We will show that, with this new quantile regression model, we can obtain novel estimators of the quantile function, which are shown to be consistent and asymptotically normal. We also establish the asymptotic efficiency bound, and show by means of a simulation study and the analysis of data on Covid-19 patients that the proposed method works well in practice compared to competing estimators.

In this paper we study the properties of the Lasso estimator of the drift component in the diffusion setting. More specifically, we consider a multivariate parametric diffusion model X observed continuously over the interval [0,T] and investigate drift estimation under sparsity constraints. We allow the dimensions of the model and the parameter space to be large. We obtain an oracle inequality for the Lasso estimator and derive an error bound for the L2-distance using concentration inequalities for linear functionals of diffusion processes. The probabilistic part is based upon elements of empirical processes theory and, in particular, on the chaining method.

We consider the infinite dimensional case for Gaussian fields having a compact set of Rd as their index set. For Hilbert space valued Gaussian fields, we prove conditions for equivalence of their Gaussian measures, in the sense that the Gaussian measures induced by two given fields are equivalent on their paths. Such conditions are proved to depend on the difference between the correlation operators that characterize the two Gaussian measures. Further, we provide the analogue of the celebrated Bochner’s characterisation for correlation operators that represents the infinite dimensional extension of matrix-valued kernels.

We present a method to construct signatures of periodic-like data. Based on topological considerations, our construction encodes information about the order and values of local extrema. Its main strength is robustness to reparametrization of the observed signal, so that it depends only on the form of the periodic function. The signature converges as the observation contains increasingly many periods. We show that it can be estimated from the observation of a single time series using bootstrap techniques.

Physics-informed neural networks (PINNs) are a promising approach that combines the power of neural networks with the interpretability of physical modeling. PINNs have shown good practical performance in solving partial differential equations (PDEs) and in hybrid modeling scenarios, where physical models enhance data-driven approaches. However, it is essential to establish their theoretical properties in order to fully understand their capabilities and limitations. In this study, we highlight that classical training of PINNs can suffer from systematic overfitting. This problem can be addressed by adding a ridge regularization to the empirical risk, which ensures that the resulting estimator is risk-consistent for both linear and nonlinear PDE systems. However, the strong convergence of PINNs to a solution satisfying the physical constraints requires a more involved analysis using tools from functional analysis and calculus of variations. In particular, for linear PDE systems, an implementable Sobolev-type regularization allows to reconstruct a solution that not only achieves statistical accuracy but also maintains consistency with the underlying physics.

Numerous studies have been devoted to the estimation and inference problems for functional linear models (FLM). However, few works focus on model checking problem that ensures the reliability of results. Limited tests in this area do not have tractable null distributions or asymptotic analysis under alternatives. Also, the functional predictor is usually assumed to be fully observed, which is impractical. To address these problems, we propose an adaptive model checking test for FLM. It combines regular moment-based and conditional moment-based tests, and achieves model adaptivity via the dimension of a residual-based subspace. The advantages of our test are manifold. First, it has a tractable chi-squared null distribution and higher powers under the alternatives than its components. Second, asymptotic properties under different underlying models are developed, including the unvisited local alternatives. Third, the test statistic is constructed upon finite grid points, which incorporates the discrete nature of collected data. We develop the desirable relationship between sample size and number of grid points to maintain the asymptotic properties. Besides, we provide a data-driven approach to estimate the dimension leading to model adaptivity, which is promising in sufficient dimension reduction. We conduct comprehensive numerical experiments to demonstrate the advantages the test inherits from its two simple components.

We propose a covariance stationarity test for an otherwise dependent and possibly globally non-stationary time series. We work in a generalized version of the new setting in Jin, Wang and Wang (J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 893–922), who exploit Walsh (Amer. J. Math. 45 (1923) 5–24) functions in order to compare sub-sample covariances with the full sample counterpart. They impose strict stationarity under the null, only consider linear processes under either hypothesis in order to achieve a parametric estimator for an inverted high dimensional asymptotic covariance matrix, and do not consider any other orthonormal basis. Conversely, we work with a general orthonormal basis under mild conditions that include Haar wavelet and Walsh functions, and we allow for linear or nonlinear processes with possibly non-iid innovations. This is important in macroeconomics and finance where nonlinear feedback and random volatility occur in many settings. We completely sidestep asymptotic covariance matrix estimation and inversion by bootstrapping a max-correlation difference statistic, where the maximum is taken over the correlation lag h and basis generated sub-sample counter k (the number of systematic samples). We achieve a higher feasible rate of increase for the maximum lag and counter HT and KT. Of particular note, our test is capable of detecting breaks in variance, and distant, or very mild, deviations from stationarity.

We introduce a hull operator on Poisson point processes, the easiest example being the convex hull of the support of a point process in Euclidean space. Assuming that the intensity measure of the process is known on the set generated by the hull operator, we discuss estimation of an expected linear statistic built on the Poisson process. In special cases, our general scheme yields an estimator of the volume of a convex body or an estimator of an integral of a Hölder function. We show that the estimation error is given by the Kabanov–Skorohod integral with respect to the underlying Poisson process. A crucial ingredient of our approach is a spatial strong Markov property of the underlying Poisson process with respect to the hull. We derive the rate of normal convergence for the estimation error, and illustrate it on an application to estimators of integrals of a Hölder function. We also discuss estimation of higher order symmetric statistics.