Abstract Variable selection comprises an important step in many modern statistical inference procedures. In the regression setting, when estimators cannot shrink irrelevant signals to zero, covariates without relationships to the response often manifest small but nonzero regression coefficients. The ad hoc procedure of discarding variables whose coefficients are smaller than some threshold is often employed in practice. We formally analyze a version of such thresholding procedures and develop a simple thresholding method that consistently estimates the set of relevant variables under mild regularity assumptions. Using this thresholding procedure, we propose a sparse, ‐consistent and asymptotically normal estimator whose nonzero elements do not exhibit shrinkage. The performance and applicability of our approach are examined via numerical studies of simulated and real data.

Abstract Despite the wide usage of parametric point processes in theory and applications, a sound goodness‐of‐fit procedure to test whether a given parametric model is appropriate for data coming from a self‐exciting point process has been missing in the literature. In this work, we establish a bootstrap‐based goodness‐of‐fit test that empirically works for all kinds of self‐exciting point processes (and even beyond). In an infill‐asymptotic setting, we also prove its asymptotic consistency, albeit only in the particular case that the underlying point process is inhomogeneous Poisson.

Abstract Univariate marked Hawkes processes are used to model a range of real‐world phenomena including earthquake aftershock sequences, contagious disease spread, content diffusion on social media platforms, and order book dynamics. This paper illustrates a fundamental connection between univariate marked Hawkes processes and multivariate Hawkes processes. Exploiting this connection renders a framework that can be built upon for expressive and flexible inference on diverse data. Specifically, multivariate unmarked Hawkes representations are introduced as a tool to parameterize univariate marked Hawkes processes. We show that such multivariate representations can asymptotically approximate a large class of univariate marked Hawkes processes, are stationary given the approximated process is stationary, and that resultant conditional intensity parameters are identifiable and, more importantly, interpretable. A simulation study provides a heuristic bound for error induced by the relatively larger parameter space of multivariate Hawkes processes, and an application to the Southern California earthquake catalogue is presented to demonstrate the efficacy of our novel approach.

Abstract We present new estimators for generalized tail distortion (GTD) risk measures to assess extreme risks. Proposed estimators are based on the first‐order asymptotic expansions of the risk measure. They are simple to apply, and they are shown through simulation experiments to provide performance that is comparable or even better than that of existing estimation methods from the literature. A reinsurance premium principle based on the GTD risk measure is proposed. It is tested on car insurance claims data. We propose to use the GTD risk measure and the corresponding reinsurance premium to embed a safety loading in pricing, protecting against statistical uncertainty.

Abstract The Gamma process stands as a prevalent model for monotonic degradation data. However, its statistical inference faces complexity due to the intricate parameter structure within the likelihood function. This paper addresses this challenge by deriving a conjugate prior specifically for the homogeneous gamma process, investigating the properties of this prior distribution. To facilitate posterior inference, three meticulously designed algorithms (Gibbs sampling, discrete grid sampling, and sampling importance resampling) are employed to generate posterior samples of the model parameters. Through extensive simulation studies, these algorithms demonstrate notably high computational efficiency and precise estimation. Extending the conjugate prior to encompass the gamma process with heterogeneous effects, this study enables recursive updates to the posterior distribution of parameters when the inspection time epochs are evenly spaced. An innovative online algorithm is consequently developed, empowering the prediction of remaining useful life for multiple systems. The effectiveness of this online algorithm is demonstrated through comprehensive illustrations from two real‐world cases and a simulated dataset under a high‐frequency monitoring scenario.

ABSTRACT A regression model for a circular response variable depending on a linear or a circular predictor is presented in this paper. The conditional density belongs to a parametric flexible family that allows for asymmetry and varying peakedness around the modal direction. The modal direction and concentration depend on the covariate and are nonparametrically modeled via local polynomial fitting with a kernel weight. The asymptotic normality of the estimators for the conditional modal direction and concentration is established. Furthermore, from these theoretical results, the expression of the optimal smoothing parameter and a proposed data‐driven estimator are derived. An application concerns the orientation of migratory birds according to the flight altitude and the wind direction.

Abstract This paper introduces a novel semiparametric model‐assisted estimation method that integrates data from both probability and nonprobability samples, thereby facilitating robust and efficient inferences regarding finite population parameters. To mitigate selection bias—whether ignorable or nonignorable—associated with the nonprobability sample, we propose a flexible semiparametric propensity score model that extends beyond the missing at random assumption. Our approach employs a pseudo‐profile‐likelihood method to estimate the propensity score model. Subsequently, a difference estimator is constructed utilizing the probability sample as a foundation, where the proxy values of the study variable for the finite population are derived from the nonprobability sample using the estimated propensity score model. We present the asymptotic properties of the proposed estimators and provide formulae for variance estimation. Through a series of simulations and a real data application, we validate our proposed estimation procedure and demonstrate its superiority over some existing estimators.

ABSTRACT This paper investigates some theoretical properties of the Partial Least Squares method. We focus our attention on the single‐component case, which provides a useful framework to understand the underlying mechanism. We provide a non‐asymptotic upper bound on the quadratic loss in prediction with high probability in a high‐dimensional regression context. The bound is attained thanks to a preliminary regularization on the first PLS component. In a second time, we extend these results to the sparse Partial Least Squares approach. In particular, we exhibit upper bounds similar to those obtained with the lasso algorithm, up to an additional restricted eigenvalue constraint on the design matrix.

Abstract For multivariate regularly random vectors of dimension , the dependence structure of the extremes is modeled by the so‐called angular measure. When the dimension is high, estimating the angular measure is challenging because of its complexity. In this paper, we use Principal Component Analysis (PCA) as a method for dimension reduction and estimate the number of significant principal components of the empirical covariance matrix of the angular measure under the assumption of a spiked covariance structure. Therefore, we develop Akaike Information Criteria () and Bayesian Information Criteria () to estimate the location of the spiked eigenvalue of the covariance matrix, reflecting the number of significant components, and explore these information criteria on consistency. On the one hand, we investigate the case where the dimension is fixed, and on the other hand, where the dimension converges to under different high‐dimensional scenarios. When the dimension is fixed, we establish that the is not consistent, whereas the is weakly consistent. In high‐dimensional contexts, we utilize methods from random matrix theory to establish sufficient conditions ensuring the consistency of the AIC and BIC. Finally, the performance of the different information criteria is compared in a simulation study and applied to high‐dimensional precipitation data.