The max-stable process is an asymptotically justified model for spatial extremes. In particular, we focus on the hierarchical extreme-value process (HEVP), which is a particular max-stable process that is conducive to Bayesian computing. The HEVP and all max-stable process models are parametric and impose strong assumptions including that all marginal distributions belong to the generalized extreme value family and that nearby sites are asymptotically dependent. We generalize the HEVP by relaxing these assumptions to provide a wider class of marginal distributions via a Dirichlet process prior for the spatial random effects distribution. In addition, we present a hybrid max-mixture model that combines the strengths of the parametric and semi-parametric models. We show that this versatile max-mixture model accommodates both asymptotic independence and dependence and can be fit using standard Markov chain Monte Carlo algorithms. The utility of our model is evaluated in Monte Carlo simulation studies and application to Netherlands wind gust data.
The study of mixture models constitutes a large domain of research in statistics. In the first part of this work, we present phi-divergences and the existing methods which produce robust estimators. We are more particularly interested in the so-called dual formula of phi-divergences. We build a new robust estimator based on this formula. We study its asymptotic properties and give a numerical comparison with existing methods on simulated data. We also introduce a proximal-point algorithm whose aim is to calculate divergence-based estimators. We give some of the convergence properties of this a…
We present a method for modelling the covariance structure of tensor-variate data, with the ulterior aim of learning an unknown model parameter vector using such data. We express the high-dimensional observable as a function of this sought model parameter vector, and attempt to learn such a high-dimensional function given training data, by modelling it as a realisation from a tensor-variate Gaussian Process (GP). The likelihood of the unknowns given training data, is then tensor-normal. We choose vague priors on the unknown GP parameters (mean tensor and covariance matrices) and write the post…
How to include censored data in a statistical analysis is a recur-rent issue in statistics. In multivariate extremes, the dependence structure of large observations can be characterized in terms of a non parametric angular measure, while marginal excesses above asymptotically large thresholds have a parametric distribution. In this work, a flexible semi-parametric Dirichlet mix-ture model for angular measures is adapted to the context of censored data and missing components. One major issue is to take into account censoring intervals overlapping the extremal threshold, without knowing whether …
We propose a structure of a semiparametric two-component mixture model when one component is parametric and the other is defined through linear constraints on its distribution function. Estimation of a two-component mixture model with an unknown component is very difficult when no particular assumption is made on the structure of the unknown component. A symmetry assumption was used in the literature to simplify the estimation. Such method has the advantage of producing consistent and asymptotically normal estimators, and identifiability of the semiparametric mixture model becomes tractable. S…
We consider a sparse linear regression model with unknown symmetric error under the high-dimensional setting. The true error distribution is assumed to belong to the locally $β$-Hölder class with an exponentially decreasing tail, which does not need to be sub-Gaussian. We obtain posterior convergence rates of the regression coefficient and the error density, which are nearly optimal and adaptive to the unknown sparsity level. Furthermore, we derive the semi-parametric Bernstein-von Mises (BvM) theorem to characterize asymptotic shape of the marginal posterior for regression coefficients. Under…
Tensor-valued data are being encountered increasingly more commonly, in the biological, natural as well as the social sciences. The learning of the unknown model parameter vector given such data, involves covariance modelling of such data, though this can be difficult owing to the high-dimensional nature of the data, where the numerical challenge of such modelling can only be compounded by the largeness of the available data set. Assuming such data to be modelled using a correspondingly high-dimensional Gaussian Process (${\cal GP}$), the joint density of a finite set of such data sets is then…
We consider the problem of estimating the expected value of information (the knowledge gradient) for Bayesian learning problems where the belief model is nonlinear in the parameters. Our goal is to maximize some metric, while simultaneously learning the unknown parameters of the nonlinear belief model, by guiding a sequential experimentation process which is expensive. We overcome the problem of computing the expected value of an experiment, which is computationally intractable, by using a sampled approximation, which helps to guide experiments but does not provide an accurate estimate of the …
In this paper, a shrinkage estimator for the population mean is proposed under known quadratic loss functions with unknown covariance matrices. The new estimator is non-parametric in the sense that it does not assume a specific parametric distribution for the data and it does not require the prior information on the population covariance matrix. Analytical results on the improvement of the proposed shrinkage estimator are provided and some corresponding asymptotic properties are also derived. Finally, we demonstrate the practical improvement of the proposed method over existing methods through…
We consider a combined state and drift estimation problem for the linear stochastic heat equation. The infinite-dimensional Bayesian inference problem is formulated in terms of the Kalman-Bucy filter over an extended state space, and its long-time asymptotic properties are studied. Asymptotic posterior contraction rates in the unknown drift function are the main contribution of this paper. Such rates have been studied before for stationary non-parametric Bayesian inverse problems, and here we demonstrate the consistency of our time-dependent formulation with these previous results building upo…
We provide a comprehensive semi-parametric study of Bayesian partially identified econometric models. While the existing literature on Bayesian partial identification has mostly focused on the structural parameter, our primary focus is on Bayesian credible sets (BCS's) of the unknown identified set and the posterior distribution of its support function. We construct a (two-sided) BCS based on the support function of the identified set. We prove the Bernstein-von Mises theorem for the posterior distribution of the support function. This powerful result in turn infers that, while the BCS and the…
Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, and other quantities.
In this paper, we present an infinite hierarchical non-parametric Bayesian model to extract the hidden factors over observed data, where the number of hidden factors for each layer is unknown and can be potentially infinite. Moreover, the number of layers can also be infinite. We construct the model structure that allows continuous values for the hidden factors and weights, which makes the model suitable for various applications. We use the Metropolis-Hastings method to infer the model structure. Then the performance of the algorithm is evaluated by the experiments. Simulation results show tha…
We study full Bayesian procedures for sparse linear regression when errors have a symmetric but otherwise unknown distribution. The unknown error distribution is endowed with a symmetrized Dirichlet process mixture of Gaussians. For the prior on regression coefficients, a mixture of point masses at zero and continuous distributions is considered. We study behavior of the posterior with diverging number of predictors. Conditions are provided for consistency in the mean Hellinger distance. The compatibility and restricted eigenvalue conditions yield the minimax convergence rate of the regression…
Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, moments, and other quantities. We prove asymptotic convergence and consiste…
In many settings it is critical to accurately model the extreme tail behaviour of a random process. Non-parametric density estimation methods are commonly implemented as exploratory data analysis techniques for this purpose as they possess excellent visualisation properties, and can naturally avoid the model specification biases implied by using parametric estimators. In particular, kernel-based estimators place minimal assumptions on the data, and provide improved visualisation over scatterplots and histograms. However kernel density estimators are known to perform poorly when estimating extr…
This paper develops Bayesian econometric methods for posterior inference in non-parametric mixed frequency VARs using additive regression trees. We argue that regression tree models are ideally suited for macroeconomic nowcasting in the face of extreme observations, for instance those produced by the COVID-19 pandemic of 2020. This is due to their flexibility and ability to model outliers. In an application involving four major euro area countries, we find substantial improvements in nowcasting performance relative to a linear mixed frequency VAR.
General relativity can be tested by comparing the binary-inspiral signals found in LIGO--Virgo data against waveform models that are augmented with artificial degrees of freedom. This approach suffers from a number of logical and practical pitfalls. 1) It is difficult to ascribe meaning to the stringency of the resultant constraints. 2) It is doubtful that the Bayesian model comparison of relativity against these artificial models can offer actual validation for the former. 3) It is unknown to what extent these tests might detect alternative theories of gravity for which there are no computed …
The evaluation of a match between the DNA profile of a stain found on a crime scene and that of a suspect (previously identified) involves the use of the unknown parameter $p=(p_1, p_2, ...)$, (the ordered vector which represents the proportions of the different DNA profiles in the population of potential donors) and the names of the different DNA types. We propose a Bayesian non parametric method which considers $P$ as a random variable distributed according to the two-parameter Poisson Dirichlet distribution, and discard information about names of DNA types. The ultimate goal of this model i…
In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations usi…
Nonparametric resampling methods such as Direct Sampling are powerful tools to simulate new datasets preserving important data features such as spatial patterns from observed datasets while using only minimal assumptions. However, such methods cannot generate extreme events beyond the observed range of data values. We here propose using tools from extreme value theory for stochastic processes to extrapolate observed data towards yet unobserved high quantiles. Original data are first enriched with new values in the tail region, and then classical resampling algorithms are applied to enriched da…
Generalized extreme value (GEV) regression is often more adapted when we investigate a relationship between a binary response variable $Y$ which represents a rare event and potentiel predictors $\mathbf{X}$. In particular, we use the quantile function of the GEV distribution as link function. Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, test of hypothesis) to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Bootstrapping estimates the properties of an est…
Preferential attachment is an appealing edge generating mechanism for modeling social networks. It provides both an intuitive description of network growth and an explanation for the observed power laws in degree distributions. However, there are often limitations in fitting parametric network models to data due to the complex nature of real-world networks. In this paper, we consider a semi-parametric estimation approach by looking at only the nodes with large in- or out-degrees of the network. This method examines the tail behavior of both the marginal and joint degree distributions and is ba…
In environmental science applications, extreme events frequently exhibit a complex spatio-temporal structure, which is difficult to describe flexibly and estimate in a computationally efficient way using state-of-art parametric extreme-value models. In this paper, we propose a computationally-cheap non-parametric approach to investigate the probability distribution of temporal clusters of spatial extremes, and study within-cluster patterns with respect to various characteristics. These include risk functionals describing the overall event magnitude, spatial risk measures such as the size of th…
Parametric max-stable processes are increasingly used to model spatial extremes. Starting from the fact that the dependence structure of a max-stable process is completely characterized by an extreme-value copula, a class of goodness-of-fit tests is proposed based on the comparison between a nonparametric and a parametric estimator of the corresponding unknown multivariate Pickands dependence function. Because of the high-dimensional setting under consideration, these functional estimators are only compared at a specific set of points at which they coincide, up to a multiplicative constant, wi…
Statistical inference for extreme values of random events is difficult in practice due to low sample sizes and inaccurate models for the studied rare events. If prior knowledge for extreme values is available, Bayesian statistics can be applied to reduce the sample complexity, but this requires a known probability distribution. By working with the quantiles for extremely low probabilities (in the order of $10^{-2}$ or lower) and relying on their asymptotic normality, inference can be carried out without assuming any distributions. Despite relying on asymptotic results, it is shown that a Bayes…
We propose a new method for estimating the extreme quantiles for a function of several dependent random variables. In contrast to the conventional approach based on extreme value theory, we do not impose the condition that the tail of the underlying distribution admits an approximate parametric form, and, furthermore, our estimation makes use of the full observed data. The proposed method is semiparametric as no parametric forms are assumed on all the marginal distributions. But we select appropriate bivariate copulas to model the joint dependence structure by taking the advantage of the recen…
Multivariate extreme value models are used to estimate joint risk in a number of applications, with a particular focus on environmental fields ranging from climatology and hydrology to oceanography and seismic hazards. The semi-parametric conditional extreme value model of Heffernan and Tawn (2004) involving a multivariate regression provides the most suitable of current statistical models in terms of its flexibility to handle a range of extremal dependence classes. However, the standard inference for the joint distribution of the residuals of this model suffers from the curse of dimensionalit…
Parametric and semiparametric tests of circular reflective symmetry about an unknown central direction are developed that are locally and asymptotically optimal in the Le Cam sense against asymmetric $k$-sine-skewed alternatives. The results from Monte Carlo studies comparing the rejection rates of tests with those of previously proposed tests lead to recommendations regarding the use of the various tests with small- to medium-sized samples. Analyses of data on the directions of cracks in cemented femoral components and the times of gun crimes in Pittsburgh illustrate the proposed methodology …
A simple variogram model with two parameters is presented that includes the power variogram for the fractional Brownian motion, a modified De Wijsian model, the generalized Cauchy model and the multiquadrics model. One parameter controls the smoothness of the process. The other parameter allows for a smooth parametrization between stationary and intrinsically stationary second order processes in a Gaussian framework, or between mixing and non-ergodic max-stable processes when modeling spatial extremes by a Brown-Resnick process.
An important research topic in survival analysis is related to the modeling and estimation of the cure rate, i.e. the proportion of subjects that will never experience the event of interest. However, most estimation methods proposed so far in the literature do not handle the case of insufficient follow-up, that is when the right end point of the support of the censoring time is strictly less than that of the survival time of the susceptible subjects, and consequently these estimators overestimate the cure rate in that case. We fill this gap by proposing a new estimator of the cure rate that ma…
Many environmental processes exhibit weakening spatial dependence as events become more extreme. Well-known limiting models, such as max-stable or generalized Pareto processes, cannot capture this, which can lead to a preference for models that exhibit a property known as asymptotic independence. However, weakening dependence does not automatically imply asymptotic independence, and whether the process is truly asymptotically (in)dependent is usually far from clear. The distinction is key as it can have a large impact upon extrapolation, i.e., the estimated probabilities of events more extreme…
Semiparametric models are useful in econometrics, social sciences and medicine application. In this paper, a new estimator based on least square methods is proposed to estimate the direction of unknown parameters in semi-parametric models. The proposed estimator is consistent and has asymptotic distribution under mild conditions without the knowledge of the form of link function. Simulations show that the proposed estimator is significantly superior to maximum score estimator given by Manski (1975) for binary response variables. When the error term is long-tailed distributions or distribution …
The estimation of extreme flood quantiles is challenging due to the relative scarcity of extreme data compared to typical target return periods. Several approaches have been developed over the years to face this challenge, including regional estimation and the use of historical flood data. This paper investigates the combination of both approaches using a multivariate peaks-over-threshold model, that allows estimating altogether the intersite dependence structure and the marginal distributions at each site. The joint distribution of extremes at several sites is constructed using a semi-paramet…
Estimation of extreme value copulas is often required in situations where available data are sparse. Parametric methods may then be the preferred approach. A possible way of defining parametric families that are simple and, at the same time, cover a large variety of multivariate extremal dependence structures is to build models based on spectral measures. This approach is considered here. Parametric families of spectral measures are defined as convex hulls of suitable basis elements, and parameters are estimated by projecting an initial nonparametric estimator on these finite-dimensional space…
In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Log Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an applicat…
A recurrent question in climate risk analysis is determining how climate change will affect heavy precipitation patterns. Dividing the globe into homogeneous sub-regions should improve the modelling of heavy precipitation by inferring common regional distributional parameters. In addition, in the detection and attribution (D&A) field, biases due to model errors in global climate models (GCMs) should be considered to attribute the anthropogenic forcing effect. Within this D&A context, we propose an efficient clustering algorithm that, compared to classical regional frequency analysis (RFA) tech…
The problem of quickest detection of a change in the distribution of a $n\times p$ random matrix based on a sequence of observations having a single unknown change point is considered. The forms of the pre- and post-change distributions of the rows of the matrices are assumed to belong to the family of elliptically contoured densities with sparse dispersion matrices but are otherwise unknown. We propose a non-parametric stopping rule that is based on a novel summary statistic related to k-nearest neighbor correlation between columns of each observed random matrix. In the large scale regime of …
Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as observing a large value in all variables simultaneously. However, there is growing evidence that asymptotic independence is equally important in real world applications. Available statistical methodology in the latter setting is scarce and not well understood theoretically. We revisit non-parametric estimation and introduce rank-bas…
Recently a distribution free approach for testing parametric hypotheses based on unitary transformations has been suggested in \cite{Khm13, Khm16, Khm17} and further studied in \cite{Ngu17} and \cite{Rob19}. In this note we show that the transformation takes extremely simple form in distribution free testing of linear regression. Then we extend it to general parametric regression with vector-valued covariates.
Multivariate extreme-value analysis is concerned with the extremes in a multivariate random sample, that is, points of which at least some components have exceptionally large values. Mathematical theory suggests the use of max-stable models for univariate and multivariate extremes. A comprehensive account is given of the various ways in which max-stable models are described. Furthermore, a construction device is proposed for generating parametric families of max-stable distributions. Although the device is not new, its role as a model generator seems not yet to have been fully exploited.
Conventional methods for extreme event estimation rely on well-chosen parametric models asymptotically justified from extreme value theory (EVT). These methods, while powerful and theoretically grounded, could however encounter a difficult bias-variance tradeoff that exacerbates especially when data size is too small, deteriorating the reliability of the tail estimation. In this paper, we study a framework based on the recently surging literature of distributionally robust optimization. This approach can be viewed as a nonparametric alternative to conventional EVT, by imposing general shape be…
We consider efficient estimation of flexible transformation models with interval-censored data. To reduce the dimension of semi-parametric models, the unknown monotone transformation function is approximated via monotone splines. A penalization technique is used to provide more computationally efficient estimation of all parameters. To accomplish model fitting, a computationally efficient nested iterative expectation-maximization (EM) based algorithm is developed for estimation, and an easily implemented variance-covariance approach is proposed for inference on regression parameters. Theoretic…
Much of the work in the literature on optimal discrimination designs assumes that the models of interest are fully specified, apart from unknown parameters in some models. Recent work allows errors in the models to be non-normally distributed but still requires the specification of the mean structures. This research is motivated by the interesting work of Otsu (2008) to discriminate among semi-parametric models by generalizing the KL-optimality criterion proposed by López-Fidalgo et al. (2007) and Tommasi and López-Fidalgo (2010). In our work we provide further important insights in this inter…
