The classical approach to analyzing extreme value data is the generalized Pareto distribution (GPD). When the GPD is used to explain a target variable with the large dimension of covariates, the shape and scale function of covariates included in GPD are sometimes modeled using the generalized additive models (GAM). In contrast to many results of application, there are no theoretical results on the hybrid technique of GAM and GPD, which motivates us to develop its asymptotic theory. We provide the rate of convergence of the estimator of shape and scale functions, as well as its local asymptotic normality.
This paper discusses asymptotic theory for penalized spline estimators in generalized additive models. The purpose of this paper is to establish the asymptotic bias and variance as well as the asymptotic normality of the penalized spline estimators proposed by Marx and Eilers (1998). Furthermore, the asymptotics for the penalized quasi likelihood fit in mixed models are also discussed.
In multivariate regression estimation, the rate of convergence depends on the dimension of the regressor. This fact, known as the curse of the dimensionality, motivated several works. The additive model, introduced by Stone (10), offers an efficient response to this problem. In the setting of continuous time processes, using the marginal integration method, we obtain the quadratic convergence rate and the asymptotic normality of the components of the additive model.
We establish asymptotic normality for estimators of the additive regression components under random censorship. To build our estimators, we couple the marginal integration method (Newey (1994)) with an initial Inverse Probability of Censoring Weighted estimator of the multivariate censored regression function introduced by Carbonez et al. (1995) and Kohler et al. (2002). Asymptotic confidence bands are derived from our result.
This paper is concerned with asymptotic theory for penalized spline estimator in bivariate additive model. The focus of this paper is put upon the penalized spline estimator obtained by the backfitting algorithm. The convergence of the algorithm as well as the uniqueness of its solution are shown. The asymptotic bias and variance of penalized spline estimator are derived by an efficient use of the asymptotic results for the penalized spline estimator in marginal univariate model. Asymptotic normality of estimator is also developed, by which an approximate confidence interval can be obtained. S…
This paper describes an estimator of the additive components of a nonparametric additive model with a known link function. When the additive components are twice continuously differentiable, the estimator is asymptotically normally distributed with a rate of convergence in probability of n^{-2/5}. This is true regardless of the (finite) dimension of the explanatory variable. Thus, in contrast to the existing asymptotically normal estimator, the new estimator has no curse of dimensionality. Moreover, the estimator has an oracle property. The asymptotic distribution of each additive component is…
We study the problem of detection of a high-dimensional signal function in the white Gaussian noise model. As well as a smoothness assumption on the signal function, we assume an additive sparse condition on the latter. The detection problem is expressed in terms of a nonparametric hypothesis testing problem and it is solved according to the asymptotical minimax approach. The minimax test procedures are adaptive in the sparsity parameter for high sparsity case. We extend to the functional case the known results in the detection of sparse high-dimensional vectors. In particular, our asymptotic …
A parameter estimation problem is considered for a diagonaliazable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter $H\geq 1/2$. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation.
We study the simple hypothesis testing problem for the drift coefficient for stochastic fractional heat equation driven by additive noise. We introduce the notion of asymptotically the most powerful test, and find explicit forms of such tests in two asymptotic regimes: large time asymptotics, and increasing number of Fourier modes. The proposed statistics are derived based on Maximum Likelihood Ratio. Additionally, we obtain a series of important technical results of independent interest: we find the cumulant generating function of the log-likelihood ratio; obtain sharp large deviation type re…
We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, \textbf{28(2)}, 386--400, 2019). Simplified smooth backfitting achieves oracle properties under regularity conditions and provides closed-form expressions of the estimators that are useful for deriving asymptotic properties. We develop a generalized likelihood ratio (GLR) and a loss function (LF) based testing framework for inference. Under the null hypothesis, both the GLR and LF tests have asymptotically rescaled ch…
In this paper we discuss the estimation of a nonparametric component $f_1$ of a nonparametric additive model $Y=f_1(X_1) + ...+ f_q(X_q) + ε$. We allow the number $q$ of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating $f_1$ in the oracle model $Z= f_1(X_1) + ε$, for which the additive components $f_2,\dots,f_q$ are known. We construct a two-step presmoothing-and-resmoothing estimator of $f_1$ and state finite-sample bounds for the difference between our estimator…
Max-stable distributions and processes are important models for extreme events and the assessment of tail risks. The full, multivariate likelihood of a parametric max-stable distribution is complicated and only recent advances enable its use. The asymptotic properties of the maximum likelihood estimator in multivariate extremes are mostly unknown. In this paper we provide natural conditions on the exponent function and the angular measure of the max-stable distribution that ensure asymptotic normality of the estimator. We show the effectiveness of this result by applying it to popular parametr…
We consider efficient estimation of the Euclidean parameters in a generalized partially linear additive models for longitudinal/clustered data when multiple covariates need to be modeled nonparametrically, and propose an estimation procedure based on a spline approximation of the nonparametric part of the model and the generalized estimating equations (GEE). Although the model in consideration is natural and useful in many practical applications, the literature on this model is very limited because of challenges in dealing with dependent data for nonparametric additive models. We show that the…
We propose to approximate the conditional expectation of a spatial random variable given its nearest-neighbour observations by an additive function. The setting is meaningful in practice and requires no unilateral ordering. It is capable of catching nonlinear features in spatial data and exploring local dependence structures. Our approach is different from both Markov field methods and disjunctive kriging. The asymptotic properties of the additive estimators have been established for $α$-mixing spatial processes by extending the theory of the backfitting procedure to the spatial case. This fac…
An additive growth curve model with orthogonal design matrices is proposed in which observations may have different profile forms. The proposed model allows us to fit data and then estimate parameters in a more parsimonious way than the traditional growth curve model. Two-stage generalized least-squares estimators for the regression coefficients are derived where a quadratic estimator for the covariance of observations is taken as the first-stage estimator. Consistency, asymptotic normality and asymptotic independence of these estimators are investigated. Simulation studies and a numerical exa…
In this paper we investigate the Burr distributions family which contains twelve members. Second order expansions of quantiles of the Burr's distributions are provided on which may be based statistical methods, in particular in extreme value theory. Beyond the proper interest of these expansions, we apply them to characterize the asymptotic laws of their records of Burr's distributions, lead to new statistical tests.
We propose a two-step estimating procedure for generalized additive partially linear models with clustered data using estimating equations. Our proposed method applies to the case that the number of observations per cluster is allowed to increase with the number of independent subjects. We establish oracle properties for the two-step estimator of each function component such that it performs as well as the univariate function estimator by assuming that the parametric vector and all other function components are known. Asymptotic distributions and consistency properties of the estimators are ob…
We consider an additive partially linear framework for modelling massive heterogeneous data. The major goal is to extract multiple common features simultaneously across all sub-populations while exploring heterogeneity of each sub-population. We propose an aggregation type of estimators for the commonality parameters that possess the asymptotic optimal bounds and the asymptotic distributions as if there were no heterogeneity. This oracle result holds when the number of sub-populations does not grow too fast and the tuning parameters are selected carefully. A plug-in estimator for the heterogen…
Extrapolation of extreme covariates in generalized additive regression using extreme-value theory
INRAE（法国国家农业食品环境研究院）；University of Geneva（日内瓦大学）；Univ Brest, CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique（布雷斯特大学、法国国家科学研究中心第6205联合研究单位、布列塔尼大西洋数学实验室）
The paper studies the asymptotic behaviour of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given by these functionals. It is shown that in the discrete sampling case additive functionals have the same asymptotic distribution as the corresponding integral functionals for the continuous functional data case. These results are applied to obtain non-central limit theorems for weighted additive functionals of random fields. As the majority of known results c…
In many complex applications, data heterogeneity and homogeneity exist simultaneously. Ignoring either one will result in incorrect statistical inference. In addition, coping with complex data that are non-Euclidean becomes more common. To address these issues we consider a distributional data response additive model in which the response is a distributional density function and the individual effect curves are homogeneous within a group but heterogeneous across groups, the covariates capturing the variation share common additive bivariate functions. A transformation approach is first utilized…
We consider pricing weather derivatives for use as protection against weather extremes. The method described utilizes results from spatial statistics and extreme value theory to first model extremes in the weather as a max-stable process, and then use these models to simulate payments for a general collection of weather derivatives. These simulations capture the spatial dependence of payments. Incorporating results from catastrophe ratemaking, we show how this method can be used to compute risk loads and premiums for weather derivatives which are renewal-additive.
Asymptotic equivalence for nonparametric additive regression
Institut f\"ur Statistik und Operations Research, Universit\"at Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria e1；Department, University or Company Name [Additional affiliations should be put in the Acknowledgments section] e1；Department, University or Company Name [Additional affiliations should be put in the Acknowledgments section] e2,e3
A framework for estimation and hypothesis testing of functional restrictions against general alternatives is proposed. The parameter space is a reproducing kernel Hilbert space (RKHS). The null hypothesis does not necessarily define a parametric model. The test allows us to deal with infinite dimensional nuisance parameters. The methodology is based on a moment equation similar in spirit to the construction of the efficient score in semiparametric statistics. The feasible version of such moment equation requires to consistently estimate projections in the space of RKHS and it is shown that thi…
We tackle the modeling of threshold exceedances in asymptotically independent stochastic processes by constructions based on Laplace random fields. These are defined as Gaussian random fields scaled with a stochastic variable following an exponential distribution. This framework yields useful asymptotic properties while remaining statistically convenient. Univariate distribution tails are of the half exponential type and are part of the limiting generalized Pareto distributions for threshold exceedances. After normalizing marginal tail distributions in data, a standard Laplace field can be use…
The maximum correlation of functions of a pair of random variables is an important measure of stochastic dependence. It is known that this maximum nonlinear correlation is identical to the absolute value of the Pearson correlation for a pair of Gaussian random variables or a pair of finite sums of iid random variables. This paper extends these results to pairwise Gaussian vectors and processes, nested sums of iid random variables, and permutation symmetric functions of sub-groups of iid random variables. It also discusses applications to additive regression models.
The approximation of fixed-interval smoothing distributions is a key issue in inference for general state-space hidden Markov models (HMM). This contribution establishes non-asymptotic bounds for the Forward Filtering Backward Smoothing (FFBS) and the Forward Filtering Backward Simulation (FFBSi) estimators of fixed-interval smoothing functionals. We show that the rate of convergence of the Lq-mean errors of both methods depends on the number of observations T and the number of particles N only through the ratio T/N for additive functionals. In the case of the FFBS, this improves recent result…
We study generalized additive partial linear models, proposing the use of polynomial spline smoothing for estimation of nonparametric functions, and deriving quasi-likelihood based estimators for the linear parameters. We establish asymptotic normality for the estimators of the parametric components. The procedure avoids solving large systems of equations as in kernel-based procedures and thus results in gains in computational simplicity. We further develop a class of variable selection procedures for the linear parameters by employing a nonconcave penalized quasi-likelihood, which is shown to…
A $d$-dimensional nonparametric additive regression model with dependent observations is considered. Using the marginal integration technique and wavelets methodology, we develop a new adaptive estimator for a component of the additive regression function. Its asymptotic properties are investigated via the minimax approach under the $\mathbb{L}_2$ risk over Besov balls. We prove that it attains a sharp rate of convergence which turns to be the one obtained in the $\iid$ case for the standard univariate regression estimation problem.
This paper studies the joint limiting behavior of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model, where the asymptotic regime is such that the dimension and sample size grow proportionally. The form of the joint limiting distribution is applied to conduct Johnson-Graybill-type tests, a family of approaches testing for signals in a statistical model. For this, higher order correction is further made, helping alleviate the impact of finite-sample bias. The proof rests on determining the joint asymptotic behavior of two classes of spectral…
Consider a standard white Wishart matrix with parameters $n$ and $p$. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the $m \times m$ principal minors, under the asymptotic regime that $n,p,m$ go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing…
