A theoretical expression is derived for the mean squared error of a nonparametric estimator of the tail dependence coefficient, depending on a threshold that defines which rank delimits the tails of a distribution. We propose a new method to optimally select this threshold. It combines the theoretical mean squared error of the estimator with a parametric estimation of the copula linking observations in the tails. Using simulations, we compare this semiparametric method with other approaches proposed in the literature, including the plateau-finding algorithm.
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We study the problem of estimating the diagonal of an implicitly given matrix $A$. For such a matrix we have access to an oracle that allows us to evaluate the matrix vector product $Av$. For random variable $v$ drawn from an appropriate distribution, this may be used to return an estimate of the diagonal of the matrix $A$. Whilst results exist for probabilistic guarantees relating to the error of estimates of the trace of $A$, no such results have yet been derived for the diagonal. We analyse the number of queries $s$ required to guarantee that with probability at least $1-\delta$ the estimates of the relative error of the diagonal entries is at most $\varepsilon$. We extend this analysis to the 2-norm of the difference between the estimate and the diagonal of $A$. We prove, discuss and experiment with bounds on the number of queries $s$ required to guarantee a probabilistic bound on the estimates of the diagonal by employing Rademacher and Gaussian random variables. Two sufficient upper bounds on the minimum number of query vectors are proved, extending the work of Avron and Toledo [JACM 58(2)8, 2011], and later work of Roosta-Khorasani and Ascher [FoCM 15, 1187-1212, 2015]. We find that, generally, there is little difference between the two, with convergence going as $O(\log(1/\delta)/\varepsilon^2)$ for individual diagonal elements. However for small $s$, we find that the Rademacher estimator is superior. These results allow us to then extend the ideas of Meyer, Musco, Musco and Woodruff [SOSA, 142-155, 2021], suggesting algorithm Diag++, to speed up the convergence of diagonal estimation from $O(1/\varepsilon^2)$ to $O(1/\varepsilon)$ and make it robust to the spectrum of any positive semi-definite matrix $A$.
Bayesian models based on the Dirichlet process and other stick-breaking priors have been proposed as core ingredients for clustering, topic modeling, and other unsupervised learning tasks. However, due to the flexibility of these models, the consequences of prior choices can be opaque. And so prior specification can be relatively difficult. At the same time, prior choice can have a substantial effect on posterior inferences. Thus, considerations of robustness need to go hand in hand with nonparametric modeling. In the current paper, we tackle this challenge by exploiting the fact that variational Bayesian methods, in addition to having computational advantages in fitting complex nonparametric models, also yield sensitivities with respect to parametric and nonparametric aspects of Bayesian models. In particular, we demonstrate how to assess the sensitivity of conclusions to the choice of concentration parameter and stick-breaking distribution for inferences under Dirichlet process mixtures and related mixture models. We provide both theoretical and empirical support for our variational approach to Bayesian sensitivity analysis.
In this paper we propose a flexible nested error regression small area model with high dimensional parameter that incorporates heterogeneity in regression coefficients and variance components. We develop a new robust small area specific estimating equations method that allows appropriate pooling of a large number of areas in estimating small area specific model parameters. We propose a parametric bootstrap and jackknife method to estimate not only the mean squared errors but also other commonly used uncertainty measures such as standard errors and coefficients of variation. We conduct both modelbased and design-based simulation experiments and real-life data analysis to evaluate the proposed methodology
In this paper we consider $L_p$-approximation, $p \in \{2,\infty\}$, of periodic functions from weighted Korobov spaces. In particular, we discuss tractability properties of such problems, which means that we aim to relate the dependence of the information complexity on the error demand $\varepsilon$ and the dimension $d$ to the decay rate of the weight sequence $(\gamma_j)_{j \ge 1}$ assigned to the Korobov space. Some results have been well known since the beginning of this millennium, others have been proven quite recently. We give a survey of these findings and will add some new results on the $L_\infty$-approximation problem. To conclude, we give a concise overview of results and collect a number of interesting open problems.
This paper studies the approximation capacity of ReLU neural networks with norm constraint on the weights. We prove upper and lower bounds on the approximation error of these networks for smooth function classes. The lower bound is derived through the Rademacher complexity of neural networks, which may be of independent interest. We apply these approximation bounds to analyze the convergence of regression using norm constrained neural networks and distribution estimation by GANs. In particular, we obtain convergence rates for over-parameterized neural networks. It is also shown that GANs can achieve optimal rate of learning probability distributions, when the discriminator is a properly chosen norm constrained neural network.
We study the off-policy evaluation (OPE) problem in an infinite-horizon Markov decision process with continuous states and actions. We recast the $Q$-function estimation into a special form of the nonparametric instrumental variables (NPIV) estimation problem. We first show that under one mild condition the NPIV formulation of $Q$-function estimation is well-posed in the sense of $L^2$-measure of ill-posedness with respect to the data generating distribution, bypassing a strong assumption on the discount factor $\gamma$ imposed in the recent literature for obtaining the $L^2$ convergence rates of various $Q$-function estimators. Thanks to this new well-posed property, we derive the first minimax lower bounds for the convergence rates of nonparametric estimation of $Q$-function and its derivatives in both sup-norm and $L^2$-norm, which are shown to be the same as those for the classical nonparametric regression (Stone, 1982). We then propose a sieve two-stage least squares estimator and establish its rate-optimality in both norms under some mild conditions. Our general results on the well-posedness and the minimax lower bounds are of independent interest to study not only other nonparametric estimators for $Q$-function but also efficient estimation on the value of any target policy in off-policy settings.
We consider the problem of finding tuned regularized parameter estimators for linear models. We start by showing that three known optimal linear estimators belong to a wider class of estimators that can be formulated as a solution to a weighted and constrained minimization problem. The optimal weights, however, are typically unknown in many applications. This begs the question, how should we choose the weights using only the data? We propose using the covariance fitting SPICE-methodology to obtain data-adaptive weights and show that the resulting class of estimators yields tuned versions of known regularized estimators - such as ridge regression, LASSO, and regularized least absolute deviation. These theoretical results unify several important estimators under a common umbrella. The resulting tuned estimators are also shown to be practically relevant by means of a number of numerical examples.
Continuous determinantal point processes (DPPs) are a class of repulsive point processes on $\mathbb{R}^d$ with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behaviour when the observation window grows towards $\mathbb{R}^d$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on $\mathbb{R}^d$ and compare favourably to common alternative estimation methods for continuous DPPs.
An error estimate for the Gauss-Lobatto quadrature formula for integration over the interval $[-1, 1]$, relative to the Jacobi weight function $w^{\alpha,\beta}(t)=(1-t)^\alpha(1+t)^\beta$, $\alpha,\beta>-1$, is obtained. This estimate holds true for functions belonging to some Sobolev-type subspaces of the weighted space $L_{w^{\alpha,\beta}}^1([-1,1])$.
The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As non-parametric model, it offers a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We describe an efficient algorithm to find the maximum likelihood estimator. The method is contrasted with the ordinary least squares approach in a simulation study, and the method is illustrated on a realistic data set.
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