In this article we propose a general class of risk measures which can be used for data based evaluation of parametric models. The loss function is defined as generalized quadratic distance between the true density and the proposed model. These distances are characterized by a simple quadratic form structure that is adaptable through the choice of a nonnegative definite kernel and a bandwidth parameter. Using asymptotic results for the quadratic distances we build a quick-to-compute approximation for the risk function. Its derivation is analogous to the Akaike Information Criterion (AIC), but unlike AIC, the quadratic risk is a global comparison tool. The method does not require resampling, a great advantage when point estimators are expensive to compute. The method is illustrated using the problem of selecting the number of components in a mixture model, where it is shown that, by using an appropriate kernel, the method is computationally straightforward in arbitrarily high data dimensions. In this same context it is shown that the method has some clear advantages over AIC and BIC.
Vine copulas allow to build flexible dependence models for an arbitrary number of variables using only bivariate building blocks. The number of parameters in a vine copula model increases quadratically with the dimension, which poses new challenges in high-dimensional applications. To alleviate the computational burden and risk of overfitting, we propose a modified Bayesian information criterion (BIC) tailored to sparse vine copula models. We show that the criterion can consistently distinguish between the true and alternative models under less stringent conditions than the classical BIC. The …
Asymptotic efficiency is proved for the constructed in part 1 procedure, i.e. Pinsker's constant is found in the asymptotic lower bound for the minimax quadratic risk. It is shown that the asymptotic minimax quadratic risk of the constructed procedure coincides with this constant.
In this paper, we deal with the data-driven selection of multidimensional and possibly anisotropic bandwidths in the general framework of kernel empirical risk minimization. We propose a universal selection rule, which leads to optimal adaptive results in a large variety of statistical models such as nonparametric robust regression and statistical learning with errors in variables. These results are stated in the context of smooth loss functions, where the gradient of the risk appears as a good criterion to measure the performance of our estimators. The selection rule consists of a comparison …
The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper, 2007, for estimating a unknown nonparametric regression. We prove that this procedure is asymptotically efficient for a quadratic risk, i.e. the asymptotic quadratic risk for this procedure coincides with the Pinsker constant which gives a sharp lower bound for the quadratic risk over all possible estimators.
The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper, 2007, for estimating an unknown nonparametric regression. %\cite{GaPe1}. We prove that this procedure is asymptotically efficient for a quadratic risk, i.e. the asymptotic quadratic risk for this procedure coincides with the Pinsker constant which gives a sharp lower bound for the quadratic risk over all possible estimates
The problems of model and variable selections for classification trees are jointly considered. A penalized criterion is proposed which explicitly takes into account the number of variables, and a risk bound inequality is provided for the tree classifier minimizing this criterion. This penalized criterion is compared to the one used during the pruning step of the CART algorithm. It is shown that the two criteria are similar under some specific margin assumptions. In practice, the tuning parameter of the CART penalty has to be calibrated by hold-out. Simulation studies are performed which confir…
We study the switch distribution, introduced by Van Erven et al. (2012), applied to model selection and subsequent estimation. While switching was known to be strongly consistent, here we show that it achieves minimax optimal parametric risk rates up to a $\log\log n$ factor when comparing two nested exponential families, partially confirming a conjecture by Lauritzen (2012) and Cavanaugh (2012) that switching behaves asymptotically like the Hannan-Quinn criterion. Moreover, like Bayes factor model selection but unlike standard significance testing, when one of the models represents a simple h…
Model selection is often performed by empirical risk minimization. The quality of selection in a given situation can be assessed by risk bounds, which require assumptions both on the margin and the tails of the losses used. Starting with examples from the 3 basic estimation problems, regression, classification and density estimation, we formulate risk bounds for empirical risk minimization under successively weakening conditions and prove them at a very general level, for general margin and power tail behavior of the excess losses.
In this paper for the first time the nonparametric autoregression estimation problem for the quadratic risks is considered. To this end we develop a new adaptive sequential model selection method based on the efficient sequential kernel estimators proposed by Arkoun and Pergamenshchikov (2016). Moreover, we develop a new analytical tool for general regression models to obtain the non asymptotic sharp or- acle inequalities for both usual quadratic and robust quadratic risks. Then, we show that the constructed sequential model selection proce- dure is optimal in the sense of oracle inequalities.
We observe a random measure $N$ and aim at estimating its intensity $s$. This statistical framework allows to deal simultaneously with the problems of estimating a density, the marginals of a multivariate distribution, the mean of a random vector with nonnegative components and the intensity of a Poisson process. Our estimation strategy is based on estimator selection. Given a family of estimators of $s$ based on the observation of $N$, we propose a selection rule, based on $N$ as well, in view of selecting among these. Little assumption is made on the collection of estimators. The procedure o…
We first study the properties of solutions of quadratic programs with linear equality constraints whose parameters are estimated from data in the high-dimensional setting where p, the number of variables in the problem, is of the same order of magnitude as n, the number of observations used to estimate the parameters. The Markowitz problem in Finance is a subcase of our study. Assuming normality and independence of the observations we relate the efficient frontier computed empirically to the "true" efficient frontier. Our computations show that there is a separation of the errors induced by es…
Portfolio selection in the periodic investment of securities modeled by a multivariate Merton model with dependent jumps is considered. The optimization framework is designed to maximize expected terminal wealth when portfolio risk is measured by the Condition-Value-at-Risk ($CVaR$). Solving the portfolio optimization problem by Monte Carlo simulation often requires intensive and time-consuming computation; hence a faster and more efficient portfolio optimization method based on closed-form comonotonic bounds for the risk measure $CVaR$ of the terminal wealth is proposed.
We propose new model selection criteria based on generalized ridge estimators dominating the maximum likelihood estimator under the squared risk and the Kullback-Leibler risk in multivariate linear regression. Our model selection criteria have the following favorite properties: consistency, unbiasedness, uniformly minimum variance. Consistency is proven under an asymptotic structure $\frac{p}{n}\to c$ where $n$ is the sample size and $p$ is the parameter dimension of the response variables. In particular, our proposed class of estimators dominates the maximum likelihood estimator under the squ…
We show that if the exposure and the outcome affect the selection indicator in the same direction and have non-positive interaction on the risk difference, risk ratio or odds ratio scale, the exposure-outcome odds ratio in the selected population is a lower bound for true odds ratio.
This paper describes several approaches for estimating the benchmark dose (BMD) in a risk assessment study with quantal dose-response data and when there are competing model classes for the dose-response function. Strategies involving a two-step approach, a model-averaging approach, a focused-inference approach, and a nonparametric approach based on a PAVA-based estimator of the dose-response function are described and compared. Attention is raised to the perils involved in data "double-dipping" and the need to adjust for the model-selection stage in the estimation procedure. Simulation result…
We consider in this paper a Gaussian sequence model of observations $Y_i$, $i\geq 1$ having mean (or signal) $θ_i$ and variance $σ_i$ which is growing polynomially like $i^γ$, $γ>0$. This model describes a large panel of inverse problems. We estimate the quadratic functional of the unknown signal $\sum_{i\geq 1}θ_i^2$ when the signal belongs to ellipsoids of both finite smoothness functions (polynomial weights $i^α$, $α>0$) and infinite smoothness (exponential weights $e^{βi^r}$, $β>0$, $0<r \leq 2$). We propose a Pinsker type projection estimator in each case and study its quadratic risk. Whe…
In this article we propose a new variable selection method for analyzing data collected from longitudinal sample surveys. The procedure is based on the survey-weighted quadratic inference function, which was recently introduced as an alternative to the survey-weighted generalized estimating function. Under the joint model-design framework, we introduce the penalized survey-weighted quadratic inference estimator and obtain sufficient conditions for the existence, weak consistency, sparsity and asymptotic normality. To illustrate the finite sample performance of the model selection procedure, we…
The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper (2007) for estimation of unknown nonparametric regression. We prove that this procedure is asymptotically efficient for a quadratic risk. It means that the asymptotic quadratic risk for this procedure coincides with a sharp lower bound.
Quadratic regression (QR) models naturally extend linear models by considering interaction effects between the covariates. To conduct model selection in QR, it is important to maintain the hierarchical model structure between main effects and interaction effects. Existing regularization methods generally achieve this goal by solving complex optimization problems, which usually demands high computational cost and hence are not feasible for high dimensional data. This paper focuses on scalable regularization methods for model selection in high dimensional QR. We first consider two-stage regulari…
We study the Cox models with semiparametric relative risk, which can be partially linear with one nonparametric component, or multiple additive or nonadditive nonparametric components. A penalized partial likelihood procedure is proposed to simultaneously estimate the parameters and select variables for both the parametric and the nonparametric parts. Two penalties are applied sequentially. The first penalty, governing the smoothness of the multivariate nonlinear covariate effect function, provides a smoothing spline ANOVA framework that is exploited to derive an empirical model selection tool…
We study the problem of identifying the set of \emph{active} variables, termed in the literature as \emph{variable selection} or \emph{multiple hypothesis testing}, depending on the pursued criteria. For a general \emph{robust setting} of non-normal, possibly dependent observations and a generalized notion of \emph{active set}, we propose a procedure that is used simultaneously for the both tasks, variable selection and multiple testing. The procedure is based on the \emph{risk hull minimization} method, but can also be obtained as a result of an empirical Bayes approach or a penalization stra…
In the pivotal variable selection problem, we derive the exact non-asymptotic minimax selector over the class of all $s$-sparse vectors, which is also the Bayes selector with respect to the uniform prior. While this optimal selector is, in general, not realizable in polynomial time, we show that its tractable counterpart (the scan selector) attains the minimax expected Hamming risk to within factor 2, and is also exact minimax with respect to the probability of wrong recovery. As a consequence, we establish explicit lower bounds under the monotone likelihood ratio property and we obtain a tigh…
We consider the teacher-student setting of learning shallow neural networks with quadratic activations and planted weight matrix $W^*\in\mathbb{R}^{m\times d}$, where $m$ is the width of the hidden layer and $d\le m$ is the data dimension. We study the optimization landscape associated with the empirical and the population squared risk of the problem. Under the assumption the planted weights are full-rank we obtain the following results. First, we establish that the landscape of the empirical risk admits an "energy barrier" separating rank-deficient $W$ from $W^*$: if $W$ is rank deficient, th…
