We discuss a general approach to hypothesis testing. The main “building block” of the proposed construction is a test for a pair of hypotheses in the situation where each particular hypothesis states that the vector of parameters identifying the distribution of observations belongs to a convex compact set associated with the hypothesis. This test, under appropriate assumptions, is nearly optimal and is yielded by a solution to a convex optimization problem, so that the construction admits computationally efficient implementation. We further demonstrate that our assumptions are satisfied in several important and interesting applications. Finally, we show how our approach can be applied to a rather general testing problems encompassing several classical statistical settings.

We consider a problem of linear model selection in the presence of both continuous and categorical predictors. Feasible models consist of subsets of numerical variables and partitions of levels of factors. A new algorithm called delete or merge regressors (DMR) is presented which is a stepwise backward procedure involving ranking the predictors according to squared t-statistics and choosing the final model minimizing BIC. We prove consistency of DMR when the number of predictors tends to infinity with the sample size and describe a simulation study using a pertaining R package. The results indicate significant advantage in time complexity and selection accuracy of our algorithm over Lasso-based methods described in the literature. Moreover, a version of DMR for generalized linear models is proposed.

For functional time series with physical dependence, we construct confidence bands for its mean function. The physical dependence is a general dependence framework, and it slightly relaxes the conditions of m-approximable dependence. We estimate functional time series mean functions via a spline smoothing technique. Confidence bands have been constructed based on a long-run variance and a strong approximation theorem, which is satisfied with mild regularity conditions. Simulation experiments provide strong evidence that corroborates the asymptotic theories. Additionally, an application to S&P500 index data demonstrates a non-constant volatility mean function at a certain significance level.

We consider the Langevin equation which contains an unknown drift parameter $\theta$ and where the noise is modeled as fractional Brownian motion with Hurst index $H\in(0,\frac{1}{2})$. The solution corresponds to the fractional Ornstein–Uhlenbeck process. We construct an estimator, based on discrete observations in time, of the unknown drift parameter, that is similar in form to the maximum likelihood estimator for the drift parameter in Langevin equation with standard Brownian motion. It is assumed that the interval between observations is $n^{-1}$, i.e. tends to zero (high-frequency data) and the number of observations increases to infinity as $n^{m}$ with $m>1$. It is proved that for strictly positive $\theta$ the estimator is strongly consistent for any $m>1$, while for $\theta\leq0$ it is consistent when $m>\frac{1}{2H}$.

In this paper we are interested in checking whether the conditional variances are equal in $k\ge2$ location-scale regression models. Our procedure is fully nonparametric and is based on the comparison of the error distributions under the null hypothesis of equality of variances and without making use of this null hypothesis. We propose four test statistics based on empirical distribution functions (Kolmogorov-Smirnov and Cramér-von Mises type test statistics) and two test statistics based on empirical characteristic functions. The limiting distributions of these six test statistics are established under the null hypothesis and under local alternatives. We show how to approximate the critical values using either an estimated version of the asymptotic null distribution or a bootstrap procedure. Simulation studies are conducted to assess the finite sample performance of the proposed tests. We also apply our tests to data on household expenditures.

We provide a novel treatment of the ability of the standard (wavelet-tensor) and of the hyperbolic (tensor product) wavelet bases to build nonparametric estimators of multivariate functions. First, we give new results about the limitations of wavelet estimators based on the standard wavelet basis regarding their inability to optimally reconstruct functions with anisotropic smoothness. Next, we provide optimal or near optimal rates at which both linear and non-linear hyperbolic wavelet estimators are well-suited to reconstruct functions from anisotropic Besov spaces and subsequently we characterize the set of all the functions that are well reconstructed by these methods with respect to these rates. As a first main result, we furnish novel arguments to understand the primordial role of sparsity and thresholding in multivariate contexts, in particular by showing a stronger exposure of linear methods to the curse of dimensionality. Second, we propose an adaptation of the well known block thresholding method to a hyperbolic wavelet basis and show its ability to estimate functions with anisotropic smoothness at the optimal minimax rate. Therefore, we prove the pertinence of horizontal information pooling even in high dimensional settings. Numerical experiments illustrate the finite samples properties of the studied estimators.

We formalise and generalise the definition of the family of univariate double two–piece distributions, obtained by using a density–based transformation of unimodal symmetric continuous distributions with a shape parameter. The resulting distributions contain five interpretable parameters that control the mode, as well as the scale and shape in each direction. Four-parameter subfamilies of this class of distributions that capture different types of asymmetry are discussed. We propose interpretable scale and location-invariant benchmark priors and derive conditions for the propriety of the corresponding posterior distribution. The prior structures used allow for meaningful comparisons through Bayes factors within flexible families of distributions. These distributions are applied to data from finance, internet traffic and medicine, comparing them with appropriate competitors.

Sharp majorization is extended to the multivariate case. To achieve this, the notions of $\sigma$-strong convexity, monotonicity, and one-sided Lipschitz continuity are extended to $\mathbf{\Sigma}$-strong convexity, monotonicity, and Lipschitz continuity, respectively. The connection between a convex function and its Fenchel-Legendre transform is then developed. Sharp majorization is illustrated in single and multiple dimensions, and we show that these extensions yield improvements on bounds given within the literature. The new methodology introduced herein is used to develop a variational approximation for the Bayesian multinomial regression model.

Many processes in biology, chemistry, physics, medicine, and engineering are modeled by a system of differential equations. Such a system is usually characterized via unknown parameters and estimating their ‘true’ value is thus required. In this paper we focus on the quite common systems for which the derivatives of the states may be written as sums of products of a function of the states and a function of the parameters.

For such a system linear in functions of the unknown parameters we present a necessary and sufficient condition for identifiability of the parameters. We develop an estimation approach that bypasses the heavy computational burden of numerical integration and avoids the estimation of system states derivatives, drawbacks from which many classic estimation methods suffer. We also suggest an experimental design for which smoothing can be circumvented. The optimal rate of the proposed estimators, i.e., their $\sqrt{n}$-consistency, is proved and simulation results illustrate their excellent finite sample performance and compare it to other estimation approaches.

In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type \[X_{t}=e^{-\xi_{t}}\left(X_{0}+\int_{0}^{t}e^{\xi_{u-}}du \right), \] where $\xi_{s}$ is a Lévy process. Our primal goal is to estimate the characteristics of the Lévy process $\xi$ from the low-frequency observations of the process $X$. We present a novel approach towards estimating the Lévy triplet of $\xi$, which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.

This article considers the problem of multi-group classification in the setting where the number of variables $p$ is larger than the number of observations $n$. Several methods have been proposed in the literature that address this problem, however their variable selection performance is either unknown or suboptimal to the results known in the two-group case. In this work we provide sharp conditions for the consistent recovery of relevant variables in the multi-group case using the discriminant analysis proposal of Gaynanova et al. [7]. We achieve the rates of convergence that attain the optimal scaling of the sample size $n$, number of variables $p$ and the sparsity level $s$. These rates are significantly faster than the best known results in the multi-group case. Moreover, they coincide with the minimax optimal rates for the two-group case. We validate our theoretical results with numerical analysis.

By virtue of long-memory time series, it is illustrated in this paper that white noise calculus can be used to handle subtle issues of stochastic integral convergence that often arise in the asymptotic theory of time series. A main difficulty of such an issue is that the limiting stochastic integral cannot be defined path-wise in general. As a result, continuous mapping theorem cannot be directly applied to deduce the convergence of stochastic integrals $\int^{1}_{0}H_{n}(s)\,dZ_{n}(s)$ to $\int^{1}_{0}H(s)\,dZ(s)$ based on the convergence of $(H_{n},Z_{n})$ to $(H,Z)$ in distribution. The white noise calculus, in particular the technique of $\mathcal{S}$-transform, allows one to establish the asymptotic results directly.

Applying the standard stochastic approximation algorithm of Robbins and Monro (1951) to calculating confidence limits leads to poor efficiency and difficulties in estimating the appropriate governing constants as well as the standard error.

We suggest sampling instead from an alternative importance distribution and modifying the Robbins-Monro recursion accordingly. This can reduce the asymptotic variance by the usual importance sampling factor. It also allows the standard error and optimal step length to be estimated from the simulation. The methodology is applied to computing almost exact confidence limits in a generalised linear model.

We consider the problem of estimating the context tree of a stationary ergodic process with finite alphabet without imposing additional conditions on the process. As a starting point we introduce a Hamming metric in the space of irreducible context trees and we use the properties of the weak topology in the space of ergodic stationary processes to prove that if the Hamming metric is unbounded, there exist no consistent estimators for the context tree. Even in the bounded case we show that there exist no two-sided confidence bounds. However we prove that one-sided inference is possible in this general setting and we construct a consistent estimator that is a lower bound for the context tree of the process with an explicit formula for the coverage probability. We develop an efficient algorithm to compute the lower bound and we apply the method to test a linguistic hypothesis about the context tree of codified written texts in European Portuguese.

Motivated by applications in genetic fields, we propose to estimate the heritability in high-dimensional sparse linear mixed models. The heritability determines how the variance is shared between the different random components of a linear mixed model. The main novelty of our approach is to consider that the random effects can be sparse, that is may contain null components, but we do not know either their proportion or their positions. The estimator that we consider is strongly inspired by the one proposed by Pirinen, Donnelly and Spencer (2013), and is based on a maximum likelihood approach. We also study the theoretical properties of our estimator, namely we establish that our estimator of the heritability is $\sqrt{n}$-consistent when both the number of observations $n$ and the number of random effects $N$ tend to infinity under mild assumptions. We also prove that our estimator of the heritability satisfies a central limit theorem which gives as a byproduct a confidence interval for the heritability. Some Monte-Carlo experiments are also conducted in order to show the finite sample performances of our estimator.

In this paper we develop a formal dynamic version of Chain Event Graphs (CEGs), a particularly expressive family of discrete graphical models. We demonstrate how this class links to semi-Markov models and provides a convenient generalization of the Dynamic Bayesian Network (DBN). In particular we develop a repeating time-slice Dynamic CEG providing a useful and simpler model in this family. We demonstrate how the Dynamic CEG’s graphical formulation exhibits asymmetric conditional independence statements and also how each model can be estimated in a closed form enabling fast model search over the class. The expressive power of this model class together with its estimation is illustrated throughout by a variety of examples that include the risk of childhood hospitalization and the efficacy of a flu vaccine.

We consider the problem of detecting a sparse Poisson mixture. Our results parallel those for the detection of a sparse normal mixture, pioneered by Ingster (1997) and Donoho and Jin (2004), when the Poisson means are larger than logarithmic in the sample size. In particular, a form of higher criticism achieves the detection boundary in the whole sparse regime. When the Poisson means are smaller than logarithmic in the sample size, a different regime arises in which simple multiple testing with Bonferroni correction is enough in the sparse regime. We present some numerical experiments that confirm our theoretical findings.

We introduce a special type of graphs, which we use as tools for designing and improving survey questionnaires. While the idea of representing questionnaires as graphs is not new, our graphs constitute novel theoretical and practical tools, which could turn a complex questionnaire into a questionnaire that is easier to visualize, test and analyze.

The concept of pointwise Fisher consistency (or classification calibration) states necessary and sufficient conditions to have Bayes consistency when a classifier minimizes a surrogate loss function instead of the 0-1 loss. We present a family of multiclass hinge loss functions defined by a continuous control parameter $\lambda$ representing the margin of the positive points of a given class. The parameter $\lambda$ allows shifting from classification uncalibrated to classification calibrated loss functions. Though previous results suggest that increasing the margin of positive points has positive effects on the classification model, other approaches have failed to give increasing weight to the positive examples without losing the classification calibration property. Our $\lambda$-based loss function can give unlimited weight to the positive examples without breaking the classification calibration property. Moreover, when embedding these loss functions into the Support Vector Machine’s framework ($\lambda$-SVM), the parameter $\lambda$ defines different regions for the Karush—Kuhn—Tucker conditions. A large margin on positive points also facilitates faster convergence of the Sequential Minimal Optimization algorithm, leading to lower training times than other classification calibrated methods. $\lambda$-SVM allows easy implementation, and its practical use in different datasets not only supports our theoretical analysis, but also provides good classification performance and fast training times.

This paper is about computations for linear mixed models having two variances, $\sigma^{2}_{e}$ for residuals and $\sigma^{2}_{s}$ for random effects, though the ideas can be extended to some linear mixed models having more variances. Researchers are often interested in either the restricted (residual) likelihood $\text{RL}(\sigma_{e}^{2},\sigma_{s}^{2})$ or the joint posterior $\pi(\sigma_{e}^{2},\sigma_{s}^{2}\,|\,y)$ or their logarithms. Both $\log\text{RL}$ and $\log\pi$ can be multimodal and computations often rely on either a general purpose optimization algorithm or MCMC, both of which can fail to find regions where the target function is high. This paper presents an alternative. Letting $f$ stand for either $\text{RL}$ or $\pi$, we show how to find a box $B$ in the $(\sigma_{e}^{2},\sigma_{s}^{2})$ plane such that

1. all local and global maxima of $\log f$ lie within $B$;

2. $\sup_{(\sigma_{e}^{2},\sigma_{s}^{2})\in B^{c}}\log f(\sigma_{e}^{2},\sigma_{s}^{2})\le \sup_{(\sigma_{e}^{2},\sigma_{s}^{2})\in B}\log f(\sigma_{e}^{2},\sigma_{s}^{2})-M$ for a prespecified $M>0$; and

3. $\log f$ can be estimated to within a prespecified tolerance $\epsilon$ everywhere in $B$ with no danger of missing regions where $\log f$ is large.

Taken together these conditions imply that the $(\sigma_{e}^{2},\sigma_{s}^{2})$ plane can be divided into two parts: $B$, where we know $\log f$ as accurately as we wish, and $B^{c}$, where $\log f$ is small enough to be safely ignored. We provide algorithms to find $B$ and to evaluate $\log f$ as accurately as desired everywhere in $B$.

In this manuscript, we study the statistical properties of convex clustering. We establish that convex clustering is closely related to single linkage hierarchical clustering and $k$-means clustering. In addition, we derive the range of the tuning parameter for convex clustering that yields a non-trivial solution. We also provide an unbiased estimator of the degrees of freedom, and provide a finite sample bound for the prediction error for convex clustering. We compare convex clustering to some traditional clustering methods in simulation studies.

We consider the matrix completion problem where the aim is to estimate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative thresholding methods. In spite of their empirical success, the theoretical guarantees of such iterative thresholding methods are poorly understood. The goal of this paper is to provide strong theoretical guarantees, similar to those obtained for nuclear-norm penalization methods and one step thresholding methods, for an iterative thresholding algorithm which is a modification of the softImpute algorithm. An important consequence of our result is the exact minimax optimal rates of convergence for matrix completion problem which were know until now only up to a logarithmic factor.

Given a large dataset and an estimation task, it is common to pre-process the data by reducing them to a set of sufficient statistics. This step is often regarded as straightforward and advantageous (in that it simplifies statistical analysis). I show that –on the contrary– reducing data to sufficient statistics can change a computationally tractable estimation problem into an intractable one. I discuss connections with recent work in theoretical computer science, and implications for some techniques to estimate graphical models.

In this paper we consider the semiparametric transformation model $\Lambda_{\theta_{o}}(Y)=m(X)+\varepsilon$, where $\theta_{o}$ is an unknown finite dimensional parameter, the function $m(\cdot)=\mathbb{E}(\Lambda_{\theta_{o}}(Y)|X=\cdot)$ is “smooth”, but otherwise unknown, and the covariate $X$ is independent of the error $\varepsilon$. An estimator of the distribution function of $\varepsilon$ is investigated and its weak convergence is proved. The proposed estimator depends on a profile likelihood estimator of $\theta_{o}$ and a nonparametric kernel estimator of $m$. We also evaluate the practical performance of our estimator in a simulation study for several models and sample sizes. Finally, the method is applied to a data set on the scattering of sunlight in the atmosphere.

For a pair $(Y_{1},Y_{2})$ of random variables there exist several measures of association that characterize the dependence between $Y_{1}$ and $Y_{2}$ by means of one single value. Classical examples are Pearson’s correlation coefficient, Kendall’s tau and Spearman’s rho. For the situation where next to the pair $(Y_{1},Y_{2})$ there is also a third variable $X$ present, so-called partial association measures, such as a partial Pearson’s correlation coefficient and a partial Kendall’s tau, have been proposed in the 1940’s. Following criticism on e.g. partial Kendall’s tau, better alternatives to these original partial association measures appeared in the literature: the conditional association measures, e.g. conditional Kendall’s tau, and conditional Spearman’s rho. Both, unconditional and conditional association measures can be expressed in terms of copulas. Even in case the dependence structure between $Y_{1}$ and $Y_{2}$ is influenced by a third variable $X$, we still want to be able to summarize the level of dependence by one single number. In this paper we discuss two different ways to do so, leading to two relatively new concepts: the (new concept of) partial Kendall’s tau, and the average Kendall’s tau. We provide a unifying framework for the diversity of concepts: global (or unconditional) association measures, conditional association measures, and partial and average association measures. The main contribution is that we discuss estimation of the newly-defined concepts: the partial and average copulas and association measures, and establish theoretical results for the estimators. The various concepts of association measures are illustrated on a real data example.

We investigate two empirical Bayes methods and a hierarchical Bayes method for adapting the scale of a Gaussian process prior in a nonparametric regression model. We show that all methods lead to a posterior contraction rate that adapts to the smoothness of the true regression function. Furthermore, we show that the corresponding credible sets cover the true regression function whenever this function satisfies a certain extrapolation condition. This condition depends on the specific method, but is implied by a condition of self-similarity. The latter condition is shown to be satisfied with probability one under the prior distribution.

We apply the results of Andresen et. al. (2014) on finite sample properties of sieve M-estimators and Andresen et. al. (2015) on the convergence of an alternating maximization procedure to analyse a sieve profile maximization estimator in the single index model with linear index function. The link function is approximated with $C^{3}$-Daubechies-wavelets with compact support. We derive results like Wilks phenomenon and Fisher Theorem in a finite sample setup even when the model is miss-specified. Furthermore we show that an alternating maximization procedure converges to the global maximizer and we assess the performance of Friedman’s projection pursuit procedure. The approach is based on showing that the conditions of Andresen et. al. (2014) and (2015) can be satisfied under a set of mild regularity and moment conditions on the link function, the regressors and the additive noise. The results allow to construct non-asymptotic confidence sets and to derive asymptotic bounds for the estimator as corollaries.

We consider a finite mixture of Gaussian regression models for high-dimensional data, where the number of covariates may be much larger than the sample size. We propose to estimate the unknown conditional mixture density by a maximum likelihood estimator, restricted on relevant variables selected by an $\ell_{1}$-penalized maximum likelihood estimator. We get an oracle inequality satisfied by this estimator with a Jensen-Kullback-Leibler type loss. Our oracle inequality is deduced from a general model selection theorem for maximum likelihood estimators on a random model subcollection. We can derive the penalty shape of the criterion, which depends on the complexity of the random model collection.

We consider the signal detection problem in the Gaussian design trace regression model with low rank alternative hypotheses. We derive the precise (Ingster-type) detection boundary for the Frobenius and the nuclear norm. We then apply these results to show that honest confidence sets for the unknown matrix parameter that adapt to all low rank sub-models in nuclear norm do not exist. This shows that recently obtained positive results in [5] for confidence sets in low rank recovery problems are essentially optimal.

In this paper we explore the problem of constructing two-level Minimum Generalized Aberration (MGA) orthogonal arrays with strength $t$, $n$ runs and $q>t$ columns, using a method that employs the $J$-characteristics of a two-level design. General results for the construction of MGA orthogonal arrays with $t+1$, $t+2$ and $t+3$ columns are given, while all MGA designs with strength $t\ge 2$, $n \equiv$ 0 mod 4 runs and $q\le 6$ are constructed. Results are also given for two-level orthogonal arrays with $q=7$ factors, but with strength greater than two. Projection properties of the MGA designs that have been identified, are also discussed.

We introduce a new copula-based correction for generalized linear mixed models (GLMMs) within the integrated nested Laplace approximation (INLA) approach for approximate Bayesian inference for latent Gaussian models. While INLA is usually very accurate, some (rather extreme) cases of GLMMs with e.g. binomial or Poisson data have been seen to be problematic. Inaccuracies can occur when there is a very low degree of smoothing or “borrowing strength” within the model, and we have therefore developed a correction aiming to push the boundaries of the applicability of INLA. Our new correction has been implemented as part of the R-INLA package, and adds only negligible computational cost. Empirical evaluations on both real and simulated data indicate that the method works well.

We consider a novel sub-class of log-location-scale models for survival and reliability data formed by restricting the density of the underlying location-scale distribution to be log-concave. These models display a number of attractive properties. We particularly explore the shapes of the hazard functions of these, LLSLC, models. A relatively elegant, if partial, theory of hazard shape arises under a further minor constraint on the hazard function of the underlying log-concave distribution. Perhaps the most useful LLSLC models are contained in a class of three-parameter distributions which allow constant, increasing, decreasing, bathtub and upside-down bathtub shapes for their hazard functions.

This paper presents an adaptive version of the Hill estimator based on Lespki’s model selection method. This simple data-driven index selection method is shown to satisfy an oracle inequality and is checked to achieve the lower bound recently derived by Carpentier and Kim. In order to establish the oracle inequality, we derive non-asymptotic variance bounds and concentration inequalities for Hill estimators. These concentration inequalities are derived from Talagrand’s concentration inequality for smooth functions of independent exponentially distributed random variables combined with three tools of Extreme Value Theory: the quantile transform, Karamata’s representation of slowly varying functions, and Rényi’s characterisation for the order statistics of exponential samples. The performance of this computationally and conceptually simple method is illustrated using Monte-Carlo simulations.

To model heteroscedasticity in a broad class of additive partial linear models, we allow the variance function to be an additive partial linear model as well and the parameters in the variance function to be different from those in the mean function. We develop a two-step estimation procedure, where in the first step initial estimates of the parameters in both the mean and variance functions are obtained and then in the second step the estimates are updated using the weights based on the initial estimates. We use polynomial splines to approximate the additive nonparametric components in both the mean and variation functions and derive their convergence rates. The resulting weighted estimators of the linear coefficients in both the mean and variance functions are shown to be asymptotically normal and more efficient than the initial un-weighted estimators. Simulation experiments are conducted to examine the numerical performance of the proposed procedure, which is also applied to analyze the dataset from a nutritional epidemiology study.

Gaussian concentration graphical models are one of the most popular models for sparse covariance estimation with high-dimensional data. In recent years, much research has gone into development of methods which facilitate Bayesian inference for these models under the standard $G$-Wishart prior. However, convergence properties of the resulting posteriors are not completely understood, particularly in high-dimensional settings. In this paper, we derive high-dimensional posterior convergence rates for the class of decomposable concentration graphical models. A key initial step which facilitates our analysis is transformation to the Cholesky factor of the inverse covariance matrix. As a by-product of our analysis, we also obtain convergence rates for the corresponding maximum likelihood estimator.

The literature on statistical learning for time series often assumes asymptotic independence or “mixing” of the data-generating process. These mixing assumptions are never tested, nor are there methods for estimating mixing coefficients from data. Additionally, for many common classes of processes (Markov processes, ARMA processes, etc.) general functional forms for various mixing rates are known, but not specific coefficients. We present the first estimator for beta-mixing coefficients based on a single stationary sample path and show that it is risk consistent. Since mixing rates depend on infinite-dimensional dependence, we use a Markov approximation based on only a finite memory length $d$. We present convergence rates for the Markov approximation and show that as $d\rightarrow\infty$, the Markov approximation converges to the true mixing coefficient. Our estimator is constructed using $d$-dimensional histogram density estimates. Allowing asymptotics in the bandwidth as well as the dimension, we prove $L^{1}$ concentration for the histogram as an intermediate step. Simulations wherein the mixing rates are calculable and a real-data example demonstrate our methodology.

A Bayesian nonparametric methodology has been recently introduced for estimating, given an initial observed sample, the species variety featured by an additional unobserved sample of size $m$. Although this methodology led to explicit posterior distributions under the general framework of Gibbs-type priors, there are situations of practical interest where $m$ is required to be very large and the computational burden for evaluating these posterior distributions makes impossible their concrete implementation. In this paper we present a solution to this problem for a large class of Gibbs-type priors which encompasses the two parameter Poisson-Dirichlet prior and, among others, the normalized generalized Gamma prior. Our solution relies on the study of the large $m$ asymptotic behaviour of the posterior distribution of the number of new species in the additional sample. In particular we introduce a simple characterization of the limiting posterior distribution in terms of a scale mixture with respect to a suitable latent random variable; this characterization, combined with the adaptive rejection sampling, leads to derive a large $m$ approximation of any feature of interest from the exact posterior distribution. We show how to implement our results through a simulation study and the analysis of a dataset in linguistics.

Ordinary Differential Equations are widespread tools to model chemical, physical, biological process but they usually rely on parameters which are of critical importance in terms of dynamic and need to be estimated directly from the data. Classical statistical approaches (nonlinear least squares, maximum likelihood estimator) can give unsatisfactory results because of computational difficulties and ill-posed statistical problem. New estimation methods that use some nonparametric devices have been proposed to circumvent these issues. We present a new estimator that shares properties with Two-Step estimators and Generalized Smoothing (introduced by Ramsay et al. [37]). Our estimation method relies on a relaxation and penalization scheme to regularize the inverse problem. We introduce a perturbed model and we use optimal control theory for constructing a criterion that aims at minimizing the discrepancy between data and the original model. Here, we focus on the case of linear Ordinary Differential Equations as our criterion has a closed-form expression that permits a detailed analysis. Our approach avoids the use of a nonparametric estimator of the derivative, which is one of the main causes of inaccuracy in Two-Step estimators. Regarding the theoretical asymptotic behavior of our estimator, we show its consistency and that we reach the parametric $\sqrt{n}$-rate when regression splines are used in the first step. We consider the estimation of two models possessing sloppy parameters, which usually makes the estimation of ODE models an ill-posed problem in applications [20, 41] and shows the efficiency of the Tracking estimator. Quite interestingly, our relaxation scheme makes the estimator robust to some kind of model misspecification, as shown in simulations.

The task of estimating a matrix given a sample of observed entries is known as the matrix completion problem. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of its entries. Here, we investigate the case of highly quantized observations when the measurements can take only a small number of values. These quantized outputs are generated according to a probability distribution parametrized by the unknown matrix of interest. This model corresponds, for example, to ratings in recommender systems or labels in multi-class classification. We consider a general, non-uniform, sampling scheme and give theoretical guarantees on the performance of a constrained, nuclear norm penalized maximum likelihood estimator. One important advantage of this estimator is that it does not require knowledge of the rank or an upper bound on the nuclear norm of the unknown matrix and, thus, it is adaptive. We provide lower bounds showing that our estimator is minimax optimal. An efficient algorithm based on lifted coordinate gradient descent is proposed to compute the estimator. A limited Monte-Carlo experiment, using both simulated and real data is provided to support our claims.

The aim of this paper is to show the existence of drift estimators dominating the standard one in continuous-time models of the form $X_{t}=u_{t}+Z_{t}$, where $u_{t}$ is the drift and $Z_{t}$ is either a Brownian martingale or a non-martingale noise living in the second Wiener chaos. Our results are based on the use of Malliavin calculus techniques, and extend previous findings of Privault and Réveillac (2008).

This paper establishes consistency of the weighted bootstrap for quadratic forms $\left(n^{-1/2}\sum_{i=1}^{n}Z_{i,n} \right)^{T}\left(n^{-1/2}\sum_{i=1}^{n}Z_{i,n}\right)$ where $(Z_{i,n})_{i=1}^{n}$ are mean zero, independent $\mathbb{R}^{d}$-valued random variables and $d=d(n)$ is allowed to grow with the sample size $n$, slower than $n^{1/4}$. The proof relies on an adaptation of Lindeberg interpolation technique whereby we simplify the original problem to a Gaussian approximation problem. We apply our bootstrap results to model-specification testing problems when the number of moments is allowed to grow with the sample size.

In this paper, we consider a nonlinear inverse problem occuring in nuclear science. Gamma rays randomly hit a semiconductor detector which produces an impulse response of electric current. Because the sampling period of the measured current is larger than the mean interarrival time of photons, the impulse responses associated to different gamma rays can overlap: this phenomenon is known as pileup. In this work, it is assumed that the impulse response is an exponentially decaying function. We propose a novel method to infer the distribution of gamma photon energies from the indirect measurements obtained from the detector. This technique is based on a formula linking the characteristic function of the photon density to a function involving the characteristic function and its derivative of the observations. We establish that our estimator converges to the mark density in uniform norm at a polynomial rate. A limited Monte-Carlo experiment is provided to support our findings.

Ordinary differential equations (ODEs) are used to model dynamic systems appearing in engineering, physics, biomedical sciences and many other fields. These equations contain an unknown vector of parameters of physical significance, say $\boldsymbol{\theta}$ which has to be estimated from the noisy data. Often there is no closed form analytic solution of the equations and hence we cannot use the usual non-linear least squares technique to estimate the unknown parameters. The two-step approach to solve this problem involves fitting the data nonparametrically and then estimating the parameter by minimizing the distance between the nonparametrically estimated derivative and the derivative suggested by the system of ODEs. The statistical aspects of this approach have been studied under the frequentist framework. We consider this two-step estimation under the Bayesian framework. The response variable is allowed to be multidimensional and the true mean function of it is not assumed to be in the model. We induce a prior on the regression function using a random series based on the B-spline basis functions. We establish the Bernstein-von Mises theorem for the posterior distribution of the parameter of interest. Interestingly, even though the posterior distribution of the regression function based on splines converges at a rate slower than $n^{-1/2}$, the parameter vector $\boldsymbol{\theta}$ is nevertheless estimated at $n^{-1/2}$ rate.

We consider the problem of inferring the total causal effect of a single continuous variable intervention on a (response) variable of interest. We propose a certain marginal integration regression technique for a very general class of potentially nonlinear structural equation models (SEMs) with known structure, or at least known superset of adjustment variables: we call the procedure S-mint regression. We easily derive that it achieves the convergence rate as for nonparametric regression: for example, single variable intervention effects can be estimated with convergence rate $n^{-2/5}$ assuming smoothness with twice differentiable functions. Our result can also be seen as a major robustness property with respect to model misspecification which goes much beyond the notion of double robustness. Furthermore, when the structure of the SEM is not known, we can estimate (the equivalence class of) the directed acyclic graph corresponding to the SEM, and then proceed by using S-mint based on these estimates. We empirically compare the S-mint regression method with more classical approaches and argue that the former is indeed more robust, more reliable and substantially simpler.