We derive, in the classical framework of Bayesian sensitivity analysis, optimal lower and upper bounds on posterior values obtained from Bayesian models that exactly capture an arbitrarily large number of finite-dimensional marginals of the data-generating distribution and/or that are as close as desired to the data-generating distribution in the Prokhorov or total variation metrics; these bounds show that such models may still make the largest possible prediction error after conditioning on an arbitrarily large number of sample data measured at finite precision. These results are obtained through the development of a reduction calculus for optimization problems over measures on spaces of measures. We use this calculus to investigate the mechanisms that generate brittleness/robustness and, in particular, we observe that learning and robustness are antagonistic properties. It is now well understood that the numerical resolution of PDEs requires the satisfaction of specific stability conditions. Is there a missing stability condition for using Bayesian inference in a continuous world under finite information?

Approximate Bayesian Computation (ABC) is a popular computational method for likelihood-free Bayesian inference. The term “likelihood-free” refers to problems where the likelihood is intractable to compute or estimate directly, but where it is possible to generate simulated data $X$ relatively easily given a candidate set of parameters $\theta$ simulated from a prior distribution. Parameters which generate simulated data within some tolerance $\delta$ of the observed data $x^{*}$ are regarded as plausible, and a collection of such $\theta$ is used to estimate the posterior distribution $\theta |X=x^{*}$. Suitable choice of $\delta$ is vital for ABC methods to return good approximations to $\theta$ in reasonable computational time.

While ABC methods are widely used in practice, particularly in population genetics, rigorous study of the mathematical properties of ABC estimators lags behind practical developments of the method. We prove that ABC estimates converge to the exact solution under very weak assumptions and, under slightly stronger assumptions, quantify the rate of this convergence. In particular, we show that the bias of the ABC estimate is asymptotically proportional to $\delta^{2}$ as $\delta\downarrow 0$. At the same time, the computational cost for generating one ABC sample increases like $\delta^{-q}$ where $q$ is the dimension of the observations. Rates of convergence are obtained by optimally balancing the mean squared error against the computational cost. Our results can be used to guide the choice of the tolerance parameter $\delta$.

In this paper, we focus on the problem of a multivariate density estimation under an $\mathbb{L}_{p}$-loss. We provide a data-driven selection rule from a family of kernel estimators and derive for it $\mathbb{L}_{p}$-risk oracle inequalities depending on the value of $p\geq1$. The proposed estimator permits us to take into account approximation properties of the underlying density and its independence structure simultaneously. Specifically, we obtain adaptive upper bounds over a scale of anisotropic Nikolskii classes when the smoothness is also measured with the $\mathbb{L}_{p}$-norm. It is important to emphasize that the adaptation to unknown independence structure of the estimated density allows us to improve significantly the accuracy of estimation (curse of dimensionality). The main technical tools used in our derivation are uniform bounds on the $\mathbb{L}_{p}$-norms of empirical processes developed in Goldenshluger and Lepski [13].

The likelihood for the parameters of a generalized linear mixed model involves an integral which may be of very high dimension. Because of this intractability, many approximations to the likelihood have been proposed, but all can fail when the model is sparse, in that there is only a small amount of information available on each random effect. The sequential reduction method described in this paper exploits the dependence structure of the posterior distribution of the random effects to reduce substantially the cost of finding an accurate approximation to the likelihood in models with sparse structure.

Consider a non–linear function $G(X_{t})$ where $X_{t}$ is a stationary Gaussian sequence with long–range dependence. The usual reduction principle states that the partial sums of $G(X_{t})$ behave asymptotically like the partial sums of the first term in the expansion of $G$ in Hermite polynomials. In the context of the wavelet estimation of the long–range dependence parameter, one replaces the partial sums of $G(X_{t})$ by the wavelet scalogram, namely the partial sum of squares of the wavelet coefficients. Is there a reduction principle in the wavelet setting, namely is the asymptotic behavior of the scalogram for $G(X_{t})$ the same as that for the first term in the expansion of $G$ in Hermite polynomial? The answer is negative in general. This paper provides a minimal growth condition on the scales of the wavelet coefficients which ensures that the reduction principle also holds for the scalogram. The results are applied to testing the hypothesis that the long-range dependence parameter takes a specific value.

Conventional principal component analysis is highly susceptible to outliers. In particular, a sufficiently outlying single data point, can draw the leading principal component toward itself. In this paper, we study the effects of outliers for high dimension and low sample size data, using asymptotics. The non-robust nature of conventional principal component analysis is verified through inconsistency under multivariate Gaussian assumptions with a single spike in the covariance structure, in the presence of a contaminating outlier. In the same setting, the robust method of spherical principal components is consistent with the population eigenvector for the spike model, even in the presence of contamination.

This article considers inference for a common coefficient of variation (CV) shared by several normal populations. The confidence distributions (CD) are used to combine the information about each CV from different sources. A new procedure for constructing a confidence interval for the common CV is developed based on a combined confidence distribution for the inverse of the CV. The new derived CD interval has a theoretical exact frequentist property. Simulation results demonstrate that the new confidence intervals perform very well in terms of empirical coverage probability and average interval length. Finally, the proposed new procedure is illustrated on a real data example.

This paper deals with the estimation of a probability measure on the real line from data observed with an additive noise. We are interested in rates of convergence for the Wasserstein metric of order $p\geq1$. The distribution of the errors is assumed to be known and to belong to a class of supersmooth or ordinary smooth distributions. We obtain in the univariate situation an improved upper bound in the ordinary smooth case and less restrictive conditions for the existing bound in the supersmooth one. In the ordinary smooth case, a lower bound is also provided, and numerical experiments illustrating the rates of convergence are presented.

This paper deals with nonparametric estimation of conditional densities in mixture models in the case when additional covariates are available. The proposed approach consists of performing a preliminary clustering algorithm on the additional covariates to guess the mixture component of each observation. Conditional densities of the mixture model are then estimated using kernel density estimates applied separately to each cluster. We investigate the expected $L_{1}$-error of the resulting estimates and derive optimal rates of convergence over classical nonparametric density classes provided the clustering method is accurate. Performances of clustering algorithms are measured by the maximal misclassification error. We obtain upper bounds of this quantity for a single linkage hierarchical clustering algorithm. Lastly, applications of the proposed method to mixture models involving electricity distribution data and simulated data are presented.

We study the problem of detecting planted solutions in a random satisfiability formula. Adopting the formalism of hypothesis testing in statistical analysis, we describe the minimax optimal rates of detection. Our analysis relies on the study of the number of satisfying assignments, for which we prove new results. We also address algorithmic issues, and give a computationally efficient test with optimal statistical performance. This result is compared to an average-case hypothesis on the hardness of refuting satisfiability of random formulas.

This paper is motivated by the comparison of genetic networks inferred from high-dimensional datasets originating from high-throughput Omics technologies. The aim is to test whether the differences observed between two inferred Gaussian graphical models come from real differences or arise from estimation uncertainties. Adopting a neighborhood approach, we consider a two-sample linear regression model with random design and propose a procedure to test whether these two regressions are the same. Relying on multiple testing and variable selection strategies, we develop a testing procedure that applies to high-dimensional settings where the number of covariates $p$ is larger than the number of observations $n_{1}$ and $n_{2}$ of the two samples. Both type I and type II errors are explicitly controlled from a non-asymptotic perspective and the test is proved to be minimax adaptive to the sparsity. The performances of the test are evaluated on simulated data. Moreover, we illustrate how this procedure can be used to compare genetic networks on Hess et al. breast cancer microarray dataset.

Non-parametric assessment of extreme dependence structures between an arbitrary number of variables, though quite well-established in dimension $2$ and recently extended to moderate dimensions such as $5$, still represents a statistical challenge in larger dimensions. Here, we propose a novel approach that combines clustering techniques with angular/spectral measure analysis to find groups of variables (not necessarily disjoint) exhibiting asymptotic dependence, thereby reducing the dimension of the initial problem. A heuristic criterion is proposed to choose the threshold over which it is acceptable to consider observations as extreme and the appropriate number of clusters. When empirically evaluated through numerical experiments, the approach we promote here is found to be very efficient under some regularity constraints, even in dimension $20$. For illustration purpose, we also carry out a case study in dietary risk assessment.

Traditional methods for the analysis of failure time data are often employed in the analysis of waiting times of transient states from multistate models. However, such methods can exhibit bias when waiting times among model states are dependent, even when censoring is random. Furthermore, right-censoring can occur prior to entry into the transient state of interest, preventing the observation of transitions from the state and providing another potential source of bias. We introduce a nonparametric linear hazards model for waiting times from multistate models, analogous to Aalen’s linear hazards model for failure time data, where proper estimation can be carried out via reweighting, a method flexible enough to incorporate general forms of induced and other dependent censoring. We illustrate the approximate unbiasedness of the proposed regression coefficient estimators through a simulation study, while also demonstrating the bias arising from traditional Aalen’s linear hazards model estimators obtained from correlated waiting time data. Theoretical results for the parameter estimators are provided. The reweighted estimators are used in the analysis of two data sets, to identify predictors of ambulatory recovery in a data set of spinal cord injury patients receiving activity-based rehabilitation and to identify prognostic indicators for patients receiving bone marrow transplant.

We study a two-component semiparametric mixture model where one component distribution belongs to a parametric class, while the other is symmetric but otherwise arbitrary. This semiparametric model has wide applications in many areas such as large-scale simultaneous testing/multiple testing, sequential clustering, and robust modeling. We develop a class of estimators that are surprisingly simple and are unique in terms of their construction. A unique feature of these methods is that they do not rely on the estimation of the nonparametric component of the model. Instead, the methods only require a working model of the unspecified distribution, which may or may not reflect the true distribution. In addition, we establish connections between the existing estimator and the new methods and further derive a semiparametric efficient estimator. We compare our estimators with the existing method and investigate the advantages and cost of the relatively simple estimation procedure.

We provide results demonstrating the smoothness of some marginal log-linear parameterizations for distributions on multi-way contingency tables. First we give an analytical relationship between log-linear parameters defined within different margins, and use this to prove that some parameterizations are equivalent to ones already known to be smooth. Second we construct an iterative method for recovering joint probability distributions from marginal log-linear pieces, and prove its correctness in particular cases. Finally we use Markov chain theory to prove that certain cyclic conditional parameterizations are also smooth. These results are applied to show that certain conditional independence models are curved exponential families.

In this paper, we study theoretical properties of the non-parametric Cox proportional hazards model in a high dimensional non-asymptotic setting. We establish the finite sample oracle $l_{2}$ bounds for a general class of group penalties that allow possible hierarchical and overlapping structures. We approximate the log partial likelihood with a quadratic functional and use truncation arguments to reduce the error. Unlike the existing literature, we exemplify differences between bounded and possibly unbounded non-parametric covariate effects. In particular, we show that bounded effects can lead to prediction bounds similar to the simple linear models, whereas unbounded effects can lead to larger prediction bounds. In both situations we do not assume that the true parameter is necessarily sparse. Lastly, we present new theoretical results for hierarchical and smoothed estimation in the non-parametric Cox model. We provide two examples of the proposed general framework: a Cox model with interactions and an ANOVA type Cox model.

In this paper we introduce randomized $t$-type statistics that will be referred to as randomized pivots. We show that these randomized pivots yield central limit theorems with a significantly smaller error as compared to that of their classical counterparts under the same conditions. This constitutes a desirable result when a relatively small number of data is available. When a data set is too big to be processed, or when it constitutes a random sample from a super-population, we use our randomized pivots to infer about the mean based on significantly smaller sub-samples. The approach taken is shown to relate naturally to estimating distributions of both small and big data sets.

We consider the use of Bayesian information criteria for selection of the graph underlying an Ising model. In an Ising model, the full conditional distributions of each variable form logistic regression models, and variable selection techniques for regression allow one to identify the neighborhood of each node and, thus, the entire graph. We prove high-dimensional consistency results for this pseudo-likelihood approach to graph selection when using Bayesian information criteria for the variable selection problems in the logistic regressions. The results pertain to scenarios of sparsity, and following related prior work the information criteria we consider incorporate an explicit prior that encourages sparsity.

Regularized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Usually the low-dimensional structure is encoded by the presence of the (unknown) parameters in some low-dimensional model subspace. In such settings, it is desirable for estimates of the model parameters to be model selection consistent: the estimates also fall in the model subspace. We develop a general framework for establishing consistency and model selection consistency of regularized M-estimators and show how it applies to some special cases of interest in statistical learning. Our analysis identifies two key properties of regularized M-estimators, referred to as geometric decomposability and irrepresentability, that ensure the estimators are consistent and model selection consistent.

We consider testing the significance of a subset of covariates in a nonparametric regression. These covariates can be continuous and/or discrete. We propose a new kernel-based test that smoothes only over the covariates appearing under the null hypothesis, so that the curse of dimensionality is mitigated. The test statistic is asymptotically pivotal and the rate of which the test detects local alternatives depends only on the dimension of the covariates under the null hypothesis. We show the validity of wild bootstrap for the test. In small samples, our test is competitive compared to existing procedures.

Inequalities are key tools to prove FDR control of a multiple test. The present paper studies upper and lower bounds for the FDR under various dependence structures of $p$-values, namely independence, reverse martingale dependence and positive regression dependence on the subset (PRDS) of true null hypotheses. The inequalities are based on exact finite sample formulas which are also of interest for independent uniformly distributed $p$-values under the null. As applications the asymptotic worst case FDR of step up and step down tests coming from an non-decreasing rejection curve is established. In addition, new step up tests are established and necessary conditions for the FDR control are discussed. The reverse martingale models yield sharper FDR results than the PRDS models. Already in certain multivariate normal dependence models the familywise error rate of the Benjamini Hochberg step up test can be different from the desired level $\alpha$. The second part of the paper is devoted to adaptive step up tests under dependence. The well-known Storey estimator is modified so that the corresponding step up test has finite sample control for various block wise dependent $p$-values. These results may be applied to dependent genome data. Within each chromosome the $p$-values may be reverse martingale dependent while the chromosomes are independent.

In this paper we study posterior consistency for different topologies on the parameters for hidden Markov models with finite state space. We first obtain weak and strong posterior consistency for the marginal density function of finitely many consecutive observations. We deduce posterior consistency for the different components of the parameter. We also obtain posterior consistency for marginal smoothing distributions in the discrete case. We finally apply our results to independent emission distributions, translated emission distributions and discrete HMMs, under various types of priors.

A new methodology for optimal linear prediction of a stationary time series is introduced. Given a sample $X_{1},\ldots,X_{n}$, the optimal linear predictor of $X_{n+1}$ is $\tilde{X}_{n+1}=\phi_{1}(n)X_{n}+\phi_{2}(n)X_{n-1}+\cdots+\phi_{n}(n)X_{1}$. In practice, the coefficient vector $\phi(n)\equiv(\phi_{1}(n),\phi_{2}(n),\ldots,\phi_{n}(n))'$ is routinely truncated to its first $p$ components in order to be consistently estimated. By contrast, we employ a consistent estimator of the $n\times n$ autocovariance matrix $\Gamma_{n}$ in order to construct a consistent estimator of the optimal, full-length coefficient vector $\phi(n)$. Asymptotic convergence of the proposed predictor to the oracle is established, and finite sample simulations are provided to support the applicability of the new method. As a by-product, new insights are gained on the subject of estimating $\Gamma_{n}$ via a positive definite matrix, and four ways to impose positivity are introduced and compared. The closely related problem of spectral density estimation is also addressed.

In this note I provide some discussion on the paper by “High-dimensional auto covariance matrices and optimal linear prediction” by T. McMurry and D. Politis.

I propose new ACF and PACF plots based on the autocovariance estimators of McMurry and Politis. I also show that the forecasting methods they propose perform poorly compared to some relatively simple autoregression algorithms already available.

Bayesian methods for low-rank matrix completion with noise have been shown to be very efficient computationally [3, 18, 19, 24, 28]. While the behaviour of penalized minimization methods is well understood both from the theoretical and computational points of view (see [7, 9, 16, 23] among others) in this problem, the theoretical optimality of Bayesian estimators have not been explored yet. In this paper, we propose a Bayesian estimator for matrix completion under general sampling distribution. We also provide an oracle inequality for this estimator. This inequality proves that, whatever the rank of the matrix to be estimated, our estimator reaches the minimax-optimal rate of convergence (up to a logarithmic factor). We end the paper with a short simulation study.

We extend the statistical toolbox for inferring the extreme value behavior of stochastic processes by defining the spectrogram. It is based on a pseudo-polar representation of bivariate data and represents a collection of spectral measures that characterize the bivariate extremal dependence structure in terms of a univariate distribution. Based on threshold exceedances in the original event data, estimation of the spectrogram is flexible and efficient. The virtues of the spectrogram for exploratory studies and for parametric inference are highlighted. We propose a variance reduction technique that can be applied to the estimation of summary statistics like the extremogram. Parametric inference based on distance measures between the empirical and the model spectrogram, as for instance pairwise likelihoods or least squares distances, is developed. Simulated data illustrate gains in parametric estimation efficiency compared to a standard estimation approach. An application to precipitation data collected in the Cévennes region in France shows the practical utility of the introduced notions.

It is common knowledge that Akaike’s information criterion (AIC) is not a consistent model selection criterion, and Bayesian information criterion (BIC) is. These have been confirmed from an asymptotic selection probability evaluated from a large-sample framework. However, when a high-dimensional asymptotic framework, such that the dimension of the response variables and the sample size are approaching $\infty$, is used for evaluating the selection probability, there are cases that the AIC for selecting variables in multivariate linear models is consistent, but the BIC is not. The AIC and BIC are included in a family of information criteria defined by adding a penalty term expressing the complexity of the model to a negative twofold maximum log-likelihood. By clarifying the condition of the penalty term to ensure the consistency, we derive conditions for consistency of the AIC, BIC and other information criteria under the high-dimensional asymptotic framework.

We propose a novel procedure to test whether the immigration process of a discretely observed age-dependent branching process with immigration is time-homogeneous. The construction of the test is motivated by the behavior of the coefficient of variation of the population size. When immigration is time-homogeneous, we find that this coefficient converges to a constant, whereas when immigration is time-inhomogeneous it is time-dependent, at least transiently. Thus, we test the assumption that the immigration process is time-homogeneous by verifying that the sample coefficient of variation does not vary significantly over time. The test is simple to run and does not require specification or fitting any branching process to the data. Its implementation is identical whether the process is sub-, super-, or critical. Simulations and an application to real data on the progression of leukemia are presented to illustrate the approach.

We propose punctual and functional estimators for the local variance of pseudo-diffusions driven by Gaussian noises. The consistency and asymptotic normality are shown. The proofs are simplified by using the Central Limit Theorem for non-linear functionals belonging to Itô-Wiener’s Chaos, of Peccati-Nualart-Tudor. Besides, a simulation study is made to assess the performance of those estimators. This study reveals, through various examples, that the estimators give good approximations for the true local variance.

The minimum divergence estimators have proved to be useful tools in the area of robust inference. The robustness of such estimators are often measured using the classical Influence Function analysis. However, in many complex situations like that of testing a composite null hypothesis require the estimators to be restricted over some proper subspace of the parameter space. The robustness of these restricted minimum divergence estimators are very crucial in order to have overall robust inference. In this paper we provide a comprehensive description of the robustness of such restricted estimators in terms of their influence function for a general class of density based divergences along with their unrestricted versions. In particular, the robustness of some popular minimum divergence estimators are also demonstrated under certain usual restrictions through examples. Thus the paper provides a general framework for the influence function analysis of a large class of minimum divergence estimators with or without restrictions on the parameters and provides theoretical solutions for measuring the impact of the parameter restrictions on the robustness of the corresponding estimators.

The Integrated Completed Likelihood (ICL) criterion was introduced by Biernacki, Celeux and Govaert (2000) in the model-based clustering framework to select a relevant number of classes and has been used by statisticians in various application areas. A theoretical study of ICL is proposed.

A contrast related to the clustering objective is introduced: the conditional classification likelihood. An estimator and model selection criteria are deduced. The properties of these new procedures are studied and ICL is proved to be an approximation of one of these criteria. We contrast these results with the current leading point of view about ICL, that it would not be consistent. Moreover these results give insights into the class notion underlying ICL and feed a reflection on the class notion in clustering.

General results on penalized minimum contrast criteria and upper-bounds of the bracketing entropy in parametric situations are derived, which can be useful per se.

Practical solutions for the computation of the introduced procedures are proposed, notably an adapted EM algorithm and a new initialization method for EM-like algorithms which helps to improve the estimation in Gaussian mixture models.

We present a linear regression method for predictions on a small data set making use of a second possibly biased data set that may be much larger. Our method fits linear regressions to the two data sets while penalizing the difference between predictions made by those two models. The resulting algorithm is a shrinkage method similar to those used in small area estimation. We find a Stein-type result for Gaussian responses: when the model has $5$ or more coefficients and $10$ or more error degrees of freedom, it becomes inadmissible to use only the small data set, no matter how large the bias is. We also present both plug-in and AICc-based methods to tune our penalty parameter. Most of our results use an $L_{2}$ penalty, but we obtain formulas for $L_{1}$ penalized estimates when the model is specialized to the location setting. Ordinary Stein shrinkage provides an inadmissibility result for only $3$ or more coefficients, but we find that our shrinkage method typically produces much lower squared errors in as few as $5$ or $10$ dimensions when the bias is small and essentially equivalent squared errors when the bias is large.

In this paper, we consider the observation of $n$ i.i.d. mixed Poisson processes with random intensity having an unknown density $f$ on $\mathbb{R}^{+}$. For fixed observation time $T$, we propose a nonparametric adaptive strategy to estimate $f$. We use an appropriate Laguerre basis to build adaptive projection estimators. Non-asymptotic upper bounds of the $\mathbb{L}^{2}$-integrated risk are obtained and a lower bound is provided, which proves the optimality of the estimator. For large $T$, the variance of the previous method increases, therefore we propose another adaptive strategy. The procedures are illustrated on simulated data.

Sign consistency of the Lasso requires the stringent irrepresentable condition. This paper examines whether preconditioning can circumvent this condition. Let $\mathbf{X}\in\mathbb{R}^{n\times p}$ and $Y\in\mathbb{R}^{n}$ satisfy the standard linear regression equation. Instead of computing the Lasso with $(\mathbf{X},Y)$, preconditioning first left multiplies by $F\in\mathbb{R}^{n\times n}$ and then computes the Lasso with $(F\mathbf{X},FY)$.

While others have proposed preconditioning for other purposes, we provide the first results that show $F\mathbf{X}$ can satisfy the irrepresentable condition even when $\mathbf{X}$ fails to satisfy the condition. Preconditioning the Lasso creates a new estimator that is sign consistent in a wider variety of settings. Importantly, left multiplying the regression equation by $F$ does not change $\beta$, the vector of unknown coefficients. However, left multiplying this equation by $F$ often inflates the variance of the errors. We propose a class of preconditioners to balance these costs and benefits.

In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in $(\mathcal{D}_{p},W_{p})$, the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards.

We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each $N$ a map $(\mathcal{D}_{p})^{N}\to\P(\mathcal{D}_{p})$. We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.

We propose methodology for statistical inference for low-dimensional parameters of sparse precision matrices in a high-dimensional setting. Our method leads to a non-sparse estimator of the precision matrix whose entries have a Gaussian limiting distribution. Asymptotic properties of the novel estimator are analyzed for the case of sub-Gaussian observations under a sparsity assumption on the entries of the true precision matrix and regularity conditions. Thresholding the de-sparsified estimator gives guarantees for edge selection in the associated graphical model. Performance of the proposed method is illustrated in a simulation study.

Exact sampling methods have been recently developed for generating random variates for exponentially tilted $\alpha$-stable distributions. In this paper we show how to generate, exactly, random variates for a more general class of tilted $\alpha$-stable distributions, which is referred to as the class of Laguerre-type exponentially tilted $\alpha$-stable distributions. Beside the exponentially tilted $\alpha$-stable distribution, such a class includes also the Erlang tilted $\alpha$-stable distribution. This is a special case of the so-called gamma tilted $\alpha$-stable distribution, for which an efficient exact random variate generator is currently not available in the literature. Our result fills this gap.

Empirical Bayes methods are designed in selecting massive variables, which may be inter-connected following certain hierarchical structures, because of three attributes: taking prior information on model parameters, allowing data-driven hyperparameter values, and free of tuning parameters. We propose an iterated conditional modes/medians (ICM/M) algorithm to implement empirical Bayes selection of massive variables, while incorporating sparsity or more complicated a priori information. The iterative conditional modes are employed to obtain data-driven estimates of hyperparameters, and the iterative conditional medians are used to estimate the model coefficients and therefore enable the selection of massive variables. The ICM/M algorithm is computationally fast, and can easily extend the empirical Bayes thresholding, which is adaptive to parameter sparsity, to complex data. Empirical studies suggest competitive performance of the proposed method, even in the simple case of selecting massive regression predictors.

We consider here together the inference questions and the change-point problem in a large class of Poisson autoregressive models (see Tjøstheim, 2012 [34]). The conditional mean (or intensity) of the process is involved as a non-linear function of it past values and the past observations. Under Lipschitz-type conditions, it can be written as a function of lagged observations. For the latter model, assume that the link function depends on an unknown parameter $\theta_{0}$. The consistency and the asymptotic normality of the maximum likelihood estimator of the parameter are proved. These results are used to study change-point problem in the parameter $\theta_{0}$. From the likelihood of the observations, two tests are proposed. Under the null hypothesis (i.e. no change), each of these tests statistics converges to an explicit distribution. Consistencies under alternatives are proved for both tests. Simulation results show how those procedures work in practice, and applications to real data are also processed.

The quadratic inference function approach is able to provide a consistent and efficient estimator if valid moment conditions are available. However, the QIF estimator is unstable when the dimension of moment conditions is large compared to the sample size, due to the singularity problem for the estimated weighting matrix. We propose a new estimation procedure which combines all valid moment conditions optimally via the spectral decomposition of the weighting matrix. In theory, we show that the proposed method yields a consistent and efficient estimator which follows an asymptotic normal distribution. In addition, Monte Carlo studies indicate that the proposed method performs well in the sense of reducing bias and improving estimation efficiency. A real data example of Fortune 500 companies is used to compare the performance of the new method with existing methods.

We reconsider the definition of the Dantzig estimator and show that, in contrast to the LASSO, standardization of an experimental matrix leads in general to a different estimator than in the case when it is based on the original data. The properties of the first method, resulting in what is called here the normalized Dantzig estimator are studied and the results on its estimation and prediction error are compared with similar results for the standard version. It is shown that in general the normalized version yields tighter estimation and prediction bounds than the other approach. In the correct specification case tighter bounds are obtained for the normalized Dantzig estimator than for the LASSO. Numerical examples indicate that in the case of imbalanced data the normalized estimator also performs better than the standard version.

Non-linear mixed Poisson autoregressive models are studied for the analysis of count time series. Given a correct mean specification of the model, we discuss quasi maximum likelihood estimation based on Poisson log-likelihood function. A score testing procedure for checking linearity of the mean process is developed. We consider the cases of identifiable and non identifiable parameters under the null hypothesis. When the parameters are identifiable then a chi-square approximation to the distribution of the score test is obtained. In the case of non identifiable parameters, a supremum score type test statistic is employed for checking linearity of the mean process. The methodology is applied to simulated and real data.

In this paper we develop an inverse Bayesian approach to find the value of the unknown model parameter vector that supports the real (or test) data, where the data comprises measurements of a matrix-variate variable. The method is illustrated via the estimation of the unknown Milky Way feature parameter vector, using available test and simulated (training) stellar velocity data matrices. The data is represented as an unknown function of the model parameters, where this high-dimensional function is modelled using a high-dimensional Gaussian Process ($\mathcal{GP}$). The model for this function is trained using available training data and inverted by Bayesian means, to estimate the sought value of the model parameter vector at which the test data is realised. We achieve a closed-form expression for the posterior of the unknown parameter vector and the parameters of the invoked $\mathcal{GP}$, given test and training data. We perform model fitting by comparing the observed data with predictions made at different summaries of the posterior probability of the model parameter vector. As a supplement, we undertake a leave-one-out cross validation of our method.

We consider the Grenander estimator that is the maximum likelihood estimator for non-increasing densities. We prove uniform central limit theorems for certain subclasses of bounded variation functions and for Hölder balls of smoothness $s>1/2$. We do not assume that the density is differentiable or continuous. The proof can be seen as an adaptation of the method for the parametric maximum likelihood estimator to the nonparametric setting. Since nonparametric maximum likelihood estimators lie on the boundary, the derivative of the likelihood cannot be expected to equal zero as in the parametric case. Nevertheless, our proofs rely on the fact that the derivative of the likelihood can be shown to be small at the maximum likelihood estimator.

This paper is concerned with the study of the signed-rank estimator of the regression coefficients under the assumption that some responses are missing at random in the regression model. Strong consistency and asymptotic normality of the proposed estimator are established under mild conditions. To demonstrate the performance of the signed-rank estimator, a simulation study is conducted under different settings of model error’s distributions, and shows that the proposed estimator is more efficient than the least squares estimator whenever the error distribution is heavy-tailed or contaminated. When the model error follows a normal distribution, the simulation experiment shows that the signed-rank estimator is more efficient than its least squares counterpart whenever a large proportion of the responses are missing.

We consider high-dimensional inference when the assumed linear model is misspecified. We describe some correct interpretations and corresponding sufficient assumptions for valid asymptotic inference of the model parameters, which still have a useful meaning when the model is misspecified. We largely focus on the de-sparsified Lasso procedure but we also indicate some implications for (multiple) sample splitting techniques. In view of available methods and software, our results contribute to robustness considerations with respect to model misspecification.

Frequentist inference about a parameter of interest in presence of a nuisance parameter can be based on an integrated likelihood function. We analyze the behaviour of inferential quantities based on such a pseudo-likelihood in a two-index-asymptotics framework, in which both sample size and dimension of the nuisance parameter may diverge to infinity. We show that a properly chosen integrated likelihood largely outperforms standard likelihood methods, such as those based on the profile likelihood. These results are confirmed by simulation studies, in which comparisons with modified profile likelihood are also considered.

We introduce the partial martingale difference correlation, a scalar-valued measure of conditional mean dependence of $Y$ given $X$, adjusting for the nonlinear dependence on $Z$, where $X$, $Y$ and $Z$ are random vectors of arbitrary dimensions. At the population level, partial martingale difference correlation is a natural extension of partial distance correlation developed recently by Székely and Rizzo [14], which characterizes the dependence of $Y$ and $X$, after controlling for the nonlinear effect of $Z$. It extends the martingale difference correlation first introduced in Shao and Zhang [10] just as partial distance correlation extends the distance correlation in Székely, Rizzo and Bakirov [13]. Sample partial martingale difference correlation is also defined building on some new results on equivalent expressions of sample martingale difference correlation. Numerical results demonstrate the effectiveness of these new dependence measures in the context of variable selection and dependence testing.

We establish a general uniform in bandwidth consistency result for kernel estimators of the unconditional and conditional joint density of a distribution, which is defined by a mixed discrete and continuous random variable.

Over the last twenty-five years, various $\sqrt{n}$-consistent estimators have been devised for the coefficient vector in the popular semiparametric single-index model. In this paper, we prove under general assumptions that the kernel estimator of the link function by a univariate regression on the index variable is oracally efficient, namely, the estimator with the true single-index coefficient vector is asymptotically indistinguishable from that with any $\sqrt{n}$-consistent coefficient vector estimator. As a mathematical byproduct of the oracle efficiency, a simultaneous confidence band is constructed for the link function based on the oracally efficient kernel estimator. Simulation experiments corroborate the theoretical results. The proposed simultaneous confidence band is applied to analyze and test hypothesis about the Boston housing data.

This paper deals with minimax rates of convergence for manifold estimation. A new lower bound is obtained by a novel construction of two sets of manifolds and an application of convex hull testing method of Le Cam (1973). The minimax lower bound matches the upper bound up to a constant factor considered by Genovese et al. (2012b).

In binary classification problems, the area under the ROC curve (AUC) is commonly used to evaluate the performance of a prediction model. Often, it is combined with cross-validation in order to assess how the results will generalize to an independent data set. In order to evaluate the quality of an estimate for cross-validated AUC, we obtain an estimate of its variance. For massive data sets, the process of generating a single performance estimate can be computationally expensive. Additionally, when using a complex prediction method, the process of cross-validating a predictive model on even a relatively small data set can still require a large amount of computation time. Thus, in many practical settings, the bootstrap is a computationally intractable approach to variance estimation. As an alternative to the bootstrap, we demonstrate a computationally efficient influence curve based approach to obtaining a variance estimate for cross-validated AUC.

Suppose that a sequence of treatments are assigned to influence an outcome of interest that occurs after the last treatment. Between treatments, there are time-dependent covariates that may be post-treatment variables of the earlier treatments and confounders of the subsequent treatments. In this article, we study identification and estimation of the net effect of each treatment in the treatment sequence. We construct a point parametrization for the joint distribution of treatments, time-dependent covariates and the outcome, in which the point parameters of interest are the point effects of treatments considered as single-point treatments. We identify net effects of treatments by their expressions in terms of point effects of treatments and express patterns of net effects of treatments by constraints on point effects of treatments. We estimate net effects of treatments through their point effects under the constraint by maximum likelihood and reduce the number of point parameters in the estimation by the treatment assignment condition. As a result, we obtain an unbiased consistent maximum-likelihood estimate for the net effect of treatment even in a long treatment sequence. We also show by simulation that the interval estimation of the net effect of treatment achieves the nominal coverage probability.