We study posterior contraction behaviors for parameters of interest in the context of Bayesian mixture modeling, where the number of mixing components is unknown while the model itself may or may not be correctly specified. Two representative types of prior specification will be considered: one requires explicitly a prior distribution on the number of mixture components, while the other places a nonparametric prior on the space of mixing distributions. The former is shown to yield an optimal rate of posterior contraction on the model parameters under minimal conditions, while the latter can be utilized to consistently recover the unknown number of mixture components, with the help of a fast probabilistic post-processing procedure. We then turn the study of these Bayesian procedures to the realistic settings of model misspecification. It will be shown that the modeling choice of kernel density functions plays perhaps the most impactful roles in determining the posterior contraction rates in the misspecified situations. Drawing on concrete posterior contraction rates established in this paper we wish to highlight some aspects about the interesting tradeoffs between model expressiveness and interpretability that a statistical modeler must negotiate in the rich world of mixture modeling.

Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most papers in the literature indeed are dealing with the null hypothesis of elliptical symmetry with specified location and actually address location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam’s asymptotic theory of statistical experiments, and optimality is to be understood locally and asymptotically within the family of generalized skew-elliptical distributions. The tests we are proposing are enjoying all the desirable properties of a “good” test of elliptical symmetry: they have simple asymptotic distributions under the entire null hypothesis of elliptical symmetry with unspecified radial density and shape parameter; they are affine-invariant, computationally fast, intuitively understandable, and not too demanding in terms of moments. While achieving optimality against generalized skew-elliptical alternatives, they remain quite powerful under a much broader class of non-elliptical distributions and significantly outperform the available competitors.

Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance – in fact, given samples of a copula C the corresponding empirical copulas do not converge weakly conditional to C with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and weak conditional convergence can even be proved to be equivalent. Moreover, it can be shown that every copula C is the weak conditional limit of a sequence of checkerboard copulas. After proving these three main results and pointing out some consequences, we sketch some implications for two recently introduced dependence measures and for the nonparametric estimation of Archimedean and Extreme Value copulas.

We study an adaptive estimation procedure called the Goldenshluger–Lepski method in the context of reproducing kernel Hilbert space (RKHS) regression. Adaptive estimation provides a way of selecting tuning parameters for statistical estimators using only the available data. This allows us to perform estimation without making strong assumptions about the estimand. In contrast to procedures such as training and validation, the Goldenshluger–Lepski method uses all of the data to produce non-adaptive estimators for a range of values of the tuning parameters. An adaptive estimator is selected by performing pairwise comparisons between these non-adaptive estimators. Applying the Goldenshluger–Lepski method is non-trivial as it requires a simultaneous high-probability bound on all of the pairwise comparisons. In the RKHS regression context, we choose our non-adaptive estimators to be clipped least-squares estimators constrained to lie in a ball in an RKHS. Applying the Goldenshluger–Lepski method in this context is made more complicated by the fact that we cannot use the ${\mathit{L}}^2$ norm for performing the pairwise comparisons as it is unknown. We use the method to address two regression problems. In the first problem the RKHS is fixed, while in the second problem we adapt over a collection of RKHSs.

The linear conditional expectation (LCE) provides a best linear (or rather, affine) estimate of the conditional expectation and hence plays an important rôle in approximate Bayesian inference, especially the Bayes linear approach. This article establishes the analytical properties of the LCE in an infinite-dimensional Hilbert space context. In addition, working in the space of affine Hilbert–Schmidt operators, we establish a regularisation procedure for this LCE. As an important application, we obtain a simple alternative derivation and intuitive justification of the conditional mean embedding formula, a concept widely used in machine learning to perform the conditioning of random variables by embedding them into reproducing kernel Hilbert spaces.

Suppose that we wish to estimate a finite-dimensional summary of one or more function-valued features of an underlying data-generating mechanism under a nonparametric model. One approach to estimation is by plugging in flexible estimates of these features. Unfortunately, in general, such estimators may not be asymptotically efficient, which often makes these estimators difficult to use as a basis for inference. Though there are several existing methods to construct asymptotically efficient plug-in estimators, each such method either can only be derived using knowledge of efficiency theory or is only valid under stringent smoothness assumptions. Among existing methods, sieve estimators stand out as particularly convenient because efficiency theory is not required in their construction, their tuning parameters can be selected data adaptively, and they are universal in the sense that the same fits lead to efficient plug-in estimators for a rich class of estimands. Inspired by these desirable properties, we propose two novel universal approaches for estimating function-valued features that can be analyzed using sieve estimation theory. Compared to traditional sieve estimators, these approaches are valid under more general conditions on the smoothness of the function-valued features by utilizing flexible estimates that can be obtained, for example, using machine learning.

This is a complementary proof of partial generalized 4 moment theorem (PG4MT) mentioned and described in “Generalized Four Moment Theorem (G4MT) and its Application to CLT for Spiked Eigenvalues of High-dimensional Covariance Matrices”. Since the G4MT proposed in that paper requires both the matrices X and Y satisfying the assumption ${max}_{\mathit{t},\mathit{s}}|{\mathit{u}}_{\mathit{t}\mathit{s}}{|}^2\mathrm{E}\{|{\mathit{x}}_{11}{|}^4\mathit{I}(|{\mathit{x}}_{11}|<\sqrt{\mathit{n}})-\mathit{\mu }\}\to 0$ with the same μ which maybe restrictive in real applications, we proposed a new G4MT, called PG4MT, without proof. After the manuscript posed in ArXiv, the authors received high interests in the proof of PG4MT through private communications and find the PG4MT more general than G4MT, it is necessary to give a detailed proof of it. Moreover, it is found that the PG4MT derives a CLT of spiked eigenvalues of sample covariance matrices which covers the work in Bai and Yao (J. Multivariate Anal. 106 (2012) 167–177) as a special case.

Hermon and Hutchcroft have recently proved the long-standing conjecture that in $Bernoulli(\mathit{p})$ bond percolation on any nonamenable transitive graph G, at any $\mathit{p}>{\mathit{p}}_{\mathit{c}}(\mathit{G})$, the probability that the cluster of the origin is finite but has a large volume n decays exponentially in n. A corollary is that all infinite clusters have anchored expansion almost surely. They have asked if these results could hold more generally, for any finite energy ergodic invariant percolation. We give a counterexample, an invariant percolation on the 4-regular tree.

In this paper, we consider a unidimensional piecewise deterministic Markov process (PDMP), with homogeneous jump rate $\mathit{\lambda }(\mathit{x})$. This process is observed continuously, so the flow ϕ is known. To estimate nonparametrically the jump rate, we first construct an adaptive estimator of the stationary density, then we derive a quotient estimator ${\hat{\mathit{\lambda }}}_{\mathit{n}}$ of λ. Under some ergodicity conditions, we bound the risk of these estimators (and give a uniform bound on a small class of functions), and prove that the estimator of the jump rate is nearly minimax (up to a ${ln}^2(\mathit{n})$ factor). The simulations illustrate our theoretical results.

We study supervised and semi-supervised algorithms in the set-valued classification framework with controlled expected size. While the former methods can use only n labeled samples, the latter are able to make use of N additional unlabeled data. We obtain semi-supervised minimax rates of convergence under the α-margin assumption and a β-Hölder condition on the conditional distribution of labels. Our analysis implies that if no further assumption is made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work – the best achievable rate for any supervised method is $\mathit{O}({\mathit{n}}^{-1/2})$, even if the margin assumption is extremely favorable; on the contrary, the developed semi-supervised estimator can achieve faster $\mathit{O}({(\mathit{n}/log\mathit{n})}^{-(1+\mathit{\alpha })\mathit{\beta }/(2\mathit{\beta }+\mathit{d})})$ rate of convergence provided that sufficiently many unlabeled samples are available. We also show that under additional smoothness assumption, supervised methods are able to achieve faster rates and the unlabeled sample cannot improve the rate of convergence. Finally, a numerical study supports our theory and emphasizes the relevance of the assumptions we required from an empirical perspective.

We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predominance of the Brownian component, the presence of exchangeable structures, and fast convergence of normal empirical quantile functions. The second procedure amounts to filtering the increments and compensating with the final value. It requires a carefully chosen threshold, in which case both methods yield the same rate of convergence. This rate depends on the small-jump activity and is given in terms of the Blumenthal–Getoor index. Finally, we discuss possible extensions, including the multidimensional case, and provide numerical illustrations.

The log-concave projection is an operator that maps a d-dimensional distribution P to an approximating log-concave density. It is known that, with suitable metrics on the underlying spaces, this projection is continuous, but not uniformly continuous. In this work, we prove a local uniform continuity result for log-concave projection – in particular, establishing that this map is locally Hölder-(1/4) continuous. A matching lower bound verifies that this exponent cannot be improved. We also examine the implications of this continuity result for the empirical setting – given a sample drawn from a distribution P, we bound the squared Hellinger distance between the log-concave projection of the empirical distribution of the sample, and the log-concave projection of P. In particular, this yields interesting statistical results for the misspecified setting, where P is not itself log-concave.

Consider n nodes ${\{{\mathit{X}}_{\mathit{i}}\}}_{1\le \mathit{i}\le \mathit{n}}$ independently distributed in the unit square S, each according to a distribution f. Nodes ${\mathit{X}}_{\mathit{i}}$ and ${\mathit{X}}_{\mathit{j}}$ are joined by an edge if the Euclidean distance $\mathit{d}({\mathit{X}}_{\mathit{i}},{\mathit{X}}_{\mathit{j}})$ is less than ${\mathit{r}}_{\mathit{n}}$, the adjacency distance and the resulting random graph ${\mathit{G}}_{\mathit{n}}$ is called a random geometric graph (RGG). We now assign a location dependent weight to each edge of ${\mathit{G}}_{\mathit{n}}$ and define ${\mathit{MST}}_{\mathit{n}}$ to be the sum of the weights of the minimum spanning trees of all components of ${\mathit{G}}_{\mathit{n}}$. For values of ${\mathit{r}}_{\mathit{n}}$ above the connectivity regime, we obtain upper and lower bound deviation estimates for ${\mathit{MST}}_{\mathit{n}}$ and ${\mathit{L}}^2$-convergence of ${\mathit{MST}}_{\mathit{n}}$ appropriately scaled and centred.

We show the convergence of an online stochastic gradient descent estimator to obtain the drift parameter of a continuous-time jump-diffusion process. The stochastic gradient descent follows a stochastic path in the gradient direction of a function to find a minimum, which in our case determines the estimate of the unknown drift parameter. We decompose the deviation of the stochastic descent direction from the deterministic descent direction into four terms: the weak solution of the non-local Poisson equation, a Riemann integral, a stochastic integral, and a covariation term. This decomposition is employed to prove the convergence of the online estimator and we use simulations to illustrate the performance of the online estimator.

In this note, we show how to provide sharp control on the least singular value of a certain translated linearization matrix arising in the study of the local universality of products of independent random matrices. This problem was first considered in a recent work of Kopel, O’Rourke, and Vu, and compared to their work, our proof is substantially simpler and established in much greater generality. In particular, we only assume that the entries of the ensemble are centered, and have second and fourth moments uniformly bounded away from 0 and infinity, whereas previous work assumed a uniform subgaussian decay condition and that the entries within each factor of the product are identically distributed.

A consequence of our least singular value bound is that the four moment matching universality results for the products of independent random matrices, recently obtained by Kopel, O’Rourke, and Vu, hold under much weaker hypotheses. Our proof technique is also of independent interest in the study of structured sparse matrices.

In this paper, we generalize the property of local asymptotic normality (LAN) to an enlarged neighborhood, under the name of rescaled local asymptotic normality (RLAN). We obtain sufficient conditions for a regular parametric model to satisfy RLAN. We show that RLAN supports the construction of a statistically efficient estimator which maximizes a cubic approximation to the log-likelihood on this enlarged neighborhood. In the context of Monte Carlo inference, we find that this maximum cubic likelihood estimator can maintain its statistical efficiency in the presence of asymptotically increasing Monte Carlo error in likelihood evaluation.

This article develops a two-sample nonparametric goodness-of-fit (GOF) test for uniform stochastic ordering (USO) when observations are taken in pairs. We propose a data-driven critical value that controls the type I error and yields a consistent test. A simulation study illustrates the finite-sample performance of our test. All the proofs are included in the supplemental file.

Recently it was shown in several papers that backpropagation is able to find the global minimum of the empirical risk on the training data using over-parametrized deep neural networks. In this paper, a similar result is shown for deep neural networks with the sigmoidal squasher activation function in a regression setting, and a lower bound is presented which proves that these networks do not generalize well on a new data in the sense that networks which minimize the empirical risk do not achieve the optimal minimax rate of convergence for estimation of smooth regression functions.

We consider a random walk ${\mathit{S}}_{\mathit{k}}$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum ${\sum }_{\mathit{k}=1}^{\mathit{N}}\mathit{f}({\mathit{S}}_{\mathit{k}})$ with Hölder continuous test functions f, including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided that the distribution of ${\mathit{S}}_{\mathit{k}}$ converges to the Haar measure in the p-Wasserstein metric fast enough. As an example, we construct discrete random walks on an irrational lattice on the torus ${\mathbb{R}}^{\mathit{d}}/{\mathbb{Z}}^{\mathit{d}}$, and find their precise rate of convergence to uniformity in the p-Wasserstein metric. The proof uses a new Berry–Esseen type inequality for the p-Wasserstein metric on the torus, and the simultaneous Diophantine approximation properties of the lattice. These results complement the first part of this paper on random walks with an absolutely continuous component and quantitative ergodic theorems for Borel measurable test functions.

A new integral with respect to an integer-valued random measure is introduced. In contrast to the finite variation integral ubiquitous in semimartingale theory, the new integral is closed under stochastic integration, composition, and smooth transformations. The new integral gives rise to a previously unstudied class of pure-jump processes – the sigma-locally finite variation pure-jump processes. As an application, it is shown that every semimartingale X has a unique decomposition

$$\mathit{X}={\mathit{X}}_0+{\mathit{X}}^{\mathrm{qc}}+{\mathit{X}}^{\mathrm{dp}},$$

where ${\mathit{X}}^{\mathrm{qc}}$ is quasi-left-continuous and ${\mathit{X}}^{\mathrm{dp}}$ is a sigma-locally finite variation pure-jump process that jumps only at predictable times, both starting at zero. The decomposition mirrors the classical result for local martingales and gives a rigorous meaning to the notions of continuous-time and discrete-time components of a semimartingale. Against this backdrop, the paper investigates a wider class of processes that are equal to the sum of their jumps in the semimartingale topology and constructs a taxonomic hierarchy of pure-jump semimartingales.

We construct an estimator of the Lévy density of a pure jump Lévy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the Lévy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the Lévy–Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of ${\mathit{L}}_{\mathit{p}}$ loss functions, $\mathit{p}\ge 1$, over Besov balls. We recover classical nonparametric rates for finite variation Lévy processes and for a nonparametric class of symmetric infinite variation Lévy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.

This work focuses on the nonparametric estimation of a drift function from N discrete repeated independent observations of a diffusion process over a fixed time interval $[0,\mathit{T}]$. We study a ridge estimator obtained by the minimization of a constrained least squares contrast. The resulting projection estimator is based on the B-spline basis. Under mild assumptions, this estimator is universally consistent with respect to an integrate norm. We establish that, up to a logarithmic factor and when the estimation is performed on a compact interval, our estimation procedure reaches the best possible rate of convergence. Furthermore, we build an adaptive estimator that achieves this rate. Finally, we illustrate our procedure through an intensive simulation study which highlights the good performance of the proposed estimator in various models.

The strong ${\mathit{L}}^2$-approximation of occupation time functionals is studied with respect to discrete observations of a d-dimensional càdlàg process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes, such as fractional Brownian motion. The results are used to approximate occupation times and local times. For Brownian motion, the upper bounds are shown to be sharp up to a log-factor.

We consider the eigenvalues of sample covariance matrices of the form $\mathcal{Q}=({\mathrm{\Sigma }}^{1/2}\mathit{X}){({\mathrm{\Sigma }}^{1/2}\mathit{X})}^{\ast }$. The sample X is an $\mathit{M}\times \mathit{N}$ rectangular random matrix with real independent entries and the population covariance matrix Σ is a positive definite diagonal matrix independent of X. Assuming that the limiting spectral density of Σ exhibits convex decay at the right edge of the spectrum, in the limit $\mathit{M},\mathit{N}\to \infty$ with $\mathit{N}/\mathit{M}\to \mathit{d}\in (0,\infty )$, we find a certain threshold ${\mathit{d}}_{+}$ such that for $\mathit{d}>{\mathit{d}}_{+}$ the limiting spectral distribution of $\mathcal{Q}$ also exhibits convex decay at the right edge of the spectrum. In this case, the largest eigenvalues of $\mathcal{Q}$ are determined by the order statistics of the eigenvalues of Σ, and in particular, the limiting distribution of the largest eigenvalue of $\mathcal{Q}$ is given by a Weibull distribution. In case $\mathit{d}<{\mathit{d}}_{+}$, we also prove that the limiting distribution of the largest eigenvalue of $\mathcal{Q}$ is Gaussian if the entries of Σ are i.i.d. random variables. While Σ is considered to be random mostly, the results also hold for deterministic Σ with some additional assumptions.

In this paper, we first provide several conditional limit theorems for Lévy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of heavy-tailed Lévy processes. As the key point, we observe that the asymptotic mainly depends on the sample paths with early arrival of large jump. Both the polynomial decay rate and the exact expression of the limit coefficients are given. As an application, we give an exact description for the extinction speed of continuous-state branching processes in heavy-tailed Lévy random environment with stable branching mechanism.

Superposition of renewal processes is common in practice, and it is challenging to estimate the distribution of the individual inter-occurrence time associated with the renewal process. This is because with only aggregated event history, the link between the observed recurrence times and the respective renewal processes are completely missing, rendering existing theory and methods inapplicable. In this article, we propose a nonparametric procedure to estimate the inter-occurrence time distribution by properly deconvoluting the renewal equation with the empirical renewal function. By carefully controlling the discretization errors and properly handling challenges due to implicit and non-smooth mapping via the renewal equation, our theoretical analysis establishes the consistency and asymptotic normality of the nonparametric estimators. The proposed nonparametric distribution estimators are then utilized for developing theoretically valid and computationally efficient inferences when a parametric family is assumed for the individual renewal process. Comprehensive simulations show that compared with the existing maximum likelihood method, the proposed parametric estimation procedure is much faster, and the proposed estimators are more robust to round-off errors in the observed data.