Consider the problem of learning undirected graphical models on trees from corrupted data. Recently Katiyar, Shah, and Caramanis showed that it is possible to recover trees from noisy binary data up to a small equivalence class of possible trees. Another paper by Katiyar, Hoffmann, and Caramanis follows a similar pattern for the Gaussian case. By framing this as a special phylogenetic recovery problem we largely generalize these two settings. Using the framework of linear latent tree models we discuss tree identifiability for binary data under a continuous corruption model (e.g. black/white images with greyscale corruption). For the Ising and the Gaussian tree model we also provide a characterisation of when the Chow-Liu algorithm consistently learns the underlying tree from the noisy data.

We investigate here a generalized construction of spherical wavelets/needlets which admits extra-flexibility in the harmonic space, i.e., it allows the corresponding support in multipole (frequency) space to vary in more general forms than in the standard constructions. We study the analytic properties of this system and we investigate its behaviour when applied to isotropic random fields: more precisely, we establish asymptotic localization and uncorrelation properties (in the high-frequency sense) under broader assumptions than typically considered in the literature.

Einmahl, de Haan and Zhou (2016, Journal of the Royal Statistical Society: Series B, 78(1), 31–51) recently introduced a stochastic model that allows for heteroscedasticity of extremes. The model is extended to the situation where the observations are serially dependent, which is crucial for many practical applications. We prove a local limit theorem for a kernel estimator for the scedasis function, and a functional limit theorem for an estimator for the integrated scedasis function. We further prove consistency of a bootstrap scheme that allows to test for the null hypothesis that the extremes are homoscedastic. Finally, we propose an estimator for the extremal index governing the dynamics of the extremes and prove its consistency. All results are illustrated by Monte Carlo simulations. An important intermediate result concerns the sequential tail empirical process under serial dependence.

Let ${\{X_t\}}_{t\ge 0}$ be a d-dimensional critical super-Brownian motion started from a Poisson random measure whose intensity is the Lebesgue measure. Denote by $R_t:=sup\{u>0:X_t(\{x\in {\mathbb{R}}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $X_t$. In this work, we prove that for $r>0$,

$$\underset{t\to \mathrm{\infty }}{lim}\mathbb{P}\left(\frac{R_t}{t^{(1∕d)\wedge {(3-d)}^{+}}}\ge r\right)=e^{-A_d(r)},$$

where $A_d(r)$ satisfies

$$\underset{r\to \mathrm{\infty }}{lim}\frac{A_d(r)}{r^{|d-2|+d{\mathbf{1}}_{\{d=2\}}}}=C$$

for some $C\in (0,\mathrm{\infty })$ depending only on d.

We investigate the problem of estimating the drift parameter of a high-dimensional Lévy-driven Ornstein–Uhlenbeck process under sparsity constraints. It is shown that both Lasso and Slope estimators achieve the minimax optimal rate of convergence (up to numerical constants), for tuning parameters chosen independently of the confidence level, which improves the previously obtained results for standard Ornstein–Uhlenbeck processes.

Functional auto-regressive moving average (FARMA or ARMAH) models allow for flexible and natural modelling of functional time series. While there are many results on pure autoregressive (FAR) models in Hilbert spaces, results on estimation and prediction of FARMA models are considerably more scarce. We devise a simple two-step method to estimate ARMA models in separable Hilbert spaces. Estimation is based on dimension-reduction using principal components analysis of the functional time series. We explore two different approaches to selecting principal component subspaces for regularization and establish consistency of the proposed estimators both under minimal assumptions and in a practical setting. The empirical performance of the estimation algorithm is evaluated in a simulation study, where it performs better than competing methods.

For two measures μ and ν that are in convex-decreasing order, Nutz and Stebegg (Canonical supermartingale couplings, Ann. Probab. 46(6) 3351–3398, 2018) studied the optimal transport problem with supermartingale constraints and introduced two canonical couplings, namely the increasing and decreasing transport plans, that are optimal for a large class of cost functions. In the present paper we provide an explicit construction of the decreasing coupling ${\mathrm{\pi }}^D$ by establishing a Brenier-type result: (a generalised version of) ${\mathrm{\pi }}^D$ concentrates on the graphs of two functions. Our construction is based on the concept of the supermartingale shadow measure and requires a suitable extension of the results by Juillet (Stability of the shadow projection and the left-curtain coupling, Ann. Inst. H. Poincaré Probab. Statist. 52(4) 1823–1843, November 2016) and Beiglböck and Juillet (Shadow couplings, Trans. Amer. Math. Soc. 374 4973–5002, 2021) established in the martingale setting. In particular, we prove the stability of the supermartingale shadow measure with respect to initial and target measures $\mathrm{\mu },\mathrm{\nu }$, introduce an infinite family of lifted supermartingale couplings that arise via shadow measure, and show how to explicitly determine the martingale points of each such coupling.

In this paper, we consider two-component mixture models having one single known component. This type of model is of particular interest when a known random phenomenon is contaminated by an unknown random effect. We propose in this setup to test the equality in distribution of the unknown random sources involved in two separate samples generated from such a model. For this purpose, we introduce the so-called IBM (Inversion-Best Matching) approach resulting in a tuning-free relaxed semiparametric Cramér-von Mises type two-sample test requiring minimal assumptions about the unknown distributions. The accomplishment of our work lies in the fact that we establish, under some natural and interpretable mutual-identifiability conditions specific to the two-sample case, a functional central limit theorem about the proportion parameters along with the unknown cumulative distribution functions of the model. An intensive numerical study is carried out from a large range of simulation setups to illustrate the asymptotic properties of our test. Finally, our testing procedure, implemented in the $\mathtt{admixRpackage}$, is applied to a real-life situation through pairwise post COVID-19 mortality excess profile testing across a panel of European countries.

We consider the following data perturbation model, where the covariates incur multiplicative errors. For two $n\times m$ random matrices $U,X$, we denote by $U\circ X$ the Hadamard or Schur product, which is defined as ${(U\circ X)}_{ij}=(U_{ij})\cdot (X_{ij})$. In this paper, we study the subgaussian matrix variate model, where we observe the matrix variate data X through a random mask U:

$$\mathcal{X}=U\circ X\text{where}X=B^{1∕2}\mathbb{Z}{A}^{1∕2},$$

where $\mathbb{Z}$ is a random matrix with independent subgaussian entries, and U is a mask matrix with either zero or positive entries, where $\mathbb{E}{U}_{ij}\in [0,1]$ and all entries are mutually independent. Under the assumption of independence between U and X, we introduce componentwise unbiased estimators for estimating covariance A and B, and prove the concentration of measure bounds in the sense of guaranteeing the restricted eigenvalue ($\mathsf{RE}$) conditions to hold on the unbiased estimator for B, when columns of data matrix X are sampled with different rates. We further develop multiple regression methods for estimating the inverse of B and show statistical rate of convergence. Our results provide insight for sparse recovery for relationships among entities (samples, locations, items) when features (variables, time points, user ratings) are present in the observed data matrix $\mathcal{X}$ with heterogeneous rates. Our proof techniques can certainly be extended to other scenarios. We provide simulation evidence illuminating the theoretical predictions.

Denote as L the local time at 0 of an Itô-McKean diffusion X. We present a new explicit description of the distribution of $L_t$ in terms of convolution exponent and, using the excursion theory, we describe the transition density of the pair $(X,L)$. We provide a simple connection formula for the distribution of excursions of a bivariate Itô-McKean diffusion from a hyperplane. Examples involving the distribution of a local time are presented, including a formula for the distribution of $(X_t,L_{\mathrm{\infty }})$ for a transient diffusion.

The infinite Viterbi alignment is the limiting maximum a-posteriori estimate of the unobserved path in a hidden Markov model as the length of the time horizon grows. For models on state-space ${\mathbb{R}}^d$ satisfying a new “decay-convexity” condition, we develop an approach to existence of the infinite Viterbi alignment in an infinite dimensional Hilbert space. Quantitative bounds on the distance to the Viterbi process, which are the first of their kind, are derived and used to illustrate how approximate estimation via parallelization can be accurate and scaleable to high-dimensional problems because the rate of convergence to the infinite Viterbi alignment does not necessarily depend on d. The results are applied to approximate estimation via parallelization and a model of neural population activity.

In this article, we are interested in the normal approximation of

$$T_n=(\sum _{i=1}^n{X}_{i1}∕(\sqrt{\sum _{i=1}^n{X}_{i1}^2}),\dots ,\sum _{i=1}^n{X}_{ip}∕(\sqrt{\sum _{i=1}^n{X}_{ip}^2}\left)\right)$$

in ${\mathcal{R}}^p$ uniformly over the class of hyper-rectangles ${\mathcal{A}}^{re}=\{{\prod }_{j=1}^p[a_j,b_j]\cap \mathcal{R}:-\mathrm{\infty }\le a_j\le b_j\le \mathrm{\infty },j=1,\dots ,p\}$, where $X_1,\dots ,X_n$ are non-degenerate independent p-dimensional random vectors. We assume that the components of $X_i$ are independent and identically distributed (iid) and investigate the optimal cut-off rate of $logp$ in the uniform central limit theorem (UCLT) for $T_n$ over ${\mathcal{A}}^{re}$. The aim is to reduce the exponential moment conditions, generally assumed for exponential growth of the dimension with respect to the sample size in high dimensional CLT, to some polynomial moment conditions. Indeed, we establish that only the existence of some polynomial moment of order $\in [2,4]$ is sufficient for exponential growth of p. However the rate of growth of $logp$ cannot further be improved from $o\left(n^{1∕2}\right)$ as a power of n even if $X_{ij}$’s are iid across $(i,j)$ and $X_{11}$ is bounded. We also establish near$-n^{-\mathrm{\kappa }∕2}$ Berry-Esseen rate for $T_n$ in high dimension under the existence of $(2+\mathrm{\kappa })$th absolute moments of $X_{ij}$ for $0<\mathrm{\kappa }\le 1$. When $\mathrm{\kappa }=1$, the obtained Berry-Esseen rate is also shown to be optimal. As an application, we find respective versions for componentwise Student’s t-statistic, which may be useful in high dimensional statistical inference.

We propose a reproducing kernel Hilbert space approach for statistical inference regarding the slope in a function-on-function linear regression via penalised least squares, regularized by the thin-plate spline smoothness penalty. We derive a Bahadur expansion for the slope surface estimator and prove its weak convergence as a process in the space of all continuous functions. As a consequence of these results, we construct minimax optimal estimates, simultaneous confidence regions for the slope surface and simultaneous prediction bands. Moreover, we derive new tests for the hypothesis that the maximum deviation between the “true” slope surface and a given surface is less than or equal to a given threshold. In other words, we are not trying to test for exact equality (because in many applications this hypothesis is hard to justify), but rather for pre-specified deviations under the null hypothesis. To ensure practicability, non-standard bootstrap procedures are developed addressing particular features that arise in these testing problems. We also demonstrate that the new methods have good finite sample properties by means of a simulation study and illustrate their practicability by analyzing a data example.

This paper establishes a Freidlin-Wentzell large deviation principle for stochastic differential equations(SDEs) under locally weak monotonicity conditions and Lyapunov conditions. We illustrate the main result of the paper by showing that it can be applied to SDEs with non-Lipschitzian coefficients, which can not be covered in the existing literature. These include the interesting biological models like stochastic Duffing-van der Pol oscillator model, stochastic SIR model, etc.

Consider a random vector $\mathbf{y}={\mathrm{\Sigma }}^{1∕2}\mathbf{x}$, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and ${\mathrm{\Sigma }}^{1∕2}$ is a deterministic $p\times p$ matrix such that the eigenvalues of the population correlation matrix R of y are uniformly bounded away from zero and infinity. In this paper, we find that the log determinant of the sample correlation matrix $\hat{\mathbf{R}}$ based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for $p∕n\to \mathit{\gamma }\in (0,1]$ and $p\le n$. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of y is unknown, we show that after re-centering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. Finally, the obtained findings are applied for testing of uncorrelatedness of p random variables. Surprisingly, in the null case $\mathbf{R}=\mathbf{I}$, the test statistic becomes distribution-free and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.

Conditioned limit theorems as $n\to \mathrm{\infty }$ are given for the increments $X_1,\dots ,X_n$ of a random walk $S_n=X_1+\cdots +X_n$, subject to the conditionings $S_n\ge nb$ or $S_n=nb$ with $b>\mathbb{E}X$. The probabilities of these conditioning events are given by saddlepoint approximations, corresponding to the exponential tilting $f_{\mathit{\theta }}(x)=$ ${\mathrm{e}}^{\mathit{\theta }x-\mathit{\psi }(\mathit{\theta })}f(x)$ of the increment density $f(x)$, with θ satisfying $b={\mathbb{E}}_{\mathit{\theta }}X={\mathit{\psi }}^{\prime }(\mathit{\theta })$ where $\mathit{\psi }(\mathit{\theta })=log\mathbb{E}{\mathrm{e}}^{\mathit{\theta }X}$. It has been noted in various formulations that conditionally, the increment density somehow is close to $f_{\mathit{\theta }}(x)$. Sharp versions of such statements are given, including correction terms for segments $(X_1,\dots ,X_k)$ with k fixed. Similar correction terms are given for the mean and variance of ${\hat{F}}_n(x)-F_{\mathit{\theta }}(x)$ where ${\hat{F}}_n$ is the empirical c.d.f. of $X_1,\dots ,X_n$. Also a result on the total variation distance for segments with $k∕n\to c\in (0,1)$ is derived. Further functional limit theorems for ${({\hat{F}}_k(x),S_k)}_{k\le n}$ are given, involving a bivariate conditioned Brownian limit.

We establish a finite-sample Berry-Esseen theorem for the entrywise limits of the eigenvectors for a broad collection of signal-plus-noise random matrix models under challenging weak signal regimes. The signal strength is characterized by a scaling factor ${\mathit{\rho }}_n$ through $n{\mathit{\rho }}_n$, where n is the dimension of the random matrix, and we allow $n{\mathit{\rho }}_n$ to grow at the rate of $logn$. The key technical contribution is a sharp finite-sample entrywise eigenvector perturbation bound. The existing error bounds on the two-to-infinity norms of the higher-order remainders are not sufficient when $n{\mathit{\rho }}_n$ is proportional to $logn$. We apply the general entrywise eigenvector analysis results to the symmetric noisy matrix completion problem, random dot product graphs, and two subsequent inference tasks for random graphs: the estimation of pure nodes in mixed membership stochastic block models and the hypothesis testing of the equality of latent positions in random graphs.

For $1\le p\le \mathrm{\infty }$, the Fréchet p-mean of a probability measure on a metric space is an important notion of central tendency that generalizes the usual notions in the real line of mean ($p=2$) and median ($p=1$). In this work we prove a collection of limit theorems for Fréchet means and related objects, which, in general, constitute a sequence of random closed sets. On the one hand, we show that many limit theorems (a strong law of large numbers, an ergodic theorem, and a large deviations principle) can be simply descended from analogous theorems on the space of probability measures via purely topological considerations. On the other hand, we provide the first sufficient conditions for the strong law of large numbers to hold in a $T_2$ topology (in particular, the Fell topology), and we show that this condition is necessary in some special cases. We also discuss statistical and computational implications of the results herein.

Let $\mathbf{S}=\mathbf{R}+\mathbf{X}$ be an $M\times N$ matrix where R is the signal matrix and X is the noise matrix consisting of i.i.d. standardized entries. The signal matrix R is allowed to be full rank, which is rarely studied in literature compared with the low rank cases. Under a regularity condition of R that assures the square root behaviour of the spectral density near the edge, we prove that the largest eigenvalue of $\mathbf{S}{\mathbf{S}}^{\ast }$ has the Tracy-Widom distribution under a tail condition on the entries of X. Moreover, such a condition is proved to be necessary and sufficient to assure the Tracy-Widom law.

In this paper we analyze the $L_2$ error of neural network regression estimates with one hidden layer. Under the assumption that the Fourier transform of the regression function decays suitably fast, we show that an estimate, where all initial weights are chosen according to proper uniform distributions and where the weights are learned by gradient descent, achieves a rate of convergence of $1∕\sqrt{n}$ (up to a logarithmic factor). Our statistical analysis implies that the key aspect behind this result is the proper choice of the initial inner weights and the adjustment of the outer weights via gradient descent. This indicates that we can also simply use linear least squares to choose the outer weights. We prove a corresponding theoretical result and compare our new linear least squares neural network estimate with standard neural network estimates via simulated data. Our simulations show that our theoretical considerations lead to an estimate with an improved performance in many cases.

We consider the problem of supervised dimension reduction with a particular focus on extreme values of the target $Y\in \mathbb{R}$ to be explained by a covariate vector $X\in {\mathbb{R}}^p$. The general purpose is to define and estimate a projection on a lower dimensional subspace of the covariate space which is sufficient for predicting exceedances of the target above high thresholds. We propose an original definition of Tail Conditional Independence which matches this purpose. Inspired by Sliced Inverse Regression (SIR) methods, we develop a novel framework (TIREX, Tail Inverse Regression for EXtreme response) in order to estimate an extreme sufficient dimension reduction (SDR) space of potentially smaller dimension than that of a classical SDR space. We prove the weak convergence of tail empirical processes involved in the estimation procedure and we illustrate the relevance of the proposed approach on simulated and real world data.

The Chebyshev or ${\ell }_{\mathrm{\infty }}$ estimator is an unconventional alternative to the ordinary least squares in solving linear regressions. It is defined as the minimizer of the ${\ell }_{\mathrm{\infty }}$ objective function

$$\hat{\mathit{\beta }}:=\underset{\mathit{\beta }}{argmin}\Vert \mathbf{Y}-\mathbf{X}\mathit{\beta }{\Vert }_{\mathrm{\infty }}.$$

The asymptotic distribution of the Chebyshev estimator under fixed number of covariates was recently studied (Knight (2020)), yet finite-sample guarantees and generalizations to high-dimensional settings remain open. In this paper, we develop non-asymptotic upper bounds on the estimation error $\Vert \hat{\mathit{\beta }}-{\mathit{\beta }}^{\ast }\Vert$ for a Chebyshev estimator $\hat{\mathit{\beta }}$, in a regression setting with uniformly distributed noise ${\mathit{\epsilon }}_i\sim U([-a,a])$ where a is either known or unknown. With relatively mild assumptions on the (random) design matrix X, we can bound the error rate by $C_p∕n$ with high probability, for some constant $C_p$ depending on the dimension p and the law of the design. Furthermore, we illustrate that there exist designs for which the Chebyshev estimator is (nearly) minimax optimal. On the other hand we also argue that there exist designs for which this estimator behaves sub-optimally in terms of the constant $C_p$’s dependence on p. Finally, we show that “Chebyshev’s LASSO” has advantages over the regular LASSO in high dimensional situations, provided that the noise is uniform. Specifically, we argue that it achieves a much faster rate of estimation under certain assumptions on the growth rate of the sparsity level and the ambient dimension with respect to the sample size.

We prove a Central Limit Theorem for the empirical optimal transport cost, $\sqrt{\frac{nm}{n+m}}\{{\mathcal{T}}_c(P_n,Q_m)-{\mathcal{T}}_c(P,Q)\}$, in the semi-discrete case, i.e when the distribution P is supported in N points, but without assumptions on Q. We show that the asymptotic distribution is the sup of a centered Gaussian process, which is Gaussian under some additional conditions on the probability Q and on the cost. Such results imply the central limit theorem for the p-Wassertein distance, for $p\ge 1$. This means that, for fixed N, the curse of dimensionality is avoided. To better understand the influence of such N, we provide bounds of $E|{\mathcal{W}}_p^p(P,Q_m)-{\mathcal{W}}_p^p(P,Q)|$ depending on m and N. Finally, the semi-discrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials and Laguerre cells. The results are supported by simulations that help to visualize the given limits and bounds. We analyse also the cases where classical bootstrap works.

In this paper, we deal with a Markov chain on a measurable state space $(\mathbb{X},\mathcal{X})$ which has a transition kernel P admitting an aperiodic small-set S and satisfying the standard geometric-drift condition. Under these assumptions, there exists ${\mathit{\alpha }}_0\in (0,1]$ such that $P{V}^{{\mathit{\alpha }}_0}\le {\mathit{\delta }}^{{\mathit{\alpha }}_0}{V}^{{\mathit{\alpha }}_0}+\mathrm{\nu }(V^{{\mathit{\alpha }}_0})1_S$. Hence P is $V^{{\mathit{\alpha }}_0}$-geometrically ergodic and its “second eigenvalue” ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}$ provides the best rate of convergence. Setting $R:=P-\mathrm{\nu }(\cdot )1_S$ and $\mathrm{\Gamma }:=\{\mathit{\lambda }\in \mathbb{C},{\mathit{\delta }}^{{\mathit{\alpha }}_0}<|\mathit{\lambda }|<1\}$, ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}$ is shown to satisfy, either ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}=max\{|\mathit{\lambda }|:\mathit{\lambda }\in \mathrm{\Gamma },{\sum }_{k=1}^{+\mathrm{\infty }}{\mathit{\lambda }}^{-k}\mathrm{\nu }(R^{k-1}{1}_S)=1\}$ if this set is not empty, or ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}\le {\mathit{\delta }}^{{\mathit{\alpha }}_0}$. Actually the set is finite in the first case and is composed by the spectral values of P in Γ. The second case occurs when P has no spectral value in Γ. Moreover, a bound of the operator-norm of ${(zI-P)}^{-1}$ allows us to derive an explicit formula for the multiplicative constant in the rate of convergence, which can be evaluated provided that any information of the “second eigenvalue” is available. Such numerical computation is carried out for a classical family of reflected random walks. Moreover we obtain a simple and explicit bound of the operator-norm of ${(I-P+\mathrm{\pi }(\cdot )1_{\mathbb{X}})}^{-1}$ involved in the definition of the so-called fundamental solution to Poisson’s equation. This allows us to specify the location of the eigenvalues of P and, then, to obtain a general bound on ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}$. The reversible case is also discussed. In particular, the bound of ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}$ obtained for positive reversible Markov kernels is the expected one, and numerical illustrations are proposed for the Metropolis-Hastings algorithm and for the Gaussian autoregressive Markov chain. The bound for the operator-norm of ${(I-P+\mathrm{\pi }(\cdot )1_{\mathbb{X}})}^{-1}$ is derived from an estimate, only depending on ${\mathit{\delta }}^{{\mathit{\alpha }}_0}$, of the operator-norm of ${(I-R)}^{-1}$ which provides another way to get a solution to Poisson’s equation. This estimate is also shown to be of greatest interest to generalize the error bounds obtained for perturbed discrete and atomic Markov chains in Liu and Li (Adv. in Appl. Probab. 50 (2018) 645–669) to the case of general geometrically ergodic Markov chains. These error estimates are the simplest that can be expected in this context. All the estimates in this work are expressed in the standard $V^{{\mathit{\alpha }}_0}$-weighted operator norm.

In classical statistics, a well known paradigm consists in establishing asymptotic equivalence between an experiment of i.i.d. observations and a Gaussian shift experiment, with the aim of obtaining optimal estimators in the former complicated model from the latter simpler model. In particular, a statistical experiment consisting of n i.i.d. observations from d-dimensional multinomial distributions can be well approximated by an experiment consisting of $d-1$ dimensional Gaussian distributions. In a quantum version of the result, it has been shown that a collection of n qudits (d-dimensional quantum states) of full rank can be well approximated by a quantum system containing a classical part, which is a $d-1$ dimensional Gaussian distribution, and a quantum part containing an ensemble of $d(d-1)∕2$ shifted thermal states. In this paper, we obtain a generalization of this result when the qudits are not of full rank. We show that when the rank of the qudits is r, then the limiting experiment consists of an $r-1$ dimensional Gaussian distribution and an ensemble of both shifted pure and shifted thermal states. For estimation purposes, we establish an asymptotic minimax result in the limiting Gaussian model. Analogous results are then obtained for estimation of a low rank qudit from an ensemble of identically prepared, independent quantum systems, using the local asymptotic equivalence result. We also consider the problem of estimation of a linear functional of the quantum state. We construct an estimator for the functional, analyze the risk and use quantum local asymptotic equivalence to show that our estimator is also optimal in the minimax sense.

The partially linear model provides an effective tool to combat the curse of dimensionality in nonparametric regression. Its applicability is, however, compromised by the need for correct distinction between linear and nonlinear components. Existing solutions are restricted to low dimensions or regression functions endowed with special structures. This paper considers a general nonparametric regression framework under which signal strength is embedded in a continuous spectrum scaled by asymptotic orders. Under a weak sparsity condition which allows for the presence of many weak, non-detectable, signals, a novel penalised regression procedure is proposed for detection and estimation of strong linear and nonlinear signals under high dimensions. The procedure applies bandwidth regularisation and SCAD penalisation to select nonlinear and linear signals, respectively, under a partially linear model setting. Theoretical results are established for its consistency in detecting strong signals and its error rate in estimating the regression function. Numerical examples are presented to illustrate its performance.

We consider smooth linear statistics of determinantal point processes on the complex plane, and their large scale asymptotics. We prove asymptotic normality in the finite variance case, where Soshnikov’s theorem is not applicable. The setting is similar to that of Rider and Virág [Electron. J. Probab., 12, no. 45, 1238–1257, (2007)] for the complex plane, but replaces analyticity conditions by the assumption that the correlation kernel is reproducing. Our proof is a streamlined version of that of Ameur, Hedenmalm and Makarov [Duke Math J., 159, 31–81, (2011)] for eigenvalues of normal random matrices. In our case, the reproducing property is brought to bear to compensate for the lack of analyticity and radial symmetries.

Classical asymptotic theory for statistical inference usually involves calibrating a statistic by fixing the dimension d while letting the sample size n increase to infinity. Recently, much effort has been dedicated towards understanding how these methods behave in high-dimensional settings, where d and n both increase to infinity together. This often leads to different inference procedures, depending on the assumptions about the dimensionality, leaving the practitioner in a bind: given a dataset with 100 samples in 20 dimensions, should they calibrate by assuming $n\gg d$, or $d∕n\approx 0.2$? This paper considers the goal of dimension-agnostic inference—developing methods whose validity does not depend on any assumption on d versus n. We introduce an approach that uses variational representations of existing test statistics along with sample splitting and self-normalization to produce a refined test statistic with a Gaussian limiting distribution, regardless of how d scales with n. The resulting statistic can be viewed as a careful modification of degenerate U-statistics, dropping diagonal blocks and retaining off-diagonal blocks. We exemplify our technique for some classical problems including one-sample mean and covariance testing, and show that our tests have minimax rate-optimal power against appropriate local alternatives. In most settings, our cross U-statistic matches the high-dimensional power of the corresponding (degenerate) U-statistic up to a $\sqrt{2}$ factor.

Motivated by statistical inference problems in high-dimensional time series data analysis, we first derive non-asymptotic error bounds for Gaussian approximations of sums of high-dimensional dependent random vectors on hyper-rectangles, simple convex sets and sparsely convex sets. We investigate the quantitative effect of temporal dependence on the rates of convergence to a Gaussian random vector over three different dependency frameworks (α-mixing, m-dependent, and physical dependence measure). In particular, we establish new error bounds under the α-mixing framework and derive faster rate over existing results under the physical dependence measure. To implement the proposed results in practical statistical inference problems, we also derive a data-driven parametric bootstrap procedure based on a kernel-type estimator for the long-run covariance matrices. The unified Gaussian and parametric bootstrap approximation results can be used to test mean vectors with combined ${\ell }^2$ and ${\ell }^{\mathrm{\infty }}$ type statistics, do change point detection, and construct confidence regions for covariance and precision matrices, all for time series data.

The Benjamini-Hochberg (BH) procedure remains widely popular despite having limited theoretical guarantees in the commonly encountered scenario of correlated test statistics. Of particular concern is the possibility that the method could exhibit bursty behavior, meaning that it might typically yield no false discoveries while occasionally yielding both a large number of false discoveries and a false discovery proportion (FDP) that far exceeds its own well controlled mean. In this paper, we investigate which test statistic correlation structures lead to bursty behavior and which ones lead to well controlled FDPs. To this end, we develop a central limit theorem for the FDP in a multiple testing setup where the test statistic correlations can be either short-range or long-range as well as either weak or strong. The theorem and our simulations from a data-driven factor model suggest that the BH procedure exhibits severe burstiness when the test statistics have many strong, long-range correlations, but does not otherwise.

Two-sample tests utilizing a similarity graph on observations are useful for high-dimensional and non-Euclidean data due to their flexibility and good performance under a wide range of alternatives. Existing works mainly focused on sparse graphs, such as graphs with the number of edges in the order of the number of observations, and their asymptotic results imposed strong conditions on the graph that can easily be violated by commonly constructed graphs they suggested. Moreover, the graph-based tests have better performance with denser graphs under many settings. In this work, we establish the theoretical ground for graph-based tests with graphs ranging from those recommended in current literature to much denser ones.

The Horvitz-Thompson (HT), the Rao-Hartley-Cochran (RHC) and the generalized regression (GREG) estimators of the finite population mean are considered, when the observations are from an infinite dimensional space. We compare these estimators based on their asymptotic distributions under some commonly used sampling designs and some superpopulations satisfying linear regression models. We show that the GREG estimator is asymptotically at least as efficient as any of the other two estimators under different sampling designs considered in this paper. Further, we show that the use of some well known sampling designs utilizing auxiliary information may have an adverse effect on the performance of the GREG estimator, when the degree of heteroscedasticity present in linear regression models is not very large. On the other hand, the use of those sampling designs improves the performance of this estimator, when the degree of heteroscedasticity present in linear regression models is large. We develop methods for determining the degree of heteroscedasticity, which in turn determines the choice of appropriate sampling design to be used with the GREG estimator. We also investigate the consistency of the covariance operators of the above estimators. We carry out some numerical studies using real and synthetic data, and our theoretical results are supported by the results obtained from those numerical studies.

The main contribution of this paper is to find a representation of the class ${\mathcal{F}}_d(p)$ of multivariate Bernoulli distributions with the same mean p that allows us to find its generators analytically in any dimension. We map ${\mathcal{F}}_d(p)$ to an ideal of points and we prove that the class ${\mathcal{F}}_d(p)$ can be generated from a finite set of simple polynomials. We present two applications. Firstly, we show that polynomial generators help to find extremal points of the convex polytope ${\mathcal{F}}_d(p)$ in high dimensions. Secondly, we solve the problem of determining the lower bounds in the convex order for sums of multivariate Bernoulli distributions with given margins, but with an unspecified dependence structure.