We propose two least-squares estimators of a discrete probability under the constraint of $k$-monotonicity and study their statistical properties. We give a characterization of these estimators based on the decomposition on a spline basis of $k$-monotone sequences. We develop an algorithm derived from the Support Reduction Algorithm and we finally present a simulation study to illustrate their properties.

Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in quantum state tomography (QST) is to estimate the unknown low rank density matrix of a quantum system by conducting Pauli measurements. Our main contribution is twofold. First, we establish the minimax lower bounds in Schatten $p$-norms with $1\leq p\leq+\infty$ for low rank density matrices estimation by Pauli measurements. In our previous paper [14], these minimax lower bounds are proved under the trace regression model with Gaussian noise and the noise is assumed to have common variance. In this paper, we prove these bounds under the Binomial observation model which meets the actual model in QST.

Second, we study the Dantzig estimator (DE) for estimating the unknown low rank density matrix under the Binomial observation model by using Pauli measurements. In our previous papers [14] and [25], we studied the least squares estimator and the projection estimator, where we proved the optimal convergence rates for the least squares estimator in Schatten $p$-norms with $1\leq p\leq2$ and, under a stronger condition, the optimal convergence rates for the projection estimator in Schatten $p$-norms with $1\leq p\leq+\infty$. In this paper, we show that the results of these two distinct estimators can be simultaneously obtained by the Dantzig estimator. Moreover, better convergence rates in Schatten norm distances can be proved for Dantzig estimator under conditions weaker than those needed in [14] and [25]. When the objective function of DE is replaced by the negative von Neumann entropy, we obtain sharp convergence rate in Kullback-Leibler divergence.

This paper presents a new estimator of the global regularity index of a multifractional Brownian motion. Our estimation method is based upon a ratio statistic, which compares the realized global quadratic variation of a multifractional Brownian motion at two different frequencies. We show that a logarithmic transformation of this statistic converges in probability to the minimum of the Hurst functional parameter, which is, under weak assumptions, identical to the global regularity index of the path.

In this paper we consider a continuous almost periodically correlated process $\{X(t),t\in\mathbb{R}\}$ that is observed at the jump moments of a stationary Poisson point process $\{N(t),t\geq0\}$. The processes $\{X(t),t\in\mathbb{R}\}$ and $\{N(t),t\geq0\}$ are assumed to be independent. We define the kernel estimators of the Fourier coefficients of the autocovariance function of $X(t)$ and investigate their asymptotic properties. Moreover, we propose a bootstrap method that provides consistent pointwise and simultaneous confidence intervals for the considered coefficients. Finally, to illustrate our results we provide a simulated data example.

Suppose the random vector $(X,Y)$ satisfies the regression model $Y=m(X)+\sigma (X)\varepsilon$, where $m(\cdot)=E(Y|\cdot),$ $\sigma^{2}(\cdot)=\mbox{Var}(Y|\cdot)$ belongs to some parametric class $\{\sigma _{\theta}(\cdot):\theta \in \Theta\}$ and $\varepsilon$ is independent of $X$. The response $Y$ is subject to random right censoring and the covariate $X$ is completely observed. A new estimation procedure is proposed for $\sigma _{\theta}(\cdot)$ when $m(\cdot)$ is unknown. It is based on nonlinear least squares estimation extended to conditional variance in the censored case. The consistency and asymptotic normality of the proposed estimator are established. The estimator is studied via simulations and an important application is devoted to fatigue life data analysis.

The Bayesian probit regression model (Albert and Chib [1]) is popular and widely used for binary regression. While the improper flat prior for the regression coefficients is an appropriate choice in the absence of any prior information, a proper normal prior is desirable when prior information is available or in modern high dimensional settings where the number of coefficients ($p$) is greater than the sample size ($n$). For both choices of priors, the resulting posterior density is intractable and a Data Augmentation (DA) Markov chain is used to generate approximate samples from the posterior distribution. Establishing geometric ergodicity for this DA Markov chain is important as it provides theoretical guarantees for constructing standard errors for Markov chain based estimates of posterior quantities. In this paper, we first show that in case of proper normal priors, the DA Markov chain is geometrically ergodic for all choices of the design matrix $X$, $n$ and $p$ (unlike the improper prior case, where $n\geq p$ and another condition on $X$ are required for posterior propriety itself). We also derive sufficient conditions under which the DA Markov chain is trace-class, i.e., the eigenvalues of the corresponding operator are summable. In particular, this allows us to conclude that the Haar PX-DA sandwich algorithm (obtained by inserting an inexpensive extra step in between the two steps of the DA algorithm) is strictly better than the DA algorithm in an appropriate sense.

This short note argues that 95% confidence intervals for MCMC estimates can be obtained even without establishing a CLT, by multiplying their widths by 2.3.

Bump-hunting or mode identification is a fundamental problem that arises in almost every scientific field of data-driven discovery. Surprisingly, very few data modeling tools are available for automatic (not requiring manual case-by-case investigation), objective (not subjective), and nonparametric (not based on restrictive parametric model assumptions) mode discovery, which can scale to large data sets. This article introduces LPMode–an algorithm based on a new theory for detecting multimodality of a probability density. We apply LPMode to answer important research questions arising in various fields from environmental science, ecology, econometrics, analytical chemistry to astronomy and cancer genomics.

We propose a new procedure for optimally estimating high dimensional Gaussian graphical models. Our approach is asymptotically tuning-free and non-asymptotically tuning-insensitive: It requires very little effort to choose the tuning parameter in finite sample settings. Computationally, our procedure is significantly faster than existing methods due to its tuning-insensitive property. Theoretically, the obtained estimator simultaneously achieves minimax lower bounds for precision matrix estimation under different norms. Empirically, we illustrate the advantages of the proposed method using simulated and real examples. The R package camel implementing the proposed methods is also available on the Comprehensive R Archive Network.

Minimum disparity estimation in controlled branching processes is dealt with by assuming that the offspring law belongs to a general parametric family. Under some regularity conditions it is proved that the minimum disparity estimators proposed -based on the nonparametric maximum likelihood estimator of the offspring law when the entire family tree is observed- are consistent and asymptotic normally distributed. Moreover, the robustness of the estimators proposed is discussed. Through a simulated example, focusing on the minimum Hellinger and negative exponential disparity estimators, it is shown that both are robust against outliers, and the minimum negative exponential estimator is also robust against inliers.

We consider the intractable posterior density that results when the one-way logistic analysis of variance model is combined with a flat prior. We analyze Polson, Scott and Windle’s (2013) data augmentation (DA) algorithm for exploring the posterior. The Markov operator associated with the DA algorithm is shown to be trace-class.

Assuming that data are collected sequentially from independent streams, we consider the simultaneous testing of multiple binary hypotheses under two general setups; when the number of signals (correct alternatives) is known in advance, and when we only have a lower and an upper bound for it. In each of these setups, we propose feasible procedures that control, without any distributional assumptions, the familywise error probabilities of both type I and type II below given, user-specified levels. Then, in the case of i.i.d. observations in each stream, we show that the proposed procedures achieve the optimal expected sample size, under every possible signal configuration, asymptotically as the two error probabilities vanish at arbitrary rates. A simulation study is presented in a completely symmetric case and supports insights obtained from our asymptotic results, such as the fact that knowledge of the exact number of signals roughly halves the expected number of observations compared to the case of no prior information.

Different from the standard treatment discovery framework which is used for finding single treatments for a homogenous group of patients, personalized medicine involves finding therapies that are tailored to each individual in a heterogeneous group. In this paper, we propose a new semiparametric additive single-index model for estimating individualized treatment strategy. The model assumes a flexible and nonparametric link function for the interaction between treatment and predictive covariates. We estimate the rule via monotone B-splines and establish the asymptotic properties of the estimators. Both simulations and an real data application demonstrate that the proposed method has a competitive performance.

We consider the fractional Ornstein–Uhlenbeck process with an unknown drift parameter and known Hurst parameter $H$. We propose a new method to test the hypothesis of the sign of the parameter and prove the consistency of the test. Contrary to the previous works, our approach is applicable for all $H\in(0,1)$.

We consider the class of all stationary Gaussian process with explicit parametric spectral density. Under some conditions on the autocovariance function, we defined a GMM estimator that satisfies consistency and asymptotic normality, using the Breuer-Major theorem and previous results on ergodicity. This result is applied to the joint estimation of the three parameters of a stationary Ornstein-Uhlenbeck (fOU) process driven by a fractional Brownian motion. The asymptotic normality of its GMM estimator applies for any $H$ in $(0,1)$ and under some restrictions on the remaining parameters. A numerical study is performed in the fOU case, to illustrate the estimator’s practical performance when the number of datapoints is moderate.

We study the problem of estimating the mean of a multivariate distribution based on independent samples. The main result is the proof of existence of an estimator with a non-asymptotic sub-Gaussian performance for all distributions satisfying some mild moment assumptions.

We prove the consistency and asymptotic normality of the Laplacian Quasi-Maximum Likelihood Estimator (QMLE) for a general class of causal time series including ARMA, AR($\infty$), GARCH, ARCH($\infty$), ARMA-GARCH, APARCH, ARMA-APARCH,..., processes. We notably exhibit the advantages (moment order and robustness) of this estimator compared to the classical Gaussian QMLE. Numerical simulations confirms the accuracy of this estimator.

We introduce a general single index semiparametric measurement error model for the case that the main covariate of interest is measured with error and modeled parametrically, and where there are many other variables also important to the modeling. We propose a semiparametric bias-correction approach to estimate the effect of the covariate of interest. The resultant estimators are shown to be root-$n$ consistent, asymptotically normal and locally efficient. Comprehensive simulations and an analysis of an empirical data set are performed to demonstrate the finite sample performance and the bias reduction of the locally efficient estimators.

We present a Bayesian model for pairwise nonlinear registration of functional data. We use the Riemannian geometry of the space of warping functions to define appropriate prior distributions and sample from the posterior using importance sampling. A simple square-root transformation is used to simplify the geometry of the space of warping functions, which allows for computation of sample statistics, such as the mean and median, and a fast implementation of a $k$-means clustering algorithm. These tools allow for efficient posterior inference, where multiple modes of the posterior distribution corresponding to multiple plausible alignments of the given functions are found. We also show pointwise 95% credible intervals to assess the uncertainty of the alignment in different clusters. We validate this model using simulations and present multiple examples on real data from different application domains including biometrics and medicine.

We develop a unified approach for classification and regression support vector machines for when the responses are subject to right censoring. We provide finite sample bounds on the generalization error of the algorithm, prove risk consistency for a wide class of probability measures, and study the associated learning rates. We apply the general methodology to estimation of the (truncated) mean, median, quantiles, and for classification problems. We present a simulation study that demonstrates the performance of the proposed approach.

Sequential quantile estimation refers to incorporating observations into quantile estimates in an incremental fashion thus furnishing an online estimate of one or more quantiles at any given point in time. Sequential quantile estimation is also known as online quantile estimation. This area is relevant to the analysis of data streams and to the one-pass analysis of massive data sets. Applications include network traffic and latency analysis, real time fraud detection and high frequency trading. We introduce new techniques for online quantile estimation based on Hermite series estimators in the settings of static quantile estimation and dynamic quantile estimation. In the static quantile estimation setting we apply the existing Gauss-Hermite expansion in a novel manner. In particular, we exploit the fact that Gauss-Hermite coefficients can be updated in a sequential manner. To treat dynamic quantile estimation we introduce a novel expansion with an exponentially weighted estimator for the Gauss-Hermite coefficients which we term the Exponentially Weighted Gauss-Hermite (EWGH) expansion. These algorithms go beyond existing sequential quantile estimation algorithms in that they allow arbitrary quantiles (as opposed to pre-specified quantiles) to be estimated at any point in time. In doing so we provide a solution to online distribution function and online quantile function estimation on data streams. In particular we derive an analytical expression for the CDF and prove consistency results for the CDF under certain conditions. In addition we analyse the associated quantile estimator. Simulation studies and tests on real data reveal the Gauss-Hermite based algorithms to be competitive with a leading existing algorithm.

When providing probabilistic forecasts for uncertain future events, it is common to strive for calibrated forecasts, that is, the predictive distribution should be compatible with the observed outcomes. Often, there are several competing forecasters of different skill. We extend common notions of calibration where each forecaster is analyzed individually, to stronger notions of cross-calibration where each forecaster is analyzed with respect to the other forecasters. In particular, cross-calibration distinguishes forecasters with respect to increasing information sets. We provide diagnostic tools and statistical tests to assess cross-calibration. The methods are illustrated in simulation examples and applied to probabilistic forecasts for inflation rates by the Bank of England. Computer code and supplementary material (Strähl and Ziegel, 2017a,b) are available online.

Shrinkage estimators that possess the ability to produce sparse solutions have become increasingly important to the analysis of today’s complex datasets. Examples include the LASSO, the Elastic-Net and their adaptive counterparts. Estimation of penalty parameters still presents difficulties however. While variable selection consistent procedures have been developed, their finite sample performance can often be less than satisfactory. We develop a new strategy for variable selection using the adaptive LASSO and adaptive Elastic-Net estimators with $p_{n}$ diverging. The basic idea first involves using the trace paths of their LARS solutions to bootstrap estimates of maximum frequency (MF) models conditioned on dimension. Conditioning on dimension effectively mitigates overfitting, however to deal with underfitting, these MFs are then prediction-weighted, and it is shown that not only can consistent model selection be achieved, but that attractive convergence rates can as well, leading to excellent finite sample performance. Detailed numerical studies are carried out on both simulated and real datasets. Extensions to the class of generalized linear models are also detailed.

In the multidimensional setting, we consider the errors-in- variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimators based on projection kernels on wavelets and a deconvolution operator. We propose an automatic and fully data driven procedure to select the wavelet level resolution. We obtain an oracle inequality and optimal rates of convergence over anisotropic Hölder classes. Our theoretical results are illustrated by some simulations.

We investigate properties of a hybrid bootstrap procedure for general, strictly stationary sequences, called the autoregressive-aided block bootstrap which combines a parametric autoregressive bootstrap with a nonparametric moving block bootstrap. The autoregressive-aided block bootstrap consists of two main steps, namely an autoregressive model fit and an ensuing (moving) block resampling of residuals. The linear parametric model-fit prewhitenes the time series so that the dependence structure of the remaining residuals gets closer to that of a white noise sequence, while the moving block bootstrap applied to these residuals captures nonlinear features that are not taken into account by the linear autoregressive fit. We establish validity of the autoregressive-aided block bootstrap for the important class of statistics known as generalized means which includes many commonly used statistics in time series analysis as special cases. Numerical investigations show that the hybrid bootstrap procedure considered in this paper performs quite well, it behaves as good as or it outperforms in many cases the ordinary moving block bootstrap and it is robust against mis-specifications of the autoregressive order, a substantial advantage over the autoregressive bootstrap.

For the problem of high-dimensional sparse linear regression, it is known that an $\ell_{0}$-based estimator can achieve a $1/n$ “fast” rate for prediction error without any conditions on the design matrix, whereas in the absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarantee the $1/\sqrt{n}$ “slow” rate. In this paper, we show that the slow rate is intrinsic to a broad class of M-estimators. In particular, for estimators based on minimizing a least-squares cost function together with a (possibly nonconvex) coordinate-wise separable regularizer, there is always a “bad” local optimum such that the associated prediction error is lower bounded by a constant multiple of $1/\sqrt{n}$. For convex regularizers, this lower bound applies to all global optima. The theory is applicable to many popular estimators, including convex $\ell_{1}$-based methods as well as M-estimators based on nonconvex regularizers, including the SCAD penalty or the MCP regularizer. In addition, we show that bad local optima are very common, in that a broad class of local minimization algorithms with random initialization typically converge to a bad solution.

We consider the segmentation problem of univariate distributions from the exponential family with multiple parameters. In segmentation, the choice of the number of segments remains a difficult issue due to the discrete nature of the change-points. In this general exponential family distribution framework, we propose a penalized $\log$-likelihood estimator where the penalty is inspired by papers of L. Birgé and P. Massart. The resulting estimator is proved to satisfy some oracle inequalities. We then further study the particular case of categorical variables by comparing the values of the key constants when derived from the specification of our general approach and when obtained by working directly with the characteristics of this distribution. Finally, simulation studies are conducted to assess the performance of our criterion and to compare our approach to other existing methods, and an application on real data modeled using the categorical distribution is provided.

In the present paper, we deal with a stationary isotropic random field $X:{\mathbb{R}}^{d}\to{\mathbb{R}}$ and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level functionals are given by the Euler characteristic of the excursion sets above a finite number of levels. On the one hand, we study the properties of these level functionals under the hypothesis that the random field $X$ is Gaussian. In particular, we focus on the mapping that associates to any level $u$ the expected Euler characteristic of the excursion set above level $u$. On the other hand, we study the same level functionals under alternative distributions of $X$, such as chi-square, harmonic oscillator and shot noise. In order to validate our methodology, a part of the work consists in numerical experimentations. We generate Monte-Carlo samples of Gaussian and non-Gaussian random fields and compare, from a statistical point of view, their level functionals. Goodness-of-fit $p-$values are displayed for both cases. Simulations are performed in one dimensional case ($d=1$) and in two dimensional case ($d=2$), using R.

We study a Bayesian approach to estimating a smooth function in the context of regression or classification problems on large graphs. We derive theoretical results that show how asymptotically optimal Bayesian regularisation can be achieved under an asymptotic shape assumption on the underlying graph and a smoothness condition on the target function, both formulated in terms of the graph Laplacian. The priors we study are randomly scaled Gaussians with precision operators involving the Laplacian of the graph.

We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local Hölder densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a log n term) and since the priors are independent of the smoothness, the rates are adaptive to the smoothness.

One of the novel feature of the paper is that these results hold for densities with polynomial tails. Similar results are obtained using a hierarchical Bayesian model based on the mixture of inverse gamma densities which can be used to estimate adaptively densities with very heavy tails, including Cauchy density.

This paper is concerned with a central limit theorem for quadratic variation when observations come as exit times from a regular grid. We discuss the special case of a semimartingale with deterministic characteristics and finite activity jumps in detail and illustrate technical issues in more general situations.

We study the ${\mathbb{L}}^{p}$-integrated risk of some classical estimators of the density, when the observations are drawn from a strictly stationary sequence. The results apply to a large class of sequences, which can be non-mixing in the sense of Rosenblatt and long-range dependent. The main probabilistic tool is a new Rosenthal-type inequality for partial sums of $BV$ functions of the variables. As an application, we give the rates of convergence of regular Histograms, when estimating the invariant density of a class of expanding maps of the unit interval with a neutral fixed point at zero. These Histograms are plotted in the section devoted to the simulations.

Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression function, and (iii) the adaptability (or qualification) of the regularization. A simple consequence of this upper bound is that the risk of the regularization-based estimators matches the minimax rate in a variety of settings. The general bound also illustrates how some regularization techniques are more adaptable than others to favorable regularity properties that the true regression function may possess. This, in particular, demonstrates a striking difference between kernel ridge regression and kernel principal component regression. Our theoretical results are supported by numerical experiments.

Multiple testing based on chi-squared test statistics is common in many scientific fields such as genomics research and brain imaging studies. However, the challenges of designing a formal testing procedure when there exists a general dependence structure across the chi-squared test statistics have not been well addressed. To address this gap, we first adopt a latent factor structure ([14]) to construct a testing framework for approximating the false discovery proportion ($\mathrm{FDP}$) for a large number of highly correlated chi-squared test statistics with a finite number of degrees of freedom $k$. The testing framework is then used to simultaneously test $k$ linear constraints in a large dimensional linear factor model with some observable and unobservable common factors; the result is a consistent estimator of the $\mathrm{FDP}$ based on the associated factor-adjusted $p$-values. The practical utility of the method is investigated through extensive simulation studies and an analysis of batch effects in a gene expression study.

We investigate a geometric computational framework, called the “scaling-rotation framework”, on $\mathrm{Sym}^{+}(p)$, the set of $p\times p$ symmetric positive-definite (SPD) matrices. The purpose of our study is to lay geometric foundations for statistical analysis of SPD matrices, in situations in which eigenstructure is of fundamental importance, for example diffusion-tensor imaging (DTI). Eigen-decomposition, upon which the scaling-rotation framework is based, determines both a stratification of $\mathrm{Sym}^{+}(p)$, defined by eigenvalue multiplicities, and fibers of the “eigen-composition” map $SO(p)\times\mathrm{Diag}^{+}(p)\to\mathrm{Sym}^{+}(p)$. This leads to the notion of scaling-rotation distance [Jung et al. (2015)], a measure of the minimal amount of scaling and rotation needed to transform an SPD matrix, $X$, into another, $Y$, by a smooth curve in $\mathrm{Sym}^{+}(p)$. Our main goal in this paper is the systematic characterization and analysis of minimal smooth scaling-rotation (MSSR) curves, images in $\mathrm{Sym}^{+}(p)$ of minimal-length geodesics connecting two fibers in the “upstairs” space $SO(p)\times\mathrm{Diag}^{+}(p)$. The length of such a geodesic connecting the fibers over $X$ and $Y$ is what we define to be the scaling-rotation distance from $X$ to $Y.$ For the important low-dimensional case $p=3$ (the home of DTI), we find new explicit formulas for MSSR curves and for the scaling-rotation distance, and identify the set ${\mathcal{M}}(X,Y)$ of MSSR curves from $X$ to $Y$ in all “nontrivial” cases. The quaternionic representation of $SO(3)$ is used in these computations. We also provide closed-form expressions for scaling-rotation distance and MSSR curves for the case $p=2$.

It is well known that, under general regularity conditions, the distribution of the maximum likelihood estimator (MLE) is asymptotically normal. Very recently, bounds of the optimal order $O(1/\sqrt{n})$ on the closeness of the distribution of the MLE to normality in the so-called bounded Wasserstein distance were obtained [2, 1], where $n$ is the sample size. However, the corresponding bounds on the Kolmogorov distance were only of the order $O(1/n^{1/4})$. In this paper, bounds of the optimal order $O(1/\sqrt{n})$ on the closeness of the distribution of the MLE to normality in the Kolmogorov distance are given, as well as their nonuniform counterparts, which work better in tail zones of the distribution of the MLE. These results are based in part on previously obtained general optimal-order bounds on the rate of convergence to normality in the multivariate delta method. The crucial observation is that, under natural conditions, the MLE can be tightly enough bracketed between two smooth enough functions of the sum of independent random vectors, which makes the delta method applicable. It appears that the nonuniform bounds for MLEs in general have no precedents in the existing literature; a special case was recently treated by Pinelis and Molzon [20]. The results can be extended to $M$-estimators.

In this paper we provide some general asymptotic properties of covariate-adaptive (CA) randomized designs aimed at balancing the allocations of two treatments across a set of chosen covariates. In particular, we establish the central limit theorem for a vast class of covariate-adaptive procedures characterized by i) a different allocation function for each covariate profile and ii) sequences of allocation rules instead of a pre-fixed one. This result allows one to derive theoretically the asymptotic expressions of the loss of information induced by imbalance and the selection bias due to the lack of randomness, that are the fundamental properties for estimation of every CA rule, widely used in order to compare different CA procedures. Besides providing the proofs of unsolved conjectures about some CA designs suggested in the literature, explored up to now almost exclusively through simulations, our results provide substantial insight for future suggestions and represent an accurate tool for the large sample comparisons between CA designs. A numerical study is also performed to assess the validity of the suggested approach.

We consider the task of learning the structure of the graph underlying a mutually-exciting multivariate Hawkes process in the high-dimensional setting. We propose a simple and computationally inexpensive edge screening approach. Under a subset of the assumptions required for penalized estimation approaches to recover the graph, this edge screening approach has the sure screening property: with high probability, the screened edge set is a superset of the true edge set. Furthermore, the screened edge set is relatively small. We illustrate the performance of this new edge screening approach in simulation studies.

Large crossed data sets, often modeled by generalized linear mixed models, have become increasingly common and provide challenges for statistical analysis. At very large sizes it becomes desirable to have the computational costs of estimation, inference and prediction (both space and time) grow at most linearly with sample size.

Both traditional maximum likelihood estimation and numerous Markov chain Monte Carlo Bayesian algorithms take superlinear time in order to obtain good parameter estimates in the simple two-factor crossed random effects model. We propose moment based algorithms that, with at most linear cost, estimate variance components, measure the uncertainties of those estimates, and generate shrinkage based predictions for missing observations. When run on simulated normally distributed data, our algorithm performs competitively with maximum likelihood methods.

We introduce a nonparametric prior on the conditional distribution of a (univariate or multivariate) response given a set of predictors. The prior is constructed in the form of a two-stage generative procedure, which in the first stage recursively partitions the predictor space, and then in the second stage generates the conditional distribution by a multi-scale nonparametric density model on each predictor partition block generated in the first stage. This design allows adaptive smoothing on both the predictor space and the response space, and it results in the full posterior conjugacy of the model, allowing exact Bayesian inference to be completed analytically through a forward-backward recursive algorithm without the need of MCMC, and thus enjoying high computational efficiency (scaling linearly with the sample size). We show that this prior enjoys desirable theoretical properties such as full $L_{1}$ support and posterior consistency. We illustrate how to apply the model to a variety of inference problems such as conditional density estimation as well as hypothesis testing and model selection in a manner similar to applying a parametric conjugate prior, while attaining full nonparametricity. Also provided is a comparison to two other state-of-the-art Bayesian nonparametric models for conditional densities in both model fit and computational time. A real data example from flow cytometry containing 455,472 observations is given to illustrate the substantial computational efficiency of our method and its application to multivariate problems.

The divide and conquer strategy, which breaks a massive data set into a series of manageable data blocks, and combines the independent results of data blocks to obtain a final decision, has been recognized as a state-of-the-art method to overcome challenges of massive data analysis. In this paper, we equip the classical local average regression with some divide and conquer strategies to infer the regressive relationship of input-output pairs from a massive data set. When the average mixture, a widely used divide and conquer approach, is adopted, we prove that the optimal learning rate can be achieved under some restrictive conditions on the number of data blocks. We then propose two variants to relax (or remove) these conditions and derive the same optimal learning rates as that for the average mixture local average regression. Our theoretical assertions are verified by a series of experimental studies.

We present a new methodology for simultaneous variable selection and parameter estimation in function-on-scalar regression with an ultra-high dimensional predictor vector. We extend the LASSO to functional data in both the dense functional setting and the sparse functional setting. We provide theoretical guarantees which allow for an exponential number of predictor variables. Simulations are carried out which illustrate the methodology and compare the sparse/functional methods. Using the Framingham Heart Study, we demonstrate how our tools can be used in genome-wide association studies, finding a number of genetic mutations which affect blood pressure and are therefore important for cardiovascular health.

The Morse-Smale complex of a function $f$ decomposes the sample space into cells where $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare multivariate functions. In this paper, we present some statistical results on estimating Morse-Smale complexes. This allows us to derive new results for two existing methods: mode clustering and Morse-Smale regression. We also develop two new methods based on the Morse-Smale complex: a visualization technique for multivariate functions and a two-sample, multivariate hypothesis test.

In this paper, we mainly study the asymptotic properties of least square (LS, for short) estimators in the simple linear errors-in-variables (EV, for short) regression model with negatively orthant dependent (NOD, for short) errors. Under some suitable conditions, the strong consistency, weak consistency and complete consistency of the LS estimators in the EV regression model with NOD errors are obtained, which generalize or improve the corresponding ones for independent random variables and negatively associated random variables in some sense.

Misspecified models often provide useful information about the true data generating distribution. For example, if $y$ is a non–linear function of $x$ the least squares estimator $\widehat{\beta}$ is an estimate of $\beta$, the slope of the best linear approximation to the non–linear function. Motivated by problems in astronomy, we study how to incorporate observation measurement error variances into fitting parameters of misspecified models. Our asymptotic theory focuses on the particular case of linear regression where often weighted least squares procedures are used to account for heteroskedasticity. We find that when the response is a non–linear function of the independent variable, the standard procedure of weighting by the inverse of the observation variances can be counter–productive. In particular, ordinary least squares may have lower asymptotic variance. We construct an adaptive estimator which has lower asymptotic variance than either OLS or standard WLS. We demonstrate our theory in a small simulation and apply these ideas to the problem of estimating the period of a periodic function using a sinusoidal model.

This paper is devoted to implementing model checking for parametric single-index models with missing responses at random. Two dimension reduction adaptive-to-model tests applying to the missing responses situation are proposed. Unlike the existing smoothing tests, our methods can greatly alleviate the curse of dimensionality in the sense that the tests behave like a test with only one covariate. It results in better significance level maintenance and higher power than the classical tests. The finite sample performance is evaluated through several simulation studies and a comparison with other popularly used tests. A real data analysis is conducted for illustration.

In this paper, we are concerned with the independence test for $k$ high-dimensional sub-vectors of a normal vector, with fixed positive integer $k$. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is proposed for this purpose. We then construct the so-called Schott type statistic as our test statistic, which turns out to be a particular linear spectral statistic of the block correlation matrix. Interestingly, the limiting behavior of the Schott type statistic can be figured out with the aid of the Free Probability Theory and the Random Matrix Theory. Specifically, we will bring the so-called real second order freeness for Haar distributed orthogonal matrices, derived in Mingo and Popa (2013)[10], into the framework of this high-dimensional testing problem. Our test does not require the sample size to be larger than the total or any partial sum of the dimensions of the $k$ sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those statistics proposed in Jiang and Yang (2013)[8] and Jiang, Bai and Zheng (2013)[7], is satisfactory. Real data analysis is also used to illustrate our method.

We study partially linear models for asynchronous longitudinal data to incorporate nonlinear time trend effects. Local and global estimating equations are developed for estimating the parametric and nonparametric effects. We show that with a proper choice of the kernel bandwidth parameter, one can obtain consistent and asymptotically normal parameter estimates for the linear effects. Asymptotic properties of the estimated nonlinear effects are established. Extensive simulation studies provide numerical support for the theoretical findings. Data from an HIV study are used to illustrate our methodology.

In this paper, we propose a novel modeling and a new methodology for estimating the location of block boundaries in a random matrix consisting of a block-wise constant matrix corrupted with white noise. Our method consists in rewriting this problem as a variable selection issue. A penalized least-squares criterion with an $\ell_{1}$-type penalty is used for dealing with this problem. Firstly, some theoretical results ensuring the consistency of our block boundaries estimators are provided. Secondly, we explain how to implement our approach in a very efficient way. This implementation is available in the R package blockseg which can be found in the Comprehensive R Archive Network. Thirdly, we provide some numerical experiments to illustrate the statistical and numerical performance of our package, as well as a thorough comparison with existing methods. Fourthly, an empirical procedure is proposed for estimating the number of blocks. Finally, our approach is applied to HiC data which are used in molecular biology for better understanding the influence of the chromosomal conformation on the cells functioning.

An important problem in time series analysis is the discrimination between non-stationarity and long-range dependence. Most of the literature considers the problem of testing specific parametric hypotheses of non-stationarity (such as a change in the mean) against long-range dependent stationary alternatives. In this paper we suggest a simple approach, which can be used to test the null-hypothesis of a general non-stationary short-memory against the alternative of a non-stationary long-memory process. The test procedure works in the spectral domain and uses a sequence of approximating tvFARIMA models to estimate the time varying long-range dependence parameter. We prove uniform consistency of this estimate and asymptotic normality of an averaged version. These results yield a simple test (based on the quantiles of the standard normal distribution), and it is demonstrated in a simulation study that - despite of its semi-parametric nature - the new test outperforms the currently available methods, which are constructed to discriminate between specific parametric hypotheses of non-stationarity short- and stationarity long-range dependence.

When facing multivariate covariates, general semiparametric regression techniques come at hand to propose flexible models that are unexposed to the curse of dimensionality. In this work a semiparametric copula-based estimator for conditional quantiles is investigated for both complete or right-censored data. In spirit, the methodology is extending the recent work of Noh, El Ghouch and Bouezmarni [34] and Noh, El Ghouch and Van Keilegom [35], as the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to an easily implementable estimator expressed, for both contexts with or without censoring, as a weighted quantile of the observed response variable. In addition, and contrary to the initial suggestion in the literature, a semiparametric estimation scheme for the multivariate copula density is studied, motivated by the possible shortcomings of a purely parametric approach and driven by the regression context. The resulting quantile regression estimator has the valuable property of being automatically monotonic across quantile levels. Additionally, the copula-based approach allows the analyst to spontaneously take account of common regression concerns such as interactions between covariates or possible transformations of the latter. From a theoretical prospect, asymptotic normality for both complete and censored data is obtained under classical regularity conditions. Finally, numerical examples as well as a real data application are used to illustrate the validity and finite sample performance of the proposed procedure.

Suppose that we observe $y\in\mathbb{R}^{n}$ and $X\in\mathbb{R}^{n\times m}$ in the following errors-in-variables model: \begin{eqnarray*}y&=&X_{0}\beta^{*}+\epsilon\\X&=&X_{0}+W\end{eqnarray*} where $X_{0}$ is an $n\times m$ design matrix with independent subgaussian row vectors, $\epsilon\in\mathbb{R}^{n}$ is a noise vector and $W$ is a mean zero $n\times m$ random noise matrix with independent subgaussian column vectors, independent of $X_{0}$ and $\epsilon$. This model is significantly different from those analyzed in the literature in the sense that we allow the measurement error for each covariate to be a dependent vector across its $n$ observations. Such error structures appear in the science literature when modeling the trial-to-trial fluctuations in response strength shared across a set of neurons.

Under sparsity and restrictive eigenvalue type of conditions, we show that one is able to recover a sparse vector $\beta^{*}\in\mathbb{R}^{m}$ from the model given a single observation matrix $X$ and the response vector $y$. We establish consistency in estimating $\beta^{*}$ and obtain the rates of convergence in the $\ell_{q}$ norm, where $q=1,2$ for the Lasso-type estimator, and for $q\in [1,2]$ for a Dantzig-type Conic programming estimator. We show error bounds which approach that of the regular Lasso and the Dantzig selector in case the errors in $W$ are tending to 0. We analyze the convergence rates of the gradient descent methods for solving the nonconvex programs and show that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to a neighborhood of the global minimizers: the size of the neighborhood is bounded by the statistical error in the $\ell_{2}$ norm. Our analysis reveals interesting connections between computational and statistical efficiency and the concentration of measure phenomenon in random matrix theory. We provide simulation evidence illuminating the theoretical predictions.

Casella and Hwang, 1983, JASA, introduced a broad class of recentered confidence spheres for the mean $\boldsymbol{\theta}$ of a multivariate normal distribution with covariance matrix $\sigma^{2}\boldsymbol{I}$, for $\sigma^{2}$ known. Both the center and radius functions of these confidence spheres are flexible functions of the data. For the particular case of confidence spheres centered on the positive-part James-Stein estimator and with radius determined by empirical Bayes considerations, they show numerically that these confidence spheres have the desired minimum coverage probability $1-\alpha$ and dominate the usual confidence sphere in terms of scaled volume. We shift the focus from the scaled volume to the scaled expected volume of the recentered confidence sphere. Since both the coverage probability and the scaled expected volume are functions of the Euclidean norm of $\boldsymbol{\theta}$, it is feasible to optimize the performance of the recentered confidence sphere by numerically computing both the center and radius functions so as to optimize some clearly specified criterion. We suppose that we have uncertain prior information that $\boldsymbol{\theta}=\boldsymbol{0}$. This motivates us to determine the center and radius functions of the confidence sphere by numerical minimization of the scaled expected volume of the confidence sphere at $\boldsymbol{\theta}=\boldsymbol{0}$, subject to the constraints that (a) the coverage probability never falls below $1-\alpha$ and (b) the radius never exceeds the radius of the standard $1-\alpha$ confidence sphere. Our results show that, by focusing on this clearly specified criterion, significant gains in performance (in terms of this criterion) can be achieved. We also present analogous results for the much more difficult case that $\sigma^{2}$ is unknown.

Much effort has been made to improve the famous step up procedure of Benjamini and Hochberg given by linear critical values $\frac{i\alpha}{n}$. It is pointed out by Gavrilov, Benjamini and Sarkar that step down multiple testing procedures based on the critical values $\beta_{i}=\frac{i\alpha}{n+1-i(1-\alpha)}$ still control the false discovery rate (FDR) at the upper bound $\alpha$ under basic independence assumptions. Since that result is no longer true for step up procedures and for step down procedures, if the p-values are dependent, a big discussion about the corresponding FDR starts in the literature. The present paper establishes finite sample formulas and bounds for the FDR and the expected number of false rejections for multiple testing procedures using critical values $\beta_{i}$ under martingale and reverse martingale dependence models. It is pointed out that martingale methods are natural tools for the treatment of local FDR estimators which are closely connected to the present coefficients $\beta_{i}$. The martingale approach also yields new results and further inside for the special basic independence model.

In stochastic decision problems, one often wants to estimate the underlying probability measure statistically, and then to use this estimate as a basis for decisions. We shall consider how the uncertainty in this estimation can be explicitly and consistently incorporated in the valuation of decisions, using the theory of nonlinear expectations.

The objective of this work is to propose a new algorithm to fit a sphere on a noisy 3D point cloud distributed around a complete or a truncated sphere. More precisely, we introduce a projected Robbins-Monro algorithm and its averaged version for estimating the center and the radius of the sphere. We give asymptotic results such as the almost sure convergence of these algorithms as well as the asymptotic normality of the averaged algorithm. Furthermore, some non-asymptotic results will be given, such as the rates of convergence in quadratic mean. Some numerical experiments show the efficiency of the proposed algorithm on simulated data for small to moderate sample sizes and for modeling an object in 3D.

We propose methods to estimate the distribution functions for multiple populations from mixture data that are only known to belong to a specific population with certain probabilities. The problem is motivated from kin-cohort studies collecting phenotype data in families for various diseases such as the Huntington’s disease (HD) or breast cancer. Relatives in these studies are not genotyped hence only their probabilities of carrying a known causal mutation (e.g., BRCA1 gene mutation or HD gene mutation) can be derived. In addition, phenotype observations from the same family may be correlated due to shared life style or other genes associated with disease, and the observations are subject to censoring. Our estimator does not assume any parametric form of the distributions, and does not require modeling of the correlation structure. It estimates the distributions through using the optimal base estimators and then optimally combine them. The optimality implies both estimation consistency and minimum estimation variance. Simulations and real data analysis on an HD study are performed to illustrate the improved efficiency of the proposed methods.

The $k$-core decomposition is a widely studied summary statistic that describes a graph’s global connectivity structure. In this paper, we move beyond using $k$-core decomposition as a tool to summarize a graph and propose using $k$-core decomposition as a tool to model random graphs. We propose using the shell distribution vector, a way of summarizing the decomposition, as a sufficient statistic for a family of exponential random graph models. We study the properties and behavior of the model family, implement a Markov chain Monte Carlo algorithm for simulating graphs from the model, implement a direct sampler from the set of graphs with a given shell distribution, and explore the sampling distributions of some of the commonly used complementary statistics as good candidates for heuristic model fitting. These algorithms provide first fundamental steps necessary for solving the following problems: parameter estimation in this ERGM, extending the model to its Bayesian relative, and developing a rigorous methodology for testing goodness of fit of the model and model selection. The methods are applied to a synthetic network as well as the well-known Sampson monks dataset.

We study a stylized multiple testing problem where the test statistics are independent and assumed to have the same distribution under their respective null hypotheses. We first show that, in the normal means model where the test statistics are normal Z-scores, the well-known method of Benjamini and Hochberg [4] is optimal in some asymptotic sense. We then show that this is also the case of a recent distribution-free method proposed by Barber and Candès [14]. The method is distribution-free in the sense that it is agnostic to the null distribution — it only requires that the null distribution be symmetric. We extend these optimality results to other location models with a base distribution having fast-decaying tails.

For a normally distributed $X\sim N(\mu,\sigma^{2})$ and for estimating $\mu$ when restricted to an interval $[-m,m]$ under general loss $F(|d-\mu|)$ with strictly increasing and absolutely continuous $F$, we establish the inadmissibility of the restricted maximum likelihood estimator $\delta_{\hbox{mle}}$ for a large class of $F$’s and provide explicit improvements. In particular, we give conditions on $F$ and $m$ for which the Bayes estimator $\delta_{BU}$ with respect to the boundary uniform prior $\pi(-m)=\pi(m)=1/2$ dominates $\delta_{\hbox{mle}}$. Specific examples include $L^{s}$ loss with $s>1$, as well as reflected normal loss. Connections and implications for predictive density estimation are outlined, and numerical evaluations illustrate the results.

In several applications, ultimately at the largest data, truncation effects can be observed when analysing tail characteristics of statistical distributions. In some cases truncation effects are forecasted through physical models such as the Gutenberg-Richter relation in geophysics, while at other instances the nature of the measurement process itself may cause under recovery of large values, for instance due to flooding in river discharge readings. Recently, Beirlant, Fraga Alves and Gomes (2016) discussed tail fitting for truncated Pareto-type distributions. Using examples from earthquake analysis, hydrology and diamond valuation we demonstrate the need for a unified treatment of extreme value analysis for truncated heavy and light tails. We generalise the classical Peaks over Threshold approach for the different max-domains of attraction with shape parameter $\xi >-1/2$ to allow for truncation effects. We use a pseudo maximum likelihood approach to estimate the model parameters and consider extreme quantile estimation and reconstruction of quantile levels before truncation whenever appropriate. We report on some simulation experiments and provide some basic asymptotic results.

A new class of rank statistics is proposed to assess that the copula of a multivariate population is radially symmetric. The proposed test statistics are weighted $L_{2}$ functional distances between a nonparametric estimator of the characteristic function that one can associate to a copula and its complex conjugate. It will be shown that these statistics behave asymptotically as degenerate V-statistics of order four and that the limit distributions have expressions in terms of weighted sums of independent chi-square random variables. A suitably adapted and asymptotically valid multiplier bootstrap procedure is proposed for the computation of $p$-values. One advantage of the proposed approach is that unlike methods based on the empirical copula, the partial derivatives of the copula need not be estimated. The good properties of the tests in finite samples are shown via simulations. In particular, the superiority of the proposed tests over competing ones based on the empirical copula investigated by [6] in the bivariate case is clearly demonstrated.

We analytically investigate size and power properties of a popular family of procedures for testing linear restrictions on the coefficient vector in a linear regression model with temporally dependent errors. The tests considered are autocorrelation-corrected F-type tests based on prewhitened nonparametric covariance estimators that possibly incorporate a data-dependent bandwidth parameter, e.g., estimators as considered in Andrews and Monahan (1992), Newey and West (1994), or Rho and Shao (2013). For design matrices that are generic in a measure theoretic sense we prove that these tests either suffer from extreme size distortions or from strong power deficiencies. Despite this negative result we demonstrate that a simple adjustment procedure based on artificial regressors can often resolve this problem.

We investigate the power of the CUSUM test and the Wilcoxon change-point tests for a shift in the mean of a process with long-range dependent noise. We derive analytic formulas for the power of these tests under local alternatives. These results enable us to calculate the asymptotic relative efficiency (ARE) of the CUSUM test and the Wilcoxon change point test. We obtain the surprising result that for Gaussian data, the ARE of these two tests equals $1$, in contrast to the case of i.i.d. noise when the ARE is known to be $3/\pi$.

The use of an Ornstein-Uhlenbeck (OU) process is ubiquitous in business, economics and finance to capture various price processes and evolution of economic indicators exhibiting mean-reverting properties. The time at which structural transition representing drastic changes in the economic dynamics occur are of particular interest to policy makers, investors and financial product providers. This paper addresses the change-point problem under a generalised OU model and investigates the associated statistical inference. We propose two estimation methods to locate multiple change points and show the asymptotic properties of the estimators. An informational approach is employed in detecting the change points, and the consistency of our methods is also theoretically demonstrated. Estimation is considered under the setting where both the number and location of change points are unknown. Three computing algorithms are further developed for implementation. The practical applicability of our methods is illustrated using simulated and observed financial market data.

We study the problem of aggregation of estimators with respect to the Kullback-Leibler divergence for various probabilistic models. Rather than considering a convex combination of the initial estimators $f_{1},\ldots,f_{N}$, our aggregation procedures rely on the convex combination of the logarithms of these functions. The first method is designed for probability density estimation as it gives an aggregate estimator that is also a proper density function, whereas the second method concerns spectral density estimation and has no such mass-conserving feature. We select the aggregation weights based on a penalized maximum likelihood criterion. We give sharp oracle inequalities that hold with high probability, with a remainder term that is decomposed into a bias and a variance part. We also show the optimality of the remainder terms by providing the corresponding lower bound results.

This paper introduces a local optimization-based approach to test statistical hypotheses and to construct confidence intervals. This approach can be viewed as an extension of bootstrap, and yields asymptotically valid tests and confidence intervals as long as there exist consistent estimators of unknown parameters. We present simple algorithms including a neighborhood bootstrap method to implement the approach. Several examples in which theoretical analysis is not easy are presented to show the effectiveness of the proposed approach.

We consider the problem of recovery of an unknown multivariate signal $f$ observed in a $d$-dimensional Gaussian white noise model of intensity $\varepsilon $. We assume that $f$ belongs to a class of smooth functions in $L_{2}([0,1]^{d})$ and has an additive sparse structure determined by the parameter $s$, the number of non-zero univariate components contributing to $f$. We are interested in the case when $d=d_{\varepsilon }\to \infty $ as $\varepsilon \to 0$ and the parameter $s$ stays “small” relative to $d$. With these assumptions, the recovery problem in hand becomes that of determining which sparse additive components are non-zero.

Attempting to reconstruct most, but not all, non-zero components of $f$, we arrive at the problem of almost full variable selection in high-dimensional regression. For two different choices of a class of smooth functions, we establish conditions under which almost full variable selection is possible, and provide a procedure that achieves this goal. Our procedure is the best possible (in the asymptotically minimax sense) for selecting most non-zero components of $f$. Moreover, it is adaptive in the parameter $s$. In addition to that, we complement the findings of [17] by obtaining an adaptive exact selector for the class of infinitely-smooth functions. Our theoretical results are illustrated with numerical experiments.

Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case.

We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods.

While popular, single index models and additive models have potential limitations, a fact that leads us to propose SiAM, a novel hybrid combination of these two models. We first address model identifiability under general assumptions. The result is of independent interest. We then develop an estimation procedure by using splines to approximate unknown functions and establish the asymptotic properties of the resulting estimators. Furthermore, we suggest a two-step procedure for establishing confidence bands for the nonparametric additive functions. This procedure enables us to make global inferences. Numerical experiments indicate that SiAM works well with finite sample sizes, and are especially robust to model structures. That is, when the model reduces to either single-index or additive scenario, the estimation and inference results are comparable to those based on the true model, while when the model is misspecified, the superiority of our method can be very great.

The matrix variate elliptical generalization of [30] is presented in this work. The published Gaussian case is revised and modified. Then, new aspects of identifiability and consistent estimation of mean form and mean form difference are considered under elliptical laws. For example, instead of using the Euclidean distance matrix for the consistent estimates, exact formulae are derived for the moments of the matrix $\mathbf{B}=\mathbf{X}^{c}\left(\mathbf{X}^{c}\right)^{T}$; where $\mathbf{X}^{c}$ is the centered landmark matrix. Finally, a complete application in Biology is provided; it includes estimation, model selection and hypothesis testing.

We consider the change-point problem for the marginal distribution of subordinated Gaussian processes that exhibit long-range dependence. The asymptotic distributions of Kolmogorov-Smirnov- and Cramér-von Mises type statistics are investigated under local alternatives. By doing so we are able to compute the asymptotic relative efficiency of the mentioned tests and the CUSUM test. In the special case of a mean-shift in Gaussian data it is always $1$. Moreover, our theory covers the scenario where the Hermite rank of the underlying process changes.

In a small simulation study, we show that the theoretical findings carry over to the finite sample performance of the tests.

This paper studies high-dimensional linear discriminant analysis (LDA). First, we review the $\ell_{1}$ penalized least square LDA proposed in [10], which could circumvent estimation of the annoying high-dimensional covariance matrix. Then detailed theoretical analyses of this sparse LDA are established. To be specific, we prove that the penalized estimator is $\ell_{2}$ consistent in high-dimensional regime and the misclassification error rate of the penalized LDA is asymptotically optimal under a set of reasonably standard regularity conditions. The theoretical results are complementary to the results to [10], together with which we have more understanding of the $\ell_{1}$ penalized least square LDA (or called Lassoed LDA).

High-dimensional prediction typically comprises two steps: variable selection and subsequent least-squares refitting on the selected variables. However, the standard variable selection procedures, such as the lasso, hinge on tuning parameters that need to be calibrated. Cross-validation, the most popular calibration scheme, is computationally costly and lacks finite sample guarantees. In this paper, we introduce an alternative scheme, easy to implement and both computationally and theoretically efficient.

The primal problem of multinomial likelihood maximization restricted to a convex closed subset of the probability simplex is studied. A solution of this problem may assign a positive mass to an outcome with zero count. Such a solution cannot be obtained by the widely used, simplified Lagrange and Fenchel duals. Related flaws in the simplified dual problems, which arise because the recession directions are ignored, are identified and the correct Lagrange and Fenchel duals are developed.

The results permit us to specify linear sets and data such that the empirical likelihood-maximizing distribution exists and is the same as the multinomial likelihood-maximizing distribution. The multinomial likelihood ratio reaches, in general, a different conclusion than the empirical likelihood ratio.

Implications for minimum discrimination information, Lindsay geometry, compositional data analysis, bootstrap with auxiliary information, and Lagrange multiplier test, which explicitly or implicitly ignore information about the support, are discussed.

A solution of the primal problem can be obtained by the PP (perturbed primal) algorithm, that is, as the limit of a sequence of solutions of perturbed primal problems. The PP algorithm may be implemented by the simplified Lagrange or Fenchel dual.

In this paper, the asymptotic and permutation testing procedures are developed in general factorial designs without assuming homoscedasticity or a particular error distribution. The one-way layout, crossed and hierarchically nested designs are contained in our general framework. New test statistics are modifications of Wald-type statistic, where a weight matrix is a certain diagonal matrix. Asymptotic properties of the new solutions are also investigated. In particular, the consistency of the tests under fixed alternatives or asymptotic validity of the permutation procedures are proved in many cases. Simulation studies show that, in the case of small sample sizes, some of the proposed methods perform comparably to or even better in certain situations than the Wald-type permutation test of Pauly et al. (2015). Illustrative real data examples of the use of the tests in practice are also given.

We consider the problems of detection and support recovery of a contiguous block of weak activation in a large matrix, from noisy, possibly adaptively chosen, compressive (linear) measurements. We precisely characterize the tradeoffs between the various problem dimensions, the signal strength and the number of measurements required to reliably detect and recover the support of the signal, both for passive and adaptive measurement schemes. In each case, we complement algorithmic results with information-theoretic lower bounds. Analogous to the situation in the closely related problem of noisy compressed sensing, we show that for detection neither adaptivity, nor structure reduce the minimax signal strength requirement. On the other hand we show the rather surprising result that, contrary to the situation in noisy compressed sensing, the signal strength requirement to recover the support of a contiguous block-structured signal is strongly influenced by both the signal structure and the ability to choose measurements adaptively.

Given a pair of random vectors $(X,Y)$, we consider the problem of approximating $Y$ by $\mathbf{c}(X)=\{\mathbf{c}_{1}(X),\dots ,\mathbf{c}_{M}(X)\}$ where $\mathbf{c}$ is a measurable set-valued function. We give meaning to the approximation by using the principles of vector quantization which leads to the definition of a multifunction regression problem. The formulated problem amounts at quantizing the conditional distributions of $Y$ given $X$. We propose a nonparametric estimate of the solutions of the multifunction regression problem by combining the method of $M$-means clustering with the nonparametric smoothing technique of $k$-nearest neighbors. We provide an asymptotic analysis of the estimate and we derive a convergence rate for the excess risk of the estimate. The proposed methodology is illustrated on simulated examples and on a speed-flow traffic data set emanating from the context of road traffic forecasting.