Moment Assumptions Research Articles

Robust estimation of high-dimensional covariance and precision matrices.

High-dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components. Covariance and precision matrices provide a useful summary of such structure, yet the performance of popular matrix estimators typically hinges upon a sub-Gaussianity assumption. This paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions. The proposed estimators, under a bounded fourth moment assumption, achieve the same minimax convergence rates as do existing methods under a sub-Gaussianity assumption. Consistency of the proposed estimators is also established under the weak assumption of bounded 2 + ε moments for ε ∈ (0, 2). The associated convergence rates depend on ε.

Perfect sampling of GI/GI/c queues

We introduce the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and develop an algorithm to simulate the vacation system backwards in time from stationarity at time zero. The algorithm has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.

Scaling limits of random bipartite planar maps with a prescribed degree sequence

We study the asymptotic behavior of uniform random maps with a prescribed face‐degree sequence, in the bipartite case, as the number of faces tends to infinity. Under mild assumptions, we show that, properly rescaled, such maps converge in distribution toward the Brownian map in the Gromov–Hausdorff sense. This result encompasses a previous one of Le Gall for uniform random q‐angulations where q is an even integer. It applies also to random maps sampled from a Boltzmann distribution, under a second moment assumption only, conditioned to be large in either of the sense of the number of edges, vertices, or faces. The proof relies on the convergence of so‐called “discrete snakes” obtained by adding spatial positions to the nodes of uniform random plane trees with a prescribed child sequence recently studied by Broutin and Marckert. This paper can alternatively be seen as a contribution to the study of the geometry of such trees.

Testing independence in high dimensions with sums of rank correlations

We treat the problem of testing independence between $m$ continuous variables when $m$ can be larger than the available sample size $n$. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank correlations. In the asymptotic regime where both $m$ and $n$ tend to infinity, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits, which are valid with no specific distributional or moment assumptions on the data. Using the framework of U-statistics, our result covers a variety of rank correlations including Kendall’s tau and a dominating term of Spearman’s rank correlation coefficient (rho), but also degenerate U-statistics such as Hoeffding’s $D$, or the $\tau^{*}$ of Bergsma and Dassios [Bernoulli 20 (2014) 1006–1028]. As in the classical theory for U-statistics, the test statistics need to be scaled differently when the rank correlations used to construct them are degenerate U-statistics. The power of the considered tests is explored in rate-optimality theory under a Gaussian equicorrelation alternative as well as in numerical experiments for specific cases of more general alternatives.

Quantitative Uniform Propagation of Chaos for Maxwell Molecules

We prove propagation of chaos at explicit polynomial rates in Wasserstein distance $${\mathcal{W}_{2}}$$ for Kac’s N-particle system associated with the spatially homogeneous Boltzmann equation for Maxwell molecules. Our approach is mainly based on novel probabilistic coupling techniques. Combining them with recent stabilization results for the particle system we obtain, under suitable moments assumptions on the initial distribution, a uniform-in-time estimate of order almost $${N^{-1/3}}$$ for $${\mathcal{W}_{2}^{2}}$$ .

Gini autocovariance function used for time series with heavy‐tail distributions

The use of covariance is limited by the need of the finite second moment. This restriction excludes the use of heavy tailed distributions and data. By developing methods that only require finite first moment assumption of the variables we allow for heavy tailed scenarios. Such methods would be valid in the heavy tailed setting, conceptually meaningful in a population model and effective in practical applications. Gini autocovariance function (Gini ACV) is defined under merely finite first moment assumption. Conceptually, it plays a similar role as the usual Pearson autocovariance function, and thus represents a new fundamental tool in time series modeling. The latest applications of Gini ACV and Gini autocorrelation function (Gini ACF) are reviewed, providing a big picture of the latest research in the area. The formulation of Gini ACV and Gini ACF is quickly reviewed and followed by the applications in linear time series modeling, unit root test and reversibility test in time series and nonlinear autoregressive Pareto process.This article is categorized under Statistical Learning and Exploratory Methods of the Data Sciences > Manifold Learning Statistical and Graphical Methods of Data Analysis > Nonparametric Methods Statistical Models > Time Series Models

Existence and continuity of the flow constant in first passage percolation

We consider the model of i.i.d. first passage percolation on $\mathbb{Z} ^d$, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution $G$ on $ [0,+\infty ]$ (including $+\infty $). Whereas the time constant is associated to the study of $1$-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of $(d-1)$-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that $G(\{+\infty \} ) < p_c(d)$ (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution $G$.

Small Deviation Probabilities for Weighted Sum of Independent Random Variables with a Common Distribution Having a Power Decrease at Zero, under Minimal Moment Assumptions

We represent a certain weight condition such that the exact formula for the asymptotic behavior of small deviation probabilities for weighted sum of independent random variables with a common distribution, which has a power decay at zero, is valid under optimal moment assumptions.

Eigenvalue Fluctuations for Lattice Anderson Hamiltonians: Unbounded Potentials

We consider random Schr\"odinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schr\"odinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials.

Fused LASSO penalized least absolute deviation estimator for high dimensional linear regression

The least absolute shrinkage and selection operator (LASSO) has been playing an important role in variable selection and dimensionality reduction for high dimensional linear regression under the zero-mean or Gaussian assumptions of the noises. However, these assumptions may not hold in practice. In this case, the least absolute deviation is a popular and useful method. In this paper, we focus on the least absolute deviation via Fused LASSO, called Robust Fused LASSO, under the assumption that the unknown vector is sparsity for both the coefficients and its successive differences. Robust Fused LASSO estimator does not need any knowledge of standard deviation of the noises or any moment assumptions of the noises. We show that the Robust Fused LASSO estimator possesses near oracle performance, i.e. with large probability, the \begin{document} $\ell_2$ \end{document} norm of the estimation error is of order \begin{document} $O(\sqrt{k(\log p)/n})$ \end{document} . The result is true for a wide range of noise distributions, even for the Cauchy distribution. In addition, we apply the linearized alternating direction method of multipliers to find the Robust Fused LASSO estimator, which possesses the global convergence. Numerical results are reported to demonstrate the efficiency of our proposed method.

Robust High-Dimensional Volatility Matrix Estimation for High-Frequency Factor Model

High-frequency financial data allow us to estimate large volatility matrices with relatively short time horizon. Many novel statistical methods have been introduced to address large volatility matrix estimation problems from a high-dimensional Ito process with microstructural noise contamination. Their asymptotic theories require sub-Gaussian or some finite high-order moments assumptions for observed log-returns. These assumptions are at odd with the heavy tail phenomenon that is pandemic in financial stock returns and new procedures are needed to mitigate the influence of heavy tails. In this paper, we introduce the Huber loss function with a diverging threshold to develop a robust realized volatility estimation. We show that it has the sub-Gaussian concentration around the volatility with only finite fourth moments of observed log-returns. With the proposed robust estimator as input, we further regularize it by using the principal orthogonal component thresholding (POET) procedure to estimate the large volatility matrix that admits an approximate factor structure. We establish the asymptotic theories for such low-rank plus sparse matrices. The simulation study is conducted to check the finite sample performance of the proposed estimation methods.

About the conditional value at risk of partial sums

In this note, we give normal approximation results for the conditional value at risk (CVaR) of partial sums of random variables satisfying moment assumptions. These results are based on Berry–Esseen-type bounds for transport costs in the central limit theorem and extensions of Cantelli's inequalities to the CVaR.

Testing Mutual Independence in High Dimension via Distance Covariance

SummaryWe introduce an L2-type test for testing mutual independence and banded dependence structure for high dimensional data. The test is constructed on the basis of the pairwise distance covariance and it accounts for the non-linear and non-monotone dependences among the data, which cannot be fully captured by the existing tests based on either Pearson correlation or rank correlation. Our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal. It exhibits excellent finite sample performance in our simulation studies even when the sample size is small albeit the dimension is high and is shown to identify non-linear dependence in empirical data analysis successfully. On the theory side, asymptotic normality of our test statistic is shown under quite mild moment assumptions and with little restriction on the growth rate of the dimension as a function of sample size. As a demonstration of good power properties for our distance-covariance-based test, we further show that an infeasible version of our test statistic has the rate optimality in the class of Gaussian distributions with equal correlation.

Change of measure technique in characterizations of the gamma and Kummer distributions

If X and Y are independent random variables with distributions μ nd ν then U=ψ(X,Y) and V=ϕ(X,Y) are also independent for some transformations ψ and ϕ. Properties of this type are known for many important probability distributions μ and ν. Also related characterization questions have been widely investigated. These are questions of the form: Let X and Y be independent and let U and V be also independent. Are the distributions of X and Y necessarily μ and ν, respectively? Recently two new properties and characterizations of this kind involving the Kummer distribution appeared in the literature. For independent X and Y with gamma and Kummer distributions Koudou and Vallois in [17] observed that U=(1+(X+Y)−1)/(1+X−1) and V=X+Y are also independent, and Hamza and Vallois in [13] observed that U=Y/(1+X) and V=X(1+Y/(1+X)) are independent. In [16] and [17] characterizations related to the first property were proved, while the characterizations in the second setting have been recently given in [29]. These results were not fully satisfactory since in both cases technical assumptions on smoothness properties of densities of X and Y were needed. In [31], the assumption of independence of U and V in the first setting was weakened to constancy of regressions of U and U−1 given V with no density assumptions. However, the additional assumption EX−1<∞ was introduced. In the present paper we provide a complete answer to the characterization question in both settings without any additional technical assumptions regarding smoothness or existence of moments. The approach is, first, via characterizations exploiting some conditions imposed on regressions of U given V, which are weaker than independence, but for which moment assumptions are necessary. Second, using a technique of change of measure we show that the moment assumptions can be avoided.

Asymptotics for $2D$ critical first passage percolation

We consider first passage percolation on $\mathbb{Z}^{2}$ with i.i.d. weights, whose distribution function satisfies $F(0)=p_{c}=1/2$. This is sometimes known as the “critical case” because large clusters of zero-weight edges force passage times to grow at most logarithmically, giving zero time constant. Denote $T(\mathbf{0},\partial B(n))$ as the passage time from the origin to the boundary of the box $[-n,n]\times[-n,n]$. We characterize the limit behavior of $T(\mathbf{0},\partial B(n))$ by conditions on the distribution function $F$. We also give exact conditions under which $T(\mathbf{0},\partial B(n))$ will have uniformly bounded mean or variance. These results answer several questions of Kesten and Zhang from the 1990s and, in particular, disprove a conjecture of Zhang from 1999. In the case when both the mean and the variance go to infinity as $n\to\infty$, we prove a CLT under a minimal moment assumption. The main tool involves a new relation between first passage percolation and invasion percolation: up to a constant factor, the passage time in critical first passage percolation has the same first-order behavior as the passage time of an optimal path constrained to lie in an embedded invasion cluster.

Cramér-type moderate deviations for intermediate trimmed means

ABSTRACTIn this article, we establish Cramér-type moderate deviation results for (intermediate) trimmed means Tn = n− 1∑n − mni = kn + 1Xi: n, where Xi: n's are the order statistics corresponding to the first n observations in a sequence X1, X2, … of independent identically distributed random variables with F. We consider two cases of intermediate and heavy trimming. In the former case, when max (αn, βn) → 0 (αn = kn/n, βn = mn/n) and min (kn, mn) → ∞ as n → ∞, we obtain our results under a natural moment assumption and a mild condition on the rate at which αn and βn tend to zero. In the latter case, we do not impose any moment conditions on F, instead, we require some smoothness of F− 1 in an open set containing the limit points of the trimming sequences αn, 1 − βn.

Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process based on continuous time observations

We consider a jump-type Cox–Ingersoll–Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak consistency and asymptotic normality, and, under an additional moment assumption, strong consistency as well. In the supercritical case, we prove strong consistency and mixed normal (but non-normal) asymptotic behavior, while in the critical case, weak consistency and non-standard asymptotic behavior are described. We specialize our results to so-called basic affine jump–diffusions as well. Concerning the asymptotic behavior of the MLE in the supercritical case, we derive a stochastic representation of the limiting mixed normal distribution, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case, where a diffusion-type critical CIR process plays a role.

Studentized U-quantile processes under dependence with applications to change-point analysis

Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantiles, which do not rely on bootstrapping procedures. We demonstrate this approach in detail at the example of the Hodges-Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good robustness and efficiency properties of the test. Two real-life data sets are analyzed.

On Asymptotic Theory for ARCH (∞) Models

Autoregressive conditional heteroskedasticity (ARCH)() models nest a wide range of ARCH and generalized ARCH models including models with long memory in volatility. Existing work assumes the existence of second moments. However, the fractionally integrated generalized ARCH model, one version of a long memory in volatility model, does not have finite second moments and rarely satisfies the moment conditions of the existing literature. This article weakens the moment assumptions of a general ARCH( ) class of models and develops the theory for consistency and asymptotic normality of the quasi‐maximum likelihood estimator.

A note on general quadratic forms of nonstationary stochastic processes

ABSTRACTIn this paper, general quadratic forms of nonstationary, α-mixing time series are considered. Under mixing and moment assumptions, asymptotically normality of these forms are derived. These results do not assume that the variance of the generalized quadratic form has a limit, thus allowing for general types of nonstationarity. However, without well-defined limits, it is not possible to understand the differences in sampling properties of quadratic forms of nonstationary and stationary processes. To understand these differences, the nonstationary process is placed within the locally stationary framework. Under the assumption that the nonstationary process is locally stationary the asymptotic expectation and variance of the weighted sample covariance of the discrete Fourier transforms (an important class of quadratic forms) is derived and shown to be very different to its stationary counterpart.