Multivariate Central Limit Theorem Research Articles

Central limit theorems and the geometry of polynomials

Let X \in \{0,\ldots,n \} be a random variable with mean \mu , standard deviation \sigma , and let f_{X}(z) = \sum_{k}\mathbb{P}(X = k) z^{k} be its probability generating function. Pemantle conjectured that if \sigma is large and f_{X} has no roots close to 1\in \mathbb{C} , then X must be approximately normal. We completely resolve this conjecture in the following strong quantitative form, obtaining sharp bounds. If \delta = \min_{\zeta}|\zeta-1| over the complex roots \zeta of f_{X} , and X^{\ast}:= (X-\mu)/\sigma , then \sup_{t \in \mathbb{R}}|\mathbb{P}(X^{\ast}\leq t) - \mathbb{P}( Z \leq t) | = O(({\log n})/({\delta\sigma})) , where Z \sim \mathcal{N}(0,1) is a standard normal variable. This gives the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. We also show that if f_{X} has no roots with small argument , then X must be approximately normal, again in a sharp quantitative form: if we set \delta = \min_{\zeta}|{\arg(\zeta)}| , then \sup_{t \in \mathbb{R}}|\mathbb{P}(X^{\ast}\leq t) - \mathbb{P}( Z \leq t)| = O({1}/({\delta\sigma})) . Using this result, we answer a question of Ghosh, Liggett and Pemantle by proving a sharp multivariate central limit theorem for random variables with real-stable probability generating functions.

Randomized limit theorems for stationary ergodic random processes and fields

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Zm and Rm,m≥1.

Improved rates of convergence for the multivariate Central Limit Theorem in Wasserstein distance

Improved rates of convergence for the multivariate Central Limit Theorem in Wasserstein distance

Multivariate central limit theorems for random clique complexes

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: the vector of critical simplex counts attained by a lexicographical Morse matching,the vector of simplex counts in the link of a fixed simplex, andthe vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein’s method.

On the perimeter estimation of pixelated excursion sets of two‐dimensional anisotropic random fields

AbstractWe are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets. To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover , the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite‐sample data setting.

Hypothesis testing procedure for binary and multi-class F1 -scores in the paired design.

In modern medicine, medical tests are used for various purposes including diagnosis, disease screening, prognosis, and risk prediction. To quantify the performance of the binary medical test, we often use sensitivity, specificity, and negative and positive predictive values as measures. Additionally, the -score, which is defined as the harmonic mean of precision (positive predictive value) and recall (sensitivity), has come to be used in the medical field due to its favorable characteristics. The -score has been extended for multi-class classification, and two types of -scores have been proposed for multi-class classification: a micro-averaged -score and a macro-averaged -score. The micro-averaged -score pools per-sample classifications across classes and then calculates the overall -score, whereas the macro-averaged -score computes an arithmetic mean of the -scores for each class. Additionally, Sokolova and Lapalme gave an alternative definition of the macro-averaged -score as the harmonic mean of the arithmetic means of the precision and recall over classes. Although some statistical methods of inference for binary and multi-class -scores have been proposed, the methodology development of hypothesis testing procedure for them has not been fully progressing yet. Therefore, we aim to develop hypothesis testing procedure for comparing two -scores in paired study design based on the large sample multivariate central limit theorem.

A nonparametric approach for setting safety stock levels

In practice, lead time demand (LTD) can be skewed, multi‐modal or highly variable, and these factors compromise the validity of typical approaches used for setting safety stock levels. Motivated by encountering this problem at our industry partner, we develop an approach for setting safety stock levels using the bootstrap, a widely used statistical procedure. Existing bootstrap approaches for inventory management either operate directly on observed LTD or assume deterministic lead times, permitting direct application of the bootstrap approach for univariate quantile estimation. As LTD is a convolution of multiple random demands over a random lead time, a multivariate bootstrap approach is required. As we demonstrate, when lead times are stochastic, our multivariate approach provides improved safety stock estimates. We develop a multivariate central limit theorem for the bootstrap mean and bootstrap quantile—components of the safety stock calculation—highlighting why the generalization of these bootstrap methods is critical for inventory management. These results provide a theoretical underpinning for the bootstrap estimator of safety stock and permit the construction of confidence intervals for safety stock estimates, allowing decision makers to understand the reliability with which the desired service level will be achieved. Building on our theoretical results, and supported by numerical experiments, we provide insights on the behavior of the bootstrap for various LTD distributions, which our results demonstrate are critical when employing the bootstrap method. Implementation of our approach with our industry partner resulted in an inventory investment reduction of $1.17 million combined with an overall increase in service level. Our approach is general and can be implemented without modification in other settings.

Randomized multivariate Central Limit Theorems for ergodic homogeneous random fields

We present new versions of the CLT which are valid for each ergodic homogeneous multivariate random field (X1(⋅),…,Xd(⋅)) on Rm or Zm(m≥1) (in particular, for each ergodic stationary random process and for each ergodic stationary random sequence) such that for all l E[|Xl(0)|2+δ]<∞ for some δ>0(l=1,...,d); in some statements ergodicity is not assumed. Randomization made it possible to significantly weaken the strong mixing conditions and other restrictions of dependence, that are imposed in the conventional CLTs. These results pave the way to consistent statistical inference for homogeneous random fields and stationary processes with strong dependence.

Confidence sets for dynamic poverty indexes

ABSTRACT In this study, we consider different poverty indexes in a dynamic framework where individuals change their rate of income randomly in time. The primary objective of this paper is to assess the accuracy of the approximation of the indexes that can be obtained by applying the strong law of large numbers to an economic system composed of an infinite number of agents. The main result is a multivariate central limit theorem for dynamic poverty measures, which is obtained applying the theory of U-statistics. We also show how to get the confidence sets for the considered dynamic indexes, which show the appropriateness of the model. An application to the Italian income data from 1998 to 2012 confirms the effectiveness of the considered approach and the possibility to determine the evolution of poverty and inequality in real economies.

Confidence interval for micro-averaged F1 and macro-averaged F1 scores

A binary classification problem is common in medical field, and we often use sensitivity, specificity, accuracy, negative and positive predictive values as measures of performance of a binary predictor. In computer science, a classifier is usually evaluated with precision (positive predictive value) and recall (sensitivity). As a single summary measure of a classifier’s performance, F1 score, defined as the harmonic mean of precision and recall, is widely used in the context of information retrieval and information extraction evaluation since it possesses favorable characteristics, especially when the prevalence is low. Some statistical methods for inference have been developed for the F1 score in binary classification problems; however, they have not been extended to the problem of multi-class classification. There are three types of F1 scores, and statistical properties of these F1 scores have hardly ever been discussed. We propose methods based on the large sample multivariate central limit theorem for estimating F1 scores with confidence intervals.

Multidimensional parameter estimation of heavy‐tailed moving averages

AbstractIn this article we present a parametric estimation method for certain multiparameter heavy‐tailed Lévy‐driven moving averages. The theory relies on recent multivariate central limit theorems obtained via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to previous papers, which propose to use the marginal empirical characteristic function to estimate the one‐dimensional parameter of the kernel function and the stability index of the driving Lévy motion. We extend their work to allow for a multiparametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein–Uhlenbeck process, certain CARMA(2, 1) models, and Ornstein–Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Poisson processes in the space of (d-1)-dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature -1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all dge 2, it is shown that in case (ii) the central limit theorem holds for din {2,3} and fails if dge 4 and k=d-1 or if dge 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

QANOVA: quantile-based permutation methods for general factorial designs

Population means and standard deviations are the most common estimands to quantify effects in factorial layouts. In fact, most statistical procedures in such designs are built toward inferring means or contrasts thereof. For more robust analyses, we consider the population median, the interquartile range (IQR) and more general quantile combinations as estimands in which we formulate null hypotheses and calculate compatible confidence regions. Based upon simultaneous multivariate central limit theorems and corresponding resampling results, we derive asymptotically correct procedures in general, potentially heteroscedastic, factorial designs with univariate endpoints. Special cases cover robust tests for the population median or the IQR in arbitrary crossed one-, two- and higher-way layouts with potentially heteroscedastic error distributions. In extensive simulations, we analyze their small sample properties and also conduct an illustrating data analysis comparing children’s height and weight from different countries.

A quantum-mechanical derivation of the multivariate central limit theorem for Markov chains

Abstract The central limit theorem for Markov chains is widely used, in particular in its pristine univariate form. As far as the multivariate case is concerned, a few proofs exist, which depend on different assumptions and require advanced mathematical and statistical tools. Here a novel proof is presented that, starting from the standard condition of regularity only, relies on time-independent quantum-mechanical perturbation theory. The result, which is obtained by using techniques that are typical of physics, is expected to enhance the usability of this cornerstone theorem, especially in nonlinear dynamics and physics of complex systems.

Assessing and Visualizing Simultaneous Simulation Error

Monte Carlo experiments produce samples to estimate features such as means and quantiles of a given distribution. However, simultaneous estimation of means and quantiles has received little attention. In this setting, we establish a multivariate central limit theorem for any finite combination of sample means and quantiles under the assumption of a strongly mixing process, which includes the standard Monte Carlo and Markov chain Monte Carlo settings. We build on this to provide a fast algorithm for constructing hyperrectangular confidence regions having the desired simultaneous coverage probability and a convenient marginal interpretation. The methods are incorporated into standard ways of visualizing the results of Monte Carlo experiments enabling the practitioner to more easily assess the reliability of the results. We demonstrate the utility of this approach in various Monte Carlo settings including simulation studies based on independent and identically distributed samples and Bayesian analyses using Markov chain Monte Carlo sampling. Supplementary materials for this article are available online.

Fluctuations of the spectrum in rotationally invariant random matrix ensembles

We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert–Schmidt inner product) within a real-linear subspace of the space of [Formula: see text] matrices. The matrices, we consider may be real or complex, and Hermitian, antihermitian, or general. We use Stein’s method to prove multivariate central limit theorems, with convergence rates, for these traces of powers, which imply central limit theorems for polynomial linear eigenvalue statistics. In contrast to the usual situation in random matrix theory, in our approach general, nonnormal matrices turn out to be easier to study than Hermitian matrices.

Stein’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem

We use Stein’s method to bound the Wasserstein distance of order 2 between a measure $$\nu $$ and the Gaussian measure using a stochastic process $$(X_t)_{t \ge 0}$$ such that $$X_t$$ is drawn from $$\nu $$ for any $$t > 0$$ . If the stochastic process $$(X_t)_{t \ge 0}$$ satisfies an additional exchangeability assumption, we show it can also be used to obtain bounds on Wasserstein distances of any order $$p \ge 1$$ . Using our results, we provide convergence rates for the multi-dimensional central limit theorem in terms of Wasserstein distances of any order $$p \ge 2$$ under simple moment assumptions.

Multivariate central limit theorems for random simplicial complexes

Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most δ. Summing powers of the volume of all k-dimensional faces defines the volume-power functionals of these random simplicial complexes. The asymptotic behavior of the volume-power functionals of the Vietoris-Rips complex is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. Univariate and multivariate central limit theorems are proven. Analogous results for the Čech complex are given.

A Non-Parametric Approach for Setting Safety Stock Levels

In practice, lead time demand (LTD) can be non-standard: skewed, multi-modal or highly variable; factors that compromise the validity of the classic approaches for setting safety stock levels. Motivated by encountering this problem at our industry partner, we develop an approach for setting safety stock levels using the bootstrap, a widely-used statistical procedure. We extend prior research that has used the bootstrap for quantile estimation to address the multi-parameter estimation of safety stocks. We develop a multivariate central limit theorem for the bootstrap mean and bootstrap quantile -- components of the safety stock calculation -- highlighting why the generalization of these bootstrap methods is critical for inventory management. These results provide a theoretical underpinning for the bootstrap estimator of safety stock and permit the construction of confidence intervals for safety stock estimates, allowing decision makers to understand the reliability with which the desired service level will be achieved. Building on our theoretical results, and supported by numerical experiments, we provide insights on the behavior of the bootstrap for various LTD distributions, which our results demonstrate are critical when employing the bootstrap method. Implementation results with our industry partner indicate our approach is quite effective in setting safety stock levels.

A multivariate Berry–Esseen theorem with explicit constants

We provide a Lyapunov type bound in the multivariate central limit theorem for sums of independent, but not necessarily identically distributed random vectors. The error in the normal approximation is estimated for certain classes of sets, which include the class of measurable convex sets. The error bound is stated with explicit constants. The result is proved by means of Stein's method. In addition, we improve the constant in the bound of the Gaussian perimeter of convex sets.