Moment Assumptions Research Articles

On Rosenbaum’s Rank-based Matching Estimator

Summary In two influential contributions, Rosenbaum (2005, 2020a) advocated for using the distances between component-wise ranks, instead of the original data values, to measure covariate similarity when constructing matching estimators of average treatment effects. While the intuitive benefits of using covariate ranks for matching estimation are apparent, there is no theoretical understanding of such procedures in the literature. We fill this gap by demonstrating that Rosenbaum’s rank-based matching estimator, when coupled with a regression adjustment, enjoys the properties of double robustness and semiparametric efficiency without the need to enforce restrictive covariate moment assumptions. Our theoretical findings further emphasize the statistical virtues of employing ranks for estimation and inference, more broadly aligning with the insights put forth by Peter Bickel in his 2004 Rietz lecture (Bickel, 2004).

Geodesics cross any pattern in first-passage percolation without any moment assumption and with possibly infinite passage times

In first-passage percolation, one places nonnegative i.i.d. random variables (T(e)) on the edges of Zd. A geodesic is an optimal path for the passage times T(e). Consider a local property of the time environment. We call it a pattern. We investigate the number of times a geodesic crosses a translate of this pattern. When we assume that the common distribution of the passage times satisfies a suitable moment assumption, it is shown in [Antonin Jacquet. Geodesics in first-passage percolation cross any pattern, arXiv:2204.02021, 2023] that, apart from an event with exponentially small probability, this number is linear in the distance between the extremities of the geodesic. This paper completes this study by showing that this result remains true when we consider distributions with an unbounded support without any moment assumption or distributions with possibly infinite passage times. The techniques of proof differ from the preceding article and rely on a notion of penalized geodesic.

Central limit theorem in disordered Monomer‐Dimer model

AbstractWe consider the disordered monomer‐dimer model on general finite graphs with bounded degrees. Under the finite fourth moment assumption on the weight distributions, we prove a Gaussian central limit theorem for the free energy of the associated Gibbs measure with a rate of convergence. The central limit theorem continues to hold under a nearly optimal finite ‐moment assumption on the weight distributions if the underlying graphs are further assumed to have a uniformly subexponential volume growth. This generalizes a recent result by Dey and Krishnan who showed a Gaussian central limit theorem in the disordered monomer‐dimer model on cylinder graphs. Our proof relies on the idea that the disordered monomer‐dimer model exhibits a decay of correlation with high probability. We also establish a central limit theorem for the Gibbs average of the number of dimers where the underlying graph has subexponential volume growth and the edge weights are Gaussians.

Central Limit Theorem and Law of Large Numbers Analogues for the Total Progeny in the Q-Processes

In this paper we are dealing with population size growth systems called Q-processes. The class of trajectories of these systems is a subset of the family of all possible trajectories of the ordinary Galton-Watson branching system, provided that they do not decay in the remote future. We observe the total progeny of a single founder-individual, generated by the reproduction law of the Q-process up to time n. By analogy with branching systems models, this variable is of great interest in studying the deep properties of the Q-process. Our main results are analogues of Central Limit Theorem and Law of Large Numbers for Sn, denoting the total progeny of a single founder-individual, generated by the reproduction law of the Q-process up to time n. We find that the total progeny as a random variable approximates the standard normal distribution function under a second moment assumption for the initial Galton-Watson system offspring law. We estimate the speed rate of this approximation. We also prove an analogue of the law of large numbers with an estimate of the approximation rate to the degenerate distribution.

Probabilistic limit theorems via the operator perturbation method, under optimal moment assumptions

Probabilistic limit theorems via the operator perturbation method, under optimal moment assumptions

Stochastic hyperplane-based ranks and their use in multivariate portmanteau tests

The article proposes and justifies an optimal rank-based portmanteau test of multivariate elliptical strict white noise against multivariate serial dependence. It is based on new stochastic hyperplane-based ranks that are simpler and easier to compute than other usable hyperplane-based competitors and still share with them many good properties such as their distribution-free nature, affine invariance, efficiency, robustness and weak moment assumptions. The finite-sample performance of the portmanteau test is illustrated empirically in a small Monte Carlo simulation study.

Exponential bounds for regularized Hotelling’s T2 statistic in high dimension

We obtain exponential inequalities for regularized Hotelling’s Tn2 statistics, that take into account the potential high dimensional aspects of the problem. We explore the finite sample properties of the tail of these statistics by deriving exponential bounds for symmetric distributions and also for general distributions under weak moment assumptions (we never assume exponential moments). For this, we use a penalized estimator of the covariance matrix and propose an optimal choice for the penalty coefficient.

On duration effects in non-life insurance pricing

The paper discusses duration effects on the consistency of mean parameter and dispersion parameter estimators in exponential dispersion families (EDFs) that are the standard models used for non-life insurance pricing. Focus is on the standard generalised linear model assumptions where both the mean and variance, conditional on duration, are linear functions in terms of duration. We derive simple convergence results that highlight consequences when the linear conditional moment assumptions are not satisfied. These results illustrate that: (i) the resulting mean estimators always have a relevant asymptotic interpretation in terms of the duration adjusted actuarially fair premium—a premium that only agrees with the standard actuarial premium using a duration equal to one, given that the expected value is linear in the duration; (ii) deviance based estimators of the dispersion parameter in an EDF should be avoided in favour of Pearson estimators; (iii) unless the linear moment assumptions are satisfied, consistency of dispersion and plug-in variance estimators can not be guaranteed and may result in spurious over-dispersion. The results provide explicit conditions on the underlying data generating process that will lead to spurious over-dispersion that can be used for model checking. This is illustrated based on real insurance data, where it is concluded that the linear moment assumptions are violated, which results in non-negligible spurious over-dispersion.

Matrix recovery from nonconvex regularized least absolute deviations

In this paper, we consider the low-rank matrix recovery problem. We propose the nonconvex regularized least absolute deviations model via ℓ1−αℓ2 (0<α<1) minimization. We establish the theoretical analysis of the proposed model and obtain a stable error estimation. Our result is a nontrivial extension of some previous work. Different from most of the state-of-the-art methods, our method does not need any knowledge of standard deviation or any moment assumption of the noise. Numerical experiments show that our method is effective for many types of noise distributions.

On the entropy and information of Gaussian mixtures

AbstractWe establish several convexity properties for the entropy and Fisher information of mixtures of centred Gaussian distributions. Firstly, we prove that if are independent scalar Gaussian mixtures, then the entropy of is concave in , thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix‐valued function acting on densities of mixtures in . As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions.

Extrapolated Cross-Validation for Randomized Ensembles

Ensemble methods such as bagging and random forests are ubiquitous in various fields, from finance to genomics. Despite their prevalence, the question of the efficient tuning of ensemble parameters has received relatively little attention. This article introduces a cross-validation method, Extrapolated Cross-Validation (ECV), for tuning the ensemble and subsample sizes in randomized ensembles. Our method builds on two primary ingredients: initial estimators for small ensemble sizes using out-of-bag errors and a novel risk extrapolation technique that leverages the structure of prediction risk decomposition. By establishing uniform consistency of our risk extrapolation technique over ensemble and subsample sizes, we show that ECV yields δ-optimal (with respect to the oracle-tuned risk) ensembles for squared prediction risk. Our theory accommodates general predictors, only requires mild moment assumptions, and allows for high-dimensional regimes where the feature dimension grows with the sample size. As a practical case study, we employ ECV to predict surface protein abundances from gene expressions in single-cell multiomics using random forests under a computational constraint on the maximum ensemble size. Compared to sample-split and K-fold cross-validation, ECV achieves higher accuracy by avoiding sample splitting. Meanwhile, its computational cost is considerably lower owing to the use of the risk extrapolation technique. Supplementary materials for this article are available online.

Adaptive robust large volatility matrix estimation based on high-frequency financial data

Several novel statistical methods have been developed to estimate large integrated volatility matrices based on high-frequency financial data. To investigate their asymptotic behaviors, they require a sub-Gaussian or finite high-order moment assumption for observed log-returns, which cannot account for the heavy-tail phenomenon of stock-returns. Recently, a robust estimator was developed to handle heavy-tailed distributions with some bounded fourth-moment assumption. However, we often observe that log-returns have heavier tail distribution than the finite fourth-moment and that the degrees of heaviness of tails are heterogeneous across asset and over time. In this paper, to deal with the heterogeneous heavy-tailed distributions, we develop an adaptive robust integrated volatility estimator that employs pre-averaging and truncation schemes based on jump-diffusion processes. We call this an adaptive robust pre-averaging realized volatility (ARP) estimator. We show that the ARP estimator has a sub-Weibull tail concentration with only finite 2α-th moments for any α>1. In addition, we establish matching upper and lower bounds to show that the ARP estimation procedure is optimal. To estimate large integrated volatility matrices using the approximate factor model, the ARP estimator is further regularized using the principal orthogonal complement thresholding (POET) method. The numerical study is conducted to check the finite sample performance of the ARP estimator.

Sharp Stability for LSI

A fundamental tool in mathematical physics is the logarithmic Sobolev inequality. A quantitative version proven by Carlen with a remainder involving the Fourier–Wiener transform is equivalent to an entropic uncertainty principle more general than the Heisenberg uncertainty principle. In the stability inequality, the remainder is in terms of the entropy, not a metric. Recently, a stability result for H1 was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an Lp norm. Afterward, Brigati, Dolbeault, and Simonov discussed the stability problem involving a stronger norm. A full characterization with a necessary and sufficient condition to have H1 convergence is identified in this paper. Moreover, an explicit H1 bound via a moment assumption is shown. Additionally, the Lp stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.

The log‐moment formula for implied volatility

AbstractWe revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that in the absence of arbitrage, if the underlying stock price at time T admits finite log‐moments for some positive q, the arbitrage‐free growth in the left wing of the implied volatility smile for T is less constrained than Lee's bound. The result is rationalized by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log‐returns, and requires no assumption for the underlying price to admit any negative moment. In this respect, the result can be derived from a model‐independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral–Fukasawa formula expressing variance swaps in terms of the implied volatility.

Dimension-Free Empirical Entropy Estimation

We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption renders the problem feasible. We argue that the moment assumption is natural and, in some sense, {\em minimalistic} -- weaker than finite support or tail decay conditions. Under the moment assumption, we provide the first finite-sample entropy estimates for infinite alphabets, nearly recovering the known minimax rates. Moreover, we demonstrate that our empirical bounds are significantly sharper than the state-of-the-art bounds, for various natural distributions and non-trivial sample regimes. Along the way, we give a dimension-free analogue of the Cover-Thomas result on entropy continuity (with respect to total variation distance) for finite alphabets, which may be of independent interest.

Continuum Percolation in Stochastic Homogenization and the Effective Viscosity Problem

This contribution is concerned with the effective viscosity problem, that is, the homogenization of the steady Stokes system with a random array of rigid particles, for which the main difficulty is the treatment of close particles. Standard approaches in the literature have addressed this issue by making moment assumptions on interparticle distances. Such assumptions, however, prevent clustering of particles, which is not compatible with physically-relevant particle distributions. In this contribution, we take a different perspective and consider moment bounds on the size of clusters of close particles. On the one hand, assuming such bounds, we construct correctors and prove homogenization. On the other hand, based on subcritical percolation techniques, these bounds are shown to hold for various mixing particle distributions with nontrivial clustering. As a by-product of the analysis, we also obtain similar homogenization results for compressible and incompressible linear elasticity with unbounded random stiffness.

Asymptotic behaviour of critical decomposable 2-type Galton–Watson processes with immigration

In this paper the asymptotic behaviour of a critical 2-type Galton–Watson process with immigration is described when its offspring mean matrix is reducible, in other words, when the process is decomposable. It is proved that, under second or fourth order moment assumptions on the offspring and immigration distributions, a sequence of appropriately scaled random step processes formed from a critical decomposable 2-type Galton–Watson process with immigration converges weakly. The limit process can be described using one or two independent squared Bessel processes and possibly the unique stationary distribution of an appropriate single-type subcritical Galton–Watson process with immigration. Our results complete and extend the results of Foster and Ney (1978) for some strongly critical decomposable 2-type Galton–Watson processes with immigration.

Bootstrapping the operator norm in high dimensions: Error estimation for covariance matrices and sketching

Although the operator (spectral) norm is one of the most widely used metrics for covariance estimation, comparatively little is known about the fluctuations of error in this norm. To be specific, let Σˆ denote the sample covariance matrix of n i.i.d. observations in Rp that arise from a population matrix Σ, and let Tn=n‖Σˆ−Σ‖ op. In the setting where the eigenvalues of Σ have a decay profile of the form λj(Σ)≍j−2β, we analyze how well the bootstrap can approximate the distribution of Tn. Our main result shows that up to factors of log(n), the bootstrap can approximate the distribution of Tn with respect to the Kolmogorov metric at the rate of n−β−1∕2 6β+4, which does not depend on the ambient dimension p. In addition, we offer a supporting result of independent interest that establishes a high-probability upper bound for Tn based on flexible moment assumptions. More generally, we discuss the consequences of our work beyond covariance matrices, and show how the bootstrap can be used to estimate the errors of sketching algorithms in randomized numerical linear algebra (RandNLA). An illustration of these ideas is also provided with a climate data example.

Testing for parameter change epochs in GARCH time series

Summary We develop a uniform test for detecting and dating the integrated or mildly explosive behaviour of a strictly stationary generalized autoregressive conditional heteroskedasticity (GARCH) process. Namely, we test the null hypothesis of a globally stable GARCH process with constant parameters against the alternative that there is an ‘abnormal’ period with changed parameter values. During this period, the parameter-value change may lead to an integrated or mildly explosive behaviour of the volatility process. It is assumed that both the magnitude and the timing of the breaks are unknown. We develop a double-supreme test for the existence of breaks, and then provide an algorithm to identify the periods of changes. Our theoretical results hold under mild moment assumptions on the innovations of the GARCH process. Technically, the existing properties for the quasi-maximum likelihood estimation in the GARCH model need to be reinvestigated to hold uniformly over all possible periods of change. The key results involve a uniform weak Bahadur representation for the estimated parameters, which leads to weak convergence of the test statistic to the supreme of a Gaussian process. Simulations in the Appendix show that the test has good size and power for reasonably long time series. We apply the test to the conventional early-warning indicators of both the financial market and a representative of the emerging Fintech market, i.e., the Bitcoin returns.

A subsampling perspective for extending the validity of state-of-the-art bootstraps in the frequency domain

Summary Bootstrapping spectral mean statistics has been a notoriously difficult problem over the past 25 years. Many frequency domain bootstraps are valid only for certain time series structures, e.g., linear processes, or for special types of statistics, i.e., ratio statistics, because such bootstraps fail to capture the limiting variance of spectral statistics in general settings. We address this issue with a different form of resampling, namely, subsampling. While not considered previously, subsampling provides consistent variance estimation under much weaker conditions than any existing bootstrap in the frequency domain. Mixing is not used, as is often standard with subsampling. Rather, subsampling can be generally justified under the same conditions needed for original spectral mean statistics to have distributional limits in the first place. This result has impacts for other bootstrap methods. Subsampling then applies to extending the validity of recent state-of-the-art bootstraps in the frequency domain. We nontrivially link subsampling to such bootstraps, which broadens their range, as moment and block assumptions needed for these are cut by more than half. Essentially, state-of-the-art bootstraps then require no more stringent assumptions than those needed for a target limit distribution to exist, which is unusual in the bootstrap world. We also close a gap in the theory of subsampling for time series with distributional approximations, in addition to variance estimation, for frequency domain statistics.