Le Cam Research Articles

Strain-induced crystallisation of reinforced elastomers using surface calorimetry

The crystallinity of rubbers which are amenable to crystallisation upon stretching is classically measured using X-ray diffraction (XRD) techniques. Recently, an alternative method for evaluating the crystallinity in unfilled natural rubber (NR) during stretching was proposed. This method is based on surface calorimetry principles (Le Cam, 2018) and was validated by XRD measurements in Le Cam et al. (2018). For reinforced rubber composites, the calorimetric response is not only due to the elastic couplings and the crystallisation/melting process, but also due to the intrinsic dissipation and additional viscoelastic dissipation resulting from the presence of fillers. This makes estimating the crystallinity of carbon black (CB) reinforced rubber compounds from calorimetric techniques typically more challenging. In this paper, however, the calorimetric approach for measuring crystallinity for CB reinforced rubber compounds was investigated. Crystallinity measurements were performed using both an unfilled and three CB reinforced NR compounds. A comparative study was performed using the XRD technique on the same four compounds. The results confirmed the appropriateness of the calorimetric technique for determining the crystallinity in the unfilled NR. The combination of the XRD and the calorimetric techniques allowed the different heat sources under strain to be identified and therefore, the thermal energy associated with crystallisation to be determined. This was done initially using one of the CB reinforced NR compounds and validated using the two other CB reinforced NR compounds. The results show reasonable correlation between the crystallinity measured using the novel calorimetric technique and the classical XRD technique for CB reinforced NR compounds. The development of this methodology has potentially significant impact. The calorimetric approach does not require the measurement of the mechanical response during testing. This is useful for measuring crystallinity gradients from infrared thermography coupled with full kinematic field measurements, typically at crack tips, where estimating the local mechanical response is very difficult or inaccurate.

Consistent covariances estimation for stratum imbalances under minimization method for covariate‐adaptive randomization

AbstractPocock and Simon's minimization method is a popular approach for covariate‐adaptive randomization in clinical trials. Valid statistical inference with data collected under the minimization method requires the knowledge of the limiting covariance matrix of within‐stratum imbalances, whose existence is only recently established. In this work, we propose a bootstrap‐based estimator for this limit and establish its consistency, in particular, by Le Cam's third lemma. As an application, we consider in simulation studies adjustments to existing robust tests for treatment effects with survival data by the proposed estimator. It shows that the adjusted tests achieve a size close to the nominal level, and unlike other designs, the robust tests without adjustment may have an asymptotic size inflation issue under the minimization method.

Asymptotic comparison of negative multinomial and multivariate normal experiments

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean–covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.

On testing the equality of latent roots of scatter matrices under ellipticity

In the present paper, we tackle the problem of testing H0q:λq>λq+1=⋯=λp, where λ1,…,λp are the scatter matrix eigenvalues of an elliptical distribution on Rp. This is a classical problem in multivariate analysis which is very useful in dimension reduction. We analyse the problem using the Le Cam asymptotic theory of experiments and show that contrary to the testing problems on eigenvalues and eigenvectors of a scatter matrix tackled in Hallin et al. (2010), the non-specification of nuisance parameters has an asymptotic cost for testing H0q. We moreover derive signed-rank tests for the problem that enjoy the property of being asymptotically distribution-free under ellipticity. The van der Waerden rank test uniformly dominates the classical pseudo-Gaussian procedure for the problem. Numerical illustrations show the nice finite-sample properties of our tests.

Coded Estimation: Design of Backscatter Array Codes for 3D Orientation Estimation

We consider the problem of estimating the orientation of a 3D object with the assistance of configurable backscatter tags. We explore the idea of designing tag response codes to improve the accuracy of orientation estimation. To minimize the difference between the true and estimated orientation, we propose two code design criteria. We also derive a lower bound on the worst-case error using Le Cam’s method and provide simulation results for multiple scenarios including line-of-sight only and multipath, comparing the theoretical bounds to those achieved by the designs.

Le Cam–Stratonovich–Boole theory for Itô diffusions

Abstract We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {\frac{T}{n^{6/7}}\to 0} as T → ∞ {T\to\infty} and n → ∞ {n\to\infty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.

A refined continuity correction for the negative binomial distribution and asymptotics of the median

In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a $\mathrm{Negative\hspace{0.5mm}Binomial}\hspace{0.2mm}(r,p)$ random variable jittered by a $\mathrm{Uniform}\hspace{0.2mm}(0,1)$, which answers a problem left open in Coeurjolly & Tr\'epanier (2020). This is used to construct a simple, robust and consistent estimator of the parameter $p$, when $r > 0$ is known. The case where $r$ is unknown is also briefly covered. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments.

Rank-based testing for semiparametric VAR models: A measure transportation approach

We develop a class of tests for semiparametric vector autoregressive (VAR) models with unspecified innovation densities, based on the recent measure-transportation-based concepts of multivariate {\it center-outward ranks} and {\it signs}. We show that these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield novel testing procedures, which (a)~are valid under a broad class of innovation densities (possibly non-elliptical, skewed, and/or with infinite moments), (b)~are optimal (locally asymptotically maximin or most stringent) at selected ones, and (c) are robust against additive outliers. In order to do so, we establish a H\' ajek asymptotic representation result, of independent interest, for a general class of center-outward rank-based serial statistics. As an illustration, we consider the problems of testing the absence of serial correlation in multiple-output and possibly non-linear regression (an extension of the classical Durbin-Watson problem) and the sequential identification of the order $p$ of a vector autoregressive (VAR($p$)) model. A Monte Carlo comparative study of our tests and their routinely-applied Gaussian competitors demonstrates the benefits (in terms of size, power, and robustness) of our methodology; these benefits are particularly significant in the presence of asymmetric and leptokurtic innovation densities. A real data application concludes the paper.

Center-outward Rank- and Sign-based VARMA Portmanteau Tests: Chitturi, Hosking, and Li–McLeod revisited

The pseudo-Gaussian portmanteau tests of Chitturi, Hosking, and Li and McLeod for VARMA models are revisited from a Le Cam perspective, providing a precise and more rigorous description of the asymptotic behavior of the multivariate portmanteau test statistic, which depends on the dimension d of the observations, the number m of lags involved, and the length n of the observation period. Then, based on the concepts of center-outward ranks and signs recently developed (Hallin, del Barrio, Cuesta-Albertos, and Matrán, Annals of Statistics 49, 1139–1165, 2021), a class of multivariate rank- and sign-based portmanteau test statistics is proposed which, under the null hypothesis and under a broad family of innovation densities, can be approximated by an asymptotically chi-square variable. The asymptotic properties of these tests are derived; simulations demonstrate their advantages over their classical pseudo-Gaussian counterpart.

Fast calibration of weak Farima models

In this paper, we investigate the asymptotic properties of Le Cam’s one-step estimator for weak Fractionally AutoRegressive Integrated Moving-Average (FARIMA) models. For these models, noises are uncorrelated but neither necessarily independent nor martingale differences errors. We show under some regularity assumptions that the one-step estimator is strongly consistent and asymptotically normal with the same asymptotic variance as the least squares estimator. We show through simulations that the proposed estimator reduces computational time compared with the least squares estimator. An application for providing remotely computed indicators for time series is proposed.

Minimax optimal estimation of high-dimensional sparse covariance matrices with missing data

This paper considers the minimax estimation of the high-dimensional sparse covariance matrices in the presence of missing observations. Based on random missing data, the upper bounds of convergence rate about the data-driven thresholding estimator are constructed over a large class of sparse covariance matrices under the [Formula: see text] norm and the Frobenius norm. In addition, we use Le Cam’s lemma and the relation between the total variation affinity and the Kullback–Leibler divergence to establish the lower bounds which illustrate the desired upper bounds cannot be improved. It is worth mentioning that the approach we adopt to get the lower bounds is simpler than the existing ones.

DeepMoM: Robust Deep Learning With Median-of-Means

Data used in deep learning is notoriously problematic. For example, data are usually combined from diverse sources, rarely cleaned and vetted thoroughly, and sometimes corrupted on purpose. Intentional corruption that targets the weak spots of algorithms has been studied extensively under the label of “adversarial attacks.” In contrast, the arguably much more common case of corruption that reflects the limited quality of data has been studied much less. Such “random” corruptions are due to measurement errors, unreliable sources, convenience sampling, and so forth. These kinds of corruption are common in deep learning, because data are rarely collected according to strict protocols—in strong contrast to the formalized data collection in some parts of classical statistics. This article concerns such corruption. We introduce an approach motivated by very recent insights into median-of-means and Le Cam’s principle, we show that the approach can be readily implemented, and we demonstrate that it performs very well in practice. In conclusion, we believe that our approach is a very promising alternative to standard parameter training based on least-squares and cross-entropy loss.

Oracle lower bounds for stochastic gradient sampling algorithms

We consider the problem of sampling from a strongly log-concave density in Rd, and prove an information theoretic lower bound on the number of stochastic gradient queries of the log density needed. Several popular sampling algorithms (including many Markov chain Monte Carlo methods) operate by using stochastic gradients of the log density to generate a sample; our results establish an information theoretic limit for all these algorithms. We show that for every algorithm, there exists a well-conditioned strongly log-concave target density for which the distribution of points generated by the algorithm would be at least ε away from the target in total variation distance if the number of gradient queries is less than Ω(σ2d/ε2), where σ2d is the variance of the stochastic gradient. Our lower bound follows by combining the ideas of Le Cam deficiency routinely used in the comparison of statistical experiments along with standard information theoretic tools used in lower bounding Bayes risk functions. To the best of our knowledge our results provide the first nontrivial dimension-dependent lower bound for this problem.

On the Le Cam Distance Between Multivariate Hypergeometric and Multivariate Normal Experiments

In this short note, we develop a local approximation for the log-ratio of the multivariate hypergeometric probability mass function over the corresponding multinomial probability mass function. In conjunction with the bounds from Carter (2002) and Ouimet (2021) on the total variation between the law of a multinomial vector jittered by a uniform on $(-1/2,1/2)^d$ and the law of the corresponding multivariate normal distribution, the local expansion for the log-ratio is then used to obtain a total variation bound between the law of a multivariate hypergeometric random vector jittered by a uniform on $(-1/2,1/2)^d$ and the law of the corresponding multivariate normal distribution. As a corollary, we find an upper bound on the Le Cam distance between multivariate hypergeometric and multivariate normal experiments.

Neyman's C(α) test for the shape parameter of the exponential power class

Tests based on Neyman's C(α) approach are proposed for the shape parameter of the exponential power distribution (EPD). Earlier work on the maximum likelihood estimators for the EPD shows that asymptotic normality is hard to establish under the classic regularity conditions due to a discontinuity in its density functions and non-dominated third derivatives. Nevertheless, local asymptotic optimality of the test can be established via Le Cam's differentiability in quadratic mean, with local alternatives of order . We mainly focus on setting the null of the shape parameter to 1 and 2, corresponding, respectively, to the Laplace and normal distributions. Further, we extend the results to the bivariate EPD. A Monte Carlo study illustrates the theoretical results. Real data applications are also given.

Optimal tests for elliptical symmetry: Specified and unspecified location

Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most papers in the literature indeed are dealing with the null hypothesis of elliptical symmetry with specified location and actually address location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam’s asymptotic theory of statistical experiments, and optimality is to be understood locally and asymptotically within the family of generalized skew-elliptical distributions. The tests we are proposing are enjoying all the desirable properties of a “good” test of elliptical symmetry: they have simple asymptotic distributions under the entire null hypothesis of elliptical symmetry with unspecified radial density and shape parameter; they are affine-invariant, computationally fast, intuitively understandable, and not too demanding in terms of moments. While achieving optimality against generalized skew-elliptical alternatives, they remain quite powerful under a much broader class of non-elliptical distributions and significantly outperform the available competitors.

Probabilistic properties and parametric inference of small variance nonlinear self-stabilizing stochastic differential equations

We consider a process (Xt) solution of a one-dimensional nonlinear self-stabilizing stochastic differential equation, with classical drift term V(α,x) depending on an unknown parameter α, self-stabilizing term Φ(β,x) depending on another unknown parameter β and small noise amplitude ɛ. Self-Stabilization is the effect of including a mean-field interaction in addition to the state-dependent drift. Adding this term leads to a nonlinear or McKean-Vlasov Markov process with transitions depending on the distribution of Xt. We study the probabilistic properties of (Xt) as ɛ tends to 0 and exhibit a Gaussian approximating process for (Xt). Next, we study the estimation of (α,β) from a continuous observation of (Xt,t∈[0,T]). We build explicit estimators using an approximate log-likelihood function obtained from the exact log-likelihood function of a proxi-model. We prove that, for fixed T, as ɛ tends to 0, α can be consistently estimated with rate ɛ−1but notβ. Then, considering ni.i.d. sample paths (Xti,i=1,…,n), we build consistent and asymptotically Gaussian estimators of (α,β) with rates nɛ−1 for α and n for β. Finally, we prove that the statistical experiments generated by (Xt) and the proxi-model are asymptotically equivalent in the sense of the Le Cam Δ-distance both for the continuous observation of one path and for ni.i.d. paths under the condition nɛ→0, which justifies our statistical method.

A Revisit to Le Cam’s First Lemma

Le Cam’s first lemma is of fundamental importance to modern theory of statistical inference: it is a key result in the foundation of the Convolution Theorem, which implies a very general form of the optimality of the maximum likelihood estimate and any statistic that is asymptotically equivalent to it. This lemma is also important for developing asymptotically efficient tests. In this note we give a relatively simple but detailed proof of Le Cam’s first lemma. Our proof allows us to grasp the central idea by making analogies between contiguity and absolute continuity, and is particularly attractive when teaching this lemma in a classroom setting.

A precise local limit theorem for the multinomial distribution and some applications

In Siotani and Fujikoshi (1984), a precise local limit theorem for the multinomial distribution is derived by inverting the Fourier transform, where the error terms are explicit up to order N−1. In this paper, we give an alternative (conceptually simpler) proof based on Stirling’s formula and a careful handling of Taylor expansions, and we show how the result can be used to approximate multinomial probabilities on most subsets of Rd. Furthermore, we discuss a recent application of the result to obtain asymptotic properties of Bernstein estimators on the simplex, we improve the main result in Carter (2002) on the Le Cam distance bound between multinomial and multivariate normal experiments while simultaneously simplifying the proof, and we mention another potential application related to finely tuned continuity corrections.

A General Theory of Comparison of Quantum Channels (and Beyond)

We present a general theory of comparison of quantum channels, concerning with the question of simulability or approximate simulability of a given quantum channel by allowed transformations of another given channel. We introduce a modification of conditional min-entropies, with respect to the set F of allowed transformations, and show that under some conditions on F, these quantities characterize approximate simulability. If F is the set of free superchannels in a quantum resource theory of processes, the modified conditional min-entropies form a complete set of resource monotones. If the transformations in F consist of a preprocessing and a postprocessing of specified forms, approximate simulability is also characterized in terms of success probabilities in certain guessing games, where a preprocessing of a given form can be chosen and the measurements are restricted. These results are applied to several specific cases of simulability of quantum channels, including postprocessings, preprocessings and processing of bipartite channels by LOCC superchannels and by partial superchannels, as well as simulability of sets of quantum measurements. These questions are first studied in a general setting that is an extension of the framework of general probabilistic theories (GPT), suitable for dealing with channels. Here we prove a general theorem that shows that approximate simulability can be characterized by comparing outcome probabilities in certain tests. This result is inspired by the classical Le Cam randomization criterion for statistical experiments and contains its finite dimensional version as a special case.