We consider a Gaussian sequence model that contains ill-posed inverse problems as special cases. We assume that the associated operator is partially unknown in the sense that its singular functions are known and the corresponding singular values are unknown but observed with Gaussian noise. For the considered model, we study the minimax goodness-of-fit testing problem. Working with certain ellipsoids in the space of squared-summable sequences of real numbers, with a ball of positive radius removed, we obtain lower and upper bounds for the minimax separation radius in the non-asymptotic framework, i.e., for fixed values of the involved noise levels. Examples of mildly and severely ill-posed inverse problems with ellipsoids of ordinary-smooth and super-smooth sequences are examined in detail and minimax rates of goodness-of-fit testing are obtained for illustrative purposes.
We consider the identity testing problem - or goodness-of-fit testing problem - in multivariate binomial families, multivariate Poisson families and multinomial distributions. Given a known distribution $p$ and $n$ iid samples drawn from an unknown distribution $q$, we investigate how large $ρ>0$ should be to distinguish, with high probability, the case $p=q$ from the case $d(p,q) \geq ρ$, where $d$ denotes a specific distance over probability distributions. We answer this question in the case of a family of different distances: $d(p,q) = \|p-q\|_t$ for $t \in [1,2]$ where $\|\cdot\|_t$ is the…
We consider multinomial goodness-of-fit tests in the high-dimensional regime where the number of bins increases with the sample size. In this regime, Pearson's chi-squared test can suffer from low power due to the substantial bias as well as high variance of its statistic. To resolve these issues, we introduce a family of U-statistic for multinomial goodness-of-fit and study their asymptotic behaviors in high-dimensions. Specifically, we establish conditions under which the considered U-statistic is asymptotically Poisson or Gaussian, and investigate its power function under each asymptotic re…
We consider the problem of testing hypotheses on the copula density from $n$ bi-dimensional observations. We wish to test the null hypothesis characterized by a parametric class against a composite nonparametric alternative. Each density under the alternative is separated in the $L_2$-norm from any density lying in the null hypothesis. The copula densities under consideration are supposed to belong to a range of Besov balls. According to the minimax approach, the testing problem is solved in an adaptive framework: it leads to a $\log\log$ term loss in the minimax rate of testing in comparison …
Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions spanning decades, there is no mention there of any curse of dimensionality. Only more recently Ramdas et al. (2015) have discussed this issue in the context of kernel methods by showing that their performance degrades with the dimension even when the underlying distributions are isotropic Gaussians. We take a minimax perspective and follow in the footsteps of Ingster (1987) to derive the minimax rate in arbitrary dimension when the discrepancy is measured in the L2 metric. That rate is …
We consider the goodness-of fit testing problem for Hölder smooth densities over $\mathbb{R}^d$: given $n$ iid observations with unknown density $p$ and given a known density $p_0$, we investigate how large $ρ$ should be to distinguish, with high probability, the case $p=p_0$ from the composite alternative of all Hölder-smooth densities $p$ such that $\|p-p_0\|_t \geq ρ$ where $t \in [1,2]$. The densities are assumed to be defined over $\mathbb{R}^d$ and to have Hölder smoothness parameter $α>0$. In the present work, we solve the case $α\leq 1$ and handle the case $α>1$ using an additional tec…
Finding anonymization mechanisms to protect personal data is at the heart of recent machine learning research. Here, we consider the consequences of local differential privacy constraints on goodness-of-fit testing, i.e. the statistical problem assessing whether sample points are generated from a fixed density $f_0$, or not. The observations are kept hidden and replaced by a stochastic transformation satisfying the local differential privacy constraint. In this setting, we propose a testing procedure which is based on an estimation of the quadratic distance between the density $f$ of the unobs…
We consider the convolution model where i.i.d. random variables $X_i$ having unknown density $f$ are observed with additive i.i.d. noise, independent of the $X$'s. We assume that the density $f$ belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically. We consider the problem of goodness-of-fit testing in the convolution model. We prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional $\int f^2$ based o…
This paper is concerned with the problem of goodness-of-fit for curve registration, and more precisely for the shifted curve model, whose application field reaches from computer vision and road traffic prediction to medicine. We give bounds for the asymptotic minimax separation rate, when the functions in the alternative lie in Sobolev balls and the separation from the null hypothesis is measured by the l2-norm. We use the generalized likelihood ratio to build a nonadaptive procedure depending on a tuning parameter, which we choose in an optimal way according to the smoothness of the ambient s…
The aim of this paper is to establish non-asymptotic minimax rates of testing for goodness-of-fit hypotheses in a heteroscedastic setting. More precisely, we deal with sequences $(Y_j)_{j\in J}$ of independent Gaussian random variables, having mean $(θ_j)_{j\in J}$ and variance $(σ_j)_{j\in J}$. The set $J$ will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to $l_2$ and $l_{\infty}$ norms, without assumption on $(σ_j)_{j\in J}$ and on several funct…
Given observations from a circular random variable contaminated by an additive measurement error, we consider the problem of minimax optimal goodness-of-fit testing in a non-asymptotic framework. We propose direct and indirect testing procedures using a projection approach. The structure of the optimal tests depends on regularity and ill-posedness parameters of the model, which are unknown in practice. Therefore, adaptive testing strategies that perform optimally over a wide range of regularity and ill-posedness classes simultaneously are investigated. Considering a multiple testing procedure,…
Within the nonparametric regression model with unknown regression function $l$ and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis $l=0$ against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is…
The reproducing kernel Hilbert space (RKHS) embedding of distributions offers a general and flexible framework for testing problems in arbitrary domains and has attracted considerable amount of attention in recent years. To gain insights into their operating characteristics, we study here the statistical performance of such approaches within a minimax framework. Focusing on the case of goodness-of-fit tests, our analyses show that a vanilla version of the kernel-embedding based test could be suboptimal, and suggest a simple remedy by moderating the embedding. We prove that the moderated approa…
Let $(Y,(X_i)_{i\in\mathcal{I}})$ be a zero mean Gaussian vector and $V$ be a subset of $\mathcal{I}$. Suppose we are given $n$ i.i.d. replications of the vector $(Y,X)$. We propose a new test for testing that $Y$ is independent of $(X_i)_{i\in \mathcal{I}\backslash V}$ conditionally to $(X_i)_{i\in V}$ against the general alternative that it is not. This procedure does not depend on any prior information on the covariance of $X$ or the variance of $Y$ and applies in a high-dimensional setting. It straightforwardly extends to test the neighbourhood of a Gaussian graphical model. The procedure …
We consider the estimation of quadratic functionals in a Gaussian sequence model where the eigenvalues are supposed to be unknown and accessible through noisy observations only. Imposing smoothness assumptions both on the signal and the sequence of eigenvalues, we develop a minimax theory for this problem. We propose a truncated series estimator and show that it attains the optimal rate of convergence if the truncation parameter is chosen appropriately. Consequences for testing problems in inverse problems are equally discussed: in particular, the minimax rates of testing for signal detection …
This paper studies goodness of fit tests and specification tests for an extension of the log-GARCH model which is stable by scaling. A Lagrange-Multiplier test is derived for testing the null assumption of extended log-GARCH against more general formulations including the Exponential GARCH (EGARCH). The null assumption of an EGARCH is also tested. Portmanteau goodness-of-fit tests are developed for the extended log-GARCH. Simulations illustrating the theoretical results and an application to real financial data are proposed.
We consider the goodness-of-fit testing problem of distinguishing whether the data are drawn from a specified distribution, versus a composite alternative separated from the null in the total variation metric. In the discrete case, we consider goodness-of-fit testing when the null distribution has a possibly growing or unbounded number of categories. In the continuous case, we consider testing a Lipschitz density, with possibly unbounded support, in the low-smoothness regime where the Lipschitz parameter is not assumed to be constant. In contrast to existing results, we show that the minimax r…
Ill-posed inverse problems arise in various scientific fields. We consider the signal detection problem for mildly, severely and extremely ill-posed inverse problems with $l^q$-ellipsoids (bodies), $q\in(0,2]$, for Sobolev, analytic and generalized analytic classes of functions under the Gaussian white noise model. We study both rate and sharp asymptotics for the error probabilities in the minimax setup. By construction, the derived tests are, often, nonadaptive. Minimax rate-optimal adaptive tests of rather simple structure are also constructed.
In this paper we study the statistical properties of Laplacian smoothing, a graph-based approach to nonparametric regression. Under standard regularity conditions, we establish upper bounds on the error of the Laplacian smoothing estimator $\widehat{f}$, and a goodness-of-fit test also based on $\widehat{f}$. These upper bounds match the minimax optimal estimation and testing rates of convergence over the first-order Sobolev class $H^1(\mathcal{X})$, for $\mathcal{X}\subseteq \mathbb{R}^d$ and $1 \leq d < 4$; in the estimation problem, for $d = 4$, they are optimal modulo a $\log n$ factor. Ad…
Classes of coordinate-invariant omnibus goodness-of-fit tests on compact Riemannian manifolds are proposed. The tests are based on Giné's Sobolev tests of uniformity. A condition for consistency is given. The tests are illustrated by an example on the rotation group $\mathit{SO}(3)$.
We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R.~C. Liu, and B. MacGibbon, {\it Minimax risk over hyperrectangles, and implications}, Ann.~ Statist. 18, 1990. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation i…
This article proposes a class of goodness-of-fit tests for the autocorrelation function of a time series process, including those exhibiting long-range dependence. Test statistics for composite hypotheses are functionals of a (approximated) martingale transformation of the Bartlett $T_p$-process with estimated parameters, which converges in distribution to the standard Brownian motion under the null hypothesis. We discuss tests of different natures such as omnibus, directional and Portmanteau-type tests. A Monte Carlo study illustrates the performance of the different tests in practice.
The degrees are a classical and relevant way to study the topology of a network. They can be used to assess the goodness-of-fit for a given random graph model. In this paper we introduce goodness-of-fit tests for two classes of models. First, we consider the case of independent graph models such as the heterogeneous Erdös-Rényi model in which the edges have different connection probabilities. Second, we consider a generic model for exchangeable random graphs called the W-graph. The stochastic block model and the expected degree distribution model fall within this framework. We prove the asympt…
In this paper we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for nonparametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\bf Y} = (Y_1,\ldots,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighborhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density $p$, PCR-LE achieves the optimal rates for both estimation (where th…
This paper develops goodness of fit statistics that can be used to formally assess Markov random field models for spatial data, when the model distributions are discrete or continuous and potentially parametric. Test statistics are formed from generalized spatial residuals which are collected over groups of nonneighboring spatial observations, called concliques. Under a hypothesized Markov model structure, spatial residuals within each conclique are shown to be independent and identically distributed as uniform variables. The information from a series of concliques can be then pooled into good…
We consider an unknown response function $f$ defined on $Δ=[0,1]^d$, $1\le d\le\infty$, taken at $n$ random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence $r_n\to 0$ as $n\to\infty$ and a known function $f_0 \in L_2(Δ)$, we propose, under general conditions, a unified framework for the goodness-of-fit testing problem for testing the null hypothesis $H_0: f=f_0$ against the alternative $H_1: f\in\CF, \|f-f_0\|\ge r_n$, where $\CF$ is an ellipsoid in the Hilbert space $ L_2(Δ)$ with respect to the tensor product Fourier basis and $\|\cdot\|$ i…
In this paper, we are concerned with nonparametric inference on the volatility of volatility process in stochastic volatility models. We construct several estimators for its integrated version in a high-frequency setting, all based on increments of spot volatility estimators. Some of those are positive by construction, others are bias corrected in order to attain the optimal rate $n^{-1/4}$. Associated central limit theorems are proven which can be widely used in practice, as they are the key to essentially all tools in model validation for stochastic volatility models. As an illustration we g…
In materials science and many other application domains, 3D information can often only be extrapolated by taking 2D slices. In topological data analysis, persistence vineyards have emerged as a powerful tool to take into account topological features stretching over several slices. In the present paper, we illustrate how persistence vineyards can be used to design rigorous statistical hypothesis tests for 3D microstructure models based on data from 2D slices. More precisely, by establishing the asymptotic normality of suitable longitudinal and cross-sectional summary statistics, we devise goodn…
We develop a new goodness fit test for Rayleigh distribution for complete as well as right censored data. We use U-Statistic theory to derive the test statistic. First we develop a test for complete data and then discuss, how right censored observations can be incorporated in the testing procedure. The asymptotic properties of the test statistics in both uncensored and censored cases are studied in detail. Extensive Monte Carlo simulation studies are carried out to validate the performance of the proposed tests. We illustrate the procedures using real data sets. We also provide, a goodness of …
This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR($\infty$) processes, as well as the GARCH or ARCH($\infty$), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. It appears from these conditions that th…
New goodness-of-fit tests for Markovian models in time series analysis are developed which are based on the difference between a fully nonparametric estimate of the one-step transition distribution function of the observed process and that of the model class postulated under the null hypothesis. The model specification under the null allows for Markovian models, the transition mechanisms of which depend on an unknown vector of parameters and an unspecified distribution of i.i.d. innovations. Asymptotic properties of the test statistic are derived and the critical values of the test are found u…
We consider spherical data $X_i$ noised by a random rotation $\varepsilon_i\in$ SO(3) so that only the sample $Z_i=\varepsilon_iX_i$, $i=1,\dots, N$ is observed. We define a nonparametric test procedure to distinguish $H_0:$ ''the density $f$ of $X_i$ is the uniform density $f_0$ on the sphere'' and $H_1:$ ''$\|f-f_0\|_2^2\geq \Cψ_N$ and $f$ is in a Sobolev space with smoothness $s$''. For a noise density $f_\varepsilon$ with smoothness index $ν$, we show that an adaptive procedure (i.e. $s$ is not assumed to be known) cannot have a faster rate of separation than $ψ_N^{ad}(s)=(N/\sqrt{\log\log…
Under mild Markov assumptions, sufficient conditions for strict minimax optimality of sequential tests for multiple hypotheses under distributional uncertainty are derived. First, the design of optimal sequential tests for simple hypotheses is revisited and it is shown that the partial derivatives of the corresponding cost function are closely related to the performance metrics of the underlying sequential test. Second, an implicit characterization of the least favorable distributions for a given testing policy is stated. By combining the results on optimal sequential tests and least favorable…
Centre for Artificial Intelligence（人工智能中心）；Gatsby Computational Neuroscience Unit（盖茨比计算神经科学研究组）；University College London（伦敦大学学院）；Inria London（法国国家信息与自动化研究所伦敦分所）
Model checking plays an important role in linear regression as model misspecification seriously affects the validity and efficiency of regression analysis. In practice, model checking is often performed by subjectively evaluating the plot of the model's residuals. This approach is objectified by constructing a random process from the model's residuals, however due to a very complex covariance function obtaining the exact distribution of the test statistic is intractable. Several solutions to overcome this have been proposed, however the simulation and bootstrap based approaches are only asympt…
This work deals with the ill-posed inverse problem of reconstructing a function $f$ given implicitly as the solution of $g = Af$, where $A$ is a compact linear operator with unknown singular values and known eigenfunctions. We observe the function $g$ and the singular values of the operator subject to Gaussian white noise with respective noise levels $\varepsilon$ and $σ$. We develop a minimax theory in terms of both noise levels and propose an orthogonal series estimator attaining the minimax rates. This estimator requires the optimal choice of a dimension parameter depending on certain chara…
We propose a family of tests to assess the goodness-of-fit of a high-dimensional generalized linear model. Our framework is flexible and may be used to construct an omnibus test or directed against testing specific non-linearities and interaction effects, or for testing the significance of groups of variables. The methodology is based on extracting left-over signal in the residuals from an initial fit of a generalized linear model. This can be achieved by predicting this signal from the residuals using modern flexible regression or machine learning methods such as random forests or boosted tre…
We reconsider the existing kernel estimators for a copula function, as proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445--464], Fermanian, Radulovič and Wegkamp [Bernoulli 10 (2004) 847--860] and Chen and Huang [Canad. J. Statist. 35 (2007) 265--282]. All of these estimators have as a drawback that they can suffer from a corner bias problem. A way to deal with this is to impose rather stringent conditions on the copula, outruling as such many classical families of copulas. In this paper, we propose improved estimators that take care of the typical corner bias prob…
We employ a general Monte Carlo method to test composite hypotheses of goodness-of-fit for several popular multivariate models that can accommodate both asymmetry and heavy tails. Specifically, we consider weighted L2-type tests based on a discrepancy measure involving the distance between empirical characteristic functions and thus avoid the need for employing corresponding population quantities which may be unknown or complicated to work with. The only requirements of our tests are that we should be able to draw samples from the distribution under test and possess a reasonable method of esti…
We address the problem of goodness-of-fit testing for Hölder continuous densities under local differential privacy constraints. We study minimax separation rates when only non-interactive privacy mechanisms are allowed to be used and when both non-interactive and sequentially interactive can be used for privatisation. We propose privacy mechanisms and associated testing procedures whose analysis enables us to obtain upper bounds on the minimax rates. These results are complemented with lower bounds. By comparing these bounds, we show that the proposed privacy mechanisms and tests are optimal u…
We provide a minimax optimal estimation procedure for F and W in matrix valued linear models Y = F W + Z where the parameter matrix W and the design matrix F are unknown but the latter takes values in a known finite set. The proposed finite alphabet linear model is justified in a variety of applications, ranging from signal processing to cancer genetics. We show that this allows to separate F and W uniquely under weak identifiability conditions, a task which is not doable, in general. To this end we quantify in the noiseless case, that is, Z = 0, the perturbation range of Y in order to obtain …
