Suppose there are two unknown parameters, each parameter is the solution to an estimating equation, and the estimating equation of one parameter depends on the other parameter. The parameters can be jointly estimated by "stacking" their estimating equations and solving for both parameters simultaneously. Asymptotic confidence intervals are readily available for stacked estimating equations. We introduce a bootstrap-based hypothesis test for stacked estimating equations which does not rely on asymptotic approximations. Test statistics are constructed by splitting the sample in two, estimating the first parameter on a portion of the sample then plugging the result into the second estimating equation to solve for the next parameter using the remaining sample. To reduce simulation variability from a single split, we repeatedly split the sample and take the sample mean of all the estimates. For parametric models, we derive the limiting distribution of sample splitting estimator and show they are equivalent to stacked estimating equations.
In many fields of research null hypothesis significance tests and p values are the accepted way of assessing the degree of certainty with which research results can be extrapolated beyond the sample studied. However, there are very serious concerns about the suitability of p values for this purpose. An alternative approach is to cite confidence intervals for a statistic of interest, but this does not directly tell readers how certain a hypothesis is. Here, I suggest how the framework used for confidence intervals could easily be extended to derive confidence levels, or "tentative probabilities…
We propose a method for summarizing the strength of association between a set of variables and a multivariate outcome. Classical summary measures are appropriate when linear relationships exist between covariates and outcomes, while our approach provides an alternative that is useful in situations where complex relationships may be present. We utilize ensemble machine learning to detect nonlinear relationships and covariate interactions and propose a measure of association that captures these relationships. A hypothesis test about the proposed associative measure can be used to test the strong…
There has been a growing interest in statistical inference from data satisfying the so-called manifold hypothesis, assuming data points in the high-dimensional ambient space to lie in close vicinity of a submanifold of much lower dimension. In machine learning, encoder-decoder pair based generative modelling approaches have been successful in learning complicated high-dimensional distributions such as those over images and texts by explicitly imposing the low-dimensional manifold structure. In this work, we introduce a new approach for estimating distributions on unknown submanifolds via mixtu…
Network-based clustering methods frequently require the number of communities to be specified \emph{a priori}. Moreover, most of the existing methods for estimating the number of communities assume the number of communities to be fixed and not scale with the network size $n$. The few methods that assume the number of communities to increase with the network size $n$ are only valid when the average degree $d$ of a network grows at least as fast as $O(n)$ (i.e., the dense case) or lies within a narrow range. This presents a challenge in clustering large-scale network data, particularly when the …
A semi-supervised framework for diverse multiple hypothesis testing scenarios
University of Washington（华盛顿大学）；Paul G. Allen School of Computer Science and Engineering, University of Washington（华盛顿大学保罗·G·艾伦计算机科学与工程学学院）；Macquarie University（麦考瑞大学）；University of Sydney（悉尼大学）
A pervasive issue in statistical hypothesis testing is that the reported $p$-values are biased downward by data "peeking" -- the practice of reporting only progressively extreme values of the test statistic as more data samples are collected. We develop principled mechanisms to estimate such running extrema of test statistics, which directly address the effect of peeking in some general scenarios.
Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $\int Φ(g(x))\,dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Hölder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $Φ$ which are satisfied for $Φ(u)=|u|^p$, $p\ge 1$. As an application, we consider the problem of estimating the $L^p$-norm and derive the minimax separation rates…
The problem of estimating the directed information rate between two discrete processes $\{X_n\}$ and $\{Y_n\}$ via the plug-in (or maximum-likelihood) estimator is considered. When the joint process $\{(X_n,Y_n)\}$ is a Markov chain of a given memory length, the plug-in estimator is shown to be asymptotically Gaussian and to converge at the optimal rate $O(1/\sqrt{n})$ under appropriate conditions; this is the first estimator that has been shown to achieve this rate. An important connection is drawn between the problem of estimating the directed information rate and that of performing a hypoth…
We propose a simple robust hypothesis test that has the same sample complexity as that of the optimal Neyman-Pearson test up to constants, but robust to distribution perturbations under Hellinger distance. We discuss the applicability of such a robust test for estimating distributions in Hellinger distance. We empirically demonstrate the power of the test on canonical distributions.
We revisit the problem of simultaneously testing the means of $n$ independent normal observations under sparsity. We take a Bayesian approach to this problem by introducing a scale-mixture prior known as the normal-beta prime (NBP) prior. We first derive new concentration properties when the beta prime density is employed for a scale parameter in Bayesian hierarchical models. To detect signals in our data, we then propose a hypothesis test based on thresholding the posterior shrinkage weight under the NBP prior. Taking the loss function to be the expected number of misclassified tests, we show…
With recent advances in high throughput technology, researchers often find themselves running a large number of hypothesis tests (thousands+) and esti- mating a large number of effect-sizes. Generally there is particular interest in those effects estimated to be most extreme. Unfortunately naive estimates of these effect-sizes (even after potentially accounting for multiplicity in a testing procedure) can be severely biased. In this manuscript we explore this bias from a frequentist perspective: we give a formal definition, and show that an oracle estimator using this bias dominates the naive …
This work constructs a hypothesis test for detecting whether an data-generating function $h: R^p \rightarrow R$ belongs to a specific reproducing kernel Hilbert space $\mathcal{H}_0$ , where the structure of $\mathcal{H}_0$ is only partially known. Utilizing the theory of reproducing kernels, we reduce this hypothesis to a simple one-sided score test for a scalar parameter, develop a testing procedure that is robust against the mis-specification of kernel functions, and also propose an ensemble-based estimator for the null model to guarantee test performance in small samples. To demonstrate th…
In this paper, we further develop the approach, originating in [14 (arXiv:1311.6765),20 (arXiv:1604.02576)], to "computation-friendly" hypothesis testing and statistical estimation via Convex Programming. Specifically, we focus on estimating a linear or quadratic form of an unknown "signal," known to belong to a given convex compact set, via noisy indirect observations of the signal. Most of the existing theoretical results on the subject deal with precisely stated statistical models and aim at designing statistical inferences and quantifying their performance in a closed analytic form. In con…
The PC algorithm allows investigators to estimate a complete partially directed acyclic graph (CPDAG) from a finite dataset, but few groups have investigated strategies for estimating and controlling the false discovery rate (FDR) of the edges in the CPDAG. In this paper, we introduce PC with p-values (PC-p), a fast algorithm which robustly computes edge-specific p-values and then estimates and controls the FDR across the edges. PC-p specifically uses the p-values returned by many conditional independence tests to upper bound the p-values of more complex edge-specific hypothesis tests. The alg…
While statistics focusses on hypothesis testing and on estimating (properties of) the true sampling distribution, in machine learning the performance of learning algorithms on future data is the primary issue. In this paper we bridge the gap with a general principle (PHI) that identifies hypotheses with best predictive performance. This includes predictive point and interval estimation, simple and composite hypothesis testing, (mixture) model selection, and others as special cases. For concrete instantiations we will recover well-known methods, variations thereof, and new ones. PHI nicely just…
The notion that an independent central bank reduces a country's inflation is a controversial hypothesis. To date, it has not been possible to satisfactorily answer this question because the complex macroeconomic structure that gives rise to the data has not been adequately incorporated into statistical analyses. We develop a causal model that summarizes the economic process of inflation. Based on this causal model and recent data, we discuss and identify the assumptions under which the effect of central bank independence on inflation can be identified and estimated. Given these and alternative…
We introduce one-sided versions of Huber's contamination model, in which corrupted samples tend to take larger values than uncorrupted ones. Two intertwined problems are addressed: estimation of the mean of uncorrupted samples (minimum effect) and selection of corrupted samples (outliers). Regarding the minimum effect estimation, we derive the minimax risks and introduce adaptive estimators to the unknown number of contaminations. Interestingly, the optimal convergence rate highly differs from that in classical Huber's contamination model. Also, our analysis uncovers the effect of particular s…
Paul G. Allen School of Computer Science and Engineering, University of Washington（华盛顿大学保罗·G·艾伦计算机科学与工程学院）
We study the problem of estimating the distribution of effect sizes (the mean of the test statistic under the alternate hypothesis) in a multiple testing setting. Knowing this distribution allows us to calculate the power (type II error) of any experimental design. We show that it is possible to estimate this distribution using an inexpensive pilot experiment, which takes significantly fewer samples than would be required by an experiment that identified the discoveries. Our estimator can be used to guarantee the number of discoveries that will be made using a given experimental design in a fu…
We consider the problem of estimating the number of false null hypotheses among a very large number of independently tested hypotheses, focusing on the situation in which the proportion of false null hypotheses is very small. We propose a family of methods for establishing lower $100(1-α)%$ confidence bounds for this proportion, based on the empirical distribution of the $p$-values of the tests. Methods in this family are then compared in terms of ability to consistently estimate the proportion by letting $α\to 0$ as the number of hypothesis tests increases and the proportion decreases. This w…
Multiple hypothesis testing is a fundamental problem in high dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any SNPs are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In the current paper, we propose a novel method based on principal factor approximation, which successfully subtracts the common dependence and weakens significantly the correl…
When there is a suspicious figure reuse case in science, research integrity investigators often find it difficult to rebut authors claiming that "it happened by chance". In other words, when there is a "collision" of image features, it is difficult to justify whether it appears rarely or not. In this article, we provide a method to predict the rarity of an image feature by statistically estimating the chance of it randomly occurring across all scientific imagery. Our method is based on high-dimensional density estimation of ORB features using 7+ million images in the PubMed Open Access Subset …
I present a critique of the methods used in a typical paper. This leads to three broad conclusions about the conventional use of statistical methods. First, results are often reported in an unnecessarily obscure manner. Second, the null hypothesis testing paradigm is deeply flawed: estimating the size of effects and citing confidence intervals or levels is usually better. Third, there are several issues, independent of the particular statistical concepts employed, which limit the value of any statistical approach: e.g. difficulties of generalizing to different contexts, and the weakness of som…
University of California Santa Barbara（加州大学圣巴巴拉分校）
Machine learning models are known to be susceptible to adversarial attacks which can cause misclassification by introducing small but well designed perturbations. In this paper, we consider a classical hypothesis testing problem in order to develop fundamental insight into defending against such adversarial perturbations. We interpret an adversarial perturbation as a nuisance parameter, and propose a defense based on applying the generalized likelihood ratio test (GLRT) to the resulting composite hypothesis testing problem, jointly estimating the class of interest and the adversarial perturbat…
We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For $Ω\subset D = (0,1)^d$, with $d \geq 2$, we are given $n$ random i.i.d. points on $D$ whose membership in $Ω$ is known. We consider the sample as a random geometric graph with connection distance $\varepsilon>0$. We estimate the perimeter of $Ω$ (relative to $D$) by the, appropriately rescaled, graph cut between the vertices in $Ω$ and the vertices in $D \backslash Ω$. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to $n$ and $\varepsilon$. We c…
First and second kind modifications of usual confidence intervals for estimating the expectation and of usual local alternative parameter choices are introduced in a way such that the asymptotic behavior of the true non-covering probabilities and the covering probabilities under the modified local non-true parameter assumption can be asymptotically exactly controlled. The orders of convergence to zero of both types of probabilities are assumed to be suitably bounded below according to an Osipov-type condition and the sample distribution is assumed to satisfy a corresponding tail condition due …
Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be estimated. We characterize the statistical difficulty of this problem by deriving upper and lower bounds on the minimax rate for estimating the dimension. First, we consider the problem of testing the hypothesis that the support of the data-generating probability distribution is a well-behaved manifold of intrinsic dimension $d_1$ versus the alternative that it is …
We consider the related problems of estimating the $l_2$-norm and the squared $l_2$-norm in sparse linear regression with unknown variance, as well as the problem of testing the hypothesis that the regression parameter is null under sparse alternatives with $l_2$ separation. We establish the minimax optimal rates of estimation (respectively, testing) in these three problems.
In a recent paper (Efron (2004)), Efron pointed out that an important issue in large-scale multiple hypothesis testing is that the null distribution may be unknown and need to be estimated. Consider a Gaussian mixture model, where the null distribution is known to be normal but both null parameters--the mean and the variance--are unknown. We address the problem with a method based on Fourier transformation. The Fourier approach was first studied by Jin and Cai (2007), which focuses on the scenario where any non-null effect has either the same or a larger variance than that of the null effects.…
In this paper, we propose some estimators for the parameters of a statistical model based on Kullback-Leibler divergence of the survival function in continuous setting. We prove that the proposed estimators are subclass of "generalized estimating equations" estimators. The asymptotic properties of the estimators such as consistency, asymptotic normality, asymptotic confidence interval and asymptotic hypothesis testing are investigated.
University of Granada（格拉纳达大学）；University of Cambridge（剑桥大学）
In the 70s a novel branch of statistics emerged focusing its effort in selecting a function in the pattern recognition problem, which fulfils a definite relationship between the quality of the approximation and its complexity. These data-driven approaches are mainly devoted to problems of estimating dependencies with limited sample sizes and comprise all the empirical out-of sample generalization approaches, e.g. cross validation (CV) approaches. Although the latter are \emph{not designed for testing competing hypothesis or comparing different models} in neuroimaging, there are a number of the…
A multiple filter test (MFT) for the analysis and detection of rate change points in point processes on the line has been proposed recently. The underlying statistical test investigates the null hypothesis of constant rate. For that purpose, multiple filtered derivative processes are observed simultaneously. Under the null hypothesis, each process $G$ asymptotically takes the form \begin{align*} G \sim L, \end{align*} while $L$ is a zero-mean Gaussian process with unit variance. This result is used to derive a rejection threshold for statistical hypothesis testing. The purpose of this paper is…
We study the well-known problem of estimating a sparse $n$-dimensional unknown mean vector $θ= (θ_1, ..., θ_n)$ with entries corrupted by Gaussian white noise. In the Bayesian framework, continuous shrinkage priors which can be expressed as scale-mixture normal densities are popular for obtaining sparse estimates of $θ$. In this article, we introduce a new fully Bayesian scale-mixture prior known as the inverse gamma-gamma (IGG) prior. We prove that the posterior distribution contracts around the true $θ$ at (near) minimax rate under very mild conditions. In the process, we prove that the suff…
For frequentist settings in which parameter randomness represents variability rather than uncertainty, the ideal measure of the support for one hypothesis over another is the difference in the posterior and prior log odds. For situations in which the prior distribution cannot be accurately estimated, that ideal support may be replaced by another measure of support, which may be any predictor of the ideal support that, on a per-observation basis, is asymptotically unbiased. Two qualifying measures of support are defined. The first is minimax optimal with respect to the population and is equival…
Estimating conditional Mann-Whitney effects using pseudo-observation-based regression
Institute of Statistics and Mathematical Economics, RWTH Aachen University（统计与数理经济研究所，亚琛工业大学）；Institute for Medical Biometry, Informatics and Epidemiology, University of Bonn, University Hospital Bonn（波恩大学医学院生物统计学、信息学与流行病学研究所，波恩大学医院）
This paper develops a method for estimating parameters of a vector autoregression (VAR) observed in white noise. The estimation method assumes the noise variance matrix is known and does not require any iterative process. This study provides consistent estimators and shows the asymptotic distribution of the parameters required for conducting tests of Granger causality. Methods in the existing statistical literature cannot be used for testing Granger causality, since under the null hypothesis the model becomes unidentifiable. Measurement error effects on parameter estimates were evaluated by us…
Given a random sample of points from some unknown distribution, we propose a new data-driven method for estimating its probability support $S$. Under the mild assumption that $S$ is $r$-convex, the smallest $r$-convex set which contains the sample points is the natural estimator. The main problem for using this estimator in practice is that $r$ is an unknown geometric characteristic of the set $S$. A stochastic algorithm is proposed for selecting it from the data under the hypothesis that the sample is uniformly generated. The new data-driven reconstruction of $S$ is able to achieve the same c…
Several approaches to testing the hypothesis that two histograms are drawn from the same distribution are investigated. We note that single-sample continuous distribution tests may be adapted to this two-sample grouped data situation. The difficulty of not having a fully-specified null hypothesis is an important consideration in the general case, and care is required in estimating probabilities with ``toy'' Monte Carlo simulations. The performance of several common tests is compared; no single test performs best in all situations.
Score tests have the advantage of requiring estimation alone of the model restricted by the null hypothesis, which often is much simpler than models defined under the alternative hypothesis. This is typically so when the alternative hypothesis involves inequality constraints. However, existing score tests address only jointly testing all parameters of interest; a leading example is testing all ARCH parameters or variances of random coefficients being zero or not. In such testing problems rejection of the null hypothesis does not provide evidence on rejection of specific elements of parameter o…
