Motivated by global warming issues, we consider a time se- ries that consists of a nondecreasing trend observed with station- ary fluctuations, nonparametric estimation of the trend under monotonicity assumption is considered. The rescaled isotonic es- timators at an interior point are shown to converge to Chernoff's distribution under minimal conditions on the stationary errors. Since the isotonic estimators suffer from the spiking problem at the end point, two modifications are proposed. The estima- tion errors for both estimators of the boundary point are shown to have interesting limiting distributions. Approximation accu- racies are assessed through simulations. One highlight of our treatment is the proof of the weak convergence results which involve several recent techniques developed in the study of con- ditional central limit questions. These weak convergences can be shown to hold conditionally given the starting values.
We focus on the problem estimating a monotone trend function under additive and dependent noise. New point-wise confidence interval estimators under both short- and long-range dependent errors are introduced and studied. These intervals are obtained via the method of inversion of certain discrepancy statistics arising in hypothesis testing problems. The advantage of this approach is that it avoids the estimation of nuisance parameters such as the derivative of the unknown function, which existing methods are forced to deal with. While the methodology is motivated by earlier work in the indepen…
We consider Grenander type estimators for monotone functions $f$ in a very general setting, which includes estimation of monotone regression curves, monotone densities, and monotone failure rates. These estimators are defined as the left-hand slope of the least concave majorant $\hat{F}_n$ of a naive estimator $F_n$ of the integrated curve $F$ corresponding to $f$. We prove that the supremum distance between $\hat{F}_n$ and $F_n$ is of the order $O_p(n^{-1}\log n)^{2/(4-τ)}$, for some $τ\in[0,4)$ that characterizes the tail probabilities of an approximating process for $F_n$. In typical exampl…
Consider a sequence of real data points $X_1,\ldots, X_n$ with underlying means $θ^*_1,\dots,θ^*_n$. This paper starts from studying the setting that $θ^*_i$ is both piecewise constant and monotone as a function of the index $i$. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a non-trivial answer to an open problem in the shape-constrained analysis literature. The minimax rate involves an interesting iterated logarithmic dependence on the dimension, a phenomenon that is revealed through characterizing the interplay between the isotonic shape …
We propose a new estimator of a discrete monotone probability mass function with known flat regions. We analyse its asymptotic properties and compare its performance to the Grenander estimator and to the monotone rearrangement estimator.
Motivated by the problem of estimating the bacterial growth rates for genome assemblies from shotgun metagenomic data, we consider the permuted monotone matrix model $Y=ΘΠ+Z$, where $Y\in \mathbb{R}^{n\times p}$ is observed, $Θ\in \mathbb{R}^{n\times p}$ is an unknown approximately rank-one signal matrix with monotone rows, $Π\in \mathbb{R}^{p\times p}$ is an unknown permutation matrix, and $Z\in \mathbb{R}^{n\times p}$ is the noise matrix. This paper studies the estimation of the extreme values associated to the signal matrix $Θ$, including its first and last columns, as well as their differe…
A new approach for evaluating time-trends in extreme values accounting also for spatial dependence is proposed. Based on exceedances over a space-time threshold, estimators for a trend function and for extreme value parameters are given, leading to a homogenization procedure for then applying stationary extreme value processes. Extremal dependence over space is further evaluated through variogram analysis including anisotropy. We detect significant inhomogeneities and trends in the extremal behaviour of daily precipitation data over a time period of 84 years and from 68 observational weather s…
We consider the problem of estimating smooth integrated functionals of a monotone nonincreasing density $f$ on $[0,\infty)$ using the nonparametric maximum likelihood based plug-in estimator. We find the exact asymptotic distribution of this natural (tuning parameter-free) plug-in estimator, properly normalized. In particular, we show that the simple plug-in estimator is always $\sqrt{n}$-consistent, and is additionally asymptotically normal with zero mean and the semiparametric efficient variance for estimating a subclass of integrated functionals. Compared to the previous results on this top…
We study the problem of estimating piecewise monotone vectors. This problem can be seen as a generalization of the isotonic regression that allows a small number of order-violating changepoints. We focus mainly on the performance of the nearly-isotonic regression proposed by Tibshirani et al. (2011). We derive risk bounds for the nearly-isotonic regression estimators that are adaptive to piecewise monotone signals. The estimator achieves a near minimax convergence rate over certain classes of piecewise monotone signals under a weak assumption. Furthermore, we present an algorithm that can be a…
We consider three problems in high-dimensional Gaussian linear mixed models. Without any assumptions on the design for the fixed effects, we construct an asymptotic $F$-statistic for testing whether a collection of random effects is zero, derive an asymptotic confidence interval for a single random effect at the parametric rate $\sqrt{n}$, and propose an empirical Bayes estimator for a part of the mean vector in ANOVA type models that performs asymptotically as well as the oracle Bayes estimator. We support our results with numerical simulations and provide comparisons with oracle estimators. …
This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functiona…
We propose a new modeling and estimation approach to select the optimal treatment regime from different options through constructing a robust estimating equation. The method is protected against misspecification of the propensity score model, the outcome regression model for the non-treated group, or the potential non-monotonic treatment difference model. Our method also allows residual errors to depend on covariates. A single index structure is incorporated to facilitate the nonparametric estimation of the treatment difference. We then identify the optimal treatment through maximizing the val…
The development of coherent missing data models to account for nonmonotone missing at random (MAR) data by inverse probability weighting (IPW) remains to date largely unresolved. As a consequence, IPW has essentially been restricted for use only in monotone missing data settings. We propose a class of models for nonmonotone missing data mechanisms that spans the MAR model, while allowing the underlying full data law to remain unrestricted. For parametric specifications within the proposed class, we introduce an unconstrained maximum likelihood estimator for estimating the missing data probabil…
University of California, San Diego（加利福尼亚大学圣地亚哥分校）
In many applications, data is collected in batches, some of which are corrupt or even adversarial. Recent work derived optimal robust algorithms for estimating discrete distributions in this setting. We consider a general framework of robust learning from batches, and determine the limits of both classification and distribution estimation over arbitrary, including continuous, domains. Building on these results, we derive the first robust agnostic computationally-efficient learning algorithms for piecewise-interval classification, and for piecewise-polynomial, monotone, log-concave, and gaussia…
In this paper, we study the asymptotic distribution of the maxima of suprema of dependent Gaussian processes with trend. For different scales of the time horizon we obtain different normalizing functions for the convergence of the maxima. The obtained results not only have potential applications in estimating the delay of certain Gaussian fork-join queueing systems but also provide interesting insights to the extreme value theory for triangular arrays of random variables with row-wise dependence.
In observational studies, potential confounders may distort the causal relationship between an exposure and an outcome. However, under some conditions, a causal dose-response curve can be recovered using the G-computation formula. Most classical methods for estimating such curves when the exposure is continuous rely on restrictive parametric assumptions, which carry significant risk of model misspecification. Nonparametric estimation in this context is challenging because in a nonparametric model these curves cannot be estimated at regular rates. Many available nonparametric estimators are sen…
Ground level Ozone is one of the six common air-pollutants on which the EPA has set national air quality standards. In order to capture the spatio-temporal trend of 1-hour and 8-hour average ozone concentration in the US, we develop a method for spatio-temporal simultaneous quantile regression. Unlike existing procedures, in the proposed method, smoothing across the sites is incorporated within modeling assumptions thus allowing borrowing of information across locations, an essential step when the number of samples in each location is low. The quantile function has been assumed to be linear in…
We consider the problem of estimating a function from $n$ noisy samples whose discrete Total Variation (TV) is bounded by $C_n$. We reveal a deep connection to the seemingly disparate problem of Strongly Adaptive online learning (Daniely et al, 2015) and provide an $O(n \log n)$ time algorithm that attains the near minimax optimal rate of $\tilde O (n^{1/3}C_n^{2/3})$ under squared error loss. The resulting algorithm runs online and optimally adapts to the unknown smoothness parameter $C_n$. This leads to a new and more versatile alternative to wavelets-based methods for (1) adaptively estimat…
Robust Methods for Disease-Genotype Association in Genetic Association Studies: Calculate P-values Using Exact Conditional Enumeration instead of Asymptotic Approximations
Nonparametric regression is a standard statistical tool with increased importance in the Big Data era. Boundary points pose additional difficulties but local polynomial regression can be used to alleviate them. Local linear regression, for example, is easy to implement and performs quite well both at interior as well as boundary points. Estimating the conditional distribution function and/or the quantile function at a given regressor point is immediate via standard kernel methods but problems ensue if local linear methods are to be used. In particular, the distribution function estimator is no…
We consider the problem of estimating an unknown coordinate-wise monotone function given noisy measurements, known as the isotonic regression problem. Often, only a small subset of the features affects the output. This motivates the sparse isotonic regression setting, which we consider here. We provide an upper bound on the expected VC entropy of the space of sparse coordinate-wise monotone functions, and identify the regime of statistical consistency of our estimator. We also propose a linear program to recover the active coordinates, and provide theoretical recovery guarantees. We close with…
The common practice in difference-in-difference (DiD) designs is to check for parallel trends prior to treatment assignment, yet typical estimation and inference does not account for the fact that this test has occurred. I analyze the properties of the traditional DiD estimator conditional on having passed (i.e. not rejected) the test for parallel pre-trends. When the DiD design is valid and the test for pre-trends confirms it, the typical DiD estimator is unbiased, but traditional standard errors are overly conservative. Additionally, there exists an alternative unbiased estimator that is mor…
The ill-posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suffer from a very slow, logarithmic rate of convergence. In this paper, we show that restricting the problem to models with monotone regression functions and monotone instruments significantly weakens the ill-posedness of the problem. In stark contrast to the existing literature, the presence of a monotone instrument implies boundedness of our measure of ill-posedness when restricted to the space of monotone functions. Based on this result we…
Structural Nested Mean Models Under Parallel Trends Assumptions
CUNY School of Public Health（纽约市立大学公共卫生学院）；Ghent University（根特大学）；University of California Irvine（加州大学尔湾分校）；University of Pennsylvania（宾夕法尼亚大学）；Harvard TH Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）
In this paper, we consider the well known problem of estimating a density function under qualitative assumptions. More precisely, we estimate monotone non increasing densities in a Bayesian setting and derive concentration rate for the posterior distribution for a Dirichlet process and finite mixture prior. We prove that the posterior distribution based on both priors concentrates at the rate $(n/\log(n))^{-1/3}$, which is the minimax rate of estimation up to a \log(n)$ factor. We also study the behaviour of the posterior for the point-wise loss at any fixed point of the support the density an…
We consider the problem of estimating the mean vector $θ$ of a $d$-dimensional spherically symmetric distributed $X$ based on balanced loss functions of the forms: {\bf (i)} $ωρ(\|\de-\de_{0}\|^{2}) +(1-ω)ρ(\|\de - θ\|^{2})$ and {\bf (ii)} $\ell\left(ω\|\de - \de_{0}\|^{2} +(1-ω)\|\de - θ\|^{2}\right)$, where $δ_0$ is a target estimator, and where $ρ$ and $\ell$ are increasing and concave functions. For $d\geq 4$ and the target estimator $δ_0(X)=X$, we provide Baranchik-type estimators that dominate $δ_0(X)=X$ and are minimax. The findings represent extensions of those of Marchand \& Strawderm…
