Thursday, Aug 7: 10:30 AM - 12:20 PM
4224
Contributed Papers
Music City Center
Room: CC-212
Main Sponsor
IMS
Presentations
Modern statistical and causal estimation problems often require estimating high-dimensional, complicated nuisance functions, that are well-suited for using modern machine learning (ML) techniques. Recent work introduced the structure-agnostic estimation, which only relies on black-box nuisance estimates and does not impose any structural assumptions on the nuisances, thereby allowing for flexible incorporation of ML-based methods. This paper studies the impact of noise on the minimax rates of structure-agnostic estimation. Focusing on the partial linear outcome model popular in causal inference, we first show that for Gaussian treatments, the widely adopted double/debiased machine learning (DML) is optimal even with complete distributional information, resolving an open problem from [mackey2018orthogonal]. For non-Gaussian treatment, we propose a general procedure for constructing robust estimators against nuisance errors. For homoscedastic treatment, our procedure induces hocein, a structure-agnostic estimator that achieves fully higher-order robustness, which is the first estimator of such type. Experiments demonstrate the effectiveness of our approach.
Keywords
causal inference
causal machine learning
semiparametric estimation
The synthetic control method estimates the causal effect by comparing the outcomes of a treated unit to a weighted average of control units that closely match the pre-treatment outcomes of the treated unit. This method presumes that the relationship between potential outcomes of treated and control units remains consistent before and after treatment. However, this estimator may become unreliable when there are shifts in these relationships or when control units are highly correlated. To address these challenges, we introduce the Adversarially Robust Synthetic Control (ARSC). This framework enhances robustness by accommodating potential shifts in relationships and addressing high correlations among control units, thereby ensuring a more reliable causal estimand. When key assumptions for the classical synthetic control method hold, the ARSC method performs comparably to traditional methods. However, under assumption violations, ARSC produces a conservative estimate of the true effect. The ARSC approach employs a distributionally robust optimization by defining the causal estimand as the solution to a worst-case optimization problem, taking into account all possible distributions of the treated unit's potential outcomes that align with observed pre-treatment data. We derive a closed-form solution for the population ARSC estimand and develop a corresponding data-dependent estimator. The consistency of this estimator is established, and its practical utility is demonstrated through various case studies, including an analysis of the economic impact of terrorism in the Basque Country.
Keywords
Causal inference
Synthetic control method
Adversarial robustness
Distributional shift
We generalize saddlepoint approximations for tail areas of cumulative distribution functions of multivariate random variables by extending the univariate Lugannani and Rice saddlepoint tail approximation to multiple dimensions. The proposed approximation is derived by parts. The resulting approximation uses a multivariate Gaussian approximation to the distribution of the signed roots of the log-likelihood statistic. As in the univariate case, the next correction terms involve the difference between reciprocals of the signed root of the likelihood statistics and the analogous Wald statistics. This approximation also uses the curvature of the boundary of the tail region in terms of signed roots of likelihood ratio statistics. The separate versions of the approximation extended to lattice variables and conditional distributions are also provided. Numerical comparisons with other methodologies show better agreement with the exact distribution. We discuss the practical importance of the approximation for multiple dimensions, providing applications for sufficient statistics and statistical multivariate inference.
Keywords
Saddlepoint approximation
Multivariate Gaussian approximation
Complex analysis
Curvature of the boundary of tail region
Co-Author
John Kolassa, Rutgers University
First Author
Donghyun Lee, SAS - Statistics, Rutgers New Brunswick
Presenting Author
Donghyun Lee, SAS - Statistics, Rutgers New Brunswick
The central limit theorem shows that a suitably rescaled sum of random variables with finite variance converges to a Gaussian distribution. When the variance of the summands is infinite, the limiting distribution is instead a heavy tailed stable distribution, indexed by $\alpha\in (0,2]$ where $\alpha=2$ is the Gaussian special case. All nondegenerate stable distributions are absolutely continuous, but a discrete notion of stability can be obtained by replacing scaling with binomial thinning. Under this definition, it has been shown that the strictly discrete stable distributions are Poisson mixtures with maximally skewed stable mixing distributions for $\alpha\leq 1$. In previous work we have established the validity of Poisson-stable mixtures for $\alpha\in [1,2]$ despite the mixing distribution having support on negative numbers. Here we show that this mixed Poisson family is the unique set of distributions satisfying a broader definition of discrete stability with Poisson translation playing the role of subtraction. We also provide a compound Poisson representation and discuss discrete infinite divisibility.
Keywords
probability
stable distributions
discrete stability
mixed Poisson
count data
First Author
Will Townes, Carnegie Mellon University
Presenting Author
Will Townes, Carnegie Mellon University
Estimators of parameters of truncated distributions, namely the truncated normal distribution, have been widely studied for a known truncation region. There is also literature for estimating the unknown bounds for known parent distributions. However, to our knowledge, there are no works that accommodate both parameter and bound estimation of the truncated normal distribution. In this work, we develop a novel algorithm under the expectation-solution (ES) framework, which is an iterative method of solving nonlinear estimating equations, to estimate both the bounds and the location and scale parameters of the parent normal distribution utilizing theory of best linear unbiased estimates from location-scale families of distribution and unbiased minimum variance estimation of truncation regions. The conditions for the algorithm to converge to the solution of the estimating equations for a fixed sample size are discussed, and the asymptotic properties of the estimators are characterized using results on M- and Z-estimation from empirical process theory. The proposed method is then compared to methods utilizing the known truncation bounds via Monte Carlo simulation.
Keywords
Truncated Normal Distribution
Parameter Estimation
EM Algorithm
Order Statistics
In recent years, extensive work has been done into developing broader families of distributions to better capture the complexity of various data types. We will present a family of distributions, in an effort to add a more flexible family in the field of probability distributions. This family is derived from the foundation of the zero-truncated Poisson (ZTP) distribution. Some special cases of the generalized-G Poisson family will be presented to show their usefulness to model different types of real-life data sets.
Keywords
Generalized distribution
zero truncated Poisson distribution
McDonald distribution
Beta distribution
Finite mixture regression models are versatile tools for analyzing mixed regression relationships within clustered and heterogeneous populations. However, the classical normal mixture model often falls short when dealing with nonlinear regression data, especially in the presence of severe outliers. To address this, we introduce a novel generalized robust mixture regression procedure within the finite mixture regression framework. This procedure features sparse, scale dependent mean shift parameters, facilitating outlier detection and ensuring robust parameter estimation. Our approach incorporates three key innovations. (1)A penalized likelihood approach using a combination of L0 and L2 regularization to induce sparsity among mean shift parameters.(2) A close connection to the method of trimming, including explicit outlyingness parameters for all samples, which simplifies computation, aids theoretical analysis, and reduces the need for parameter tuning.(3) High scalability, allowing the implementation to handle nonlinear regression data. A threshold-based generalized Expectation-Maximization algorithm has been developed to ensure stable and efficient computation.
Keywords
Mixture regression models
EM algorithm
Thresholding
Outlier detection
Mean-shift
Robust procedures
Co-Author(s)
Weixin Yao, University of California-Riverside
Zhen Zeng, Nanjing University of Finance and Economics
First Author
Xin Shen, University of California-Riverside
Presenting Author
Xin Shen, University of California-Riverside