Mediation analysis is an increasingly popular statistical method for explaining causal pathways to inform intervention. While methods have increased, there is still a dearth of robust mediation methods for count outcomes with excess zeroes. Current mediation methods addressing this issue are computationally intensive, biased, or challenging to interpret. To overcome these limitations, we propose a new mediation methodology for zero-inflated count outcomes using the marginalized zero-inflated Poisson (MZIP) model and the counterfactual approach to mediation. This novel work gives population-average mediation effects whose variance can be estimated rapidly via delta method. This methodology is extended to cases with exposure-mediator interactions. We apply this novel methodology to explore if diabetes diagnosis can explain BMI differences in healthcare utilization and test model performance via simulations comparing the proposed MZIP method to existing zero-inflated and Poisson methods. We find that our proposed method minimizes bias and computation time compared to alternative approaches while allowing for straight-forward interpretations.
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Distinguishing two classes of candidate models is a fundamental and practically important problem in statistical inference. Error rate control is crucial to the logic but, in complex nonparametric settings, such guarantees can be difficult to achieve, especially when the stopping rule that determines the data collection process is not available. In this paper we develop a novel e-process construction that leverages the so-called predictive recursion (PR) algorithm designed to rapidly and recursively fit nonparametric mixture models. The resulting PRe-process affords anytime valid inference uniformly over stopping rules and is shown to be efficient in the sense that it achieves the maximal growth rate under the alternative relative to the mixture model being fit by PR. In the special case of testing for a log-concave density, the PRe-process test is computationally simpler and faster, more stable, and no less efficient compared to a recently proposed anytime valid test.
An overview is presented of a general theory of statistical inference that is referred to as the fiducial-Bayes fusion. This theory combines organic fiducial inference and Bayesian inference. The aim is that the reader is given a clear summary of the conceptual framework of the fiducial-Bayes fusion as well as pointers to further reading about its more technical aspects. Particular attention is paid to the issue of how much importance should be attached to the role of Bayesian inference within this framework. The appendix contains a substantive example of the application of the theory of the fiducial-Bayes fusion, which supplements various other examples of the application of this theory that are referenced in the paper.
We consider the hedging of European options when the price of the underlying asset follows a single-factor Markovian framework. By working in such a setting, Carr and Wu \cite{carr2014static} derived a spanning relation between a given option and a continuum of shorter-term options written on the same asset. In this paper, we have extended their approach to simultaneously include options over multiple short maturities. We then show a practical implementation of this with a finite set of shorter-term options to determine the hedging error using a Gaussian Quadrature method. We perform a wide range of experiments for both the \textit{Black-Scholes} and \textit{Merton Jump Diffusion} models, illustrating the comparative performance of the two methods.
Nowadays, numerical models are widely used in most of engineering fields to simulate the behaviour of complex systems, such as for example power plants or wind turbine in the energy sector. Those models are nevertheless affected by uncertainty of different nature (numerical, epistemic) which can affect the reliability of their predictions. We develop here a new method for quantifying conditional parameter uncertainty within a chain of two numerical models in the context of multiphysics simulation. More precisely, we aim to calibrate the parameters $\theta$ of the second model of the chain conditionally on the value of parameters $\lambda$ of the first model, while assuming the probability distribution of $\lambda$ is known. This conditional calibration is carried out from the available experimental data of the second model. In doing so, we aim to quantify as well as possible the impact of the uncertainty of $\lambda$ on the uncertainty of $\theta$. To perform this conditional calibration, we set out a nonparametric Bayesian formalism to estimate the functional dependence between $\theta$ and $\lambda$, denoted by $\theta(\lambda)$. First, each component of $\theta(\lambda)$ is assumed to be the realization of a Gaussian process prior. Then, if the second model is written as a linear function of $\theta(\lambda)$, the Bayesian machinery allows us to compute analytically the posterior predictive distribution of $\theta(\lambda)$ for any set of realizations $\lambda$. The effectiveness of the proposed method is illustrated on several analytical examples.
Organizations often rely on statistical algorithms to make socially and economically impactful decisions. We must address the fairness issues in these important automated decisions. On the other hand, economic efficiency remains instrumental in organizations' survival and success. Therefore, a proper dual focus on fairness and efficiency is essential in promoting fairness in real-world data science solutions. Among the first efforts towards this dual focus, we incorporate the equal opportunity (EO) constraint into the Neyman-Pearson (NP) classification paradigm. Under this new NP-EO framework, we (a) derive the oracle classifier, (b) propose finite-sample based classifiers that satisfy population-level fairness and efficiency constraints with high probability, and (c) demonstrate statistical and social effectiveness of our algorithms on simulated and real datasets.
The maximum likelihood method is the best-known method for estimating the probabilities behind the data. However, the conventional method obtains the probability model closest to the empirical distribution, resulting in overfitting. Then regularization methods prevent the model from being excessively close to the wrong probability, but little is known systematically about their performance. The idea of regularization is similar to error-correcting codes, which obtain optimal decoding by mixing suboptimal solutions with an incorrectly received code. The optimal decoding in error-correcting codes is achieved based on gauge symmetry. We propose a theoretically guaranteed regularization in the maximum likelihood method by focusing on a gauge symmetry in Kullback -- Leibler divergence. In our approach, we obtain the optimal model without the need to search for hyperparameters frequently appearing in regularization.
We propose a method to construct a joint statistical model for mixed-domain data to analyze their dependence. Multivariate Gaussian and log-linear models are particular examples of the proposed model. It is shown that the functional equation defining the model has a unique solution under fairly weak conditions. The model is characterized by two orthogonal parameters: the dependence parameter and the marginal parameter. To estimate the dependence parameter, a conditional inference together with a sampling procedure is proposed and is shown to provide a consistent estimator. Illustrative examples of data analyses involving penguins and earthquakes are presented.
I propose an alternative algorithm to compute the MMS voting rule. Instead of using linear programming, in this new algorithm the maximin support value of a committee is computed using a sequence of maximum flow problems.
Meta-analysis aims to combine effect measures from several studies. For continuous outcomes, the most popular effect measures use simple or standardized differences in sample means. However, a number of applications focus on the absolute values of these effect measures (i.e., unsigned magnitude effects). We provide statistical methods for meta-analysis of magnitude effects based on standardized mean differences. We propose a suitable statistical model for random-effects meta-analysis of absolute standardized mean differences (ASMD), investigate a number of statistical methods for point and interval estimation, and provide practical recommendations for choosing among them.
Complexity is a fundamental concept underlying statistical learning theory that aims to inform generalization performance. Parameter count, while successful in low-dimensional settings, is not well-justified for overparameterized settings when the number of parameters is more than the number of training samples. We revisit complexity measures based on Rissanen's principle of minimum description length (MDL) and define a novel MDL-based complexity (MDL-COMP) that remains valid for overparameterized models. MDL-COMP is defined via an optimality criterion over the encodings induced by a good Ridge estimator class. We provide an extensive theoretical characterization of MDL-COMP for linear models and kernel methods and show that it is not just a function of parameter count, but rather a function of the singular values of the design or the kernel matrix and the signal-to-noise ratio. For a linear model with $n$ observations, $d$ parameters, and i.i.d. Gaussian predictors, MDL-COMP scales linearly with $d$ when $d<n$, but the scaling is exponentially smaller -- $\log d$ for $d>n$. For kernel methods, we show that MDL-COMP informs minimax in-sample error, and can decrease as the dimensionality of the input increases. We also prove that MDL-COMP upper bounds the in-sample mean squared error (MSE). Via an array of simulations and real-data experiments, we show that a data-driven Prac-MDL-COMP informs hyper-parameter tuning for optimizing test MSE with ridge regression in limited data settings, sometimes improving upon cross-validation and (always) saving computational costs. Finally, our findings also suggest that the recently observed double decent phenomenons in overparameterized models might be a consequence of the choice of non-ideal estimators.
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