Inference on counterfactual distributions

Decomposition of differences in distribution using quantile regression

Recent work has documented a rising degree of wealth inequality in the United States between 1983 and 1998. In this paper we look at another dimension of the distribution: polarization. Using techniques developed by Esteban and Ray (1994) and extended by D'Ambrosia (2001), we examine whether a similar pattern exists with regard to trends in wealth polarization over this period. The approach followed provides a decomposition method, based on counterfactual distributions, that allows one to monitor which factors modified the entire distribution and precisely where on the distribution these factors had an effect. An index of polarization is provided, as are summary statistics of the observed movements and of distance and divergence among the estimated and the counterfactual distributions. The decomposition method is applied to U.S. data on the distribution of wealth between 1983 and 1998. We find that polarization between homeowners and tenants and among different educational groups continuously increased from 1983 to 1998, while polarization by income class continuously decreased. In contrast, polarization by racial group increased from 1983 to 1989 and then declined from 1989 to 1998, while polarization by age group followed the opposite pattern. We also find that most of the observed variation in the overall wealth density over the 1983?98 period can be attributed to changes in the within-group wealth densities rather than changes in household characteristics.

Convergence of averages of scaled functions of I(1) linear processes

This paper focuses on the Least Square (LS) regression using the mean and Quantile (M) regression analysis using median which is based on “well-Known” parametric estimation methodologies. Data from Oregon and California highway bridges were used for the comparison of the two methods. Relationships were developed to predict the unit cost of FRP repair work and FRP cost was found to have a high degree of correlation with FRP area for both Oregon and California. It was observed that the Cost Estimating Relationships (CERs) obtained by Quantile (M) regression method had the smaller Mean Absolute Deviation (MAD) values and lower Mean Absolute Percentage Error (MAPE) values than Least Square (LS) regression. The stuudy showed that Quantile Regression is much less sensitive to outliers than Least Squares Regression.

A functional central limit theorem is proved for the number of solutions $(p,q)$ of the inequality $|q\omega - p| < f(q){q^{ - 1}},q \leq n$ (respectively $0 < q\omega - p < f(q){q^{ - 1}},q \leq n$ for some functions $f$ having a positive limit.

We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset S of the Euclidean space Rk . We assume the maps are monotone (with respect to a suitable partial order) and a ‘topological’ condition on the maps. Then, we prove the existence of a pullback random attractor whose distribution is the unique stationary measure of the random iteration, and we obtain the synchronisation of random orbits. As a consequence of the synchronisation phenomenon, a functional central limit theorem is established.

We show that one can perform causal inference in a natural way for continuous-time scenarios using tools from stochastic analysis. This provides new alternatives to the positivity condition for inverse probability weighting. The probability distribution that would govern the frequency of observations in the counterfactual scenario can be characterized in terms of a so-called martingale problem. The counterfactual and factual probability distributions may be related through a likelihood ratio given by a stochastic differential equation. We can perform inference for counterfactual scenarios based on the original observations, re-weighted according to this likelihood ratio. This is possible if the solution of the stochastic differential equation is uniformly integrable, a property that can be determined by comparing the corresponding factual and counterfactual short-term predictions. Local independence graphs are directed, possibly cyclic, graphs that represent short-term prediction among sufficiently autonomous stochastic processes. We show through an example that these graphs can be used to identify and provide consistent estimators for counterfactual parameters in continuous time. This is analogous to how Judea Pearl uses graphical information to identify causal effects in finite state Bayesian networks.

We consider new formulations and methods for sparse quantile regression in the high-dimensional setting. Quantile regression plays an important role in many data mining applications, including outlier-robust exploratory analysis in gene selection. In addition, the sparsity consideration in quantile regression enables the exploration of the entire conditional distribution of the response variable given the predictors and therefore yields a more comprehensive view of the important predictors. We propose a generalized Orthogonal Matching Pursuit algorithm for variable selection, taking the misfit loss to be either the traditional quantile loss or a smooth version we call quantile Huber, and compare the resulting greedy approaches with convex sparsity-regularized formulations. We apply a recently proposed interior point methodology to efficiently solve all formulations, provide theoretical guarantees of consistent estimation, and demonstrate the performance of our approach using empirical studies of simulated and genomic datasets.

In this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes.

We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. Existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor X. The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on X through a single linear combination of the predictors, then extend to a multi index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and a real data application. Our results suggest that this method has a good finite sample performance and often outperforms existing methods.

In recent years, the intersection of machine learning (ML) and climate science has yielded profound insights into understanding and predicting extreme climate events, particularly heatwaves and droughts. Various approaches have been suggested to define and model extreme events, including extreme value theory (Sura, 2011), random forests (e.g., (Weirich-Benet et al., 2023) and, more recently, deep learning (e.g., (Jacques-Dumas et al., 2022)). Within this context, quantile regression (QR) is valuable for modelling the relationship between variables by estimating the conditional quantiles of the response variable. This provides insights into the entire distribution rather than just the mean but also aids in unravelling the complex relationships among climate variables (Barbosa et al., 2011; Franzke, 2015). QR has been extended in many ways to address critical issues such as nonlinear relations, nonstationary processes, compound events, and the complexities of handling spatio-temporal data.&amp;#160;This study presents a novel approach for predicting and better understanding heatwaves. We introduce an interpretable, nonlinear, non-parametric, and structured Spatio-Temporal Quantile Regression (STQR) method that incorporates the QR check function, commonly known as pinball loss, into machine learning models. We focus on analysing how the importance of predictors changes as the quantile being modelled increases. This allows us to circumvent arbitrary definitions of what constitutes a heatwave and instead observe if a natural definition of a heatwave emerges in predictor space. By analysing European heatwaves over recent decades using reanalysis and weather data, we demonstrate the advantages of our methodology over traditional extreme event modelling methods.ReferencesBarbosa, S.M., Scotto, M.G., Alonso, A.M., 2011. Summarising changes in air temperature over Central Europe by quantile regression and clustering. Nat. Hazards Earth Syst. Sci. 11, 3227&amp;#8211;3233. https://doi.org/10.5194/nhess-11-3227-2011Franzke, C.L.E., 2015. Local trend disparities of European minimum and maximum temperature extremes. Geophys. Res. Lett. 42, 6479&amp;#8211;6484. https://doi.org/10.1002/2015GL065011Jacques-Dumas, V., Ragone, F., Borgnat, P., Abry, P., Bouchet, F., 2022. Deep Learning-based Extreme Heatwave Forecast. Front. Clim. 4, 789641. https://doi.org/10.3389/fclim.2022.789641Sura, P., 2011. A general perspective of extreme events in weather and climate. Atmospheric Res. 101, 1&amp;#8211;21. https://doi.org/10.1016/j.atmosres.2011.01.012Weirich-Benet, E., Pyrina, M., Jim&amp;#233;nez-Esteve, B., Fraenkel, E., Cohen, J., Domeisen, D.I.V., 2023. Subseasonal Prediction of Central European Summer Heatwaves with Linear and Random Forest Machine Learning Models. Artif. Intell. Earth Syst. 2. https://doi.org/10.1175/AIES-D-22-0038.

Objective: To assess various marginal effects of nutrient intakes, health behaviours and nutrition knowledge on the entire distribution of body mass index (BMI) across individuals. Design: Quantitative and distributional study. Setting: Taiwan Methods: This study applies Becker’s (1965) model of health production to construct an individual’s BMI determining function. Quantile regression method is used to analyse the entire distribution of BMI across a nationwide survey sample in Taiwan. Endogeneity associated with choice variables of nutrient intakes, health behaviours and nutrition knowledge in determining BMI is corrected using a two-stage least-squares method. Results: Empirical results indicate that energy, oleic acid and cholesterol increase BMI, while fibre, calcium and sodium reveal a reverse effect. For overweight and obese females, protein lowers BMI. For underweight and slightly to severely obese males, vitamin C reduces BMI. Jogging lowers BMI, but drinking raises non-obese individuals’ BMI. For males whose BMI is in the ideal weight to slightly overweight ranges, nutrition knowledge reduces BMI, but this effect is insignificant for females. Conclusions: Several important conclusions can be drawn. (1) Having a BMI which is too high or too low is a risk factor to health. (2) Quantile regression is a useful empirical method to evaluate crucial factors determining the entire distribution (high, medium and low) of BMI. (3) The econometric problem of endogeneity in regressors has to be considered and corrected when using a quantile regression approach. (4) BMI-relevant health education and policy can be evaluated and derived using quantile regression effects of an individual’s choices of nutrient intakes, health behaviours and nutrition knowledge.

Newer statistical procedures are typically introduced to help address the limitations of those already in practice or to deal with emerging research needs. Quantile regression (QR) is introduced in this paper as a relatively new methodology, which is intended to overcome some of the limitations of least squares mean regression (LMR). QR is more appropriate when assumptions of normality and homoscedasticity are violated. Also QR has been recommended as a good alternative when the research literature suggests that explorations of the relationship between variables need to move from a focus on average performance, that is, the central tendency, to exploring various locations along the entire distribution. Although QR has long been used in other fields, it has only recently gained popularity in educational statistics. For example, in the ongoing push for accountability and the need to document student improvement, the calculation of student growth percentiles (SGP) utilizes QR to document the amount of growth a student has made. Despite its proven advantages and its utility, QR has not been utilized in areas such as language testing research. This paper seeks to introduce the field to basic QR concepts, procedures, and interpretations. Researchers familiar with LMR will find the comparisons made between the two methodologies helpful to anchor the new information. Finally, an application with real data is employed to demonstrate the various analyses (the code is also appended) and to explicate the interpretations of results.

In a state-dependent queueing network, arrival and service rates, as well as routing probabilities, depend on the vector of queue lengths. For properly normalized such networks, we derive functional laws of large numbers (FLLNs) and functional central limit theorems (FCLTs). The former support fluid approximations and the latter support diffusion refinements. The fluid limit in FLLN is the unique solution to a multidimensional autonomous ordinary differential equation with state-dependent reflection. The diffusion limit in FCLT is the unique strong solution to a stochastic differential equation with time-dependent reflection. Examples are provided that demonstrate how such approximations facilitate the design, analysis and optimization of various manufacturing, service, communication and other systems.