Nonparametric methods for comparing distribution functionals for dependent samples with application to inequality measures
Abstract
This paper proposes asymptotically distribution-free inference methods for comparing a broad range of welfare indices across dependent samples, including those employed in inequality, poverty, and risk analysis. Two distinct situations are considered. \emph{First}, we propose asymptotic and bootstrap intersection methods which are completely robust to arbitrary dependence between two samples. \emph{Second}, we focus on the common case of overlapping samples -- a special form of dependent samples where sample dependence arises solely from matched pairs -- and provide asymptotic and bootstrap methods for comparing indices. We derive consistent estimates for asymptotic variances using the influence function approach. The performance of the proposed methods is studied in a simulation experiment: we find that confidence intervals with overlapping samples exhibit satisfactory coverage rates with reasonable precision, whereas conventional methods based on an assumption of independent samples have an inferior performance in terms of coverage rates and interval widths. Asymptotic inference can be less reliable when dealing with heavy-tailed distributions, while the bootstrap method provides a viable remedy, unless the variance is substantial or nonexistent. The intersection method yields reliable results with arbitrary dependent samples, including instances where overlapping samples are not feasible. We demonstrate the practical applicability of our proposed methods in analyzing dynamic changes in household financial inequality in Italy over time.
Summary
This paper addresses the problem of comparing welfare indices (e.g., inequality, poverty, risk measures) across different samples, particularly when those samples are dependent. The authors identify that traditional methods often assume independence or complete dependence (matched pairs), while many real-world datasets, such as rotating panel surveys, exhibit partial dependence due to overlapping observations. To overcome this, the paper proposes two main approaches: (1) Asymptotic and bootstrap intersection methods (IMs) that are robust to arbitrary dependence between samples, and (2) Asymptotic and bootstrap methods specifically designed for overlapping samples (OS), where dependence arises solely from matched pairs. The authors derive consistent estimates for asymptotic variances using influence functions and evaluate the methods through simulation experiments. The simulations demonstrate that ignoring sample dependence can lead to unreliable confidence intervals, while the proposed methods, especially those tailored for OS, provide improved coverage rates and precision. Finally, the paper applies these methods to analyze changes in household financial inequality in Italy. The key finding is that properly accounting for sample dependence is crucial for accurate inference when comparing welfare indices. The OS methods generally outperform conventional methods (assuming independence) when the OS structure holds. The IMs, although somewhat conservative, offer robustness when the dependence structure is unknown or when OS assumptions are violated. The paper contributes by providing a unified framework for a broad class of welfare measures, developing methods robust to arbitrary dependence, and offering efficient inference for overlapping samples with a consistent variance estimator. This matters to the field of applied economics and statistics because it provides practitioners with reliable tools for comparing welfare indices in situations where sample dependence is a significant concern.
Key Insights
- •Novelty: The paper introduces asymptotic and bootstrap intersection methods (IMs) that are completely robust to arbitrary dependence between two samples. This is a valuable contribution as it provides a reliable inference method even when the dependence structure is unknown or complex.
- •Overlapping Samples Methodology: The authors develop asymptotic and bootstrap methods specifically tailored for overlapping samples (OS), which are common in survey data. This allows for more efficient inference compared to the more general IMs.
- •Influence Function Application: The paper utilizes influence functions to derive asymptotic distributions and estimate asymptotic variances in a computationally efficient manner. This approach extends the existing literature on influence functions to dependent samples.
- •Simulation Results: Simulations show that conventional methods assuming independent samples can have inferior performance in terms of coverage rates and interval widths when the samples are indeed dependent. In some cases, ignoring dependence can lead to confidence intervals with coverage rates much lower than the nominal 95%.
- •Variance Estimation: The authors provide a consistent estimator for the asymptotic variance in the overlapping samples case, addressing a limitation of previous estimators that were not guaranteed to be positive definite.
- •Heavy-Tailed Distributions: The paper finds that asymptotic inference can be less reliable when dealing with heavy-tailed distributions. The bootstrap method provides a viable remedy, unless the variance is substantial or nonexistent.
- •Empirical Application: The paper demonstrates the practical applicability of the proposed methods by analyzing dynamic changes in household financial inequality in Italy over time, revealing distinct patterns of internal inequality within different regions.
Practical Implications
- •Real-world Applications: The methods developed in this paper are directly applicable to a wide range of economic analyses involving comparisons of welfare indices across different groups or time periods, particularly when dealing with survey data that exhibits overlapping samples.
- •Beneficiaries: Researchers and practitioners in economics, public policy, and social sciences who work with survey data or other forms of dependent samples will benefit from the methods developed in this paper.
- •Practical Use: Practitioners can use the proposed asymptotic and bootstrap methods, along with the consistent variance estimator, to construct more reliable confidence intervals for differences in welfare indices when dealing with dependent samples. The choice between IM and OS methods depends on the knowledge about the dependence structure.
- •Future Research: The paper opens up avenues for future research, such as extending the methods to more complex survey designs (e.g., clustered data), investigating alternative bootstrap procedures, and exploring the application of these methods to other fields where dependent samples are prevalent.