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Empirical Likelihood Inference for Sen and Sen--Shorrocks--Thon Indices

This paper proposes and evaluates empirical likelihood and jackknife empirical likelihood methods for constructing reliable confidence intervals for the Sen and Sen-Shorrocks-Thon poverty indices, introducing a new U-statistic-based estimator and validating the approach through simulations and real-world data from the US and India.

Original authors: Sreelakshmi N, Saparya Suresh, Sudheesh K. Kattumannil

Published 2026-03-19

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CC BY 4.0

Original authors: Sreelakshmi N, Saparya Suresh, Sudheesh K. Kattumannil

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to measure how "poor" a group of people is. You can't just count heads; you need to know how deep the poverty is and how unfair the distribution is among the poor. Economists have built special tools for this called the Sen Index and the SST Index. Think of these as "Poverty Thermometers."

However, there's a problem with these thermometers: they are very complex. If you try to use them on a small group of people (like a specific village or a small survey), the old ways of calculating the results often give you a "foggy" reading. You might get a number, but you won't know how much to trust it. It's like trying to guess the temperature with a broken thermometer; you get a number, but you don't know if it's 20°C or 40°C.

This paper is about building a new, clearer lens to look at these poverty thermometers, so we can trust the results even when we have limited data.

Here is the breakdown of their solution using simple analogies:

1. The Problem: The "Foggy" Thermometer

The authors explain that the standard way to calculate these poverty scores (called "plug-in estimators") is like trying to balance a stack of Jenga blocks.

The Sen Index combines three things: How many people are poor, how poor they are on average, and how unequal the poor people are to each other.
Because these three parts are mixed together in a complicated math formula, small errors in the data get magnified.
When researchers tried to draw a "confidence interval" (a safety zone saying, "The real poverty score is likely between X and Y"), the old methods often made the zone too narrow or placed it in the wrong spot. It was like saying, "The temperature is definitely between 21 and 22 degrees," when it could actually be 15 or 30.

2. The Solution: The "U-Statistic" Blueprint

The authors decided to rebuild the thermometer from scratch using a mathematical tool called U-statistics.

The Analogy: Imagine you want to know the average height of a crowd. Instead of measuring everyone and doing a messy calculation, you pick two random people, measure the distance between them, and repeat this thousands of times. U-statistics is a way of breaking a complex problem down into tiny, simple pairs of data points.
By rewriting the Sen and SST indices as a collection of these simple pairs, the authors found a "clean" mathematical structure. This made it possible to understand exactly how much the result would wiggle if they took a different sample of people.

3. The New Tools: EL and JEL

Once they had the clean blueprint, they needed a way to draw the "safety zones" (confidence intervals) without making assumptions about the shape of the income data (which is often weird and lopsided, like a pyramid with a huge base).

They used two advanced statistical techniques:

Empirical Likelihood (EL): Think of this as a flexible rubber band. Instead of forcing the data into a rigid, pre-made shape (like a bell curve), the rubber band stretches and molds itself perfectly around the actual data you have. This gives a much more accurate safety zone.
Jackknife Empirical Likelihood (JEL): This is the "super-charged" version. The "Jackknife" part is like a chef tasting a soup. The chef takes a spoonful out, tastes it, then puts it back, then takes a different spoonful out, and tastes again. By seeing how the taste changes when one ingredient is missing, the chef can predict the flavor of the whole pot.
- The authors used this "tasting" method to make the rubber band (EL) even more stable and reliable, especially when the sample size is small.

4. The Proof: The Simulation Kitchen

To prove their new method worked, they ran a massive computer simulation (a "Monte Carlo study").

They created fake worlds with different types of income distributions (some where everyone is rich, some where a few are super rich and many are poor).
They tested their new "JEL rubber band" against the old "rigid ruler" method.
The Result: The old method often missed the true poverty score (low coverage) or gave a safety zone that was too narrow (overconfident). The new JEL method hit the target almost every time and gave a safety zone that was just the right size—reliable and honest.

5. Real World Testing: The US and India

Finally, they tested their new tools on real data:

USA: They looked at income data from the "Panel Study of Income Dynamics." They tracked poverty from 2017 to 2023.
- The Surprise: In 2021 (during the pandemic), their new tool showed poverty dropped. This seemed weird because the economy was crashing. But when they looked at unemployment benefits, they realized the government was sending so much money to people that, technically, poverty went down. Their tool captured this nuance perfectly.
India: They compared three states: Kerala (wealthier), Tamil Nadu (middle), and Bihar (poorer).
- Their tool correctly ranked them: Bihar had the highest poverty, and Kerala the lowest. The "safety zones" they drew were tight enough to clearly show the differences between the states.

The Bottom Line

This paper is about making poverty measurement more honest.

Before, if you had a small dataset, you might get a poverty score but you wouldn't know if it was accurate. The authors built a new mathematical "lens" (using U-statistics and Jackknife Empirical Likelihood) that lets us see the poverty score clearly, even with messy or small data.

In short: They took a complex, shaky measuring stick and replaced it with a flexible, self-adjusting ruler that tells us not just how much poverty there is, but exactly how sure we can be about that number. This helps policymakers make better decisions because they can trust the data more.

1. Problem Statement

The Sen index and the Sen-Shorrocks-Thon (SST) index are widely used, axiomatic measures of poverty that simultaneously capture incidence (headcount ratio), intensity (income gap), and inequality among the poor. Despite their theoretical importance, statistical inference for these indices is challenging due to:

Non-linear Structure: The indices involve complex interactions between the headcount ratio, mean income gap, and a Gini-type inequality component.
Analytical Complexity: Their sampling distributions are analytically intractable, especially under finite samples or complex survey designs.
Limitations of Existing Methods: Conventional inference relies on plug-in estimators combined with first-order asymptotic approximations (normal approximation). These methods often suffer from:
- Non-negligible bias in small to moderate samples.
- Unreliable coverage probabilities for confidence intervals.
- Difficulty in deriving consistent variance estimators due to the complex dependence structure.
Lack of U-Statistic Representation: Existing literature lacks a transparent representation of these indices within the classical U-statistic framework, hindering higher-order refinements and robust inference.

2. Methodology

The authors propose a unified framework based on U-statistics, Empirical Likelihood (EL), and Jackknife Empirical Likelihood (JEL) to address the inference gaps.

A. New U-Statistic Estimators

The paper establishes an explicit symmetric kernel representation for both indices, embedding them into the theory of U-statistics.

Sen Index: The authors derive a new non-parametric estimator ( $\hat{S}$ ) using a symmetric kernel of degree 2. They prove that this estimator is consistent and asymptotically unbiased. The estimator is expressed as a ratio of two U-statistics ( $\Delta_1$ and $\Delta_2$ ), which simplifies to a closed-form expression using order statistics.
SST Index: Similarly, an unbiased U-statistic estimator ( $\hat{S}_h$ ) is proposed for the SST index. The authors rewrite the integral definition of the SST index into an expectation involving $\min(X_1, X_2)$ , allowing for the construction of a degree-2 kernel.
Asymptotic Properties: The paper proves that $\sqrt{n}(\hat{S} - S)$ and $\sqrt{n}(\hat{S}_h - S_h)$ converge in distribution to Gaussian random variables with zero mean and specific variance structures derived via Hoeffding decomposition.

B. Empirical Likelihood (EL) and Jackknife Empirical Likelihood (JEL)

To overcome the computational difficulties of constructing confidence intervals for non-linear U-statistics (which involve non-linear constraints in standard EL), the authors employ the Jackknife Empirical Likelihood (JEL) approach.

JEL Construction:
1. Define the estimating equation for the U-statistic.
2. Compute Jackknife pseudo-values ( $\hat{V}_k$ ) by removing one observation at a time: $\hat{V}_k = n\hat{\theta} - (n-1)\hat{\theta}_{-k}$ .
3. Construct the empirical likelihood ratio based on these pseudo-values, which are approximately independent.
4. Maximize the likelihood subject to the constraint that the weighted sum of pseudo-values equals zero.
Theoretical Justification: The authors prove a Wilks-type theorem for both the Sen and SST indices. They demonstrate that the log-likelihood ratio statistics ( $-2\log L$ for EL and $J(S)$ for JEL) converge to a $\chi^2_1$ distribution as $n \to \infty$ . This allows for the construction of confidence intervals without explicitly estimating the asymptotic variance.

3. Key Contributions

Unified U-Statistic Framework: The paper provides the first explicit symmetric kernel representation for the Sen and SST indices, formally embedding them in U-statistic theory. This facilitates the derivation of influence functions and asymptotic variances.
Novel Estimators: Introduction of new, simpler, and unbiased U-statistic-based estimators for both indices, which are shown to have lower bias compared to existing plug-in and Davidson (2009) estimators.
Robust Inference Procedures: Development of EL and JEL-based confidence intervals that do not require explicit variance estimation. The JEL approach specifically addresses the computational challenges of non-linear constraints in standard EL.
Theoretical Proofs: Rigorous proofs of consistency, asymptotic normality, and the Wilks-type limiting distribution for the proposed likelihood ratio statistics.

4. Results

The performance of the proposed methods was evaluated through extensive Monte Carlo simulations and empirical illustrations.

A. Simulation Study

Bias and MSE: The proposed U-statistic estimators demonstrated significantly lower bias compared to the plug-in estimator (for Sen) and the Davidson/Shorrocks estimators (for SST) across various distributions (Exponential, Pareto, Log-normal) and sample sizes ( $n=20$ to $100$). Mean Squared Errors (MSE) were comparable or slightly better.
Confidence Intervals:
- Coverage Probability: The proposed EL and JEL confidence intervals achieved coverage probabilities much closer to the nominal levels (e.g., 95%) compared to the normal-approximation-based intervals (Davidson, 2009), particularly in small samples and skewed distributions.
- Interval Width: While normal-based intervals were often narrower, they suffered from under-coverage (unreliable). The JEL intervals maintained appropriate width while ensuring reliability.

B. Empirical Illustrations

US Data (PSID): Analysis of individual income data from 2017, 2019, 2021, and 2023.
- Results showed a consistent decline in poverty (Sen and SST indices) over time.
- The 2021 data, despite the pandemic, showed a drop in poverty, which the authors attribute to increased unemployment benefits and transfers, a trend validated by the proposed estimators.
Indian Data (CPHS): Analysis of household income data from Kerala, Tamil Nadu, and Bihar (2023).
- The indices correctly captured the known poverty hierarchy: Bihar (Highest Poverty) > Tamil Nadu > Kerala (Lowest Poverty).
- The confidence intervals provided statistically significant distinctions between the states, validating the method's ability to handle real-world, heterogeneous data.

5. Significance

Methodological Advancement: The paper bridges a critical gap in poverty measurement by providing a rigorous statistical framework for the Sen and SST indices, moving beyond heuristic plug-in methods.
Practical Utility: The JEL-based inference is particularly valuable for policymakers and researchers working with small to moderate sample sizes or skewed income distributions, where traditional normal approximations fail.
Reliability: By ensuring better coverage probabilities, the proposed methods reduce the risk of drawing incorrect conclusions about poverty trends or regional disparities.
Future Directions: The framework is extensible to survey-weighted settings, stochastic dominance testing, and multidimensional poverty indices, offering a robust foundation for future welfare analysis.

In conclusion, this work establishes a theoretically sound and practically superior methodology for estimating and making inferences about some of the most important poverty indices in economics.