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

Sreelakshmi N, Saparya Suresh, Sudheesh K. Kattumannil

Abstract

The Sen index and Sen-Shorrocks-Thon (SST) index are widely used measures of poverty indices. Developing reliable inference for these measures enables us to compare these measures in different populations of interest in an effective way. It is important to construct confidence intervals for the Sen index and SST index, which provide better coverage probability and shorter interval length. Motivated by this, we discuss empirical likelihood (EL) and jackknife empirical likelihood (JEL) based inference for the Sen index. To derive a JEL-based confidence interval for the Sen and SST indices, we propose a new estimator for the Sen index using the theory of U-statistics and examine its properties. The large sample properties of the EL and JEL ratio statistics are studied. We also discuss EL and JEL-based inference for the Sen-Shorrocks-Thon (SST) index. The finite sample performance of the EL and JEL-based confidence intervals of both Sen and SST indices is evaluated through a Monte Carlo simulation study. Finally, we illustrate our methods using individual-level data from the Panel Study of Income Dynamics (PSID) survey from the US as well as Indian household level income data for different states sourced from the Consumer Pyramids Household Survey (CPHS).

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

Abstract

The Sen index and Sen-Shorrocks-Thon (SST) index are widely used measures of poverty indices. Developing reliable inference for these measures enables us to compare these measures in different populations of interest in an effective way. It is important to construct confidence intervals for the Sen index and SST index, which provide better coverage probability and shorter interval length. Motivated by this, we discuss empirical likelihood (EL) and jackknife empirical likelihood (JEL) based inference for the Sen index. To derive a JEL-based confidence interval for the Sen and SST indices, we propose a new estimator for the Sen index using the theory of U-statistics and examine its properties. The large sample properties of the EL and JEL ratio statistics are studied. We also discuss EL and JEL-based inference for the Sen-Shorrocks-Thon (SST) index. The finite sample performance of the EL and JEL-based confidence intervals of both Sen and SST indices is evaluated through a Monte Carlo simulation study. Finally, we illustrate our methods using individual-level data from the Panel Study of Income Dynamics (PSID) survey from the US as well as Indian household level income data for different states sourced from the Consumer Pyramids Household Survey (CPHS).
Paper Structure (15 sections, 8 theorems, 122 equations, 4 figures, 17 tables)

This paper contains 15 sections, 8 theorems, 122 equations, 4 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Inference for the Sen index
  3. Non-parametric estimator of Sen index
  4. Empirical likelihood based confidence interval for the Sen index
  5. JEL based inference for Sen index
  6. Inference for SST index
  7. Unbiased estimator of SST index
  8. Empirical likelihood based confidence interval for the SST index
  9. JEL based inference for SST index
  10. Simulation results
  11. Comparison of Estimators
  12. Comparison of confidence intervals
  13. Illustration
  14. Conclusion
  15. Appendix

Key Result

Theorem 1

Given $z$, as $n\rightarrow \infty$, $\widehat{S}$ converges in probability to $S$.

Figures (4)

  • Figure 1: Graph showing the SST and Sen indices with jackknife empirical likelihood confidence intervals for different years
  • Figure 2: Proportion of unemployment benefit received by the individuals incomes in US for different years
  • Figure 3: Income distribution of different states in India
  • Figure 4: Graph showing the SST and Sen indices with jackknife empirical likelihood confidence intervals for different states of India

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8