# Asymptotic Theory and Statistical Decomposability gap Estimation for Takayama's Index

## TL;DR

This paper develops an asymptotic representation for Takayama's index, enabling better estimation of its decomposability gaps and demonstrating its practical decomposability in real data applications.

## Contribution

It introduces an asymptotic theory for Takayama's index and extends methods for estimating its decomposability gaps, which was previously unaddressed.

## Key findings

- Takayama's index has low decomposability gaps in real data.
- The asymptotic representation facilitates statistical analysis of Takayama's index.
- Practical applications support using Takayama's index as effectively decomposable.

## Abstract

In the spirit of recent asymptotic works on the General Poverty Index (GPI) in the field of Welfare Analysis, the asymptotic representation of the non-decomposable Takayama's index, which has failed to be incorporated in the unified GPI approach, is addressed and established here. This representation allows also to extend to it, recent results of statistical decomposability gaps estimations. The theoretical results are applied to real databases. The conclusions of the undertaken applications recommend to use Takayama's index as a practically decomposable one, in virtue of the low decomposability gaps with respect to the large values of the index.