# Parametric Inference for Index Functionals

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## Abstract

**:**

## 1. Introduction

- The (nonparametric) bootstrap is a distribution-free approach that allows to derive the sample distribution of $T({F}^{(n)})$ from which quantiles (for confidence intervals) and variance (for testing) can be estimated; for application to inequality indices, see e.g., Mills and Zandvakili (1997) and Biewen (2002).
- Another distribution-free approach consists in deriving the asymptotic variance of the index using the Influence Function ($IF$) of Hampel (1974) (see also Hampel et al. 1986) as is done in Cowell and Victoria-Feser (2003) (for different types of data features such as censoring and truncating) and estimate it directly from the sample (see also Victoria-Feser 1999; Cowell and Flachaire 2015).
- A parametric (and asymptotic) approach, given a chosen parametric model ${F}_{\theta}$ for the data generating model, consists in first consistently estimating $\theta $, say $\widehat{\theta}$, then considering its asymptotic properties such as its variance $\mathrm{var}(\widehat{\theta})$ and derive the corresponding asymptotic variance of $T({F}_{\widehat{\theta}})$ using e.g., the delta method (based on a first order Taylor series expansion).
- A parametric (finite sample) approach, given a chosen parametric model ${F}_{\theta}$ for the data generating model, consists in first consistently estimating $\theta $, say $\widehat{\theta}$, then using parametric bootstrap to derive the sample distribution of $T({F}_{\widehat{\theta}})$ from which quantiles (for confidence intervals) and variance (for testing) can be estimated.
- Refinements and combinations of these approaches.

## 2. A Target Matching Estimator

Algorithm 1: TME-percentile confidence interval |

## 3. Asymptotic Properties

**Theorem**

**1.**

## 4. Simulation Study

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

**Proof.**

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**Figure 1.**Illustration of the coverage probabilities obtained over 10,000 Monte Carlo experiments for the GB2 (

**a**) (see Table 1) and the Singh-Madalla (

**b**) (see Table 2). Each color represents a different method. The shade area around each line is the 99.9% asymptotic confidence interval for proportion. The black line is the nominal confidence level of 95%.

**Table 1.**Finite sample coverage probability with respect to a nominal confidence level (two-sided) of 95% for the Theil Index. Data are simulated under the GB2 with ${\theta}^{s}={(a=3,p=3.5,q=0.8)}^{\prime}$, ${\theta}^{c}=(b=10)$. $\nu ={[T(x),{U}_{2}(x),{U}_{3}(x)]}^{\prime}$ with $T(x)$ the Theil index. In Algorithm 1, $b=1$. The experiment is repeated ${10}^{4}$ times and $B={10}^{3}$.

Sample Size | Boot | MLE | TME |
---|---|---|---|

$n=250$ | 0.708 | 0.962 | 0.927 |

$n=500$ | 0.753 | 0.978 | 0.942 |

$n=1000$ | 0.790 | 0.990 | 0.949 |

**Table 2.**Finite sample coverage probability with respect to a nominal confidence level (two-sided) of 95% for the Theil Index. The values for Varstab, Semip and Mixture are directly reported from (Cowell and Flachaire 2015, Table 6.6). Data are simulated under the Singh-Madalla with $n=500$, ${\theta}^{s}=(a=2.8,q)$, ${\theta}^{c}=(b=0.193)$. The parameter q accounts for the shape of the upper tail of the distribution, the smaller the heavier the tail. $\nu ={[T(x),{U}_{2}(x)]}^{\prime}$ with $T(x)$ the Theil index. In Algorithm 1, $b=1$. The experiment is repeated ${10}^{4}$ times and $B={10}^{3}$.

Singh-Madalla | Varstab | Semip | Mixture | Boot | MLE | TME |
---|---|---|---|---|---|---|

$q=1.7$ | 0.933 | 0.926 | 0.928 | 0.912 | 0.962 | 0.952 |

$q=1.2$ | 0.899 | 0.905 | 0.912 | 0.859 | 0.979 | 0.957 |

$q=0.7$ | 0.796 | 0.871 | 0.789 | 0.637 | 0.994 | 0.939 |

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**MDPI and ACS Style**

Guerrier, S.; Orso, S.; Victoria-Feser, M.-P.
Parametric Inference for Index Functionals. *Econometrics* **2018**, *6*, 22.
https://doi.org/10.3390/econometrics6020022

**AMA Style**

Guerrier S, Orso S, Victoria-Feser M-P.
Parametric Inference for Index Functionals. *Econometrics*. 2018; 6(2):22.
https://doi.org/10.3390/econometrics6020022

**Chicago/Turabian Style**

Guerrier, Stéphane, Samuel Orso, and Maria-Pia Victoria-Feser.
2018. "Parametric Inference for Index Functionals" *Econometrics* 6, no. 2: 22.
https://doi.org/10.3390/econometrics6020022