# 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.**

## References

- Arvanitis, Stelios, and Antonis Demos. 2015. A class of indirect inference estimators: Higher-order asymptotics and approximate bias correction. The Econometrics Journal 18: 200–41. [Google Scholar] [CrossRef]
- Bandourian, Ripsy, James McDonald, and Robert S. Turley. 2002. A Comparison of Parametric Models of Income Distribution Across Countries and over Time. Available online: http://www.lisdatacenter.org/wps/liswps/305.pdf (accessed on 28 November 2017).
- Beirlant, Jan, Goedele Dierckx, A. Guillou, and Catalin Stărică. 2002. On exponential representations of log-spacings of extreme order statistics. Extremes 5: 157–80. [Google Scholar] [CrossRef]
- Biewen, Martin. 2002. Bootstrap inference for inequality, mobility and poverty measurement. Journal of Econometrics 108: 317–42. [Google Scholar] [CrossRef]
- Cowell, Frank A., and Emmanuel Flachaire. 2007. Income distribution and inequality measurement: The problem of extreme values. Journal of Econometrics 141: 1044–72. [Google Scholar] [CrossRef] [Green Version]
- Cowell, Frank A., and Emmanuel Flachaire. 2015. Statistical Methods for Distributional Analysis. In Handbook of Income Distribution. Edited by François Bourguignon and Anthony B. Atkinson. Amsterdam: Elsevier, vol. 2, pp. 359–465. [Google Scholar]
- Cowell, Frank A., and Maria-Pia Victoria-Feser. 1996. Robustness properties of inequality measures. Econometrica 64: 77–101. [Google Scholar] [CrossRef]
- Cowell, Frank A., and Maria-Pia Victoria-Feser. 2000. Distributional analysis: A robust approach. In Putting Economics to Work, Volume in Honour of Michio Morishima. Edited by Anthony Atkinson, Howard Glennerster and Nicholas Stern. London: STICERD. [Google Scholar]
- Cowell, Frank A., and Maria-Pia Victoria-Feser. 2002. Welfare rankings in the presence of contaminated data. Econometrica 70: 1221–33. [Google Scholar] [CrossRef]
- Cowell, Frank A., and Maria-Pia Victoria-Feser. 2003. Distribution-free inference for welfare indices under complete and incomplete information. Journal of Economic Inequality 1: 191–219. [Google Scholar] [CrossRef]
- Dagum, Camilo. 1977. A new model of personal income distribution: Specification and estimation. Economie Appliquée 30: 413–36. [Google Scholar]
- Danielsson, Jon, Lauredns de Haan, Liang Peng, and Casper G. de Vries. 2001. Using a bootstrap method to choose sample fraction in Tail index estimation. Journal of Multivariate Analysis 76: 226–48. [Google Scholar] [CrossRef]
- Davidson, Russell. 2009. Reliable inference for the Gini index. Journal of Econometrics 150: 30–40. [Google Scholar] [CrossRef]
- Davidson, Russell. 2010. Innis lecture: Inference on income distributions. Canadian Journal of Economics 43: 1122–48. [Google Scholar] [CrossRef]
- Davidson, Russell. 2012. Statistical inference in the presence of heavy tails. Econometrics Journal 15: 31–53. [Google Scholar] [CrossRef]
- Davidson, Russell, and Emmanuel Flachaire. 2007. Asymptotic and bootstrap inference for inequality and poverty measures. Journal of Econometrics 141: 141–66. [Google Scholar] [CrossRef] [Green Version]
- Dupuis, Debbie J., and Maria-Pia Victoria-Feser. 2006. A robust prediction error criterion for Pareto modeling of upper tails. Canadian Journal of Statistics 34: 639–58. [Google Scholar] [CrossRef]
- Flachaire, Emmanuel, and Olivier G. Nuñez. 2007. Estimation of income distribution and detection of subpopulations: An explanatory model. Computational Statistics & Data Analysis 51: 3368–80. [Google Scholar]
- Gallant, A. Ronald, and George Tauchen. 1996. Which moments to match? Econometric Theory 12: 657–81. [Google Scholar] [CrossRef]
- Guerrier, Stephane, Elise Dupuis, Yanyuan Ma, and Maria-Pia Victoria-Feser. 2018. Simulation based bias correction methods for complex models. Journal of the American Statistical Association (Theory & Methods). in press. [Google Scholar] [CrossRef]
- Guillou, Armelle, and Peter Hall. 2001. A diagnostic for selecting the threshold in extreme-value analysis. Journal of the Royal Statistical Society, Series B 63: 293–305. [Google Scholar] [CrossRef]
- Hall, Peter. 1992. The Bootstrap and Edgeworth Expansions. New York: Springer Verlag. [Google Scholar]
- Hampel, Frank R. 1974. The influence curve and its role in robust estimation. Journal of the American Statistical Association 69: 383–93. [Google Scholar] [CrossRef]
- Hampel, Frank R., Elvezio M. Ronchetti, Peter J. Rousseeuw, and Werner A. Stahel. 1986. Robust Statistics: The Approach Based on Influence Functions. New York: John Wiley. [Google Scholar]
- Hansen, Lars Peter. 1982. Large sample properties of generalized method of moments estimators. Econometrica 50: 1029–54. [Google Scholar] [CrossRef]
- Heggland, Knut, and Arnoldo Frigessi. 2004. Estimating functions in indirect inference. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 66: 447–62. [Google Scholar] [CrossRef]
- Kleiber, Christian, and Samuel Kotz. 2003. Statistical Size Distributions in Economics and Actuarial Sciences. New York: John Wiley & Sons, vol. 470. [Google Scholar]
- McDonald, James B. 1984. Some generalized functions for the size distribution of income. Econometrica 52: 647–64. [Google Scholar] [CrossRef]
- McDonald, James B., and Yexiao J. Xu. 1995. A generalization of the beta distribution with applications. Journal of Econometrics 66: 133–52. [Google Scholar] [CrossRef]
- McFadden, Daniel. 1989. Method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57: 995–1026. [Google Scholar] [CrossRef]
- Mills, Jeffrey A., and Sourushe Zandvakili. 1997. Statistical inference via bootstrapping for measures of inequality. Journal of Applied Econometrics 12: 133–50. [Google Scholar] [CrossRef]
- Newey, Whitney K., and Daniel McFadden. 1994. Large sample estimation and hypothesis testing. In Handbook of Econometrics. Amsterdam: Elsevier, vol. 4, pp. 2111–245. [Google Scholar]
- Phillips, Peter C. B. 2012. Folklore theorems, implicit maps, and indirect inference. Econometrica 80: 425–54. [Google Scholar]
- Rudin, Walter. 1976. Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics). New York: McGraw-Hill Education. [Google Scholar]
- Schluter, Christian. 2012. On the problem of inference for inequality measures for heavy-tailed distributions. The Econometrics Journal 15: 125–53. [Google Scholar] [CrossRef]
- Schluter, Christian, and Kees Jan van Garderen. 2009. Edgeworth expansions and normalizing transforms for inequality measures. Journal of Econometrics 150: 16–29. [Google Scholar] [CrossRef]
- Singh, S. K., and G. S. Maddala. 1976. A function for the size distribution of income. Econometrica 44: 963–70. [Google Scholar] [CrossRef]
- Storn, Rainer, and Kenneth Price. 1997. Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11: 341–59. [Google Scholar] [CrossRef]
- Van der Vaart, Aad W. 1998. Asymptotic Statistics. Cambridge: Cambridge University Press, vol. 3. [Google Scholar]
- Victoria-Feser, Maria-Pia. 1999. Comment on Giorgi’s chapter: The sampling properties of inequality indices. In Income Inequality Measurement: From Theory to Practice. Edited by J. Silber. Boston: Kluwer Academic Publisher, pp. 260–67. [Google Scholar]
- Victoria-Feser, Maria-Pia. 2000. A general robust approach to the analysis of income distribution, inequality and poverty. International Statistical Review 68: 277–93. [Google Scholar] [CrossRef]

**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