# Gini Index Estimation within Pre-Specified Error Bound: Application to Indian Household Survey Data

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## Abstract

**:**

## 1. Introduction

## 2. Survey Design and Point Estimation

## 3. Bounded Width Confidence Intervals

#### Estimation of ${\xi}^{2}$

## 4. Sequential Methodology

#### 4.1. Purely Sequential Procedure

#### 4.2. Two-Stage Procedure

#### 4.3. Pilot Cluster Size

## 5. Characteristics of the Procedures and Simulation Study

#### 5.1. Characteristics

**Theorem**

**1.**

- (i)
- $\frac{N}{C}\to 1$ in probability,
- (ii)
- $\frac{Q}{C}\to 1$ in probability, and
- (iii)
- $\frac{2{z}_{\alpha /2}{V}_{N}}{\sqrt{N}}\le \omega$.

**Proof of Theorem 1.**

- (i)
- The definition of stopping rule N associated with the purely sequential procedure in (8) yields$${\left(\frac{2{z}_{\alpha /2}}{\omega}\right)}^{2}\phantom{\rule{0.166667em}{0ex}}{V}_{N}^{2}\phantom{\rule{0.166667em}{0ex}}\le N\phantom{\rule{0.166667em}{0ex}}\le t\mathbf{1}(N=t)\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\left(\frac{2{z}_{\alpha /2}}{\omega}\right)}^{2}\left({V}_{N-1}^{2}+\frac{1}{N-1}\right).$$Furthermore, $tPr(N=t)/C\le t/C\to 0$ as $\omega \downarrow 0$. Hence, dividing all sides of (14) by C and letting $\omega \downarrow 0$, we prove $N/C\to 1$ in probability as $\omega \downarrow 0$.
- (ii)
- The definition of final cluster size Q related to the two-stage procedure in (10) yields$${\left(\frac{2{z}_{\alpha /2}}{\omega}\right)}^{2}\phantom{\rule{0.166667em}{0ex}}{V}_{t}^{2}\phantom{\rule{0.166667em}{0ex}}\le Q\phantom{\rule{0.166667em}{0ex}}\le t\mathbf{1}(Q=t)\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{\left(\frac{2{z}_{\alpha /2}}{\omega}\right)}^{2}\left({V}_{t}^{2}+\frac{1}{t}\right).$$Furthermore, $tPr(Q=t)/C\le t/C\to 0$ as $\omega \downarrow 0$. Now, ${V}_{t}^{2}\to {\xi}^{2}$ in probability as $\omega \downarrow 0$. Hence, dividing all sides of (15) by C and letting $\omega \downarrow 0$, we prove $Q/C\to 1$ in probability as $\omega \downarrow 0$.
- (iii)
- Using stopping rule N in (8) we have, for all N,$$\begin{array}{ccc}\hfill {\left(\frac{2{z}_{\alpha /2}}{\omega}\right)}^{2}{V}_{N}^{2}\le N& \Rightarrow & \frac{4{z}_{\alpha /2}^{2}}{N}{V}_{N}^{2}\le {\omega}^{2}\hfill \\ & \Rightarrow & 2{z}_{\alpha /2}\frac{{V}_{N}}{\sqrt{N}}\le \omega \hfill \end{array}$$

#### 5.2. Simulation Study

## 6. Gini Index Estimation in India

#### 6.1. Application of Purely Sequential Procedure (PSP)

#### 6.2. Application of Two-Stage Procedure

## 7. Extension: Narrow Confidence Region

## 8. Discussion

## 9. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | The survey excluded “(i) Leh (Ladakh) and Kargil districts of Jammu and Kashmir (for central sample), (ii) interior villages of Nagaland situated beyond 5 km of the bus route and (ii) villages of Andaman and Nicobar Islands which remain inaccessible throughout the year.” (National Sample Survey Office 2007). |

Region | ${\widehat{\mathit{G}}}_{\mathit{H}}$ | H | $\widehat{\mathit{C}}$ | N | ${\widehat{\mathit{G}}}_{\mathit{N}}$ | Lower CI | Upper CI | ${\mathit{w}}_{\mathit{N}}$ | $Pr({\mathit{N}}_{\mathit{s}}<{\widehat{\mathit{C}}}_{\mathit{s}})$ |
---|---|---|---|---|---|---|---|---|---|

$\mathit{se}({\widehat{\mathit{G}}}_{\mathit{H}})$ | $\left(\mathit{t}\right)$ | $\mathit{se}({\widehat{\mathit{G}}}_{\mathit{N}})$ | |||||||

Uttar Pradesh | |||||||||

All | 0.2163 | 1262 | 622 | 672 | 0.2116 | 0.2023 | 0.2209 | 0.0186 | 0.2138 |

(0.0042) | (321) | (0.0057) | |||||||

Rural | 0.1997 | 903 | 505 | 523 | 0.2024 | 0.1931 | 0.2117 | 0.0186 | 0.4 |

(0.0041) | (198) | (0.0057) | |||||||

Urban | 0.2229 | 359 | 903 | 359 | 0.2229 | 0.2077 | 0.2381 | 0.0304 | 1.0 |

(0.0092) | (180) | (0.0092) | |||||||

West Bengal | |||||||||

All | 0.2320 | 878 | 587 | 593 | 0.2334 | 0.2239 | 0.2430 | 0.0191 | 0.1282 |

(0.0051) | (190) | (0.0058) | |||||||

Rural | 0.1812 | 551 | 450 | 450 | 0.1816 | 0.1723 | 0.1909 | 0.0186 | 0.2353 |

(0.0048) | (172) | (0.0057) | |||||||

Urban | 0.2609 | 327 | 612 | 327 | 0.2609 | 0.2482 | 0.2736 | 0.0254 | 1.0 |

(0.0077) | (185) | (0.0077) |

Region | ${\widehat{\mathit{G}}}_{\mathit{H}}$ | H | $\widehat{\mathit{C}}$ | N | ${\widehat{\mathit{G}}}_{\mathit{N}}$ | Lower CI | Upper CI | ${\mathit{w}}_{\mathit{N}}$ | $Pr({\mathit{N}}_{\mathit{s}}<{\widehat{\mathit{C}}}_{\mathit{s}})$ |
---|---|---|---|---|---|---|---|---|---|

$\mathit{se}({\widehat{\mathit{G}}}_{\mathit{H}})$ | $\left(\mathit{t}\right)$ | $\mathit{se}({\widehat{\mathit{G}}}_{\mathit{N}})$ | |||||||

Uttar Pradesh | |||||||||

All | 0.2163 | 1262 | 834 | 878 | 0.2117 | 0.2022 | 0.2212 | 0.0190 | 0.2138 |

(0.0042) | (333) | (0.0048) | |||||||

Rural | 0.1997 | 903 | 643 | 667 | 0.2024 | 0.1930 | 0.2117 | 0.0187 | 0.4 |

(0.0041) | (226) | (0.0048) | |||||||

Urban | 0.2229 | 359 | 1282 | 359 | 0.2229 | 0.2048 | 0.2410 | 0.0362 | 1.0 |

(0.0092) | (254) | (0.0092) | |||||||

West Bengal | |||||||||

All | 0.2320 | 878 | 906 | 878 | 0.2320 | 0.2221 | 0.2419 | 0.0198 | 1.0 |

(0.0051) | (223) | (0.0051) | |||||||

Rural | 0.181 | 551 | 552 | 551 | 0.1812 | 0.1719 | 0.1906 | 0.01871 | 1.0 |

(0.0048) | (203) | (0.0048) | |||||||

Urban | 0.2609 | 327 | 869 | 327 | 0.2609 | 0.2458 | 0.2761 | 0.0303 | 1.0 |

(0.0077) | (207) | (0.0077) |

Region | ${\widehat{\mathit{G}}}_{\mathit{H}}$ | H | $\widehat{\mathit{C}}$ | N | ${\widehat{\mathit{G}}}_{\mathit{N}}$ | Lower CI | Upper CI | ${\mathit{w}}_{\mathit{N}}$ | $Pr({\mathit{N}}_{\mathit{s}}<{\widehat{\mathit{C}}}_{\mathit{s}})$ |
---|---|---|---|---|---|---|---|---|---|

$\mathit{se}({\widehat{\mathit{G}}}_{\mathit{H}})$ | $\left(\mathit{t}\right)$ | $\mathit{se}({\widehat{\mathit{G}}}_{\mathit{N}})$ | |||||||

Uttar Pradesh | |||||||||

All | 0.2163 | 1262 | 401 | 540 | 0.2138 | 0.2035 | 0.2242 | 0.0207 | 0.0 |

(0.0042) | (302) | (0.0063) | |||||||

Rural | 0.1997 | 903 | 386 | 400 | 0.2014 | 0.1899 | 0.2130 | 0.0231 | 0.1714 |

(0.0041) | (168) | (0.0070) | |||||||

Urban | 0.2229 | 359 | 578 | 359 | 0.2229 | 0.2077 | 0.2381 | 0.0304 | 1.0 |

(0.0092) | (168) | (0.0092) | |||||||

West Bengal | |||||||||

All | 0.2320 | 878 | 324 | 319 | 0.2288 | 0.2175 | 0.2401 | 0.0226 | 0.1795 |

(0.0051) | (158) | (0.0069) | |||||||

Rural | 0.1812 | 551 | 276 | 289 | 0.1829 | 0.1721 | 0.1937 | 0.0216 | 0.2353 |

(0.00477) | (138) | (0.0066) | |||||||

Urban | 0.2609 | 327 | 392 | 327 | 0.2609 | 0.2482 | 0.2736 | 0.0254 | 1.0 |

(0.0077) | (142) | (0.0077) |

Region | ${\widehat{\mathit{G}}}_{\mathit{H}}$ | H | $\widehat{\mathit{C}}$ | N | ${\widehat{\mathit{G}}}_{\mathit{N}}$ | Lower CI | Upper CI | ${\mathit{w}}_{\mathit{N}}$ | $Pr({\mathit{N}}_{\mathit{s}}<{\widehat{\mathit{C}}}_{\mathit{s}})$ |
---|---|---|---|---|---|---|---|---|---|

$\mathit{se}({\widehat{\mathit{G}}}_{\mathit{H}})$ | $\left(\mathit{t}\right)$ | $\mathit{se}({\widehat{\mathit{G}}}_{\mathit{N}})$ | |||||||

Uttar Pradesh | |||||||||

All | 0.2163 | 1262 | 572 | 653 | 0.2123 | 0.2010 | 0.2236 | 0.0226 | 0.2138 |

(0.0042) | (728) | (0.0058) | |||||||

Rural | 0.1997 | 903 | 496 | 510 | 0.2010 | 0.1893 | 0.2128 | 0.0234 | 0.1714 |

(0.0041) | (197) | (0.0060) | |||||||

Urban | 0.2229 | 359 | 821 | 359 | 0.2229 | 0.2048 | 0.2410 | 0.0362 | 1.0 |

(0.0092) | (717) | (0.0092) | |||||||

West Bengal | |||||||||

All | 0.2320 | 878 | 517 | 519 | 0.2318 | 0.2199 | 0.2437 | 0.0238 | 0.1538 |

(0.0051) | (186) | (0.0061) | |||||||

Rural | 0.1812 | 551 | 351 | 352 | 0.1815 | 0.1703 | 0.1927 | 0.0223 | 0.2353 |

(0.0048) | (163) | (0.0057) | |||||||

Urban | 0.2609 | 327 | 556 | 327 | 0.2609 | 0.2458 | 0.2761 | 0.0303 | 1.0 |

(0.0077) | (162) | (0.0077) |

**Table 5.**Application results for the two-stage procedure on NSS 64th round data for $\alpha =0.1$ and $\omega =0.02$.

Region | H | ${\mathit{Q}}^{\ast}$ | $\tilde{\mathit{Q}}$ | ${\widehat{\mathit{G}}}_{\mathit{H}}$ | ${\widehat{\mathit{G}}}_{\tilde{\mathit{Q}}}$ | Lower CI | Upper CI | ${\mathit{w}}_{\tilde{\mathit{Q}}}$ |
---|---|---|---|---|---|---|---|---|

$\left(\mathit{t}\right)$ | $\left(\mathit{Q}\right)$ | $\left(\mathit{se}\right({\widehat{\mathit{G}}}_{\mathit{H}}\left)\right)$ | $\left(\mathit{se}\right({\widehat{\mathit{G}}}_{\tilde{\mathit{Q}}}\left)\right)$ | |||||

Uttar Pradesh | ||||||||

All | 1262 | 1146 | 1171 | 0.2163 | 0.2137 | 0.2072 | 0.2202 | 0.0131 |

(321) | (1146) | (0.0042) | (0.0040) | |||||

Rural | 903 | 398 | 406 | 0.1997 | 0.2027 | 0.1940 | 0.2114 | 0.0174 |

(198) | (398) | (0.0041) | (0.0053) | |||||

Urban | 359 | 1177 | 359 | 0.2229 | 0.2229 | 0.2077 | 0.2381 | 0.0304 |

(180) | (359) | (0.0092) | (0.0092) | |||||

West Bengal | ||||||||

All | 878 | 624 | 626 | 0.2320 | 0.2307 | 0.2216 | 0.2398 | 0.0182 |

(190) | (624) | (0.0051) | (0.0055) | |||||

Rural | 551 | 422 | 420 | 0.1812 | 0.1785 | 0.1707 | 0.1862 | 0.0155 |

(173) | (422) | (0.0048) | (0.0047) | |||||

Urban | 327 | 857 | 327 | 0.2609 | 0.2609 | 0.2482 | 0.2736 | 0.0254 |

(185) | (327) | (0.0077) | (0.0077) |

**Table 6.**Application results for the two-stage procedure on NSS 64th round data for $\alpha =0.05$ and $\omega =0.02$.

Region | H | ${\mathit{Q}}^{\ast}$ | $\tilde{\mathit{Q}}$ | ${\widehat{\mathit{G}}}_{\mathit{H}}$ | ${\widehat{\mathit{G}}}_{\tilde{\mathit{Q}}}$ | Lower CI | Upper CI | ${\mathit{w}}_{\tilde{\mathit{Q}}}$ |
---|---|---|---|---|---|---|---|---|

$\left(\mathit{t}\right)$ | $\left(\mathit{Q}\right)$ | $\left(\mathit{se}\right({\widehat{\mathit{G}}}_{\mathit{H}}\left)\right)$ | $\left(\mathit{se}\right({\widehat{\mathit{G}}}_{\tilde{\mathit{Q}}}\left)\right)$ | |||||

Uttar Pradesh | ||||||||

All | 1262 | 1665 | 1262 | 0.2163 | 0.2163 | 0.2081 | 0.2245 | 0.0164 |

(333) | (1262) | (0.0042) | (0.0042) | |||||

Rural | 903 | 593 | 595 | 0.2000 | 0.2000 | 0.1914 | 0.2085 | 0.0171 |

(226) | (593) | (0.0041) | (0.0044) | |||||

Urban | 359 | 1712 | 359 | 0.2229 | 0.2229 | 0.2048 | 0.2410 | 0.0362 |

(254) | (359) | (0.0092) | (0.0092) | |||||

West Bengal | ||||||||

All | 878 | 874 | 878 | 0.2320 | 0.2320 | 0.2221 | 0.2419 | 0.0198 |

(223) | (874) | (0.0051) | (0.0051) | |||||

Rural | 551 | 535 | 534 | 0.1812 | 0.1814 | 0.1719 | 0.1910 | 0.0191 |

(203) | (535) | (0.0048) | (0.0049) | |||||

Urban | 327 | 1110 | 327 | 0.2609 | 0.2609 | 0.2458 | 0.2761 | 0.0303 |

(207) | (327) | (0.0077) | (0.0077) |

**Table 7.**Application results for the two-stage procedure on NSS 64th round data for $\alpha =0.1$ and $\omega =0.025$.

Region | H | ${\mathit{Q}}^{\ast}$ | $\tilde{\mathit{Q}}$ | ${\widehat{\mathit{G}}}_{\mathit{H}}$ | ${\widehat{\mathit{G}}}_{\tilde{\mathit{Q}}}$ | Lower CI | Upper CI | ${\mathit{w}}_{\tilde{\mathit{Q}}}$ |
---|---|---|---|---|---|---|---|---|

$\left(\mathit{t}\right)$ | $\left(\mathit{Q}\right)$ | $\left(\mathit{se}\right({\widehat{\mathit{G}}}_{\mathit{H}}\left)\right)$ | $\left(\mathit{se}\right({\widehat{\mathit{G}}}_{\tilde{\mathit{Q}}}\left)\right)$ | |||||

Uttar Pradesh | ||||||||

All | 1262 | 688 | 680 | 0.2163 | 0.2104 | 0.2023 | 0.2185 | 0.0162 |

(302) | (688) | (0.0042) | (0.0049) | |||||

Rural | 903 | 299 | 308 | 0.1997 | 0.2026 | 0.1927 | 0.2126 | 0.0199 |

(168) | (299) | (0.0041) | (0.0061) | |||||

Urban | 359 | 1087 | 359 | 0.2229 | 0.2229 | 0.2077 | 0.2381 | 0.0304 |

(168) | (359) | (0.0092) | (0.0092) | |||||

West Bengal | ||||||||

All | 878 | 396 | 396 | 0.2320 | 0.2293 | 0.2171 | 0.2414 | 0.0243 |

(158) | (396) | (0.0051) | (0.0074) | |||||

Rural | 551 | 275 | 275 | 0.1812 | 0.1750 | 0.1660 | 0.1840 | 0.0180 |

(138) | (275) | (0.0048) | (0.0055) | |||||

Urban | 327 | 582 | 327 | 0.2609 | 0.2609 | 0.2482 | 0.2736 | 0.0254 |

(142) | (327) | (0.0077) | (0.0077) |

**Table 8.**Application results for the two-stage procedure on NSS 64th round data for $\alpha =0.05$ and $\omega =0.025$.

Region | H | ${\mathit{Q}}^{\ast}$ | $\tilde{\mathit{Q}}$ | ${\widehat{\mathit{G}}}_{\mathit{H}}$ | ${\widehat{\mathit{G}}}_{\tilde{\mathit{Q}}}$ | Lower CI | Upper CI | ${\mathit{w}}_{\tilde{\mathit{Q}}}$ |
---|---|---|---|---|---|---|---|---|

$\left(\mathit{t}\right)$ | $\left(\mathit{Q}\right)$ | $\left(\mathit{se}\right({\widehat{\mathit{G}}}_{\mathit{H}}\left)\right)$ | $\left(\mathit{se}\right({\widehat{\mathit{G}}}_{\tilde{\mathit{Q}}}\left)\right)$ | |||||

Uttar Pradesh | ||||||||

All | 1262 | 976 | 947 | 0.2163 | 0.2124 | 0.2041 | 0.2207 | 0.0166 |

(302) | (946) | (0.0042) | (0.0042) | |||||

Rural | 903 | 364 | 353 | 0.1997 | 0.2032 | 0.1922 | 0.2142 | 0.0220 |

(197) | (364) | (0.0041) | (0.0056) | |||||

Urban | 359 | 1081 | 359 | 0.2229 | 0.2229 | 0.2048 | 0.2410 | 0.0362 |

(177) | (359) | (0.0092) | (0.0092) | |||||

West Bengal | ||||||||

All | 878 | 607 | 608 | 0.2320 | 0.2315 | 0.2204 | 0.2427 | 0.0224 |

(186) | (607) | (0.0051) | (0.0057) | |||||

Rural | 551 | 391 | 392 | 0.1812 | 0.1759 | 0.1670 | 0.1849 | 0.0178 |

(163) | (391) | (0.0048) | (0.0045) | |||||

Urban | 327 | 754 | 327 | 0.2609 | 0.2609 | 0.2458 | 0.2761 | 0.0303 |

(162) | (327) | (0.0077) | (0.0077) |

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## Share and Cite

**MDPI and ACS Style**

Bilson Darku, F.; Konietschke, F.; Chattopadhyay, B.
Gini Index Estimation within Pre-Specified Error Bound: Application to Indian Household Survey Data. *Econometrics* **2020**, *8*, 26.
https://doi.org/10.3390/econometrics8020026

**AMA Style**

Bilson Darku F, Konietschke F, Chattopadhyay B.
Gini Index Estimation within Pre-Specified Error Bound: Application to Indian Household Survey Data. *Econometrics*. 2020; 8(2):26.
https://doi.org/10.3390/econometrics8020026

**Chicago/Turabian Style**

Bilson Darku, Francis, Frank Konietschke, and Bhargab Chattopadhyay.
2020. "Gini Index Estimation within Pre-Specified Error Bound: Application to Indian Household Survey Data" *Econometrics* 8, no. 2: 26.
https://doi.org/10.3390/econometrics8020026