Linearization versus bootstrap for variance estimation of the change between Gini indexes
Section 1. Introduction

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The Gini coefficient (Gini, 1914) is one of the best known concentration measure often desired in economical studies. If ${\mathcal{Y}}_1$ denotes a quantitative positive variable such as the income and $F_1(\cdot )$ denotes its distribution function defined on $\left]-\infty \mathrm{,}\infty \right[,$ the Gini coefficient is

$$G_1=\frac{1}{2}\frac{{\displaystyle \int }{\displaystyle \int }\left|v-u\right|dF\left(u\right)dF\left(v\right)}{{\displaystyle \int }udF\left(u\right)}\mathrm{,}$$

provided ${\displaystyle \int }udF\left(u\right)\ne 0.$ The Gini coefficient measures the dispersion of a quantitative positive variable within a population. Statistical institutes generally make use of the Gini coefficient to evaluate the income inequalities of a country at different periods of time, or of different countries at the same time. In the last decades, the Gini coefficient has also been considered in economic and sociodemographic fields (see for example Navarro, Muntaner, Borrell, Benach, Quiroga, Rodriguez-Sanz, Vergès and Pasarin, 2006; Bhattacharya, 2007; Lai, Huang, Risser and Kapadia, 2008; Barrett and Donald, 2009), biology (Graczyk, 2007), environment (Druckman and Jackson, 2008; Groves-Kirkby, Denman and Phillips, 2009) or astrophysics (Lisker, 2008).

There is an extensive literature on variance estimation for the Gini coefficient with observations obtained from survey data, see Langel and Tillé (2013) for a review. Glasser (1962) and Sandström, Wretman and Waldèn (1985) considered the case of simple random sampling. Sandström, Wretman and Waldèn (1988) listed possible variance estimators for a general sampling design, including a jackknife variance estimator. This latter approach was further investigated by Yitzhaki (1991), Karagiannis and Kovačević (2000) and Berger (2008). Linearization variance estimation was studied by Kovačević and Binder (1997), and Berger (2008) demonstrated the equivalence between linearization and a generalized jackknife technique first suggested by Campbell (1980). Qin, Rao and Wu (2010) proposed bootstrap and empirical likelihood based confidence intervals for the Gini coefficient. They studied these methods both theoretically and empirically in the particular case of stratified with replacement simple random sampling. However, bootstrap variance estimation has not been compared with alternative methods for the change between Gini indexes.

In this article, we consider linearization versus bootstrap to estimate the change between Gini indexes. The paper is structured as follows. In Section 2, we first consider the estimation of the Gini coefficient in the one-sample case. The notation is defined in Section 2.1, and the substitution estimator of the Gini coefficient is presented in Section 2.2. The linearization variance estimator is given in Section 2.3, with application to the simple random sampling (SI) design and to a multistage sampling design. The main principles of the weighted bootstrap are briefly reviewed at the beginning of Section 2.4, and the without-replacement bootstrap (BWO) suitable for SI sampling is introduced in Section 2.4.1, while the bootstrap of primary sampling units (BWR) suitable for multistage sampling is introduced in Section 2.4.2. In Section 3, we consider the estimation of the change between Gini indexes in the two-sample case. The notation is defined in Section 3.1, and we briefly review the principles of composite estimation which is applied in Section 3.1.1 for the two-dimensional SI design (SI2) and in Section 3.1.2 for a two-dimensional two-stage sampling design (MULT2). The composite estimator of the change between Gini indexes is presented in Section 3.2. The linearization variance estimator by means of the partial influence functions is given in Section 3.3, with application to the SI2 design and to the MULT2 design. An extension of the BWO for the SI2 design and of the BWR for the MULT2 design are then presented in Section 3.4. Linearization and the proposed bootstrap methods are compared in Section 4 through a simulation study. Section 5 concludes.

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