Linearization versus bootstrap for variance estimation of the change between Gini indexes
Section 2. One sample case

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2.1  Notation

Let $U$ denote some finite population of size $N$ whose units may be identified by the labels $k\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}N.$ Suppose that the variable ${\mathcal{Y}}_1$ is measured on the population $U,$ and let $y_{11}\mathrm{,}\dots \mathrm{,}{y}_{1N}$ denote the values taken by ${\mathcal{Y}}_1$ on the units in the population. Let $M_1={\displaystyle {\sum }_{k\in U}}{\delta }_{y_{1k}}$ denote the discrete measure taking unit mass on any point $y_{1k}$ in the population and 0 elsewhere, with ${\delta }_{y_{1k}}$ the Dirac mass at $y_{1k}.$ Most of the parameters of interest ${\theta }_1$ studied in surveys can be written as a functional $T$ of $M_1,$ namely ${\theta }_1\mathrm{<mo>=</mo>}T\left(M_1\right).$ For instance, the total $t_{y1}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in U}}{y}_{1k}$ equals ${\displaystyle \int }{\mathcal{Y}}_{1}d{M}_1.$ In practice, a sample $s$ (with or without repetitions) is selected by means of a sampling design $p(\cdot ),$ and we observe the values $y_{1k}$ for $k\in s$ only. A substitution principle is used for estimation (see Deville, 1999, and Goga, Deville and Ruiz-Gazen, 2009). Let ${\pi }_k$ denote the expected number of draws for unit $k$ in the sample; in case of without-replacement sampling, this is the probability that unit $k$ is selected in the sample. Let ${\widehat{M}}_1\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in s}}{w}_k{\delta }_{y_{1k}}$ denote the discrete measure taking mass $w_k$ on any point in the sample and 0 elsewhere, where $w_k\mathrm{<mo>=</mo>}{\pi }_k^{-1}$ is the sampling weight. Substituting ${\widehat{M}}_1$ into ${\theta }_1$ yields the estimator ${\widehat{\theta }}_1\mathrm{<mo>=</mo>}T\left({\widehat{M}}_1\right).$

For a without-replacement sampling design, the substitution estimator for a total is the so-called Horvitz-Thompson (HT) estimator ${\widehat{t}}_{y1}^{\text{HT}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in s}}{w}_k{y}_{1k}.$ The HT variance estimator is

$$v^{\text{HT}}\left({\widehat{t}}_{y1}^{\text{HT}}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s}}{\displaystyle \sum _{l\in s}}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{y_{1k}}{{\pi }_k}\frac{y_{1l}}{{\pi }_l}\mathrm{,}(2.1)$$

where ${\pi }_{kl}\mathrm{<mo>=</mo>}\text{Pr}\left(k\mathrm{,}l\in s\right)$ denotes the probability that units $k$ and $l$ are selected jointly in the sample, and ${\Delta }_{kl}\mathrm{<mo>=</mo>}{\pi }_{kl}-{\pi }_k{\pi }_l.$ In the particular case of simple random sampling without replacement (SI) of size $n,$ we have ${\widehat{t}}_{y1}^{\text{HT}}\mathrm{<mo>=</mo>}N{\overline{y}}_{\mathrm{1,}s}$ with ${\overline{y}}_{\mathrm{1,}s}\mathrm{<mo>=</mo>}{n}^{-1}{\displaystyle {\sum }_{k\in s}}{y}_{1k},$ and formula (2.1) yields

$$v^{\text{HT}}\left({\widehat{t}}_{y1}^{\text{HT}}\right)\mathrm{<mo>=</mo>}{N}^2\left(\frac{1}{n}-\frac{1}{N}\right)S_{y_1\text{}\mathrm{,}s}^2\text{where}{S}_{y_1\text{}\mathrm{,}s}^2\mathrm{<mo>=</mo>}\frac{1}{n-1}{\displaystyle \sum _{k\in s}}{\left(y_{1k}-{\overline{y}}_{\mathrm{1,}s}\right)}^2\text{}\mathrm{.}(2.2)$$

For a with-replacement sampling design, the substitution estimator for a total is the so-called Hansen-Hurwitz (HH) estimator ${\widehat{t}}_{y1}^{\text{HH}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in s}}{w}_k{y}_{1k}.$ We consider the important case of multistage sampling, where the $N$ units are grouped inside $N_I$ non-overlapping Primary Sampling Units (PSU) $U_1\mathrm{,}\dots \mathrm{,}{U}_{N_I}\text{},$ and where a with-replacement first-stage sample $s_I$ of size $m$ is selected. Let ${\pi }_{Ii}$ denote the expected number of draws for the PSU $U_i$ in $s_I\text{}.$ A second-stage sample $s_i$ is then selected inside any $i\in s_I$ by means of some sampling design $p_i(\cdot ).$ Let ${\pi }_{k|i}$ denote the expected number of draws for unit $k$ in $s_i\text{}.$ The estimated measure is then ${\widehat{M}}_1\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s_I}}{\displaystyle {\sum }_{k\in s_i}}{\pi }_{Ii}^{-1}{\pi }_{k|i}^{-1}{\delta }_{y_{1k}}\text{}.$ We have ${\widehat{t}}_{y1}^{\text{HH}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s_I}}{\pi }_{Ii}^{-1}{\widehat{Y}}_i$ where ${\widehat{Y}}_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in s_i}}{\pi }_{k|i}^{-1}{y}_{1k}\text{},$ and an unbiased variance estimator for ${\widehat{t}}_{y1}^{\text{HH}}$ is

$$v^{\text{HH}}\left({\widehat{t}}_{y1}^{\text{HH}}\right)\mathrm{<mo>=</mo>}\frac{m}{m-1}{\displaystyle \sum _{i\in s_I}}{\left(\frac{{\widehat{Y}}_i}{{\pi }_{Ii}}-\frac{{\widehat{t}}_{y1}^{\text{HH}}}{m}\right)}^2\text{}\mathrm{.}(2.3)$$

2.2  Estimating the Gini coefficient

If the variable ${\mathcal{Y}}_1$ is measured on the population $U,$ the Gini coefficient is

$$G_1\mathrm{<mo>=</mo>}\frac{1}{2}\frac{{\displaystyle {\sum }_{k\in U}{\displaystyle {\sum }_{l\in U}\left|y_{1k}-y_{1l}\right|}}}{N{\displaystyle {\sum }_{k\in U}{y}_{1k}}}\mathrm{,}$$

see for example Nygård and Sandström (1985). It follows that $G_1$ is zero if ${\mathcal{Y}}_1$ is constant on the population, which occurs when the total of ${\mathcal{Y}}_1$ is equally distributed among all the population individuals. In the opposite case, when only one individual owns the whole amount of ${\mathcal{Y}}_1\mathrm{,}$ $G_1$ is maximized and equal to $1-1/N:$ the total of ${\mathcal{Y}}_1$ is then concentrated in one point only, which means maximum inequality among members of the population.

If all individuals $k\ne l$ have different values for the variable ${\mathcal{Y}}_1,$ the Gini coefficient $G_1$ is

$$G_1\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^N{y}_{1\left(k\right)}\left(2k/N-1\right)}}{t_{y1}}-\frac{1}{N}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in U}{y}_{1k}\left\{2F_{1N}\left(y_{1k}\right)-1\right\}}}{t_{y1}}-\frac{1}{N}(2.4)$$

with $y_{1\left(1\right)}\le \cdots \le y_{1\left(N\right)}$ the ordered values and $F_{1N}(\cdot )\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{k\in U}}{1}_{\left\{y_{1k}\le \cdot \right\}}$ the finite population distribution function; see Sandström, Wretman and Waldèn (1988) and Deville (1997) for further details on the derivation of (2.4). Nygård and Sandström (1985) called the term $-1/N$ the Gini finite population correction and gave several reasons to make this correction, such as the non-negativity of the lower bound of $G_1.$ As is frequently done in the literature (see for example Glasser, 1962), this correction is ignored in the sequel. We redefine the Gini coefficient as

$$G_1\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in U}{y}_{1k}\left\{2F_{1N}\left(y_{1k}\right)-1\right\}}}{t_{y1}}\mathrm{<mo>=</mo>}\frac{{\displaystyle \int }\left\{2F_{1N}\left(y\right)-1\right\}yd{M}_1\left(y\right)}{{\displaystyle \int }yd{M}_1\left(y\right)}(2.5)$$

where the finite population distribution function $F_{1N}(\cdot )$ is a functional family

$$F_{1N}\left(y\right)\mathrm{<mo>=</mo>}\frac{1}{{\displaystyle \int }d{M}_1\left(y\right)}{\displaystyle \int }{1}_{\left\{\xi \le y\right\}}d{M}_1\left(\xi \right)(2.6)$$

indexed by $y.$ Substituting ${\widehat{M}}_1$ into (2.5) and (2.6) yields the estimator

$${\widehat{G}}_1\mathrm{<mo>=</mo>}\frac{{\displaystyle \int }\left\{2{\widehat{F}}_{1N}\left(y\right)-1\right\}yd{\widehat{M}}_1\left(y\right)}{{\displaystyle \int }yd{\widehat{M}}_1\left(y\right)}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in s}{w}_k\left\{2{\widehat{F}}_{1N}\left(y_{1k}\right)-1\right\}{y}_{1k}}}{{\displaystyle {\sum }_{k\in s}{w}_k{y}_{1k}}}\mathrm{,}(2.7)$$

where

$${\widehat{F}}_{1N}\left(y\right)\mathrm{<mo>=</mo>}\frac{1}{{\displaystyle \int }d{\widehat{M}}_1\left(y\right)}{\displaystyle \int }{1}_{\left\{\xi \le y\right\}}d{\widehat{M}}_1\left(\xi \right)\mathrm{<mo>=</mo>}\frac{1}{{\displaystyle {\sum }_{k\in s}{w}_k}}{\displaystyle \sum _{k\in s}{w}_k{1}_{\left\{y_{1k}\le y\right\}}}(2.8)$$

is the substitution estimator of the distribution function $F_{1N}.$

2.3  Linearization variance estimation

We give below some brief details about the influence function linearization (IFL) (Deville, 1999), which consists in giving a first-order expansion of the substitution estimator ${\widehat{\theta }}_1\mathrm{<mo>=</mo>}T\left({\widehat{M}}_1\right)$ around the true value ${\theta }_1\mathrm{<mo>=</mo>}T\left(M_1\right),$ to approximate the error by a linear estimator of some artificial linearized variable. More precisely, the first derivatives of $T$ with respect to $M_1$ are the influence functions

$$\text{IT}\left(M_1\mathrm{<mo>;</mo>}y\right)\mathrm{<mo>=</mo>}\underset{h\to 0}{{\displaystyle \lim }}\frac{T\left(M_1+h{\delta }_y\right)-T\left(M_1\right)}{h}\mathrm{,}$$

and $u_{1k}\mathrm{<mo>=</mo>}\text{IT}\left(M_1\mathrm{<mo>;</mo>}{y}_{1k}\right)$ is the linearized variable for all $k\in U.$ Suppose that $T(\cdot )$ is homogeneous, namely there exists some positive number $\beta$ dependent on $T$ such that $T\left(r{M}_1\right)\mathrm{<mo>=</mo>}{r}^{\beta }T\left(M_1\right)$ for any real $r\mathrm{<mo>></mo>\; 0.}$ Assume also that ${\text{lim}}_{N\to \infty }{N}^{-\beta }T\left(M_1\right)\mathrm{<mo><</mo>}\infty .$ Under some additional regularity assumptions upon $T(\cdot )$ and the sampling design (e.g., Goga and Ruiz-Gazen, 2014), Deville (1999) establishes that

$${\widehat{\theta }}_1-{\theta }_1\mathrm{<mo>=</mo>}\left({\displaystyle \sum _{k\in s}{w}_k{u}_{1k}}-{\displaystyle \sum _{k\in U}{u}_{1k}}\right)+o_p\left(N^{\beta }{n}^{-1/2}\right)\mathrm{,}$$

so that the error ${\widehat{\theta }}_1-{\theta }_1$ can be approximated by the error of the HT estimator for the total of the linearized variable $u_{1k}.$ For a without-replacement sampling design, using a sample-based estimator ${\widehat{u}}_{1k}$ of the linearized variable $u_{1k}$ in the HT variance estimator yields the variance estimator

$$v_{\text{LIN}}^{\text{HT}}\left({\widehat{\theta }}_1\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s}}{\displaystyle \sum _{l\in s}}\frac{{\Delta }_{kl}}{{\pi }_{kl}}\frac{{\widehat{u}}_{1k}}{{\pi }_k}\frac{{\widehat{u}}_{1l}}{{\pi }_l}\mathrm{,}(2.9)$$

where ${\pi }_{kl}\mathrm{<mo>=</mo>}\text{Pr}\left(k\mathrm{,}l\in s\right)$ denotes the probability that units $k$ and $l$ are selected jointly in the sample, and ${\Delta }_{kl}\mathrm{<mo>=</mo>}{\pi }_{kl}-{\pi }_k{\pi }_l.$ Several results of asymptotic normality have been proved for specific sampling designs, see Hájek (1960, 1961, 1964), Rosén (1972), Sen (1980), Krewski and Rao (1981), Gordon (1983), Ohlsson (1986, 1989), Chen and Rao (2007), Brändén and Jonasson (2012), Saegusa and Wellner (2013) and Chauvet (2015), among others. If the sampling design is such that the substitution estimator ${\widehat{\theta }}_1$ satisfies a central-limit theorem, an approximately $\left(1-2\alpha \right)\mathrm{<mi>\%</mi>}$ confidence interval is $\left[{\widehat{\theta }}_1-z_{\alpha }\sqrt{v_{\text{lin}}\left({\widehat{\theta }}_1\right)}\mathrm{,}{\widehat{\theta }}_1+z_{\alpha }\sqrt{v_{\text{lin}}\left({\widehat{\theta }}_1\right)}\right]$ where $z_{\alpha }$ is the upper $\alpha \mathrm{<mi>\%</mi>}$ cutoff for the standard normal distribution.

In case of the Gini coefficient, we have $\beta \mathrm{<mo>=</mo>\; 0}$ and the linearized variable is

$$u_{1k}\mathrm{<mo>=</mo>}2F_{1N}\left(y_{1k}\right)\frac{y_{1k}-{\overline{y}}_{1k\mathrm{,}U\mathrm{<mo><</mo>}}}{t_{y1}}-y_{1k}\frac{G_1+1}{t_{y1}}+\frac{1-G_1}{N}\mathrm{,}(2.10)$$

where ${\overline{y}}_{1k\mathrm{,}U\mathrm{<mo><</mo>}}\mathrm{<mo>=</mo>}{\left({\displaystyle {\sum }_{l\in U}}{1}_{\mathrm{<mo>\{</mo>}{y}_{1l}\mathrm{<mo><</mo>}{y}_{1k}\mathrm{<mo>\}</mo>}}\right)}^{-1}{\displaystyle {\sum }_{j\in U}{y}_{1j}{1}_{\left\{y_{1j}\mathrm{<mo><</mo>}{y}_{1k}\right\}}}$  denotes the mean of the $y_{1j}$ lower than $y_{1k},$ see Deville (1999). Kovačević and Binder (1997) derived the same expression by means of the estimating equations linearization method; using the Demnati and Rao (2004) linearization approach also leads to the same result. The estimated linearized variable is

$${\widehat{u}}_{1k}\mathrm{<mo>=</mo>}2{\widehat{F}}_{1N}\left(y_{1k}\right)\frac{y_{1k}-{\overline{y}}_{1k\text{}\mathrm{,}s\mathrm{<mo><</mo>}}}{{\widehat{t}}_{y1}}-y_{1k}\frac{{\widehat{G}}_1+1}{{\widehat{t}}_{y1}}+\frac{1-{\widehat{G}}_1}{\widehat{N}}(2.11)$$

where ${\overline{y}}_{1k\text{}\mathrm{,}s\mathrm{<mo><</mo>}}\mathrm{<mo>=</mo>}{\left({\displaystyle {\sum }_{l\in s}{w}_l{1}_{\left\{y_{1l}\mathrm{<mo><</mo>}{y}_{1k}\right\}}}\right)}^{-1}{\displaystyle {\sum }_{j\in s}{w}_j{y}_{1j}{1}_{\left\{y_{1j}\mathrm{<mo><</mo>}{y}_{1k}\right\}}}.$

In the particular SI case, the linearization variance estimator for the Gini coefficient is

$$v_{\text{LIN}}^{\text{HT}}\left({\widehat{G}}_1\right)\mathrm{<mo>=</mo>}{N}^2\left(\frac{1}{n}-\frac{1}{N}\right)S_{{\widehat{u}}_1\text{}\mathrm{,}s}^2\text{where}{S}_{{\widehat{u}}_1\text{}\mathrm{,}s}^2\mathrm{<mo>=</mo>}\frac{1}{n-1}{{\displaystyle \sum _{k\in s}\left({\widehat{u}}_{1k}-{\overline{\widehat{u}}}_{\mathrm{1,}s}\right)}}^2\mathrm{,}(2.12)$$

and where ${\overline{\widehat{u}}}_{\mathrm{1,}s}\mathrm{<mo>=</mo>}{n}^{-1}{\displaystyle {\sum }_{k\in s}{\widehat{u}}_{1k}}.$ In the particular case of multistage sampling and with-replacement sampling of PSUs, the linearization variance estimator for the Gini coefficient is

$$v_{\text{LIN}}^{\text{HH}}\left({\widehat{G}}_1\right)\mathrm{<mo>=</mo>}\frac{m}{m-1}{\displaystyle \sum _{i\in s_I}}{\left(\frac{{\widehat{U}}_{1i}}{{\pi }_{Ii}}-\frac{{\widehat{t}}_{{\widehat{u}}_1}^{\text{HH}}}{m}\right)}^2\text{where}{\widehat{U}}_{1i}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s_i}{\pi }_{k|i}^{-1}{\widehat{u}}_{1k}}\mathrm{.}(2.13)$$

2.4  Bootstrap variance estimation

The use of bootstrap techniques in survey sampling has been extensively studied in the literature. The main bootstrap techniques may be thought as particular cases of the weighted bootstrap (Bertail and Combris, 1997; Antal and Tillé, 2011; Beaumont and Patak, 2012); see also Shao and Tu (1995, Chapter 6), Davison and Hinkley (1997, Section 3.7) and Davison and Sardy (2007) for detailed reviews. Under a weighted bootstrap procedure, the measure ${\widehat{M}}_1\mathrm{<mo>=</mo>}{\displaystyle {\sum }_s{w}_k{\delta }_{y_k}}$ is estimated, conditionally on the sample $s,$ by the bootstrap measure

$${\widehat{M}}_1^{*}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s}{w}_k{D}_k{\delta }_{y_k}}(2.14)$$

where $D\mathrm{<mo>=</mo>}{\left\{D_k\right\}}_{k\in s}$ denotes a (random) vector of resampling weights. We note $E_{*}$ and $V_{*}$ for the expectation and variance with respect to the resampling scheme. In case of without-replacement sampling, the vector $D$ is generated in such a way that

$$E_{*}\left({\displaystyle \sum _s}{w}_k{D}_k{y}_k\right)\simeq {\widehat{t}}_{y1}^{\text{HT}}\text{and}{V}_{*}\left({\displaystyle \sum _s}{w}_k{D}_k{y}_k\right)\simeq v^{\text{HT}}\left({\widehat{t}}_{y1}^{\text{HT}}\right)(2.15)$$

so that the two first moments of the HT-estimator are approximately matched. In case of with-replacement sampling, the vector $D$ is generated in such a way that

$$E_{\mathrm{<mo>*</mo>}}\left({\displaystyle \sum _s}{w}_k{D}_k{y}_k\right)\simeq {\widehat{t}}_{y1}^{\text{HH}}\text{and}{V}_{\mathrm{<mo>*</mo>}}\left({\displaystyle \sum _s}{w}_k{D}_k{y}_k\right)\simeq v^{\text{HH}}\left({\widehat{t}}_{y1}^{\text{HH}}\right)(2.16)$$

so that the two first moments of the HH-estimator are approximately matched.

Under any weighted bootstrap technique, the plug-in estimator of ${\theta }_1\mathrm{<mo>=</mo>}T\left(M_1\right)$ is ${\widehat{\theta }}_1^{*}\mathrm{<mo>=</mo>}T\left({\widehat{M}}_1^{*}\right),$ and the variance of ${\widehat{\theta }}_1\mathrm{<mo>=</mo>}T\left({\widehat{M}}_1\right)$ is estimated by

$$V_{*}\left({\widehat{\theta }}_1^{*}\right)\mathrm{<mo>=</mo>}{E}_{*}{\left\{{\widehat{\theta }}_1^{*}-E_{*}\left({\widehat{\theta }}_1^{*}\right)\right\}}^2\text{}\mathrm{.}(2.17)$$

Since the variance estimator (2.17) may be difficult to compute exactly, a simulation-based variance estimator may be used instead. More precisely, $C$ independent realizations $D_1\mathrm{,}\dots \mathrm{,}{D}_C$ of the vector $D$ are generated, and we denote ${\widehat{\theta }}_{1c}^{*}\mathrm{<mo>=</mo>}T\left({\widehat{M}}_{1c}^{*}\right)$ with ${\widehat{M}}_{1c}^{*}$ the Bootstrap measure associated to the vector $D_c.$ Then $V\left({\widehat{\theta }}_1\right)$ is estimated by

$$v_B\left({\widehat{\theta }}_1\right)\mathrm{<mo>=</mo>}\frac{1}{C-1}\mathrm{<mi>\; </mi>}{\displaystyle \sum _{c\mathrm{<mo>=</mo>\; 1}}^C}{\left\{{\widehat{\theta }}_{1c}^{*}-\frac{1}{C}\mathrm{<mi>\; </mi>}{\displaystyle \sum _{c^{\prime }\mathrm{<mo>=</mo>\; 1}}^C{\widehat{\theta }}_{1c^{\prime }}^{*}}\right\}}^2\text{}\mathrm{.}(2.18)$$

Two types of confidence intervals are usually computed. The percentile method makes use of the ordered bootstrap estimates ${\widehat{\theta }}_{\left(1c\right)}^{*}\text{}\mathrm{,}c\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}C$ to form a $\left(1-2\alpha \right)\mathrm{<mi>\%</mi>}$ confidence interval $\left[{\widehat{\theta }}_{\left(1L\right)}^{*}\mathrm{,}{\widehat{\theta }}_{\left(1U\right)}^{*}\right]$ with $L\mathrm{<mo>=</mo>}\alpha C$ and $U\mathrm{<mo>=</mo>}\left(1-\alpha \right)C.$ The bootstrap $-t$ involves the estimation of the pivotal statistic $t\mathrm{<mo>=</mo>}\left({\widehat{\theta }}_1-{\theta }_1\right)/\sqrt{v_{\text{BWO}}\left({\widehat{\theta }}_1\right)}$ by its bootstrap counterpart $t^{*}\mathrm{<mo>=</mo>}\left({\widehat{\theta }}_1^{*}-{\widehat{\theta }}_1\right)/\sqrt{v_{\text{BWO}}^{*}\left({\widehat{\theta }}_1^{*}\right)},$ where $v_{\text{BWO}}^{*}\left({\widehat{\theta }}_1^{*}\right)$ is obtained by applying the bootstrap procedure to the resample $s^{*}\text{}.$ The bootstrap $-t$ is computationally very intensive since a double bootstrap is required, and is thus less attractive for a data user. Therefore, we do not pursue this approach further and we focus on the percentile method.

Linearization methods provide variance formulas applicable to general sampling designs, but involve possibly intricate computation of derivatives for complex parameters of interest such as the Gini coefficient. Unlike the linearization, the bootstrap avoids theoretical work by re-calculating the existing estimation system repeatedly. Replicate weights are supplied with the data set, and may be easily used to produce variance estimates for a wide range of statistics. However, a bootstrap technique is usually not suitable for general sampling designs. That is, a particular sampling design usually requires a tailor made resampling scheme. In this paper, we focus on two particular bootstrap techniques, which will be generalized in Section 3 to the two-sample context.

2.4.1  Without-replacement bootstrap for SI sampling

When the sample $s$ is selected by means of SI, we consider the without replacement bootstrap (BWO) introduced by Gross (1980). The approach is readily extended to stratified simple random sampling (STSI) with a finite number of strata. Suppose that $N/n$ is an integer. Then the vector $D$ is obtained by, first creating a pseudo-population $U^{*}$ of size $N$ by duplicating $N/n$ times each unit $k$ in the original sample $s,$ and then by selecting a SI resample $s^{*}$ of size $n$ in $U^{*}\text{}.$

The bootstrap measure is given by (2.14), where the resampling weight $D_k$ is the number of times unit $k\in s$ is selected in $s^{*}\text{}.$ The building of $U^{*}$ may be avoided by noting that under the BWO procedure, the vector $D$ follows a multivariate hypergeometric distribution. Therefore, the resampling weights may be directly generated. It can be shown that the BWO procedure leads to

$$E_{\mathrm{<mo>*</mo>}}\left({\displaystyle \sum _s}{w}_k{D}_k{y}_k\right)\mathrm{<mo>=</mo>}{\widehat{t}}_{y1}^{\text{HT}}\text{and}{V}_{\mathrm{<mo>*</mo>}}\left({\displaystyle \sum _s}{w}_k{D}_k{y}_k\right)\mathrm{<mo>=</mo>}\frac{1-n^{-1}}{1-N^{-1}}{v}^{\text{HT}}\left({\widehat{t}}_{y1}^{\text{HT}}\right)\mathrm{,}(2.19)$$

where $v^{\text{HT}}\left({\widehat{t}}_{y1}^{\text{HT}}\right)$ is given in (2.2), so that equation (2.15) is approximately matched for a large sample size.

Several solutions have been proposed to handle the case when $N/n$ is not an integer, see Chao and Lo (1985), Bickel and Freedman (1984), Sitter (1992b), Booth, Butler and Hall (1994), Presnell and Booth (1994), among others. The generalization of BWO variance estimation for unequal probability sampling designs is considered in Särndal, Swensson and Wretman (1992) and Chauvet (2007).

2.4.2  With-replacement bootstrap for multistage sampling

When the sample $s$ is selected by means of multistage sampling and with-replacement unequal probability sampling of PSUs, we consider the bootstrap of PSUs (BWR) introduced by Rao and Wu (1988). A with-replacement resample $s_I^{*}$ of size $m-1$ is selected by means of simple random sampling with replacement (SIR) in the original first-stage sample $s_I.$ The bootstrap measure is

$${\widehat{M}}_1^{*}\mathrm{<mo>=</mo>}\frac{m}{m-1}{\displaystyle \sum _{i\in s_I^{*}}}{\displaystyle \sum _{k\in s_i}}{\pi }_{Ii}^{-1}{\pi }_{k|i}^{-1}{\delta }_{y_{1k}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s}}{w}_k{D}_k{\delta }_{y_k}\text{}\mathrm{,}(2.20)$$

where the resampling weight $D_k$ equals $m{\left(m-1\right)}^{-1}$ multiplied by the number of times the PSU containing $k$ is selected in $s_I^{*}.$

The resampling size $m-1$ is used to reproduce the usual unbiased variance estimator in the linear case (see Rao and Wu, 1988). It can be shown that the BWR procedure leads to

$$E_{*}\left({\displaystyle \sum _s}{w}_k{D}_k{y}_k\right)\mathrm{<mo>=</mo>}{\widehat{t}}_{y1}^{\text{HH}}\text{and}{V}_{*}\left({\displaystyle \sum _s}{w}_k{D}_k{y}_k\right)\mathrm{<mo>=</mo>}{v}^{\text{HH}}\left({\widehat{t}}_{y1}^{\text{HH}}\right)\mathrm{,}(2.21)$$

where $v^{\text{HH}}\left({\widehat{t}}_{y1}^{\text{HH}}\right)$ is given in (2.3), so that equation (2.16) is exactly matched. The BWR procedure is particulary simple, since involving a resampling for the first-stage of sampling only, the sub-samples of Secondary sampling Units (SSUs) being left unchanged inside the resampled PSUs.

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