The sequential procedure that is proposed under i.i.d. framework may be extended to non i.i.d framework where stratified sampling is used. Suppose we divide the population into S strata. In the population, stratum s contains a mass of ${H}_{s}$ households. The total number of households in the population is $H={\sum}_{s=1}^{S}{H}_{s}$. The density of the household income or expenditure, X in the ${s}^{\text{th}}$ stratum is denoted by $dF\left(x\right|s)$.

Now, a sample of

${n}_{s}$ households (indexed by

${h}_{s}$) is drawn by simple random sample with replacement from every strata so that the total number of households in the sample is

$n={\sum}_{s=1}^{S}{n}_{s}$ and

${n}_{s}=n{a}_{s}$ with

${\sum}_{s=1}^{S}{a}_{s}=1\text{,}\phantom{\rule{4.pt}{0ex}}\text{where}\phantom{\rule{4.pt}{0ex}}{a}_{s}={H}_{s}/H$. Let

${x}_{s{h}_{s}}$ be the total income of the

${h}_{s}^{\text{th}}$ household belonging to the

${s}^{\text{th}}$ stratum. If

${w}_{s{h}_{s}}=\frac{{H}_{s}}{{n}_{s}}$ is the weight of

${h}_{s}^{\text{th}}$ household in the

${s}^{\text{th}}$ stratum, then following [

21,

22], Gini index can be estimated by the estimator

where

Now, following [

21,

22] we have

where

${V}^{*}$ is the asymptotic variance given in [

22], modified to take into account of the stratified sampling only. Now,

The confidence coefficient will be approximately

$1-\alpha $ provided

$\sqrt{n}d/\sqrt{{V}^{*}}\ge {z}_{\alpha /2}$. In order to have a fixed-width confidence interval, we need sample size

n satisfying

Since

C is unknown it must be estimated in the first stage and continue sampling until the sample size

n is bigger than corresponding estimated value of

C. Note that

C is the optimum (i.e., minimum) household size to be sampled to achieve

$(1-\alpha )$ confidence level provided

${V}^{*}$ were known. The optimal number of households to be sampled in the

${s}^{\text{th}}$ stratum (

$s=1,\cdots ,S$) will be

${C}_{s}=C{a}_{s}$ which is also unknown since

C is unknown. Bhattacharya [

22] proposed an estimator of the asymptotic variance

${V}^{*}$ of the Gini index under complex household survey which can be used in Equation (

18). Then the stopping rule developed in Equation (

12) may be modified taking into account of the stratification and finite sampling scenario to find out an estimate of the optimum number of households in order to find a fixed-width confidence interval for Gini index under stratified sampling. However, we do not intend to explore this possibility in this article, and we believe that this could be a good topic of future research.