General Estimation Procedures for Population Mean in Successive Cluster Sampling in Presence of Scrambled Response Situation

Abstract

We have addressed the problems of estimation in successive cluster sampling in the presence of scrambled response situation. A generalized class of estimators for population mean in two-occasion successive sampling under two-stage cluster sampling schemes has been suggested. Numerous estimators have been shown as a member of the recommended family. We have derived the properties of the proposed strategy and also found its optimum condition. The superiority of the suggested strategy over the conventional ones has been demonstrated through empirical investigation carried over the data set of real as well as simulated population studies.

Relevance of Our Work in Broad Context

In practical surveys, often it may be seen that a complete list of all the items of study in the population is not available which indicates that drawing a simple random sample is not possible in such cases. For instance, in socioeconomic survey, a list of households is not generally available, whereas a list of residential houses each accommodating a number of households should be available with municipal and other appropriate authorities. In such cases, it may be advisable to draw a simple random sample of houses and survey all the households belonging to sample households. This procedure is known as cluster sampling. It is also a common experience in sample surveys on sensitive data that someone is agreed to response but prefers to hide the true value for avoiding possible social stigma and harassment. These people prefer to hide their true responses. Therefore, the available sample of returns is camouflaged. It is noted that the most of recently developed estimation procedures of successive sampling are based on simple random sampling only and no attempt has been taken for estimation of population parameters in successive sampling under cluster sampling scheme. It is also observed that no significant step has been taken yet to estimate population parameters under cluster sampling in the presence of scrambled response situations. Inspired with these points, we have constructed an estimation technique for population mean in successive cluster sampling in the presence of scrambled response situation.

Appendix 1

Important results relating actual and scrambled study variables:

Result 1: $S_{{O^{*} }}^{*} \, = \, (S_{O}^{{*^{2} }} + \overline{Y}_{..}^{2} ) \, S_{T}^{{*^{2} }} \, + \, S_{O}^{{*^{2} }} \overline{T}_{..}^{*}$.

Proof

We prove the above relation by using the following relationship

$$ y^{*} \, = \, yT^{*} , \, \overline{y}_{i.}^{*} \, = \, \overline{y}_{i.} T_{i.}^{*} , \, \overline{y}_{..}^{*} \, = \, \overline{y}_{..} \overline{T}_{..}^{*} $$

$$ \begin{aligned} S_{{O^{*} }}^{*} & = E(\overline{y}_{i.}^{{*^{2} }} ) - (\overline{y}_{..}^{*} )^{2} \\ & = E(\overline{y}_{i.}^{2} T_{i.}^{{*^{2} }} ) - (\overline{y}_{..} \overline{T}_{..}^{*} ) \\ & = E(\overline{y}_{i.}^{2} ) \, E(T_{i.}^{{*^{2} }} ) - (\overline{y}_{..} \overline{T}_{..}^{*} )^{2} \\ & = (S_{O}^{{*^{2} }} + \overline{Y}_{..}^{2} ) \, (S_{T}^{{*^{2} }} + T_{..}^{{*^{2} }} ) - (\overline{Y}_{..} \overline{T}_{..}^{*} )^{2} \\ & = (S_{O}^{{*^{2} }} + \overline{Y}_{..}^{2} ) \, S_{T}^{{*^{2} }} \, + \, S_{O}^{{*^{2} }} \overline{T}_{..}^{*} \\ & \Rightarrow S_{{O^{*} }}^{*} = (S_{O}^{{*^{2} }} + \overline{Y}_{..}^{2} )S_{T}^{{*^{2} }} + S_{O}^{{*^{2} }} \overline{T}_{..}^{*} . \\ \end{aligned} $$

Proof

$$ \begin{aligned} \overline{S}_{{0^{*} }}^{{*^{2} }} & = \mathop E\limits_{i} [\mathop E\limits_{j} (y_{ij}^{{*^{2} }} ) - \mathop E\limits_{j} (\overline{y}_{i.}^{*} )^{2} ] \\ & = E[\mathop E\limits_{j} (y_{ij}^{2} T_{ij}^{{*^{2} }} ) - \mathop E\limits_{j} (\overline{y}_{i.} \overline{T}_{i.}^{*} )^{2} ]_{{y^{*} = yT^{*} }} \\ & = \mathop E\limits_{i} [E(y_{ij}^{2} ) \, E(T_{ij}^{{*^{2} }} ) - \overline{y}_{i.}^{2} \overline{T}_{i.}^{{*^{2} }} ] \\ & = \, E[(S_{0i}^{2} + \overline{y}_{i.}^{2} ) \, (S_{{T_{i} }}^{2} + \overline{T}_{i.}^{{*^{2} }} ) \, - \, \overline{y}_{i.}^{2} \overline{T}_{i.}^{{*^{2} }} ] \\ & = \, E[S_{0i}^{2} S_{{T_{i} }}^{2} + S_{0i}^{2} \overline{T}_{i.}^{{*^{2} }} + \overline{y}_{i.}^{2} S_{{T_{i} }}^{2} ] \\ & = E[(S_{0i}^{2} + \overline{y}_{i.}^{2} )S_{{T_{i} }}^{2} + S_{9i}^{2} \overline{T}_{i.}^{*} ] \\ & = \, E(S_{0i}^{2} )E(S_{{T_{i} }}^{2} ) + E(\overline{y}_{i.}^{2} )E(S_{{T_{i} }}^{2} ) + E(S_{0i}^{2} )E(\overline{T}_{i.}^{*} ) \\ & = \, \overline{S}_{0}^{2} \overline{S}_{T}^{2} + (S_{o}^{2} + \overline{y}_{..}^{2} )(\overline{S}_{T}^{2} ) + \overline{S}_{0}^{2} (S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} ) \\ & {\text{So, }}\overline{S}_{{0^{*} }}^{{*^{2} }} \, = \, \overline{S}_{0}^{2} \overline{S}_{T}^{2} + (S_{o}^{2} + \overline{y}_{..}^{2} )(\overline{S}_{T}^{2} ) + \overline{S}_{0}^{2} (S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} ) \\ \end{aligned} $$

Similarly, $\overline{S}_{{1^{*} }}^{{*^{2} }} \, = \, \overline{S}_{1}^{2} \overline{S}_{T}^{2} + (S_{1}^{2} + \overline{X}_{..}^{2} )(\overline{S}_{T}^{2} ) + \overline{S}_{1}^{2} (S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} )$.

Proof

$$ \begin{aligned} S_{{01^{*} }}^{*} & = E[\overline{Y}_{i.}^{*} \overline{X}{}_{i.}^{*} ] - \overline{Y}_{..}^{*} \overline{X}_{..}^{*} \\ & = E[\overline{Y}_{i.} T_{i.}^{*} \, \overline{X}_{i.} T_{i.}^{*} ] - \overline{Y}_{..}^{*} \overline{X}_{..}^{*} \\ & = (S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} )E(\overline{Y}_{i.} \overline{X}_{i.} ) - \overline{Y}_{..} \overline{X}_{..} \overline{T}_{..}^{*2} . \\ \end{aligned} $$

So, $S_{{01^{*} }}^{*}$ = $(S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} )E(\overline{Y}_{i.} \overline{X}_{i.} ) - \overline{Y}_{..} \overline{X}_{..} \overline{T}_{..}^{*2} .$

Proof

$$ \begin{aligned} & \mathop E\limits_{i} [\mathop E\limits_{j} (y_{ij}^{*} x_{ij}^{*} ) - \mathop {E(y_{ij}^{*} ) \, E(x_{ij}^{*} )]}\limits_{j} \\ & \quad = \mathop E\limits_{i} [\mathop E\limits_{j} (Y_{ij} T_{ij}^{*} \, X_{ij} T_{ij}^{*} ) - (\overline{Y}_{i.} T_{i.}^{*} )(\overline{X}_{i.} T_{i.}^{*} )]_{{y^{*} \, = \, YT^{*} }} \\ & \quad = \mathop E\limits_{i} [\mathop E\limits_{j} (T_{ij}^{{*^{2} }} )E(Y_{ij} X_{ij} ) - \overline{Y}_{i.} \overline{X}_{i.} T_{i.}^{{*^{2} }} ] \\ & \quad = \mathop E\limits_{i} [(S_{{T_{i.} }}^{2} + \overline{T}_{i.}^{{*^{2} }} )\mathop E\limits_{j} (Y_{ij} X_{ij} ) - \overline{Y}_{i.} \overline{X}_{i.} T_{i.}^{{*^{2} }} ] \\ & \quad = \mathop E\limits_{i} [\overline{T}_{i.}^{{*^{2} }} [\mathop E\limits_{j} (Y_{ij} X_{ij} )\mathop E\limits_{j} (Y_{ij} )\mathop E\limits_{i} (X_{ij} )]] + S_{{T_{i.} }}^{2} \mathop E\limits_{j} (Y_{ij} X_{ij} ) \\ \end{aligned} $$

$$ \begin{aligned} S_{{01^{*} }} & = \mathop E\limits_{j} (Y_{ij} X_{ij} ) - \mathop E\limits_{j} (Y_{ij} )(X_{ij} ) \\ & = \, \frac{1}{M - 1}\sum\limits_{j = 1}^{M} {(Y_{ij} - \overline{Y}_{i.} )(X_{ij} - \overline{X}_{i.} )} \\ & = \mathop E\limits_{i} [T_{i.}^{*2} S_{01i} + S_{{T_{i.} }}^{2} \frac{1}{M - 1}\sum\limits_{j = 1}^{M} {Y_{ij} X_{ij} } ] \\ \overline{S}_{01}^{*} = (S_{{T^{*} }}^{2} + \overline{T}_{..}^{{*^{2} }} )E(S_{01i} ) + \overline{S}_{{T^{*} }}^{2} \frac{1}{NM}\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{M} {Y_{ij} X_{ij} } } \\ & {\text{So, }}\overline{S}_{01}^{*} = (S_{{T^{*} }}^{2} + \overline{T}_{..}^{{*^{2} }} )\overline{S}_{01} + \overline{S}_{{T^{*} }}^{2} \frac{1}{NM}\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{M} {Y_{ij} X_{ij} } } \\ & {\text{Similarly, }}\overline{S}_{02}^{*} = (S_{{T^{*} }}^{2} + \overline{T}_{..}^{{*^{2} }} )\overline{S}_{02} + \overline{S}_{{T^{*} }}^{2} \frac{1}{NM}\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{M} {Y_{ij} Z_{ij} } } \\ & \overline{S}_{12}^{*} = (S_{{T^{*} }}^{2} + \overline{T}_{..}^{{*^{2} }} )\overline{S}_{12} + \overline{S}_{{T^{*} }}^{2} \frac{1}{NM}\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{M} {X_{ij} Z_{ij} } } \\ \end{aligned} $$

Similarly, $X^{*} = XT^{*}$

$$ \begin{gathered} S_{{1^{*} }}^{*} = (S_{1}^{{*^{2} }} + \overline{X}_{..}^{2} ) \, S_{T}^{{*^{2} }} + S_{1}^{{*^{2} }} \overline{T}_{..}^{*} \hfill \\ S_{{2^{*} }}^{*} = (S_{2}^{{*^{2} }} + \overline{Z}_{..}^{2} ) \, S_{T}^{{*^{2} }} + S_{2}^{{*^{2} }} \overline{T}_{..}^{*} , \hfill \\ \end{gathered} $$

where

$$ \begin{gathered} S_{O}^{*} = E[\overline{y}_{i.}^{2} ] - \overline{Y}_{..}^{2} \hfill \\ S_{T}^{*} = E[T_{i.}^{{*^{2} }} ] - \overline{T}_{..}^{*2} . \hfill \\ \end{gathered} $$

Result 2:

$$ \overline{S}_{{0^{*} }}^{{*^{2} }} \, = \, \overline{S}_{0}^{2} \overline{S}_{T}^{2} + (S_{o}^{2} + \overline{y}_{..}^{2} )(\overline{S}_{T}^{2} ) + \overline{S}_{0}^{2} (S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} ). $$

and, $\overline{S}_{{2^{*} }}^{{*^{2} }} = \overline{S}_{2}^{2} \overline{S}_{T}^{2} + (S_{2}^{2} + \overline{z}_{..}^{2} )(\overline{S}_{T}^{2} ) + \overline{S}_{2}^{2} (S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} ),$

where $S_{0i}^{2} = \frac{1}{m - 1}\sum\limits_{j = 1}^{m} {(y_{ij} - \overline{y}_{i.} )^{2} } E[y_{ij} - \overline{y}_{i.} ]^{2}$

$= E[y_{ij}^{2} ] - [\overline{y}_{i.}^{2} ]$

$$ \begin{aligned} S_{0}^{2} & = E(y_{i.}^{2} ) - \overline{y}_{..}^{2} \\ \overline{S}_{T}^{2} & = \frac{1}{N}\sum\limits_{i = }^{N} {S_{{T_{i} }}^{2} } \\ \overline{S}_{0}^{2} & = \frac{1}{N}\sum\limits_{i = }^{N} {S_{{0_{i} }}^{2} } \\ S_{{T_{i} }}^{2} & = \frac{1}{m - 1}\sum\limits_{j = 1}^{m} {(T_{ij}^{*} - T_{i.}^{*2} )} \\ & = E[T_{ij}^{*} - \overline{T}_{i.}^{*} ]^{2} \\ & = E[T_{ij}^{*} ]^{2} - T_{i.}^{{*^{2} }} \\ S_{{T^{*} }}^{{*^{2} }} & = E(T_{i.}^{{*^{2} }} ) - \overline{T}_{i.}^{{*^{2} }} \\ \end{aligned} $$

Result 3: $S_{{01^{*} }}^{*} = \, \frac{1}{N - 1}\sum\limits_{i = 1}^{N} {(\overline{Y}_{i.} - \overline{Y}_{..} )(\overline{X}_{i.} - \overline{X}_{..} )} ,$

where

$$ \text{Cov}(\overline{Y}_{i.} \overline{X}_{i.} ) = E[\overline{Y}_{i.} \overline{X}_{i.} ] - E[\overline{Y}_{i.} ] \, E[\overline{X}_{i.} ]. $$

Similarly, $S_{{02^{*} }}^{*}$ = $(S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} )E(\overline{Y}_{i.} \overline{Z}_{i.} ) - \overline{Y}_{..} \overline{Z}_{..} \overline{T}_{..}^{*2} .$

$$ S_{{12^{*} }}^{*} = (S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} )E(\overline{X}_{i.} \overline{Z}_{i.} ) - \overline{X}_{..} \overline{Z}_{..} \overline{T}_{..}^{*2} , $$

where $S_{T}^{*} = E(T_{i.}^{{*^{2} }} ) - \overline{T}_{..}^{*2} ,E(T_{i.}^{{*^{2} }} ) = S_{T}^{{*^{2} }} + \overline{T}_{..}^{*2} ,$

$$ \begin{aligned} \overline{Y}_{..}^{*} \overline{X}_{..}^{*} & = E[\overline{Y}_{i.}^{*} ] \, E[\overline{X}_{i.}^{*} ] \, \\ & = \, E[Y_{i.} T_{i.}^{*} ] - E[X_{i.} T_{i.}^{*} ] \, \\ & = E[\overline{Y}_{i.} ] \, E[\overline{X}_{i.} ] \, E[T_{i.}^{*} ] \, E[T_{i.}^{*} ] \, \\ & = \overline{Y}_{..} \overline{X}_{..} \overline{T}_{..}^{*2} . \\ \end{aligned} $$

Result 4:$\overline{S}_{01}^{*} = \mathop E\limits_{i} [\mathop E\limits_{j} (y_{ij}^{*} x_{ij}^{*} ) - \mathop {E(y_{ij}^{*} ) \, E(x_{ij}^{*} )}\limits_{j} ]$.

$$ \begin{aligned} {\text{where }}\text{Cov}(x,y) & = \, \frac{1}{N}\sum\limits_{i = 1}^{N} {(x_{i} - \overline{x})} (y_{i} - \overline{y}) \\ S_{{T_{i.} }}^{2} & = \, \mathop E\limits_{j} [T_{ij}^{*2} ] - \mathop E\limits_{j} [T_{ij}^{*} ]^{2} \\ & = \, \frac{1}{M - 1}\sum\limits_{j = 1}^{M} {(T_{ij}^{*} - \overline{T}_{ij}^{*} )^{2} } \\ \end{aligned} $$

$$ \eqalign{& S_{{T_{i.}}}^2 = \mathop E\limits_j [T_{ij}^{*2}] - \bar T_{i.}^{*2} \cr & \quad \quad = \mathop E\limits_j {(T_{ij}^*)^2} - [T_{ij}^*] \cr & \mathop E\limits_j {(T_{ij}^*)^2} = S_{{T_{i.}}}^2 + \bar T_{i.}^{{*^2}} \cr & V(\bar y) = \frac{1}{{N - 1}}{\sum\limits_{i = 1}^N {({Y_i} - \bar Y)} ^2} = E(Y_i^2) - {{\bar Y}^2} \cr & S_{{T^*}}^2 = \frac{1}{{N - 1}}{\sum\limits_{i = 1}^N {(T_{i.}^* - \bar T_{..}^*)} ^2} = {\text{ }}\mathop E\limits_i (T_{i.}^{*2}) - T_{..}^{*2} \cr & S_{{T_{i.}}}^2 = \frac{1}{{M - 1}}\sum\limits_{j = }^M {(T_{ij}^* - \bar T_{i.}^{{*^2}})} \cr & \bar S_{{T^*}}^2 = \frac{1}{N}\sum\limits_{i = 1}^N {S_{{T_{i.}}}^2} \cr} $$