3 YR estimator

Yong You, J.N.K. Rao and Mike Hidiroglou

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WFQ applied the You and Rao (2002) method to model (2.1) and obtained an estimator of ${\theta }_i$ given by

${\widehat{\theta }}_i^{\text{YR}}={\widehat{\gamma }}_i{y}_i+(1-{\widehat{\gamma }}_i){x^{\prime }}_i{\widehat{\beta }}^{\text{YR}},$(3.1)

where ${\widehat{\beta }}^{\text{YR}}$ is obtained from

${\tilde{\beta }}^{\text{YR}}={\left\{{\displaystyle \sum _{i=1}^m{w}_i(1-{\gamma }_i)x_i}{x^{\prime }}_i\right\}}^{-1}\left\{{\displaystyle \sum _{i=1}^m{w}_i(1-{\gamma }_i)x_i{y}_i}\right\}$(3.2)

by replacing ${\gamma }_i$ by ${\widehat{\gamma }}_i.$ Note that the YR estimator (3.1) has the same form as the EBLUP ${\widehat{\theta }}_i^{\text{EBLUP}},$ but it uses a non-optimal estimator for $\beta .$ The YR estimators ${\widehat{\theta }}_i^{\text{YR}}$ are self-benchmarking, i.e., ${\displaystyle {\sum }_{i=1}^m{w}_i{\widehat{\theta }}_i^{\text{YR}}}={\displaystyle {\sum }_{i=1}^m{w}_i{y}_i},$ since by (3.2)

${\displaystyle \sum _{i=1}^m{w}_i(1-{\gamma }_i)(y_i-{x^{\prime }}_i{\tilde{\beta }}^{\text{YR}})}=0.$

However, the MSPE of ${\widehat{\theta }}_i^{\text{YR}}$ will be slightly higher than the MSPE of ${\widehat{\theta }}_i^{\text{EBLUP}}$ based on $\widehat{\beta },$ because $\widehat{\beta }$ is asymptotically more efficient than ${\widehat{\beta }}^{\text{YR}}.$

As in the case of ${\widehat{\theta }}_i^{\text{EBLUP}},$ the estimator of $\text{MSPE}({\widehat{\theta }}_i^{\text{YR}})$ has $g_{1i},g_{2i}$ and $g_{3i}$ terms. We need to estimate the variance of ${\tilde{\beta }}^{\text{YR}}$ in order to get the $g_{2i}$ term in the estimator of $\text{MSPE}({\widehat{\theta }}_i^{\text{YR}});$ the other terms $g_{1i}$ and $g_{3i}$ are not affected. It follows from (3.2) that

$\text{V}({\tilde{\beta }}^{\text{YR}})={\sigma }_v^2{\left\{{\displaystyle \sum _{i=1}^m{w}_i(1-{\gamma }_i)x_i}{x^{\prime }}_i\right\}}^{-1}\left\{{\displaystyle \sum _{i=1}^m{w}_i^2{(1-{\gamma }_i)}^2{\gamma }_i^{-1}{x}_i{x^{\prime }}_i}\right\}{\left\{{\displaystyle \sum _{i=1}^m{w}_i(1-{\gamma }_i)x_i}{x^{\prime }}_i\right\}}^{-1}.$(3.3)

The estimator $\widehat{V}({\tilde{\beta }}^{\text{YR}})$ is obtained by substituting ${\widehat{\sigma }}_v^2$ and ${\widehat{\gamma }}_i$ for ${\sigma }_v^2$ and ${\gamma }_i$ in (3.3).

The estimator of $\text{MSPE}({\widehat{\theta }}_i^{\text{YR}})$ is given by

$\text{mspe}({\widehat{\theta }}_i^{\text{YR}})=g_{1i}+g_{2i}^{\text{YR}}+2g_{3i},$(3.4)

where

$g_{2i}^{\text{YR}}={(1-{\widehat{\gamma }}_i)}^2{x^{\prime }}_i\widehat{V}({\tilde{\beta }}^{\text{YR}})x_i.$

The MSPE estimator (3.4) is nearly unbiased under the true model (2.1), similar to the MSPE estimator (2.5) of ${\widehat{\theta }}_i^{\text{EBLUP}}.$

Remark: Any estimator ${\widehat{y}}_i$ of ${\theta }_i$ may be adjusted as

${\widehat{y}}_i^a={\widehat{y}}_i+a_i\left({\displaystyle {\sum }_{i=1}^m{w}_i{y}_i}-{\displaystyle {\sum }_{i=1}^m{w}_i{\widehat{y}}_i}\right)$

for specified $a_i$ to satisfy the benchmarking constraint ${\displaystyle {\sum }_{i=1}^m{w}_i{\widehat{y}}_i^a}={\displaystyle {\sum }_{i=1}^m{w}_i{y}_i},$ where ${\displaystyle {\sum }_{i=1}^m{w}_i{a}_i}=1.$ In particular, we can use ${\widehat{y}}_i={\widehat{\theta }}_i^{\text{EBLUP}}$ to obtain the adjusted EBLUP estimator. As noted by WFQ, both ${\widehat{\theta }}_i^{\text{YR}}$ and ${\widehat{\theta }}_i^{\text{WFQ}}$ are estimators of the form ${\widehat{y}}_i^a$ because ${\displaystyle {\sum }_{i=1}^m{w}_i{y}_i}-{\displaystyle {\sum }_{i=1}^m\text{}}{w}_i{\widehat{y}}_i$ is equal to zero when ${\widehat{y}}_i$ is set equal to ${\widehat{\theta }}_i^{\text{YR}}$ or ${\widehat{\theta }}_i^{\text{WFQ}}.$ Any set of estimators $\left\{{\widehat{y}}_i\right\}$ that satisfy ${\displaystyle {\sum }_{i=1}^m{w}_i{y}_i}={\displaystyle {\sum }_{i=1}^m{w}_i{\widehat{y}}_i}$ has the self-benchmarking property.

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