4. MSE estimation

Jae-kwang Kim, Seunghwan Park and Seo-young Kim

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We now discuss mean squared error (MSE) estimation of the GLS estimator ${\widehat{\overline{X}}}_h$ which is given by (2.9). Note that the GLS estimator is a function of $\left({\beta }_0\mathrm{,}{\beta }_1\right)$ and ${\sigma }_e^2.$ If the model parameters are known, then the MSE of ${\widehat{\overline{X}}}_h$ is equal to $M_{h1}={\alpha }_{h}V\left({\overline{x}}_h\right)+\left(1-{\alpha }_h\right)\text{Cov}\left({\overline{x}}_h\mathrm{,}{\tilde{x}}_h\right),$ as discussed in Remark 1. That is, writing $\theta =\left({\beta }_0\mathrm{,}{\beta }_1\mathrm{,}{\sigma }_e^2\right)$ and ${\widehat{\overline{X}}}_h={\widehat{\overline{X}}}_h\left(\theta \right),$ the actual prediction for ${\overline{X}}_h$ is computed by ${\widehat{\overline{X}}}_{eh}={\widehat{\overline{X}}}_h\left(\widehat{\theta }\right).$ To account for the effect of estimating the model parameters, we first note the following decomposition of $\text{MSE}\left({\widehat{\overline{X}}}_h^{*}\right):$

$$\begin{array}{lll}\text{MSE}\left({\widehat{\overline{X}}}_{eh}\right)\hfill & =\hfill & \text{MSE}\left({\widehat{\overline{X}}}_h\right)+E\left\{{\left({\widehat{\overline{X}}}_{eh}-{\widehat{\overline{X}}}_h\right)}^2\right\}\hfill \\ \hfill & =:\hfill & M_{h1}+M_{h2}\mathrm{,}\hfill \end{array}$$

which was originally proved by Kackar and Harville (1984) under normality assumptions. The first term, $M_{h1},$  is of order $1/n_h,$  where $n_h$  is the size of $A_h,$  and the second term, $M_{h2},$  is of order $1/n$  with $n={\displaystyle {\sum }_{h=1}^H}{n}_h.$ The second term is often much smaller than the first term.

We consider a jackknife approach to estimate the MSE. Use of the jackknife for bias-corrected estimation was originally proposed by Quenouille (1956). Jiang, Lahiri and Wan (2002) provided a rigorous justification of the jackknife method for the MSE estimation in small area estimation. The following steps can be used for the jackknife computation.

• Step 1 Calculate the $k^{\text{th}}$  replicate ${\widehat{\theta }}^{\left(-k\right)}$  of $\widehat{\theta }$  by deleting the $k^{\text{th}}$  area data set $\left({\overline{x}}_k\mathrm{,}{\overline{y}}_{1k}\right)$ from the full data set $\left\{\left({\overline{x}}_h\mathrm{,}{\overline{y}}_{1h}\right);h=1,2,\dots \mathrm{,}H\right\}.$ This calculation is done for each $k$  to get $H$  replicates of $\theta :$  $\left\{{\widehat{\theta }}^{\left(-k\right)};k=1,\dots \mathrm{,}H\right\}$ which, in turn, provide $H$  replicates of ${\widehat{\overline{X}}}_h:$  $\left\{{\widehat{\overline{X}}}_h^{\left(-k\right)};k=1,2,\dots \mathrm{,}H\right\},$ where ${\widehat{\overline{X}}}_h^{\left(-k\right)}={\widehat{\overline{X}}}_h\left({\widehat{\theta }}^{\left(-k\right)}\right).$
• Step 2 Calculate the estimator of $M_{h2}$  as
• $${\widehat{M}}_{2h}=\frac{H-1}{H}{\displaystyle \sum _{k=1}^H}{\left({\widehat{\overline{X}}}_h^{\left(-k\right)}-{\widehat{\overline{X}}}_h\right)}^2\mathrm{.}(4.1)$$
• Step 3 Calculate the estimator of $M_{h1}$  as
• $${\widehat{M}}_{1h}={\widehat{\alpha }}_h^{\left(\text{JK}\right)}V\left({\overline{x}}_h\right)+\left(1-{\widehat{\alpha }}_h^{\left(\text{JK}\right)}\right)\text{Cov}\left({\overline{x}}_h\mathrm{,}{\tilde{x}}_h\right)(4.2)$$
• where ${\widehat{\alpha }}_h^{\left(\text{JK}\right)}$  is a bias-corrected estimator of ${\alpha }_h$ given by
• $$\begin{array}{lll}{\widehat{\alpha }}_h^{\left(\text{JK}\right)}\hfill & =\hfill & {\widehat{\alpha }}_h-\frac{H-1}{H}{\displaystyle \sum _{k=1}^H}\left({\widehat{\alpha }}_h^{\left(-k\right)}-{\widehat{\alpha }}_h\right)\mathrm{,}\hfill \\ {\widehat{\alpha }}_h\hfill & =\hfill & \frac{{\widehat{\sigma }}_e^2+V\left(b_h\right)-{\widehat{\beta }}_1\text{Cov}\left(a_h\mathrm{,}{b}_h\right)}{{\widehat{\sigma }}_e^2+V\left(b_h\right)+{\widehat{\beta }}_1^{2}V\left(a_h\right)-2{\widehat{\beta }}_1\text{Cov}\left(a_h\mathrm{,}{b}_h\right)}\mathrm{,}\hfill \end{array}$$
• and
• $${\widehat{\alpha }}_h^{\left(-k\right)}=\frac{{\widehat{\sigma }}_e^{\left(-k\right)2}+V\left(b_h\right)-{\widehat{\beta }}_1^{\left(-k\right)}\text{Cov}\left(a_h\mathrm{,}{b}_h\right)}{{\widehat{\sigma }}_e^{\left(-k\right)2}+V\left(b_h\right)+{\left({\widehat{\beta }}_1^{\left(-k\right)}\right)}^{2}V\left(a_h\right)-2{\widehat{\beta }}_1^{\left(-k\right)}\text{Cov}\left(a_h\mathrm{,}{b}_h\right)}\mathrm{.}$$

Remark 4 For the transformation in (3.13), we use the bias-corrected estimator in (3.14) and its MSE estimation method needs to be changed. Using ${\widehat{\overline{X}}}_{eh\mathrm{,}bc}$  to denote the bias-corrected estimator in (3.14) evaluated at $\widehat{\theta },$  we can have the

$$\begin{array}{lll}\text{MSE}\left({\widehat{\overline{X}}}_{eh\mathrm{,}bc}\right)\hfill & =\hfill & \text{MSE}\left({\widehat{\overline{X}}}_{eh}\right)\hfill \\ \hfill & =\hfill & \text{MSE}\left\{Q\left({\widehat{\overline{X}}}_{eh}^{*}\right)\right\}\hfill \\ \hfill & \cong \hfill & {\left\{Q^{\prime }\left({\overline{X}}_h^{*}\right)\right\}}^2\cdot \text{MSE}\left({\widehat{\overline{X}}}_{eh}^{*}\right)\hfill \\ \hfill & =\hfill & {\overline{X}}_h^2\cdot \text{MSE}\left({\widehat{\overline{X}}}_{eh}^{*}\right),\hfill \end{array}$$

where the first equality follows that ${\widehat{\overline{X}}}_{h\mathrm{,}bc}-{\widehat{\overline{X}}}_h$  is of order $O_p\left(n_h^{-1}\right).$  The MSE of ${\widehat{\overline{X}}}_h^{*},$  the EGLS estimator of ${\overline{X}}_h^{*}$  after transformation, is computed by (4.1) and (4.2). Once $\text{MSE}\left({\widehat{\overline{X}}}_{eh}^{*}\right)$  is estimated, we should multiply it by ${\widehat{\overline{X}}}_h^2$  to obtain the MSE estimator of the back-transformed EGLS estimator ${\widehat{\overline{X}}}_{eh\mathrm{,}bc}.$

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