Comparison of some positive variance estimators for the Fay-Herriot small area model 4. The MIX variance estimatorComparison of some positive variance estimators for the Fay-Herriot small area model 4. The MIX variance estimator

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4.1 Variance estimation

The MIX variance estimator is a procedure that first calculates the REML variance estimate and only substitutes it by an adjusted likelihood variance estimate if the REML estimate is negative. The MIX variance estimator is always positive and it is unbiased up to a term of order $o\left(1/m\right).$ The MIX variance estimator of ${\sigma }_v^2$ is defined by:

$${\widehat{\sigma }}_{v\text{MIX}}^2=\{\begin{array}{ll}{\widehat{\sigma }}_{v\text{REML}}^2\hfill & \text{if} {\widehat{\sigma }}_{v\text{REML}}^2>0\hfill \\ {\widehat{\sigma }}_{v\text{adj }}^2\hfill & \text{if} {\widehat{\sigma }}_{v\text{REML}}^2\text{= }0,\hfill \end{array}(4.1)$$

where ${\widehat{\sigma }}_{v\text{adj }}^2$ is one of the adjusted likelihood estimators defined in Section 3.

Remark 4.1. The MIX variance estimator automatically carries some of the common properties shared by the REML and the adjusted likelihood variance estimator. For example, it is even and translation invariant. Thus, under normality of the sampling errors, the second order approximation (2.6) of the MSE of the EBLUP is also valid: Theorem 4.1 below shows that the MSE of the EBLUP under the MIX variance estimator inherits the same asymptotic properties as the MSE under the REML variance estimator.

Theorem 4.1. Under regularity conditions 1 through 3 given in Section 2, and the assumption that ${\sigma }_v^2>0,$ the MSE of the EBLUP under the MIX variance estimator is equal to the MSE under the REML variance estimator up to the second order. The theorem follows from the fact that the asymptotic variance of ${\widehat{\sigma }}_{v\text{MIX}}^2$ coincides with the asymptotic variance of ${\widehat{\sigma }}_{\text{vREML}}^{\text{2}}$ (see Appendix A for details).

Theorem 4.2. Under the conditions of Theorem 4.1, $\text{Bias}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)=o\left(1/m\right).$ The proof is given in Appendix A.

4.2 MSE estimation

The fact that the MIX estimator, ${\widehat{\sigma }}_{v\text{MIX}}^2,$ is unbiased to the second order, is crucial to show that our proposed MSE estimator is also unbiased up to the second order.

Corollary 4.2. The MSE estimator of the EBLUP under ${\widehat{\sigma }}_{v\text{MIX}}^2$ given by:

$$\text{mse}\left[{\widehat{\theta }}_i\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)\right]=g_{1i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)+g_{2i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)+2g_{3i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)(4.2)$$

is second order unbiased. Once given that ${\widehat{\sigma }}_{v\text{MIX}}^2$ is second order unbiased, the result follows along the lines of Datta and Lahiri (2000).

4.3 Alternative MSE estimators

In the following the MIX variance estimator is the combination of REML and AM.LL.

Rubin-Bleuer and You (2012) had suggested another MSE estimator, also unbiased up to the second order: a ‘split’ MSE estimator of the form:

$$\text{mse}*\left[{\widehat{\theta }}_i\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)\right]=\{\begin{array}{ll}{g}_{1i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)+g_{2i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)+2g_{3i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)\hfill & \text{if }{\widehat{\sigma }}_{v\text{MIX}}^2={\widehat{\sigma }}_{v\text{REML}}^2,\hfill \\ \begin{array}{l}{g}_{1i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)+g_{2i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)\\ +2g_{3i}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)-{\left(1-{\widehat{\gamma }}_{i\text{MIX}}\right)}^2\cdot \text{Bias}\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)\end{array}\hfill & \text{if }{\widehat{\sigma }}_{v\text{MIX}}^2={\widehat{\sigma }}_{v\text{AM}\text{.LL}}^2.\hfill \end{array}(4.3)$$

Estimator $\text{mse*}$ has a lower average relative bias (ARB) than the MSE estimator given in (4.2). The lower ARB occurs because the MSE estimates overestimate when REML is positive and underestimate when REML is zero. The $\text{mse*}$ estimator is good on average, but for a particular data set the $\text{mse*}$ estimator might take on negative values.

Molina et al. (2015) proposed two different MSE estimators for the EBLUP under the MIX: with PT standing for their proposed preliminary test of hypothesis for zero variance these estimators are:

$${\text{mse}}_0\left\{{\widehat{\theta }}_i\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)\right\}=\{\begin{array}{ll}\text{mse}\left\{{\widehat{\theta }}_i\left({\widehat{\sigma }}_{v\text{REML}}^2\right)\right\}\hfill & \text{if }{\widehat{\sigma }}_{v\text{REML}}^2>0\hfill \\ g_{2i}\left(0\right)\hfill & \text{if }{\widehat{\sigma }}_{v\text{REML}}^2\text{= }0\hfill \end{array}(4.4)$$

and

$${\text{mse}}_{\text{PT}}\left\{{\widehat{\theta }}_i\left({\widehat{\sigma }}_{v\text{MIX}}^2\right)\right\}=\{\begin{array}{ll}\text{mse}\left\{{\widehat{\theta }}_i\left({\widehat{\sigma }}_{v\text{REML}}^2\right)\right\}\hfill & \text{if }{\widehat{\sigma }}_{v\text{REML}}^2>0\text{ and PT rejected}\hfill \\ g_{2i}\left(0\right)\hfill & \text{if }{\widehat{\sigma }}_{v\text{REML}}^2\text{= }0\text{ or PT not rejected}\text{.}\hfill \end{array}(4.5)$$

The rationale for ${\text{mse}}_{\text{0}}$ and ${\text{mse}}_{\text{PT}}$ is based on the MSE of the BLUP with ${\sigma }_v^2\text{= }0.$ Molina et al. (2015) showed in an empirical study that their proposed MSE estimators performed well on average when both ${\sigma }_v^2$ and the number of areas $m$ were small.

Remark 4.2. ${\text{mse}}_{\text{0}}$ and ${\text{mse}}_{\text{PT}}$ are also unbiased up to the second order (see Appendix for a brief proof of this property). Our argument against $\text{mse}\left\{{\widehat{\theta }}_i\left({\widehat{\sigma }}_v^2\right)\right\}$ (in 3.3) is also valid against ${\text{mse}}_{\text{0}}$ and ${\text{mse}}_{\text{PT}}:$ for a moderate number of areas, the % of populations with ${\widehat{\sigma }}_{v\text{REML}}^2=0$ may be significant even if ${\sigma }_v^2/{\psi }_i$ is not negligible. In this case, the MSE of the EBLUP should account also for the variation due to variance estimation or risk underestimation.

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