Comparison of some positive variance estimators for the Fay-Herriot small area model 5. Simulation set up and performance measuresComparison of some positive variance estimators for the Fay-Herriot small area model 5. Simulation set up and performance measures

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5.1 Simulation set-up

We conducted a model-based Monte Carlo simulation, following Rubin-Bleuer and You (2012), to examine the finite sample performance of the various methods. ‘Direct’ estimates $\left(y_1,\dots ,y_m\right)$ with $m=15,m=45$ and $m=100,$ are generated from the Fay-Herriot model in (2.3) with ${\beta }^{\prime }=\left(5,4,3,2,1\right)$ and covariates $z_{i^{}}=\left(1,z_{i2},\dots ,z_{ip}\right),$ generated once from normal distributions $z_{ik}\sim k+N\left(1,1\right),$ $k=2,\dots ,5,$ $i=1,\dots ,m,$ and held fixed over the repeated populations. The independent normal random area effects $v_i$ are generated with variance ${\sigma }_v^2=1.$ Independent sampling errors $e_i,$ are generated with sampling variances ${\psi }_i\triangleq 50/n_i,$ where $n_i$ is the sample size for area $i,i=1,\dots ,m.$ There are five sampling variance groups determined by $n_i\text{= 3, 5, 7, 10 or 15,}$ with signal to noise ratios ${\sigma }_v^2/{\psi }_i=0.06,$ $0.1,0.14,0.2\text{ and }0.3,$ respectively. Thus when $m=100$ there are 20 areas per signal to noise ratio. We first generated 50,000 sets of direct estimators for each case and computed the EBLUP and the true Monte Carlo MSE of the EBLUP using the REML, AM.LL, MIX, AM.YL and AR.YL variance estimators. We did not study AR.LL due to its poor performance reported by Li and Lahiri (2011). Next we generated 10,000 sets of direct estimators independently of the first 50,000. For each generated set, we computed the five variance estimators. For the MIX variance estimator we looked at three of the four linearization type MSE estimators discussed in Section 4. Since the linearization MSE estimators often do not estimate bias accurately, we also considered the parametric bootstrap MSE (PB MSE) estimator adjusted for bias using Pfeffermann and Glickman’s (2004) method and the naïve PB MSE estimator with 500 repetitions each (see Appendix B for the construction of the bootstrap). The Monte Carlo performance measures are defined below.

1. The MSE of the EBLUP, ${\overline{\text{MSE}}}_{\ell }\left({\widehat{\theta }}_i\right),$ per sampling variance group:

$$\text{MSE}\left({\widehat{\theta }}_i\right)=\frac{1}{\text{50,000}}{\displaystyle \sum _{r=1}^{\text{50,000}}{\left({\widehat{\theta }}_i^{\left(r\right)}-{\theta }_i^{\left(r\right)}\right)}^2},{\overline{\text{MSE}}}_{\ell }\left(\widehat{\theta }\right)=\frac{5}{m}{\displaystyle \sum _{i\in \left\{j:{\psi }_j=50/n_{\ell }\right\}}\text{MSE}\left({\widehat{\theta }}_i\right)},\ell =1,\mathrm{...},5.$$

2. $E\left({\widehat{\sigma }}_v^2\right)={\displaystyle {\sum }_{r=1}^{\text{10,000}}{\widehat{\sigma }}_v^{2\left(r\right)}}/\text{10,000}\text{, }V\left({\widehat{\sigma }}_v^2\right)={\displaystyle {\sum }_{r=1}^{\text{10,000}}{\left({\widehat{\sigma }}_v^{2\left(r\right)}-E\left({\widehat{\sigma }}_v^2\right)\right)}^2}/\text{10,000},$ where ${\widehat{\sigma }}_v^{2\left(r\right)}$ is the value of ${\widehat{\sigma }}_v^2$ for the $r^{\text{th}}$ simulation run $\left(r=1,\dots ,\text{10,000}\right).$
3. The Average Relative Bias (ARB) of the MSE per sampling variance group:

$${\text{ARB}}_{\ell }\left(\text{mse}\right)=\frac{5}{m}{\displaystyle \sum _{i\in \left\{j:{\psi }_j=50/n_{\ell }\right\}}\text{RB}\left(\text{mse}\left({\widehat{\theta }}_i\right)\right)},\ell =1,\dots 5,$$

• where $\text{RB}\left(\text{mse}\left({\widehat{\theta }}_i\right)\right)=\left[{\displaystyle {\sum }_{r=1}^{\text{10,000}}\text{mse}\left({\widehat{\theta }}_i^{\left(r\right)}\right)}/\text{10,000}-\text{MSE}\left({\widehat{\theta }}_i\right)\right]/\text{MSE}\left({\widehat{\theta }}_i\right).$

4.    The Root Relative MSE of MSE estimators per sampling variance group:

$${\text{RRMSE}}_{\ell }\left(\text{mse}\right)={\left(\frac{5}{m}{\displaystyle \sum _{i\in \left\{j:{\psi }_j={50/n}_{\ell }\right\}}\frac{{\displaystyle \sum _{r=1}^{\text{10,000}}{\left(\text{mse}\left({\widehat{\theta }}_i^{\left(r\right)}\right)-\text{MSE}\left({\widehat{\theta }}_i\right)\right)}^2/\text{10,000}}}{\text{MSE}\left({\widehat{\theta }}_i\right)}}\right)}^{1/2}.$$

• We also examine the bias of the conditional MSE estimators given that $\left\{{\widehat{\sigma }}_{v\text{REML}}^{\text{2}}=0\right\}$ because these are the populations for which the positive estimators were developed.

5. The Average Relative Bias of Conditional MSE estimators:

$$AR{B}_{ℂ}=\frac{5}{m}{\displaystyle \sum _{i\in \ell }E\left[\text{mse}\left({\widehat{\theta }}_i\right)|{\widehat{\sigma }}_{v\text{REML}}^2=0\right]}/E\left[{\left({\widehat{\theta }}_i-{\theta }_i\right)}^2|{\widehat{\sigma }}_{v\text{REML}}^2=0\right]-1.$$

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