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APPENDIX 2 EMPIRICAL BAYES ESTIMATION OF INDIGENOUS UNDERCOUNT
where this is read: " is distributed as a normal random variable with mean and variance ".
A2.5 This information is to be weighed up against a model for the likely actual variation between the true values. The information provided by this model is summarised as follows.
A2.6 This model says that, in the absence of survey information about individual regions, we would assume that the regions had similar values. The constant A determines how different regions are likely to be in their undercount adjustments.
A2.7 A model like equation A2.2 was in fact the basis for the practice in 2001 of assuming that a single undercount adjustment should be applied to all regions. This was done in the light of the large survey error associated with PES estimates of Indigenous at state/territory level in 2001.
A2.8 Assuming first that the values A and are known constants, and are provided from the PES survey. The best estimate of T given equations A2.1 and A2.2 is given by:
and the estimate of is:
A2.9 This gives a very logical outcome: where variance is high, the value is very like the overall value M, while for a region with variance low the value is close to the region's PES estimate .
EMPIRICAL BAYES AND THE MORRIS ALGORITHM
A2.10 In the Empirical Bayes approach the survey estimates themselves are used to estimate the variability between the underlying true values i.e. to estimate the constant A. The Morris algorithm gives a simple approach to this which should give a nearly optimal choice of A.
A2.11 First, note that under the model, with A known,
A2.12 Given this, set up the random variable
A2.13 This will have a chi-squared distribution with 14 degrees of freedom (there are 15 regions, but one degree of freedom is lost by substituting the estimator M for the true value T). The expected value of X for this correct value of A is then 14.
A2.14 The Morris algorithm proceeds to find the value which when substituted for A gives X = 14. A simple iterative algorithm achieves this. This value is then used in producing estimates from equations A2.3 and A2.4.
STABLE VARIANCE PARAMETERS
A2.15 A first issue in applying the EB methodology is that the PES survey estimates of the variance are quite unstable, being based on the same small sample of the Indigenous population as the PES estimates themselves. Rather than use these directly, the sample sizes in each region were used in apportioning each region a share of the overall variance.
A2.16 Suppose that the PES provided a simple random sample of size from the population (Indigenous and non-Indigenous), with turning out to be Indigenous, of whom were undercounted. Writing for the Indigenous Census count, for the whole Census count and for the sample size, we have expected Indigenous sample size of E() = /. Let the expected proportion undercounted be a constant
E(/) = .
A2.17 The PES estimate of undercounted Indigenous persons in region r would then be:
A2.18 Assuming that is small, we have:
A2.19 In practice PES is not a simple random sample, nor is its estimator as simple as that above. The above development is used to justify distributing the overall PES variance across Australia in proportion to /. Thus:
for the variance of the undercount adjustment at the Australia level, as estimated directly from the PES. Noting that = / , the value used in EB estimation is:
A2.21 Note that the resulting parameters do not depend on the observed sample of the Indigenous population in PES except via the overall variance estimate var(Sr ).
ADJUSTING TO ADD TO THE AUSTRALIAN PES ESTIMATE
A2.22 The standard EB estimates are not guaranteed to add to the PES Indigenous estimate at Australia level. To enforce additivity to this PES estimate, a constant c was added to the undercount adjustment rates in all regions. This gave the final estimates
A2.23 Setting the constraint:
gives the value of c as:
THE EB ESTIMATE AS A WEIGHTED AVERAGE OF PES REGION ESTIMATES
A2.24 Using an additive adjustment as given above to ensure additivity allows the EB estimates to be written as a simple weighted sum of the region PES estimates.
VARIANCE OF EB ESTIMATE CONDITIONAL ON A
A2.25 This and the next two sections give information about the reliability of the estimates conditional on a known value of A. The effect of using the EB estimate of A is discussed in a later section.
A2.26 Since the PES estimates for each region are almost independent, the variance of the empirical Bayes estimates follows from the linear form equation A2.15 as follows:
A2.27 The variance estimates var() are provided by the PES estimation system based upon the observed data. They do not depend on the variance model that gave the values and are unbiased estimates of variance of the estimator (conditional on A) whether or not the model given by equations A2.1 and A2.2 holds.
A2.28 Note also that state estimates can be written as a weighted sum of the component region estimates, and hence as a weighted sum similar to equation A2.15. The variance of a state estimate can thus be written in a form similar to equation A2.17.
EXPECTED BIAS UNDER THE MODEL
A2.29 Since the PES estimates are design-unbiased we have E() = , and hence:
A2.30 Clearly if the true values are treated as fixed unknown values with no underlying model, then the estimate is biased to the extent that the particular region r is different to other regions. So for a region with a high value of the estimate will tend to be biased downwards. However, for any actual region we do not know the value of ; we only observe the PES estimate . A high value of could be because is high, or because the sampling error was positive, or a combination of these. The estimate tries to balance these possibilities based on the model.
A2.31 The estimator is unbiased for in the sense of expectation across repeated drawings from the model. Thus if we were able to repeatedly draw sets of 15 regions from the model (equation A2.2) and then get PES estimates from them with variance structure given by equation A2.1, and use them to produce estimates , then on average the bias would be zero. This is not very helpful, as even the overall mean estimate M given by equation A2.3 is unbiased in this sense.
A2.32 More useful is to measure the mean squared bias (MSB) of the estimator (or its square root, the root mean squared bias or RMSB). The MSB is zero for the PES estimate, and A for the mean estimate M. Writing EM for expectation across the model, the MSB of is obtained as follows:
ESTIMATES OF MEAN SQUARED ERROR
A2.33 Adding the MSB (equation A2.19) to the variance (equation A2.17) gives the expected mean squared error (MSE) of an EB estimate . The MSE serves as a summary of the likely size of errors from using the EB estimator . Estimates of the root MSB (RMSB) and root MSE (RMSE) are presented in the following table, alongside SE of the PES and EB estimators.
A2.34 For a hypothetical region with no PES information at all, the RMSB would be sqrt(A) = 6.6% and the RMSE would be 7.2% (larger because it still gets variance from the PES Australian estimate).
EFFECT OF ESTIMATING THE SMOOTHING CONSTANT A
A2.35 The estimator can be defined for any specified value of the ratio (A / ), and the resulting SEs can be predicted. These SEs do not depend on the model being correct at all (though the model is required for analysis of the bias). Thus (A / ) could have been chosen to give estimates with a specified size of predicted SEs. The Morris algorithm could still be used to estimate for presentation of RMSB etc.
A2.36 In 2006, the ABS has chosen to use the estimated value in defining the estimator . Different estimates could have arisen, giving different estimates. Thus estimating induces additional variability in the estimates.
A2.37 An example can make this clear. Suppose that a very unusual estimate arises by chance. This will increase the estimated value , which in turn will lead to the estimates being smoothed less than they should be. Thus using the estimated makes the estimates more subject to influence of unusual estimates i.e. more variable.
A2.38 In practice, the ABS is committed to presenting stable Indigenous ERP. In the future this may lead to not using the estimated value if it would lead to unstable estimates, or conversely an unnecessarily extreme smoothing.
A2.39 In the light of this, the ABS is content to present the SEs conditional on the chosen value . Experimental estimates show that the unconditional SE of a "pure" EB estimate which always uses the estimated is somewhat increased over the SEs presented above. Even accounting for this, the unconditional RMSE will still be markedly lower than the SE of the PES estimates.
ALTERNATIVE MODELS AND ESTIMATORS
A2.40 It should be acknowledged that there are many alternative models that could have been used as the basis of an estimator, and alternative methods of producing the estimate. In the process of deciding to use Empirical Bayes techniques, a number of alternatives were investigated. These included modelling different classes of region (e.g. capital cities) separately, and looking for explanatory variables that could explain region differences. Different components of the undercount (e.g. the effect of misclassification as to whether a person is Indigenous) were also examined to see if predicting them separately could improve the estimator. The fit of these more sophisticated models was not sufficiently improved to justify choosing them over the simpler model (equation A2.2) that was used.
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3228.0.55.001 - Population Estimates: Concepts, Sources and Methods, 2009
Latest ISSUE Released at 11:30 AM (CANBERRA TIME) 12/06/2009