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The Rao-Blackwell theorem says that if we want an estimator with small MSE we can confine our search to estimators which are functions of the sufficient statistic. 0, Image CroppedIf a species is fairly prevalent, a random sample of sites might let us see plenty of animals. Notify me of new posts via email. A random sample could lead us to a bunch of unfruitful site visits, despite the fact that the species is quite common in other areas close to those sites.

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The sample mean abundance or even the mean of cluster means can be biased under this type of adaptive sampling design (since it’s no longer completely random). Adaptive SamplingBut what if we had a little information about the uneven distribution of condors? Adaptive sampling methods allow us to incorporate information about the structure that we’ve observed so far to help us decide where to sample next. We refer to sites that don’t have any condors but who are in the neighborhood of a site that does as an “edge unit. More specifically speaking, particle filter becomes inefficient when the state is high-dimensional.

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Since we often expect large variability in the abundance of a species even within clusters of sites, this is good news for ecologists. Think about how complicated our new design is; we need all the help we can get. Image Credit: Bureau of Land Management, CC BY 2. Wait a moment and try again. This becomes our new, Rao-Blackwellized, estimator.

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That sounds great, but what is a sufficient statistic? Informally, a function of the data T is sufficient discover here we can’t learn anything more about the parameter of interest from the distribution of the data if we already know what T is.
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. Let’s close with some info on the people behind this theorem. How do we do that? We take the conditional expectation of the original estimator given a sufficient statistic T. But we went through all this trouble to collect additional data using a new sampling method. setAttribute( “value”, ( new Date() ).

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Have a quantitative term or concept that mystifies you? Want it explained simply? Suggest a topic for next month →  @sastoudtFill in your details below or click an icon to log in:Connecting to %s Notify me of new comments via email. Sign us up!The Rao-Blackwell theorem tells us that if we have an estimator, then we can obtain a new estimator that is never worse than the original. David Blackwell was an American mathematician and statistician with his own set of accolades including being a member of the National Academy of Sciences (he was the first African American to be included). However, we know species investigate this site are rarely even across a given region. 0)What do we mean by an estimator being better? The new Rao-Blackwellized estimator will have a mean-squared error that is less than or equal to the original. We keep doing this until we fail to see a condor at any of the previous sites’ neighbors.

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We wouldn’t need any of the original sample data to make a decision about our estimate, hence the sample mean this post sufficient. For example, if we are trying to estimate the population mean, we could do so using only the sample mean as our T. This is especially helpful if finding even a starting point for an estimator is hard. .

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