Why a perfectly unbiased estimate might actually be the wrong choice
In short: In statistics, an 'unbiased' estimator is one whose expected value matches the true parameter. It sounds like the gold standard for accuracy, but in reality, choosing the most honest number can sometimes lead to less reliable results than accepting a calculated error.
The term bias describes how much an estimator's expected value deviates from the actual measured parameter. While 'unbiasedness' is an objective property, it is not always the most useful one for practical research. For instance, some unbiased estimators are significantly harder to compute—such as the standard deviation—leading researchers to frequently use biased alternatives that remain manageable.
A primary reason to favor a biased estimator is efficiency. In many cases, a biased estimator can yield a lower 'loss function,' particularly mean squared error, compared to its unbiased counterpart. This is notably seen in shrinkage estimators. Additionally, certain parameters cannot be estimated without bias; for instance, a binomial random variable's parameter reciprocal has no unbiased estimator.
The sample variance provides a classic illustration of these trade-offs. The 'naive' sample variance (dividing by n) is biased downward because it uses the sample mean—an estimate itself—to calculate deviations. Multiplying by n/(n-1), known as Bessel’s correction, creates an unbiased version. However, even this correction doesn't guarantee minimum error. In complex models like propensity score weighted Cox models (CoxPSW), using 'uncorrected variance' can be misleading; it may be conservative for certain weights but not others, potentially leading to incorrect conclusions about data reliability.
Finally, bias behaves unpredictably under transformation. While median-unbiasedness is preserved during non-linear transformations, mean-unbiasedness is not. This means that even if an initial estimate is perfectly centered on the truth, applying a mathematical function to it can introduce systematic error, making the choice of estimator a nuanced balance between theoretical purity and practical precision.
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- Summary by Intellect · source: Bias of an estimator
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