bias, consistency, and efficiency
/ BY-us, kun-SIS-ten-see, eff-ISH-en-see /
Think of an estimator as an archer firing at a target, where the bullseye is the true unknown value and each arrow is the estimate from one sample. Three separate questions describe how good the archer is. Do the arrows center on the bullseye, or do they systematically drift to one side? Do they close in on the bullseye as the archer takes more and more shots from a bigger bow? And among archers who all center correctly, whose arrows land in the tightest cluster? Those three questions are bias, consistency, and efficiency.
Bias is the difference between the average value an estimator produces and the true value; an unbiased estimator has zero bias, hitting the target on average even if any single shot misses. Consistency means the estimator converges to the true value as the sample size grows without limit — with infinite data it gets the right answer. Efficiency compares the spread (variance) of competing estimators: the efficient one has the smallest variance, so it pins down the truth most tightly from the same data. A subtle point: these are different virtues. The usual sample variance that divides by (n minus 1) instead of n is the famous fix that removes bias; an estimator can be biased yet consistent, or unbiased yet hopelessly noisy.
Actuaries care because real decisions ride on estimated rates and reserves, and the three properties tell you which estimator to trust. There is often a trade-off: a slightly biased estimator with much smaller variance can beat an unbiased but jumpy one, a balance captured by mean squared error (bias squared plus variance). This is exactly the logic behind credibility weighting and modern shrinkage methods, which deliberately accept a little bias to gain a lot of stability — far more useful than a 'pure' but wildly unstable estimate.
Estimating a portfolio's variance by dividing the squared deviations by n understates it on average — a small but real bias. Dividing by (n minus 1) instead removes that bias, which is why software defaults to the (n minus 1) version.
The (n minus 1) divisor is the textbook example of removing bias from an estimator.
Unbiased is not the same as best. A biased estimator with far smaller variance often has lower overall error (mean squared error), which is the whole point of credibility and shrinkage.