## Biased Estimators

An estimatorfora statistical parameterissaid to be biased if

Bias is often impossible toavoid in practice and must be taken into account when statisicalcalculations are performed.

Example: To estimate thenumber,ofnesting birds, scientists catch 100, tag them and release them. Thefraction of birds with tags isLater,they catch another 100 and count the number with tags. Supposeofthese birds have tags, so the probability of a randomly picked birdfrom this sample having a tag is

Equating these two fractionsgivessothatandIn fact isa random variable since it will vary between samples, soisan estimate forWe write

The expected value forisbut it is possible thatsothat usingthis estimator. In fact,isobviously at mostsothat the estimatorisbiased.

More subtle examples of biasare give by considering the mode and median as estimators for themean.

Suppose we have 100 people.80 of the people are labelled with a 1 and 20 are labelled with a 0(probably signifying, like me, that their net wealth is zero).

The mode is 1 but the meanis 0.8 times 1 + 0.2 times 0 = 0.8

The bias of an estimatorfora parameteris

The bias of the mode as anestimator for the mean is 1-0.8=0.2

The 100 people are lined upin numerical order. First in line are those twenty people labelledwith a zero, and then the 80 people labelled with a 1.

The median is obviously 1,but the mean is 0.8, as before.

The bias of the median as anestimate for the mean is 1-0.8=0.2