# Biased Estimators

An estimatorfor a statistical parameteris said to be biased if

Bias is often impossible to avoid in practice and must be taken into account when statisical calculations are performed.

Example: To estimate the number,of nesting birds, scientists catch 100, tag them and release them. The fraction of birds with tags isLater, they catch another 100 and count the number with tags. Supposeof these birds have tags, so the probability of a randomly picked bird from this sample having a tag is

Equating these two fractions givesso thatand In fact is a random variable since it will vary between samples, sois an estimate for We write

The expected value foris but it is possible thatso that using this estimator. In fact,is obviously at mostso that the estimatoris biased.

More subtle examples of bias are give by considering the mode and median as estimators for the mean.

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 mean is 0.8 times 1 + 0.2 times 0 = 0.8

The bias of an estimatorfor a parameteris

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

The 100 people are lined up in numerical order. First in line are those twenty people labelled with 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 an estimate for the mean is 1-0.8=0.2