Question

If two statistics are available for estimating a population characteristic, under what circumstances might you choose a biased statistic over an unbiased statistic?

Short answer

A biased statistic might be chosen over an unbiased one if it results in a lower Mean Squared Error (MSE), which takes into account both bias and variance. Also, it might be preferred due to computational simplicity, especially if the bias can be corrected.

Step-by-step solution

Define Biased and Unbiased Statistics

An unbiased statistic or estimator is one for which the mean of the statistic's sampling distribution is equal to the population parameter being estimated. That is, the statistic doesn't overestimate or underestimate the parameter on average. On the other hand, a biased statistic is one that systematically overestimates or underestimates the parameter.

Discuss scenarios for choosing biased statistic

A biased estimator might be used in preference to an unbiased one if it leads to a lower Mean Squared Error (MSE). The MSE of an estimate is the variance of the estimator plus the square of the bias. Thus, a biased estimator with low variance could have a lower MSE than an unbiased one with high variance. Also, in some cases, the computational simplicity of biased estimators might make them more attractive. This could be especially true in situations where the bias is known, and adjustments can be made to counteract it.

Illustrate with examples

For example, the sample variance (dividing by n-1 instead of n) is a biased estimator of the population variance. Despite being biased, it is preferable to the unbiased estimator (dividing by n) because it has a lower MSE.