Question

Why is an unbiased statistic generally preferred over a biased statistic for estimating a population characteristic? Does unbiasedness alone guarantee that the estimate will be close to the true value? Explain. Under what circumstances might you choose a biased statistic over an unbiased statistic if two statistics are available for estimating a population characteristic?

Short answer

An unbiased statistic is usually preferred as it doesn't consistently over/underestimate the true value. However, unbiasedness alone doesn't guarantee closeness to the true value: high variance can lead to significant differences. When a biased statistic results in lower variance and closer estimates especially with smaller sample sizes, it might be preferred.

Step-by-step solution

Define Biased and Unbiased Statistics

Firstly, it's important to understand what these terms mean: a statistic is unbiased if its expected value is equal to the population parameter being estimated. Conversely, a statisic is biased if its expected value is not equal to the parameter it estimates.

Explain Preference for Unbiased Statistics

An unbiased statistic is generally preferred over a biased one because an average taken from unbiased statistics will be close to the population parameter. Over time, it doesn't consistently underestimate or overestimate the true value, making unbiased estimators more reliable.

Discuss Limits of Unbiasedness

However, being unbiased doesn't guarantee that an estimate will always be close to the true value. For instance, if the variance of the statistic is high, the estimates can differ significantly from the population parameter despite being unbiased on average. Thus, while unbiasedness is generally preferred, it's not the only factor to consider.

Explain Choosing a Biased Statistic

There could be situations where a biased statistic might be preferred. For example, if the biased statistic consistently gives estimates closer to the population parameter with lower variance, it could be chosen over an unbiased statistic. Accuracy and precision of estimates are key, especially when sample sizes are small.