Biased estimator

Definition

A biased estimator is a statistical estimator that does not accurately estimate a parameter because its expected value does not equal the true value of that parameter. This means that if you were to take many samples and calculate the estimator for each sample, the average of those estimators would not be equal to the actual parameter you are trying to estimate. Biased estimators can lead to systematic errors in statistical inference and decision-making.

5 Must Know Facts For Your Next Test

1. Biased estimators can be useful in certain situations where systematic errors are acceptable or where they can provide more stable estimates compared to unbiased estimators.
2. The degree of bias in an estimator can be quantified, allowing researchers to assess how far off an estimate is from the true parameter value.
3. Common examples of biased estimators include using sample variance as an estimator for population variance without Bessel's correction.
4. In practice, understanding whether an estimator is biased or unbiased helps inform decisions about its reliability in estimating parameters.
5. The use of biased estimators is often justified if they have lower mean squared error compared to unbiased estimators, especially in small samples.

Review Questions

• Compare and contrast biased estimators with unbiased estimators in terms of their properties and applications.
  • Biased estimators have expected values that do not equal the true parameter they aim to estimate, while unbiased estimators maintain this equality, leading to no systematic errors. While biased estimators can offer more stable estimates in certain contexts, unbiased estimators are preferred for their accuracy over multiple samples. Each type of estimator has its own applications; for instance, biased estimators might be used when lower variance is needed despite systematic bias, while unbiased estimators are chosen for their reliability and truthfulness in estimating parameters.
• Analyze how the concept of bias in estimators affects statistical inference and decision-making processes.
  • The presence of bias in an estimator can significantly impact statistical inference, leading to incorrect conclusions about population parameters. If an estimator consistently overestimates or underestimates a parameter, decisions based on these estimates may be misguided, potentially resulting in ineffective policies or flawed predictions. Understanding the bias allows statisticians to account for it or select alternative estimators that minimize its effects, thereby improving the accuracy and reliability of analyses.
• Evaluate a scenario where using a biased estimator might be preferable over an unbiased one, considering practical implications.
  • In a situation where limited sample sizes are available, a biased estimator might be preferable if it demonstrates lower mean squared error compared to an unbiased estimator. For example, using the sample mean directly as an estimator for a population mean could be less stable due to sample variability. A biased estimator that adjusts for sample characteristics may produce estimates that are more consistent across different samples, leading to better practical outcomes. This choice underscores the importance of evaluating bias alongside other statistical properties when making decisions.

Related terms

unbiased estimator: An unbiased estimator is a statistical estimator where the expected value equals the true parameter value being estimated, resulting in no systematic error.

mean squared error (MSE): The mean squared error is a measure of the average squared difference between an estimator's predicted values and the actual values, which helps evaluate the accuracy of an estimator.

consistency: Consistency refers to the property of an estimator whereby it converges in probability to the true parameter value as the sample size increases.