An unbiased estimator is a statistical term referring to a property of an estimator where the expected value of the estimate equals the true value of the population parameter being measured. Essentially, this means that, on average, an unbiased estimator hits the target: it does not systematically overestimate or underestimate the population parameter.
Importance of Unbiased Estimators in Statistics
Unbiased estimators are crucial in statistics because they ensure the accuracy of conclusions drawn from sample data. By leveraging unbiased estimators, statisticians can make reliable inferences about a population based on sampled data.
Key Properties
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Expected Value: The expected value \( E(\hat{\theta}) \) of an unbiased estimator \( \hat{\theta} \) is equal to the parameter \( \theta \).
$$ E(\hat{\theta}) = \theta $$ -
Consistency: An unbiased estimator is consistent, meaning it converges to the true parameter value as the sample size increases.
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Efficiency: Among unbiased estimators, the one with the smallest variance is considered the most efficient.
Examples of Unbiased Estimators
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Sample Mean as an Unbiased Estimator: The sample mean \( \bar{X} \) is an unbiased estimator of the population mean \( \mu \):
$$ E(\bar{X}) = \mu $$ -
Sample Variance as an Unbiased Estimator: The sample variance \( s^2 \) (adjusted by \( n-1 \) instead of \( n \)) is an unbiased estimator of the population variance \( \sigma^2 \):
$$ E\left(\frac{1}{n-1}\sum_{i=1}^{n} (X_i - \bar{X})^2\right) = \sigma^2 $$
Special Considerations
Bias and Sampling Methods
The bias in an estimator can often result from flawed sampling methods. For instance, consider sampling only within one neighborhood in a city when estimating the average credit card balances city-wide. This would lead to a biased estimator if the financial behaviors of residents in that neighborhood differ significantly from those in other parts of the city.
Corrective Measures
To avoid biased estimators, it’s crucial to employ proper sampling techniques such as simple random sampling, stratified sampling, or systematic sampling. The goal is to ensure that every member of the population has an equal chance of being selected in the sample.
Historical Context
The concept of unbiasedness was formally introduced in the early 20th century as statistics evolved as a distinct field of study. Sir Ronald A. Fisher, a pivotal figure in statistics, heavily contributed to the understanding and formalization of estimation theory.
Practical Applicability
- Finance: Estimating average return on investment.
- Economics: Estimating average household income.
- Real Estate: Estimating average property prices.
- Healthcare: Estimating average recovery times for treatments.
Comparisons and Related Terms
Biased Estimator
A biased estimator \( \hat{\theta} \) satisfies:
Consistent Estimator
An estimator is consistent if it converges in probability to the parameter as the sample size increases.
Efficient Estimator
An estimator with the smallest variance among all unbiased estimators of the same parameter.
FAQs
Why is an unbiased estimator preferred?
Can an estimator be unbiased and inefficient?
How do we test for unbiasedness?
References
- Fisher, R. A. (1925). Statistical Methods for Research Workers.
- Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation.
Summary
The unbiased estimator is a foundational concept in statistics that ensures the accuracy of estimations by matching the expected value to the population parameter. By utilizing proper sampling methods and understanding the underlying principles, statisticians can effectively employ unbiased estimators in a wide range of applications, from finance to healthcare.
