Consistent estimator

Definition

A consistent estimator is a statistical estimator that, as the sample size increases, converges in probability to the true value of the parameter being estimated. This means that with larger samples, the estimates become increasingly accurate, making consistent estimators essential in estimation methods. Consistency is a desirable property because it ensures that as you collect more data, your estimates get closer to reality.

5 Must Know Facts For Your Next Test

1. Consistent estimators ensure that with an increasing number of observations, the estimates converge to the true parameter value with high probability.
2. For an estimator to be consistent, it does not need to be unbiased; it can be biased but should still converge to the true value as sample size increases.
3. Common examples of consistent estimators include the sample mean and sample variance under certain conditions.
4. Consistency can be established through the law of large numbers, which states that as the number of trials increases, the sample mean will converge to the expected value.
5. In practical terms, consistent estimators are critical in fields like economics and biology, where accurate parameter estimation is crucial for decision-making.

Review Questions

• What is the relationship between sample size and the accuracy of a consistent estimator?
  • The accuracy of a consistent estimator improves as the sample size increases. This means that with larger samples, the estimates produced by the estimator converge in probability to the actual value of the parameter being estimated. Therefore, collecting more data leads to more reliable and precise estimates.
• Discuss how bias affects the consistency of an estimator and provide an example.
  • Bias refers to the systematic difference between an estimator's expected value and the true parameter value. While a consistent estimator can be biased, its bias must decrease as sample size increases for consistency to hold. For instance, if an estimator consistently overestimates a parameter but does so by a diminishing amount with larger samples, it can still be considered a consistent estimator.
• Evaluate why consistent estimators are important in practical applications and their implications for data analysis.
  • Consistent estimators are crucial in practical applications because they provide confidence that estimates will accurately reflect true parameters as more data is collected. This reliability is vital in fields like finance or medicine, where decisions based on estimates can significantly impact outcomes. The implication is that researchers and analysts can trust their findings over time, allowing them to make informed decisions based on robust statistical foundations.

Related terms

Convergence in probability: A type of convergence where an estimator approaches the true parameter value in probability as the sample size increases.

Bias:

The difference between the expected value of an estimator and the true value of the parameter it estimates; a consistent estimator may still be biased but will become less biased as sample size grows.

Asymptotic normality:

A property of estimators where, as sample size increases, their distribution approaches a normal distribution, facilitating inference based on larger samples.