Lesson plan
Consistent estimator
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter θ 0 —having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ 0 . This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to θ 0 converges to one. In practice one constructs an estimator as a function of an available sample of size n, and then imagines being able to keep collecting data and expanding the sample ad infinitum. In this way one would obtain a sequence of estimates indexed by n, and consistency is a property of what occurs as the sample size “grows to infinity”. If the sequence of estimates can be mathematically shown to converge in probability to the true value θ 0 , it is called a consistent estimator; otherwise the estimator is said to be inconsistent. Consistency as defined here is sometimes referred to as weak consistency. When we replace convergence in probability with almost sure convergence, then the estimator is said to be strongly consistent. Consistency is related to bias; see bias versus consistency.
Article summary
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter θ 0 —having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ 0 . This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to θ 0 converges to one. In practice one constructs an estimator as a function of an available sample of size n, and then imagines being able to keep collecting data and expanding the sample ad infinitum. In this way one would obtain a sequence of estimates indexed by n, and consistency is a property of what occurs as the sample size “grows to infinity”. If the sequence of estimates can be mathematically shown to converge in probability to the true value θ 0 , it is called a consistent estimator; otherwise the estimator is said to be inconsistent. Consistency as defined here is sometimes referred to as weak consistency. When we replace convergence in probability with almost sure convergence, then the estimator is said to be strongly consistent. Consistency is related to bias; see bias versus consistency.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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