Why is an “unbiased estimator” called as such?

Ignacio Riggs 2022-10-21 Answered
Why is an “unbiased estimator” called as such?
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Answers (1)

rcampas4i
Answered 2022-10-22 Author has 22 answers
The bias of an estimator θ ^ for a parameter θ is defined as
B i a s ( θ ^ ) = E ( θ ^ ) θ .
Thus "unbiased" is exactly the same as having B i a s ( θ ^ ) = 0..
For an intuitive explanation, suppose we have a population of people living in a small village who have heights 65 " , 65 " , 67 " , 68 " , 78 " ,, but the person who is 78 " tall is a hermit who doesn't like being sampled. If we construct an estimate for the population mean by sampling two people who are not hermits, call this μ ^ ,, we would get
E ( μ ^ ) = 65 " + 66 " + 66 " + 66.5 " + 66.5 " + 67.5 " ( 4 2 ) = 397.5 " 6 = 66.25 " ,
while μ = 68.6 " .". Thus, this estimator has a bias of 66.25 " 68.6 "= 2.35 " .. But we should expect that this estimator would be biased, since it isn't taking a representative sample of the population, so this agrees with intuition.
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