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If I take some estimator X. Lets say that X is unbiased.

Suppose I have 100 samples and each sample has 5 points. I now calculate the value of X on each sample.

Because X is unbiased, this means that the average value of X will equal to its population value (ex: real value)?

And because X is unbiased, this will be true if the 100 samples have 4 points, 3 points, 18 points, etc for all points?

And because X is unbiased, this will be true even if the 100 samples have unequal numbers of points (ex: sample1 has 6 points, sample2 has 10 points, etc)?

So basically, unbiased estimator means that when the value of the estimator is calculated for many samples, on average the average value of the estimator will be equal to the population value?

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  • $\begingroup$ Could you re-phrase that so the Question reflected the exposition? $\endgroup$ Commented Oct 11, 2023 at 23:39

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Yes, "on average" (more precisely in expectation) the average value of the samples $X_1, \dots, X_n$ will be equal to the population parameter, regardless of the size of the samples. In the classic dartboard analogy, the estimates will not cluster away from the target, they might be inaccurate but they'll be evenly spread around the target.

That applies regardless of the number of observations in the sample -- that will of course change the variance of each sample, so with smaller sample sizes you will get very imprecise (if unbiased) estimates.

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