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Hello List
My inquiry is quite straight forward. I
require an unbiased estimate of variance using weighted samples.
There are several equations commonly used to calculate an estimate of
variance using weighted samples. But they are all slightly different and
thus, they can't all be unbiased. Currently, my favorite equation is as
follows:
1. Calculate a weighted estimate of the mean
as xbar = Sum[(w_i * x_i)] / Sum[w_i], where x_i are
sample values and w_i are the corresponding sample weights.
2. Calculate the denominator D = Sum[w] - Sum[w *
w] / Sum[w], where Sum[w] is the sum of weights and Sum[w * w] is the sum
of squared weights.
3. Now, I believe an unbiased estimate of
the variance is given by s2 = Sum[ w_i * (x_i - xbar) * (w_i - xbar)] / D where
xbar is the weighted estimate of the mean. Do you agree?
The thing that bothers me most is that JMP (an
excellent EDA stat tool put out by SAS for those of you not familiar with
JMP) calculates the weighted estimate of variance as follows: s2 =
Sum[w_i * (x_i - xbar) * (x_i - xbar)] / N-1 where xbar is the weighted
estimate of the mean. JMP Support insists that this equation is correct.
However, it doesn't make any sense to me. Can anyone explain the theoretical
basis or statistical model that might give some validity to this equation?
Thank you for your response.
Edward Isaaks.
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