![]() This is a tedious exercise to show, but it is accessible to an undergraduate level statistics student.Īs for your second comment, if you are calculating weighted absolute standard deviation as in the first paper you linked, then yes of course, $N'$ appears directly in the formula. It just so happens that by using this strange looking formula, it actually turns out that $s,s_w$ are unbiased estimators of $\sigma$. That is, we want $E=\sigma$, and we want $E = \sigma$ where $\sigma$ is the true population standard deviation, and $E$ denotes the expected value. The reason that this isn't just ignored in the first paper you cited is because we like our estimators to be unbiased. ![]() You are also correct that $(N'-1)/N'$ will be close to one if $N$ is large. If we have $w_i=0$ then we are ignoring the $i$th observation, so it doesn't really count as part of our sample. The number of nonzero weights is effectively the sample size. ![]()
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