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I am trying to track down a formula (and derivation) for an unbiased estimate of the variance of a weighted mean.

Wikipedia provides such a formula:

http://en.wikipedia.org/wiki/Weighted_mean However, the reference they cite (the GNU Scientific Library implementation comments):

a. has no derivation b. states that the implementation is designed for the circumstance where the weight is a variance normalization. The circumstance I am interested in is where the weight is an arbitrary set of numbers which are positive and sum to 1.

Does anyone know of a derivation of the formula wikipedia gives, or related work?

Thanks,

SetJmp

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Brian Gough (maintainer of the Gnu Scientific Library) pointed me towards a document within the GSL version control repository that describes their derivation. The document pretty much answer's my question. –  user3268 Jan 17 '10 at 0:30
    
I edited my answer a little bit. It seems the documents you point to make a weighted estimator of the sampled random variable, rather than estimating the weighted mean random variable. This is weird for me. –  Matus Telgarsky Jan 25 '10 at 7:00
    
Sorry, not an answer, but I've just asked a related question (re. what to do for unnormalized weights) at mathoverflow.net/questions/22203/… I'd appreciate if you were able to help me out :-) And hopefully this link will also be useful to someone at some point... Andy –  andybuckley Apr 22 '10 at 15:42
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4 Answers

First some notation. Each example is drawn from some unknown distribution $Y$ with $E[Y] = \mu$ and $\textrm{Var}[Y] = \sigma^2$. Suppose the weighted mean consists of $n$ independent draws $X_i\sim Y$, and $\{w_i\}_1^n$ is in the standard simplex. Finally define the r.v. $X = \sum_i w_i X_i$. Note that $E[X] = \sum_i w_i E[X_i] = \mu$ and $\textrm{Var}[X] = \sum_i w_i^2 \textrm{Var} [X_i] = \sigma^2\sum_i w_i^2$.

Generalizing the standard definition of sample mean, take $$ \hat \mu(\{x_i\}_1^n) := \sum_i w_i x_i. $$ Note that $E[\hat \mu(\{x_i\}_1^n)] = \sum_i w_i E[x_i] = \mu = E[X]$, so $\hat \mu$ is an unbiased estimator.

For the sample variance, generalize the sample variance as $$ \hat \sigma^2_b(\{x_i\}_1^n) := \sum_i w_i (x_i - \hat \mu(\{x_i\}_1^n))^2, $$ where the subscript foreshadows this will need a correction to be unbiased. Anyway, $$ E[\hat \sigma^2_b] = \sum_i w_i E[(x_i - \hat \mu)^2] = \sum_i w_i E\left[\left(\sum_j w_j (x_i - x_j)\right)^2\right]. $$ The term in the expectation can be written as $$ \sum_{j,k} w_j(x_i - x_j)w_k(x_i - x_k) = \sum_jw_j^2(x_i - x_j)^2 + \sum_{j\neq k} w_j w_k(x_i - x_j)(x_i - x_k). $$ Passing in the expectation, the first term (when $x_i\neq x_j$, which would yield 0) is $$ E[(x_i-x_j)^2] = 2E[x_i^2] - 2\mu^2 = 2\sigma^2, $$ whereas the second (when $x_i \neq x_j$ and $x_i \neq x_k$, which would yield 0) is $$ E[x_i^2 - x_ix_j - x_ix_k + x_jx_k] = E[x_i^2] - \mu^2 = \sigma^2. $$ Combining everything, $$ \sum_i w_i \left(2\sigma^2\sum_{j\neq i}w_j^2 + \sigma^2\sum_{j\neq k\neq i} w_j w_k\right) = \sigma^2( 1 - \sum_j w_j^2). $$ Therefore $E[\hat \sigma_b^2] - \sigma^2 = -\sigma^22\sum_j w_j^2$, i.e. this is a biased estimator. To make this an unbiased estimator of $Y$, divide by the excess term derived above: $$ \hat \sigma_u^2(\{x_i\}_1^n) := \frac {\hat \sigma_b^2(\{x_i\}_1^n)}{1- \sum_j w_j^2} = \frac {\sum_i w_i(x_i - \hat \mu)^2}{1- \sum_j w_j^2 } $$ This matches the definition you gave (and a sanity check $w_i = 1/N$, recovering the normal unbiased estimate).

Now, if one instead were to seek an unbiased estimator of $X=\sum_i X_i$, the formula would instead be $\hat \sigma_b^2(\{x_i\}_1^n)(\sum_j w_j^2) / ( 1 - \sum_j w_j^2)$.

It is very odd for me that the documents you refer to are making estimators of $Y$ and not $X$; I don't see the justification of such an estimator. Also it is not clearly how to extend it to samples that don't have length $n$, whereas for the estimator of $X$, you simply have some number $m$ of $n$-samples, and averaging everything above makes things work out. Also, I didn't check, but it's my suspicion that the weighted estimator for $Y$ has higher variance than the usual one; as such, why use this weighted estimator at all? Building an estimator for $X$ would seem to have been the intent..

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Hi Matus - I am not sure where the indices (i \neq j) come from in the combining everything section. Also, your decomposition of variance in the beginning is flawed (missing a square). –  user3268 Jan 17 '10 at 0:28
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Hi Matus,

Thanks for your comments (over a year ago!). I derived the same results for myself yesterday, but a question: why would you ever want such a weighted sample variance?

If we assume that all Yi come from same population (with mean mu, variance sigma2), and YBARwtd = 1/V1 * sum(i = 1 to n) Wi * Yi

then var(YBARwtd) = (1/V1^2) * V2 * sigma2

where V1 = sum of weights, V2 = sum of squared weights,

then surely we just want to get the best possible estimate of sigma2, the variance of the population from which the Yi come, and assuming they are independent observations from a single population, surely the best estimate is the usual unweighted one? So why would you ever want that weighted sample variance?

Not sure if you'll get alerted to this added comment, but you're the most likely person I've found on the web to discuss this, so I hope you have some comment.

Cheers, Kathy

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I also ended up deriving Kathy's result.

The weighted mean is merely a projection w.x onto a sample x drawn from the population. If the x's are IID and w.1 = 1 then Var(w.x) = V2 Var x where V2 = w.w (sum of squares of weights).

For an unnormalized mean, following the usual rules for the Variance operator:

WMean x = w.x/w.1 => Var WMean x = (1/w.1)^2 Var w.x = (w.w)/(w.1)^2 Var x = V2/V1^2 Var x

(here V1 is the sum of the weights). For Var x you should take the usual Bessel corrected sample variance estimator:

Var x = 1/(n-1) Sum (x - Mean x)^2

This then gives an unbiased estimator of the Variance of the Weighted Mean.

Graham

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There is something I don't understand in Matus' derivation. Shouldn't the expectation operator E be applied to the weights as well? For instance, isn't E[Wi] = 1/n?

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