Timeline for The variance of the weighted median and optimal weights
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 5, 2022 at 0:21 | history | edited | Wolfgang Brehm | CC BY-SA 4.0 |
add code and pdf link
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| May 8, 2022 at 12:54 | comment | added | Wolfgang Brehm | Let us continue this discussion in chat. | |
| May 8, 2022 at 12:22 | comment | added | user225256 | I have no idea what the parameters of this question are. | |
| May 8, 2022 at 11:52 | comment | added | Wolfgang Brehm | @MattF. "If the distributions (the function $p_i$) are inputs to the problem, then the minimum-variance weighting should place all the weight on one distribution" If we know the distribution of each sample, we know the median of each sample distribution. Why not take the median directly then? | |
| May 8, 2022 at 10:48 | comment | added | Wolfgang Brehm | I am not assuming that all samples come from the same distribution: "Assume all samples $x_i$ can follow a different probability distribution" I would have been happy to grant this assumption although my specific problem would not have been solved that way, but it turns out it is not needed anyways. You are correct though with the observation that the optimal weights would be inverse to the scale factor and inverse to the square root of the variance in that case. | |
| May 8, 2022 at 10:36 | comment | added | Wolfgang Brehm | @MattF. Estimating the density at the median of each sample is just as fundamental to the optimal weights of the weighted median as the variance is for the weights in the weighted mean. There is no way around it, you need to have some estimate or proxy or you are better off using the unweighted median or mean. The method of estimation can be different in different applications. | |
| S May 8, 2022 at 10:29 | review | First answers | |||
| May 8, 2022 at 15:38 | |||||
| S May 8, 2022 at 10:29 | history | edited | Wolfgang Brehm | CC BY-SA 4.0 |
add dependence on variance of samples
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| May 8, 2022 at 7:15 | comment | added | user225256 | Are you assuming that all the samples come from scaled copies of the same distribution? In that case the weights you propose would be inversely proportional to the square roots of the variances in each sample, which we can estimate well enough. That would make sense of a lot of what you say, but if that’s the assumption it’s worth highlighting and explaining. | |
| May 8, 2022 at 6:50 | comment | added | user225256 | As @whuber asked about the question: “how do you know or estimate the density at the median of each of these distributions?” If the distributions (the functions $p_i$) are inputs to the problem, then the minimum-variance weighting should place all the weight on one distribution, as in my answer; if the distributions are not inputs to the problem, then this answer is incomplete, because it doesn’t explain how to calculate the weights proposed at the end. | |
| May 7, 2022 at 23:06 | history | edited | Wolfgang Brehm | CC BY-SA 4.0 |
[Edit removed during grace period]
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| S May 7, 2022 at 22:47 | review | First answers | |||
| May 7, 2022 at 23:38 | |||||
| S May 7, 2022 at 22:47 | history | answered | Wolfgang Brehm | CC BY-SA 4.0 |