Timeline for Is this a sufficient statistic for variance?
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12 events
| when toggle format | what | by | license | comment | |
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| Feb 28 at 17:38 | vote | accept | Dave | ||
| Oct 3, 2023 at 7:05 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 23, 2022 at 7:35 | comment | added | Scortchi♦ | @krkeane: Even for parametric families, it's only in special cases that one, two, or any fixed number of statistics are sufficient regardless of the sample size. But assuming merely that observations are i.i.d. implies the order they come in is irrelevant to inference about the distribution; & the set of unordered observations is sufficient. This is clearly a paltry degree of data reduction compared with your example, but has important consequences nevertheless (I've edited my answer to mention one). | |
| Apr 22, 2022 at 18:21 | comment | added | krkeane | @Scortchi-ReinstateMonica - in the context of parametric distributions, eg the normal distribution, location and scale parameters uniquely identify the distribution, and $\sum x_i$, $\sum x_i^2$ are sufficient statistics. In the context of an empirical distribution, it seems you may not even know what statistics characterize the distribution. I'm thinking real versus deep fake images for instance. You can match any observed statistic in a synthesized distribution (eg Zhu Wu Mumford FRAME), but how do you know the number of statistics that characterize an empirical distribution? | |
| Apr 22, 2022 at 18:10 | comment | added | Scortchi♦ | @krkeane: The concept of sufficiency is the same in non-parametric families: a statistic is sufficient when the distribution of $X$ conditional on the value of the sufficient statistic doesn't depend on the particular distribution in the family from which $X$ arises. | |
| Apr 21, 2022 at 16:20 | answer | added | Scortchi♦ | timeline score: 2 | |
| Apr 21, 2022 at 16:09 | comment | added | krkeane | Okay, so I think you are trying to estimate a population central moment from a sample statistic. Your algorithm sounds adequate and perhaps optimal. I haven't done enough theoretical statistics to say if its sufficient, or if sufficiency applies to population statistics as opposed to parameters of a distribution. | |
| Apr 21, 2022 at 16:02 | comment | added | Dave | @krkeane I'm taking $\sigma^2$ to be $\mathbb E\left[\left(X -\mathbb E\left[X\right]\right)^2\right]$, not as a parameter of, say, a Gaussian distribution, so I think the answer is that I don't have a particular parametric form in mind. Why should we need a particular parametric form, though? Even if $\sigma^2$ isn't a function of the parameters (since there are no particular parameters), it is a property of the distribution that can be estimated like any other. | |
| Apr 21, 2022 at 15:57 | comment | added | krkeane | Do you have a parametric form for $F_{X}\left(x\right)$? Sufficiency depends upon the distribution you seeking to characterize. en.wikipedia.org/wiki/Sufficient_statistic | |
| Apr 21, 2022 at 15:56 | comment | added | Dave | @krkeane Part of my calculation of $S_{n+1}^2$ for estimating $\sigma^2$ involves calculating $\bar X_{n+1} = \dfrac{n\bar X_n + X_{n+1}}{n+1}$, yes. | |
| Apr 21, 2022 at 15:54 | comment | added | krkeane | Do you intend to update $\bar{X}$ with observation $n+1$? | |
| Apr 21, 2022 at 15:27 | history | asked | Dave | CC BY-SA 4.0 |