Timeline for When data has a gaussian distribution, how many samples will characterise it?
Current License: CC BY-SA 3.0
22 events
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| Oct 15, 2022 at 0:38 | answer | added | Yaroslav Bulatov | timeline score: 0 | |
| Oct 15, 2022 at 0:37 | comment | added | Yaroslav Bulatov | Look at the discussion in comments mathoverflow.net/questions/395098/… . In particular Vershynin's Theorem 9.2.4 addresses this, from High-Dimensional Probability book, it's a dimension-free sample bound for estimating Gaussian covariance matrix | |
| Jun 11, 2013 at 21:46 | vote | accept | omatai | ||
| S Jun 8, 2013 at 0:08 | history | bounty ended | CommunityBot | ||
| S Jun 8, 2013 at 0:08 | history | notice removed | CommunityBot | ||
| Jun 6, 2013 at 16:15 | answer | added | whuber♦ | timeline score: 13 | |
| Jun 4, 2013 at 18:32 | comment | added | whuber♦ | Snedecor & Cochran [Statistical Methods, 8th edition] are authorities on sampling. They describe this process in chapters 4 and 6: "we assume at first that the population standard deviation $\sigma_D$ ... is known." Later they write, "The method is therefore most useful in the early stages of a line of work. ... For example, previous small experiments have indicated that a new treatment gives an increase of around 20% and $\sigma$ is around 7%. The investigator ... [wants an] SE of $\pm$2% and thus sets $\sqrt{2}(7)/\sqrt{n}=2$, giving $n=25$ ... This ... is often helpful in later work. | |
| Jun 3, 2013 at 22:50 | comment | added | omatai | @whuber - Can you provide a reference to somewhere that details the process you describe? Better still, formulate an answer which incorporates this and any other relevant observations, and claim a bounty of 50 points :-) | |
| Jun 3, 2013 at 19:16 | comment | added | whuber♦ | You're right; it absolutely does depend on the underlying covariance matrix. The usual approach--even in the univariate case--is to obtain a small preliminary sample set, estimate the covariances from that, and then use those to figure out how many additional samples would be needed to estimate any desired parameters to specified accuracy: that's where the absolute scale comes in. If you don't have any such accuracy targets, then in principle you would be happy with the absolute minimum number of samples needed to estimate the parameters. | |
| Jun 3, 2013 at 17:53 | answer | added | EngrStudent | timeline score: 1 | |
| Jun 3, 2013 at 17:33 | comment | added | EngrStudent | I think that the details are implied, but not explicitly stated in the question. How many samples are required in a multivariate Gaussian to get the same level of uncertainty in parameter estimate as that which arises from 30 samples informing a univariate Gaussian distribution, all other things being equal. | |
| May 31, 2013 at 3:37 | comment | added | omatai | I understand the first part of your comment, but not what you mean by an "absolute" scale of desired accuracy. I am trying to answer practical questions of data acquisition in my own mind - wondering whether 10 or 100 or 1000 samples is just as effective at encapsulating data in 1-D as in 10-D or 100-D, and if not, why not. Maybe it depends on the covariance matrix and the co-dependence/independence of the data, or maybe it doesn't. Right now I'd be grateful for anything that helps me decide if constructing a problem in 200-D might simply require impractically large numbers of samples. | |
| May 30, 2013 at 23:00 | comment | added | whuber♦ | As soon as you have one more independent data value than there are parameters (whence $\binom{d+2}{2}$ values in $d$ dimensions), you can erect a 95% confidence region around them. (One can do even better using non-traditional techniques.) That's an answer--it's a definitive one--but it's probably not what you're looking for. The point is that you need to stipulate some absolute scale of desired accuracy in order to obtain an answer to this question. | |
| S May 30, 2013 at 22:31 | history | bounty started | omatai | ||
| S May 30, 2013 at 22:31 | history | notice added | omatai | Canonical answer required | |
| May 19, 2013 at 23:52 | history | edited | omatai | CC BY-SA 3.0 |
Better question specification
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| May 19, 2013 at 23:22 | comment | added | Glen_b | Ah, you now have a relatively standard type of question; you should probably edit your original question to reflect your intent more clearly. | |
| May 19, 2013 at 23:17 | comment | added | omatai | That is to say... 95% of all samples will lie within some defined distance of the mean. How many samples are required to define that distance (interval/ellipse/ellipsoid/etc) with 95% or better confidence? | |
| May 19, 2013 at 23:11 | comment | added | omatai | How about: how many samples does it take to be at least 95% confident that 95% of all samples (but only 95% of all samples) will lie within a defined interval/ellipse/ellipsoid/hyperellipsoid? | |
| May 19, 2013 at 23:00 | history | tweeted | twitter.com/#!/StackStats/status/336255042501890048 | ||
| May 19, 2013 at 22:37 | comment | added | Glen_b | Without a sufficiently precise definition of 'pin down', it's not really possible to answer this question even for a univariate Gaussian. | |
| May 19, 2013 at 22:09 | history | asked | omatai | CC BY-SA 3.0 |
