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Trying to learn statistics, haphazardly.
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Jul 30 |
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Intuition behind why Stein's paradox only applies in dimensions $\ge 3$ Actually, something like this is what I hoped to. A connection to another field of mathematics (be it differential geometry or stochastic processes) which shows that the admissibility for $n=2$ was not just a fluke. Great answer! |
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Jul 29 |
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Mathematical reference for the convergence in distribution of the Gibbs sampler In my personal experience, mathematicians also take a delight in avoiding sledge hammers (non-complex proofs of the prime number theorem comes to mind)! :) I would imagine that a finite-state Markov chain could be proven along the lines of the Bernoulli case outlined by Casella-George. So I guess Casella-George suitably adapted is a proof of convergence for distributions on finite (and perhaps discrete) sets (which is a nice, but a bit small, class of distributions admittedly). |
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Jul 29 |
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Mathematical reference for the convergence in distribution of the Gibbs sampler That is my also suspicion, which is why I am hoping that somewhere someone has written a Markov chain theory text which culminates with proving some properties of a Gibbs sampler. But perhaps using full-blown Markov chain theory is using a sledgehammer to kill a mosquito, and there is a simpler proof (in the sense that it doesn't depend on too much Markov chain machinery). |
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Jul 29 |
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Intuition behind why Stein's paradox only applies in dimensions $\ge 3$ That'd be great. I'll try to share my thoughts as well, which are not exactly enlightened, but at least somewhat clearer than a few days ago. |
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Jul 28 |
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Intuition behind why Stein's paradox only applies in dimensions $\ge 3$ As you point out, it is of course not the main condition, only one of the reasons one might suspect that shrinking the estimate is a good idea. In Stein's original paper, he takes this as the starting point for the intuitive discussion and shows that the problem gets even worse in higher dimensions. I'll update the text accordingly. |
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Jul 27 |
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Intuition behind why Stein's paradox only applies in dimensions $\ge 3$ @probabilityislogic and cardinal Would you guys mind elaborating? :) I'm curious but I don't see exactly what you mean. |
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Jul 27 |
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Beta binomial Bayesian updating over many iterations (I presume a sort of ad hoc engineering approach here) Then I would suggest dividing by an $N$ every week (or day, hour, etc..). I.e., (1) above. This will discount observations last week by $N$, observations from the week before that by $N^2$ and so on. What you are in effect doing is a weighted average where you give more weight to more recent observations. |
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Jul 26 |
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Intuition behind why Stein's paradox only applies in dimensions $\ge 3$ I agree. Do you have any good references for that (aside from the original paper). I found Stein's original paper overly computational and was hoping that someone would have developed a different method in the last fifty years. |
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Jul 26 |
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Intuition behind why Stein's paradox only applies in dimensions $\ge 3$ Sorry, I should have explicitly mentioned that Stein proved that for $N=2$, the MLE is admissible! See projecteuclid.org/… |
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Jul 26 |
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Intuition behind why Stein's paradox only applies in dimensions $\ge 3$ @mpiktas 1) Yes, the setup is purely theoretical. But still really important (for example, we can let the variables be means of i.i.ds). 2) Yes, although I don't know what you mean with required properties. 3) See above. 4) True, but the expected value of $1/S$ is not defined for $N=1$. (the inverse chi square distribution doesn't have a mean for $N=1$). |
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Jul 20 |
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What is the expected value of the sample variance under a linear regression with omitted variables of an AR(2) process? When you are saying $RSS_p = RSS_{p-1}(1-\phi^2_{pp})$, do you mean $\mathbb{E}(RSS_p) = \mathbb{E}(RSS_{p-1}(1-\phi^2_{pp}))$? It seems to me that the only sensible thing we can talk about is the expected value of the $RSS$. |
