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Nov 14, 2018 at 14:52 comment added Iosif Pinelis If $\Sigma$ is not full rank, it just means that the underlying linear space was wrongly chosen: it should be the column space (say $V$) of $\Sigma$, preferably with a good basis of it, say an orthonormal eigenbasis or somewhat close to it. Then the matrix of the linear transformation $x\mapsto\Sigma x$ in that good basis will be somewhat close to a diagonal matrix, or even to the identity matrix after appropriate re-scaling. So, if $\mu\in V$, then we will have $\mu^T\Sigma^{-1}\mu\in[0,\infty)$. If $\mu\notin V$, then $\mu^T\Sigma^{-1}\mu$ should naturally be defined as $\infty$.
Nov 14, 2018 at 9:06 comment added dohmatob What happens when $\Sigma$ is not full rank ? How should the quantity $\mu^T\Sigma^{-1}\mu$ be then interpreted ? Thanks.
Oct 29, 2018 at 20:36 comment added dohmatob Also, I was wondering if you'd know the rate of convergence in of the centered Hotelling's statistics, to a $\chi^2$ mathoverflow.net/q/314094/78539 ? Some papers of yours prove that the rate is $1/2$ for the noncentered version. Unfortunately, I've not seen paper with something similar for the centered version. Thanks in advance.
Oct 29, 2018 at 20:31 comment added dohmatob OK, moved to SE math.stackexchange.com/q/2976654/168758
Oct 29, 2018 at 20:18 comment added Iosif Pinelis I am not really active on that forum.
Oct 29, 2018 at 17:59 comment added dohmatob OK, I've opened a new question on "stats exchange" (which I think is the best venue for such ramblings) stats.stackexchange.com/q/374292/156791. Thanks in advance for your response.
Oct 24, 2018 at 20:05 history edited Iosif Pinelis CC BY-SA 4.0
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Oct 24, 2018 at 19:55 comment added Iosif Pinelis I thought the comment right after Remark 7.3 in the MLE paper should be enough. I guess I can try to state the result for $M$-estimators formally. However, I think this would rather be a separate question.
Oct 24, 2018 at 16:37 comment added dohmatob Thanks for the generous update. OK, I see . However, concerning $M$-estimators, this extension is not completely trivial though. IMO, it deserves a separate treatment (post, manuscript etc.). I don't master enough statistics to do this myself without screwing certain things badly. I can make this into a separate question, and maybe you could roughly sketch the main steps ? Thanks.
Oct 24, 2018 at 15:21 comment added Iosif Pinelis (1) I have now reproduced the explicit expression for $\tilde\sigma$. (2) Results of the first referenced paper do apply to $M$-estimators, albeit somewhat indirectly; see the comment right after Remark 7.3 in projecteuclid.org/euclid.ejs/1491897618 .
Oct 24, 2018 at 15:17 history edited Iosif Pinelis CC BY-SA 4.0
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Oct 23, 2018 at 23:06 comment added dohmatob Two short questions: (1) In Theorem 3.9 what's an explict formula for $\tilde{\sigma}$ (it's supposed to be defined in 2.21, but I kinda lost tract of the definition of $L(V)$ relative to Hotelling's statistic under study in Theorem 3.9. (2) Can the works in this paper be applied to get asymptotic distributions of $M$-estimators ? Thanks in advance.
Oct 23, 2018 at 18:39 vote accept dohmatob
Oct 23, 2018 at 16:49 history answered Iosif Pinelis CC BY-SA 4.0