Comments on: [ArXiv] Identifiability and mixtures of distributions, Aug. 3, 2007

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Comments on: [ArXiv] Identifiability and mixtures of distributions, Aug. 3, 2007 http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/ Weaving together Astronomy+Statistics+Computer Science+Engineering+Intrumentation, far beyond the growing borders Fri, 01 Jun 2012 18:47:52 +0000 hourly 1 http://wordpress.org/?v=3.4 By: hlee http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/comment-page-1/#comment-90 hlee Mon, 10 Sep 2007 03:51:06 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/#comment-90 <p>In addition to Likelihood Ratio Test (LRT), Condition 2 in the paper, although the appendix didn't explicitly state identifiability (I forgot it because generally identifiability is stated as regularity conditions), implies this identifiability. For a simple hypothesis testing, the null hypothesis is stated as Ho:Ое=Ое* (I chose the same Ое* in the paper) and the statement related to this Ое* bears the idea, the parameters to be identifiable. A textbook version of identifiability is f(Ое_1)=f(Ое_2) implies Ое_1 = Ое_2, where Ое_i are parameters and f(.) are density functions. Fitting mixture of normal distributions suffers from identifiability issues.</p> In addition to Likelihood Ratio Test (LRT), Condition 2 in the paper, although the appendix didn’t explicitly state identifiability (I forgot it because generally identifiability is stated as regularity conditions), implies this identifiability. For a simple hypothesis testing, the null hypothesis is stated as Ho:Ое=Ое* (I chose the same Ое* in the paper) and the statement related to this Ое* bears the idea, the parameters to be identifiable. A textbook version of identifiability is f(Ое_1)=f(Ое_2) implies Ое_1 = Ое_2, where Ое_i are parameters and f(.) are density functions. Fitting mixture of normal distributions suffers from identifiability issues.

]]> By: vlk http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/comment-page-1/#comment-88 vlk Sat, 08 Sep 2007 17:57:26 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/#comment-88 Re Identifiability: Protassov et al don't explicitly use that word. Which of the regularity conditions listed by them is about identifiability? If not identifiability, perhaps you can more easily say what is non-identifiability? Re Identifiability: Protassov et al don’t explicitly use that word. Which of the regularity conditions listed by them is about identifiability?

If not identifiability, perhaps you can more easily say what is non-identifiability?

]]> By: hlee http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/comment-page-1/#comment-86 hlee Fri, 07 Sep 2007 19:32:33 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/#comment-86 1.I haven't apprehended fully the idea of identifiability beyond mathematical definitions and I'm unable to illustrate its profound meaning by adopting actual astronomical/empirical examples. An ApJ paper by Protassov et.al. (<a href="http://adsabs.harvard.edu/abs/2002ApJ...571..545P" rel="nofollow">2002, ApJ, 571, p.545</a>) describes identifiability as one of the regularity conditions in the appendix. 2. Without theoretical details, if I were asked to made a short comment on H-L estimator, its asymptotic variance is smaller by 3 times compared to that of median estimator. In other words, Asymptotic relative efficiency of H-L to median is 3. I think this is related to U-statistics, which I haven't comprehended the related theories fully. 1.I haven’t apprehended fully the idea of identifiability beyond mathematical definitions and I’m unable to illustrate its profound meaning by adopting actual astronomical/empirical examples. An ApJ paper by Protassov et.al. (2002, ApJ, 571, p.545) describes identifiability as one of the regularity conditions in the appendix.

2. Without theoretical details, if I were asked to made a short comment on H-L estimator, its asymptotic variance is smaller by 3 times compared to that of median estimator. In other words, Asymptotic relative efficiency of H-L to median is 3. I think this is related to U-statistics, which I haven’t comprehended the related theories fully.

]]> By: vlk http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/comment-page-1/#comment-84 vlk Fri, 07 Sep 2007 17:54:51 +0000 http://hea-www.harvard.edu/AstroStat/slog/2007/arxiv-identifiability-and-mixtures/#comment-84 Couple of questions -- 1, what exactly is "identifiability"? 2, they use something called the Hodges-Lehmann estimator as a proxy for the median. What advantage does this H-L estimator have over the median? Couple of questions –
1, what exactly is “identifiability”?
2, they use something called the Hodges-Lehmann estimator as a proxy for the median. What advantage does this H-L estimator have over the median?

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