Testwiki:Reference desk/Archives/Mathematics/2013 February 12

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February 12

Cosets and normal subgroup equality

Hello. I recently came across an exercise, and would be very grateful if someone could help.

Given a normal subgroup H in G, with g in G, if $gH=H$, then is $g\in H$?

Neuroxic (talk) 11:31, 12 February 2013 (UTC)

Any subgroup H contains the identity. Think what happens with that. Dmcq (talk) 12:11, 12 February 2013 (UTC)

That question is a special case of the following question, which may actually be easier (intuitively) to solve: if $gH=L$, then is $g\in L$? Basically you're just asking "Is $g\in gH$ for some subgroup $H$?" and it is a very well known fact that the answer is yes, and the proof is also very well known and straightforward: $1\in H⟹g\cdot 1\in gH⟹g\in gH$ --AnalysisAlgebra (talk) 19:21, 12 February 2013 (UTC)

There's no need for $H$ to be normal, by the way. --AnalysisAlgebra (talk) 19:23, 12 February 2013 (UTC)

Estimating parameters of a distribution from censored data

Suppose you hypothesize that your data are drawn from, say, a lognormal distribution or a gamma distribution. Your data are right-censored at the value x*. That is, you have exact data whenever x≤x*, but whenever x>x* all you know is that x>x*.

(1) How do you estimate the parameters of the hypothesized distribution? I know it would be by maximum likelihood, but how do you handle the censored data?

(2) Is there a command in, say, SAS that will do this?

(3) How do you test the hypothesis that the data did in fact come from that distribution? Duoduoduo (talk) 18:37, 12 February 2013 (UTC)

(1) If your data consists of uncensored points $x_1,\dots ,x_m$ and n censored points, and your pdf with parameters $\theta$ is $f(x|\theta )$, then the likelihood of $\theta$ is ${(\underset{x^{*}}{\overset{\mathrm{\infty }}{\int }}f(t|\theta )dt)}^n\prod _{i=1}^{m}f(x_i|\theta )$. There's ostensibly an inconsistency in that you use a density for the contribution to likelihood of some points and a probability for others, but since likelihood functions are only meaningful up to a constant factor it doesn't matter. -- Meni Rosenfeld (talk) 19:49, 12 February 2013 (UTC)

Thanks, Meni! Duoduoduo (talk) 14:19, 14 February 2013 (UTC)