Timeline for Why MLEs are asymptotically efficient whereas method of moment estimators are not?
Current License: CC BY-SA 4.0
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| Nov 10, 2023 at 18:06 | comment | added | Aaron Hendrickson | It seems the way to think about this is from the perspective of how the sample space is partitioned. If we define the data reduction $\mathbf X\to L(\theta|\mathbf X)$ with the equivalence relation specifying that samples $\mathbf X$ that produce proportional likelihoods are equivalent then we do indeed get a minimal sufficient partition of the sample space. So while the likelihood is not a statistic in the usual sense, it combined with this equivalence relation serves to induce the same unique partitioning that a minimal sufficient statistic would. | |
| Nov 10, 2023 at 15:42 | comment | added | Iosif Pinelis | @MichaelHardy : If the likelihood is defined, as apparently was done in your comment, up to a factor not depending on the parameter, then you do get a minimal sufficient statistic, according to the "useful characterization of minimal sufficiency". | |
| Nov 10, 2023 at 15:27 | comment | added | Michael Hardy | mathoverflow.net/questions/140481/what-is-a-likelihood-kernel If we take the "likelihood kernel" to be what I defined it to be in the posting linked here, we then have the question of whether the likelihood kernel is a minimal sufficient statistic. | |
| Nov 10, 2023 at 15:25 | comment | added | Michael Hardy | I upvoted your answer. | |
| Nov 10, 2023 at 15:25 | comment | added | Iosif Pinelis | @MichaelHardy : Have you just downvoted my answer? If so, why? | |
| Nov 10, 2023 at 15:24 | comment | added | Iosif Pinelis | @MichaelHardy : The comment was one under, and concerning, the answer. Do you think that in every comment under the answer I have to repeat the relevant definitions given in the answer? | |
| Nov 10, 2023 at 15:21 | comment | added | Michael Hardy |
\limits seems to be intended to cause the positioning of subscripts and superscripts in an "inline" setting to behave the way they otherwise would in a "displayed" setting. Since this instance of the use of \operatorname{argmax} was already in a "displayed" setting, \limits would have no effect.
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| Nov 10, 2023 at 15:06 | comment | added | Michael Hardy |
$$ \begin{align} & \operatorname{argmax}\limits_{\theta\in\Theta} L_X(\theta) \\ {} \\ & \operatorname*{argmax}_{\theta\in\Theta}L_X(\theta) \end{align} $$ I changed the first form above, coded as \operatorname{argmax}\limits_{\theta\in\Theta} L_X(\theta), to the second, coded as \operatorname*{argmax}_{\theta\in\Theta} L_X(\theta). I'm guessing the use of \limits was intended to affect the position of the subscript, but that didn't work.
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| Nov 10, 2023 at 14:47 | comment | added | Michael Hardy | I didn't mean to say that your answer omits that. Rather, I was responding to your comment about what Aaron Hendrickson said. | |
| Nov 10, 2023 at 14:45 | comment | added | Iosif Pinelis | @MichaelHardy : As was explained in my answer, in the context of sufficiency, by "the likelihood function" I mean the statistic $L_X$. This statistic may indeed contain "some information not relevant to estimating" the parameter, in the precise sense that $L_X$ may not be minimal sufficient -- which is what I said in my latter comment. | |
| Nov 10, 2023 at 14:39 | comment | added | Michael Hardy | People who write about this don't always say which of those two conventions they have in mind. | |
| Nov 10, 2023 at 14:30 | comment | added | Michael Hardy | @IosifPinelis : I wonder whether the truth value of your latest comment depends on what you consider to be the likelihood function. For the example you mention, I might write $$ L(\lambda) \propto e^{-n\lambda} \lambda^{x_1+\cdots + x_n} $$ and say that's the likelihood function. But I imagine some (you?) might write $\text{“} {=} \text{”}$ rather than $\text{“} {\propto} \text{”}$ and include the reciprocal of the product of factorials, in which case the likelihood function would give some information not relevant to estimating $\lambda. \qquad$ | |
| Nov 10, 2023 at 14:18 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Oct 31, 2023 at 21:52 | comment | added | Iosif Pinelis | @AaronHendrickson : In general, the likelihood function is not minimal sufficient. E.g., consider the iid sample of size $n\ge2$ from the Poisson distribution with parameter $\theta$. | |
| Oct 31, 2023 at 17:05 | comment | added | Aaron Hendrickson | Is the fact that the MLE is always a function of a minimal sufficient statistic a direct consequence of the likelihood function itself being minimal sufficient? | |
| Aug 3, 2023 at 12:57 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 3, 2023 at 10:38 | vote | accept | Aaron Hendrickson | ||
| Aug 3, 2023 at 2:23 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 3, 2023 at 2:13 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 2, 2023 at 21:13 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 2, 2023 at 20:57 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 2, 2023 at 20:27 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 2, 2023 at 20:21 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |