Timeline for Measurability of maximum likelihood estimator under conditions from Lehmann's "Theory of point estimation"
Current License: CC BY-SA 4.0
11 events
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| Oct 24, 2021 at 16:42 | comment | added | Botnakov N. | @ Iosif Pinelis, I was afraid that I didn't notice some simple reason why $\hat{\theta}_n$ is measurable. Now I'm inclined to believe that the simple thing that was not noticed is that it's not a good idea to treat "Theory of point estimation" as a rigorous book) Thank you very much! | |
| Oct 24, 2021 at 16:40 | vote | accept | Botnakov N. | ||
| Oct 24, 2021 at 16:37 | comment | added | Botnakov N. | @ Iosif Pinelis, I agree that $\hat{\theta}_n$ is not an estimate (since it depends on $\theta_0$). It was interesting if $\hat{\theta}_n$ is measurable because of the corollary 3.8. This corollary states that if likelihood equation has a unique root for all $n, X_1, \dots, X_n$ and if assumptions of theorem 3.7 hold true then $\hat{\theta}_n$ is a consistent estimator. It helps to cope with the problem of dependence of $\theta_0$ and the onliest problem is measurability of $\hat{\theta}_n$. | |
| Oct 24, 2021 at 16:36 | comment | added | Botnakov N. | @ Iosif Pinelis, thank you for pointing out that there are other mistakes! I didn't notice in (i) that we use the continuity of $l'(\theta)$. In (ii) and (iii) we may somehow cope with these problems (e.g. in (ii) we may speak about root and not promise that it's maximazer, in (iii) we may take left closest root), but for (i) we need an extra assumption of continuity of $l'$. | |
| Oct 24, 2021 at 14:27 | comment | added | Iosif Pinelis | Previous comment continued: So, there are a number of problems with Lehmann's proof -- which I think are worse than the possible non-measurability. | |
| Oct 24, 2021 at 14:26 | comment | added | Iosif Pinelis | @BotnakovN. : Also, in Lehmann's proof, it is said "Let $\theta_n^*$ be the root [of (3.12)? -- I.P.] closest to $\theta_0$. [This exists because the limit of a sequence of roots is again a root by the continuity of of $l(\theta)$.]" Here one can say the following: (i) for the closest root to $\theta_0$ of $l'(\theta)=0$ to exist, one needs -- not the continuity of $l(\theta)$ -- but continuity of $l'(\theta)$; (ii) it is unclear why such a closest root must be a maximizer of $l(\theta)$; (iii) it is unclear why such a closest root is unique. | |
| Oct 24, 2021 at 13:22 | comment | added | Iosif Pinelis | @BotnakovN. : The condition that we should take the root closest to $\theta_0$ makes no sense statistically, because $\theta_0$ is unknown, and then $\hat\theta_n$ will not be an estimator in any sense, be it measurable or not -- because an estimator cannot depend on an unknown parameter. However, indeed it appears harder to get a counterexample with this additional "closest" condition. But, given what has been said here, do we really need such a counterexample? Anyhow, generally, I do not think Lehmann's book is quite rigorous. It is a statistical book, not really mathematical. | |
| Oct 24, 2021 at 13:17 | comment | added | Iosif Pinelis | @BotnakovN. : The consistency can be generally defined as the convergence of $(P_\theta)^∗(|\hat\theta_n−\theta|>\delta)$ to $0$ for each real $\delta>0$. I think this will hold in all existing theorems about MLE consistency, and this will be at least formally stronger than the statement that $P_\theta(|\hat\theta_n−\theta|>\delta)\to0$ for all measurable versions of $\hat\theta_n$ and for each real $\delta>0$. | |
| Oct 24, 2021 at 11:42 | comment | added | Botnakov N. | About $U[\theta, \theta+1]$: I tried to make counterexample with not measurable set $\{ |\theta_n - \theta| < a \}$ under conditions of theorem 3.7, using functions $f(x_i|\theta)$ such that they are constant$>0$ on some set, but I didn't successed: Lehmann's approach imply that we should take the root which is closest to $\theta_0$ and it makes the problem harder. So unfortunately, it is still not clear whether the proof from the book can be considered correct if we will not redefine consisestency for non-measurable estimator. | |
| Oct 24, 2021 at 11:41 | comment | added | Botnakov N. | Thank you! So if $MLE$ is non-measurable then we may define consistensy of non-measurable estitamor as convergence of inner $P_{\theta}$-measure of $\{ |\hat{\theta}_n - \theta| < \delta \}$ to $0$, right? Sounds interesting! I have not seen this approach. | |
| Oct 24, 2021 at 1:05 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |