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I understand that the Delta Method can be used to find asymptotic distribution of estimators.

I have a MLE Estimator with

$ E[\hat\Theta] = \frac{n\Theta_0}{n+1} $

$ Var[\hat\Theta] = \frac{\Theta^2_0}{n(n+2)} $

How can I find the asymptotic distribution of this estimator?

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  • $\begingroup$ Related? math.stackexchange.com/questions/2798941/… $\endgroup$
    – Christoph Hanck
    Commented Jan 22, 2020 at 8:10
  • $\begingroup$ Is this the only available information? You can only say $E[\hat\theta]\to \theta_0$ and $Var[\hat\theta]\to 0$, so $\hat\theta$ converges in probability (and hence in distribution) to $\theta_0$. This of course gives a degenerate asymptotic distribution. $\endgroup$
    – StubbornAtom
    Commented Jan 22, 2020 at 14:35
  • $\begingroup$ Thank you for your response. The problem provides this info as the mean and variance of MLE of a uniform distribution over [0,$\Theta$]. The problem then asks for an asymptotic distribution for such a MLE estimator. $\endgroup$
    – gts92
    Commented Jan 22, 2020 at 20:33
  • $\begingroup$ In that case see stats.stackexchange.com/q/130447/119261 for the non-degenerate asymptotic distribution. $\endgroup$
    – StubbornAtom
    Commented Jan 23, 2020 at 13:32
  • $\begingroup$ See also stats.stackexchange.com/a/96689/28746 $\endgroup$
    – Alecos Papadopoulos
    Commented Nov 2, 2020 at 20:23

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and welcome to stack exchange.

The delta method does not have anything to do with your question.

Maximum likelihood estimators are, under some regularity conditions, asymptically normally distributed.

The delta method is a way of finding the asymptotic distribution of a (differentiable) function of a random variable that is itself asymptotically normal.

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