Talk:Delta method

more comments in the binomial example
I could use more comments in the binomial example.

For example, how does this follow? "the asymptotic variance of $${\displaystyle \log \left({\frac {X_{n}}{n}}\right)} \log \left({\frac {X_{n}}{n}}\right)$$ does exist and is equal to .."

For example, how does this follow? "since p>0, $${\displaystyle \Pr \left({\frac {X_{n}}{n}}>0\right)\rightarrow 1}$$ $${\displaystyle \Pr \left({\frac {X_{n}}{n}}>0\right)\rightarrow 1}$$ as $${\displaystyle n\rightarrow \infty } n \rightarrow \infty ..$$"

For example, how does this follow? "the logarithm of the estimated relative risk $${\displaystyle {\frac {\hat {p}}{\hat {q}}}} {\frac }$$ has asymptotic variance equal to .."

I'm sure these are trivial to someone who can already see it, but I don't.

Also, more text about the practical implications of this example would be good. The relative risk thing at the end is a start, but what about the proportion itself? What is a concrete, real-world example?

dfrankow (talk) 16:48, 21 December 2018 (UTC)

normality
You do not need normality to use the delta method —Preceding unsigned comment added by 75.73.53.186 (talk) 01:28, 2 March 2011 (UTC)

Convergence in probability to $$\theta$$
At the beginning of the proof, it says "..since $$X_n \to \theta$$..". Where does this come from? — Preceding unsigned comment added by 128.40.213.241 (talk) 14:29, 5 November 2015‎ $$\forall \epsilon>0,P(|X_n-\theta|<\epsilon)=P(\sqrt{n}|X_n-\theta|<\sqrt{n}\epsilon)\to 1$$ as $$n\to\infty$$ since $$\sqrt{n}(X_n-\theta)\to N(0,\sigma^2)$$ and $$\sqrt{n}\epsilon\to\infty$$ Vinzklorthos (talk) 20:08, 5 August 2020 (UTC)

Someone commented on the variance of the log of a rv.
It does exist provided you are dealing with positive support $$X \in (0,\infty)$$. Now the average of $$X_n$$ will need n to be large enough for the density at 0 to become vanishingly small. Limit-theorem (talk) 20:57, 8 July 2015 (UTC)


 * Perhaps to avoid the subtleties of almost sure convergene, it is best to insert another example with C-infinity transformation on the real line. I can do so later. Limit-theorem (talk) 13:57, 15 July 2015 (UTC)
 * I was the one who added that comment. The support is not positive since a binomial random variable can be zero. Btyner (talk) 17:49, 23 August 2015 (UTC)


 * The example doesn't state that it is an approximation of the variance of a binomial, rather that it is an approximation of the variance of the limiting normal distribution whose variance does exist. I'd recommend removing the disclaimer. 2602:306:3844:5400:90C5:53FB:DA9D:CDDA (talk) 21:41, 20 April 2016 (UTC)
 * I went ahead and reworded it in this |this edit, and removed the disclaimer. Btyner (talk) 00:49, 28 April 2016 (UTC)