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 ${\displaystyle {\frac {\hat {p}}{\hat {q}}}}{\frac {\hat {p}}{\hat {q}}}$ 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 edit, and removed the disclaimer. Btyner (talk) 00:49, 28 April 2016 (UTC)