Delta method to find E(Y) and V(Y)

Question

I know a Poisson random variable X has E(X) = μ and also Var(X) = μ. I'm given symmetrizing chance of variables, $$ Y= X^{2/3} $$ and I need to use the delta method to approximate E(Y) and Var(Y) for large μ.

I'm saying $Y=g(x)=X^{2/3}$, so $g'(x)=\frac{2}{3}X^{-1/3}$ and $g''(x)= \frac{-2}{9}X^{-4/3}$ . Is the Delta method for $E(Y)= g(\mu)+\frac{1}{2}Var(X)g''(\mu)$?? How do I find Var(Y)?

Answer 1

We taylor expand the function $g(x)$ about the point $x=\mu$ to give $$ f(x)\approx f(\mu) + f'(\mu)(x-\mu)+\frac{1}{2}f''(\mu)(x-\mu)^2,$$ so taking the expectation termwise gives $$ E(f(X)) \approx f(\mu)+f'(\mu)E(X-\mu) +\frac{1}{2}f''(\mu)E((X-\mu)^2) =f(\mu)+\frac{1}{2}f''(\mu)\operatorname{Var}(X).$$

To approximate the variance, we can take $$ \operatorname{Var}(f(X))\approx \operatorname{Var}[f(\mu))+f'(\mu)(X-\mu)]=(f'(\mu))^2 \operatorname{Var}(X)$$

Alternatively you can use the fact that $\operatorname{Var}(Y)=E(Y^2)-E(Y)^2$ and then approximate $E(Y^2)=E(X^{4/3})$ using the method for the mean.

I can't really vouch for the accuracy of these approximations (especially the variance)... you may need to go to more terms to get something reasonable (which will require you to know higher moments of the Poisson distribution). They are uncontrolled, as far as I can tell, although one would expect better performance in terms of relative error for larger $\mu,$ where the function is closer to linear. (Edit: actually, now that I think about it, what really matters is how linear the function is over the range of likely values for $X,$ so since the variance of $X$ increases as $\mu$ increases, widening the range, this will tend to cause the accuracy to go down. Not sure which effect will win out here.)

Answer 2

Interesting question. First, let's talk about the delta method. The delta method says if you have a sequence of random variables $X_n$ (this can be an estimator) such that $$X_n \overset{d}{\to} \operatorname{Normal}(\mu, \sigma^2)$$

Then for any function $g$ such that $g^\prime(\mu)$ exists we have $$g(X_n) \overset{d}{\to} \operatorname{Normal}(g(\mu), \sigma^2 [g^\prime(\mu)]^2)$$

This version of the delta method is for limiting distributions that are normal. There are more general versions.

Here's the part I'm not fully confident on. For sufficiently large $\lambda$ (i.e. $\lambda > 1000$) we have $$X_n \overset{d}{\to} \operatorname{Normal}(\lambda, \lambda)$$

Now let $Y_n= g(X_n) = X_n^{2/3}$. Then by the delta method we have

$$Y_n \overset{d}{\to} \operatorname{Normal}(\lambda^{2/3}, \dfrac{2}{3}\lambda^{2/3})$$

So we can approximate $E[Y] = \lambda^{2/3}$ and $\operatorname{Var}(Y) = \dfrac{2}{3}\lambda^{2/3}$.