# Taylor series expansion of quantile function

Suppose $$Y$$ and $$Z$$ two random variables, $$\lambda \in \mathbb{R}$$.

We note $$F^{-1}_{Y}(\alpha)$$ the quantile function of the variable $$Y$$ at the quantile level $$\alpha \in (0,1)$$.

Do you have any idea how to obtain this Taylor series expansion formula ? $$F^{-1}_{Y+\lambda Z}(\alpha)=F^{-1}_{Y}(\alpha)+\lambda[\frac{dF^{-1}_{Y+\lambda Z}}{d\lambda}(\alpha)]_{\lambda = 0} + \frac{\lambda^2}{2!}[\frac{d^2 F^{-1}_{Y+\lambda Z}}{d\lambda^2}(\alpha)]_{\lambda = 0}+...+\frac{\lambda^n}{n!}[\frac{d^n F^{-1}_{Y+\lambda Z}}{d\lambda^n}(\alpha)]_{\lambda = 0}+...$$

• This is just the usual formula for a Taylor series. Or do you mean you want to find the coefficients? – Robert Israel Feb 20 at 15:21
• To have any hope of analyticity, I think you'll want to assume $Y$ and $Z$ have a joint distribution with a density. – Robert Israel Feb 20 at 15:29
• The approximation of a usual Taylor formula is usually at the value of $\alpha$ and not at the functions. So, this formula is not evident for me and I don't know how to prove it (or find the coefficients). Yes, we can suppose that $Y$ and $Z$ have a joint distribution with a density. – NN2 Feb 20 at 15:36
• An example may help: Suppose X and Y are both lognormal, eg modeling returns for bonds and stocks, so lambda measures the riskiness of the portfolio. What are good series for approximating the quantile function near a particular portfolio value? – Matt F. Feb 20 at 16:12
• You can have a look at en.wikipedia.org/wiki/Lagrange_inversion_theorem for that. Nevertheless, your problem seems more subtle if you do not assume a nice expansion in $\lambda$. I would also be interested in an expression of the coefficients $\frac{d^n g}{d\lambda^n}(0)$ in terms of, say, moments. This looks like an interesting way of estimating a quantile. – Synia Feb 21 at 14:24