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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 ? $$F^{-1}_{Y+\lambda Z}(\alpha)=F^{-1}_{Y}(\alpha)+\lambda[\frac{dF^{-1}_{Y+\lambda Z}}{d\lambda}(\alpha)]_{\lambda = 0} + \frac{\lambda^2}{2!}[\frac{d^2 F^{-1}_{Y+\lambda Z}}{d\lambda^2}(\alpha)]_{\lambda = 0}+...+\frac{\lambda^n}{n!}[\frac{d^n F^{-1}_{Y+\lambda Z}}{d\lambda^n}(\alpha)]_{\lambda = 0}+...$$

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  • $\begingroup$ This is just the usual formula for a Taylor series. Or do you mean you want to find the coefficients? $\endgroup$
    – Robert Israel
    Commented Feb 20, 2019 at 15:21
  • $\begingroup$ To have any hope of analyticity, I think you'll want to assume $Y$ and $Z$ have a joint distribution with a density. $\endgroup$
    – Robert Israel
    Commented Feb 20, 2019 at 15:29
  • $\begingroup$ 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. $\endgroup$
    – NN2
    Commented Feb 20, 2019 at 15:36
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    $\begingroup$ 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? $\endgroup$
    – user44143
    Commented Feb 20, 2019 at 16:12
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    $\begingroup$ 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. $\endgroup$
    – Synia
    Commented Feb 21, 2019 at 14:24

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