### Finding Quantile Given Weird Density Function

For one of my Statistical Finance homeworks, I had to solve this question.

Given a density function $f(x)$, where

$f(x) = \int_{-\infty}^{\infty} \frac{1}{Z} \frac{|x+1|}{(x^2+1)^2} = 1$

Find the quantile $y$ that satisfies the $F^{-1}(y) = 0.95$ where $F(x)$ is the cumulative ditribution function of $f(x)$ and $F^{-1}$ is the quantile function (the inverse of $F$).

Obviously that’d mean that $F(x) = \int f(x) \: dx$ and I should start by integrating $f(x)$ and then finding its inverse. However, upon visual inspection, one should see that $f(x)$ is a bitch to integrate.

Yes it is possible to integrate it by splitting the ranges for the absolute function, then splitting $(x+1) = x + 1$, then $\frac{x}{(x^2+1)^2}$ can be integrated by substitution of $u = x^2+1$ and the other portion can be done trigonometrically. Now because my integration-fu is too bad, let me solve for this numerically. So here’s the trick to (ab)use R.

First let’s solve for $Z$. Since $f(x)$ is a density function, then by definition of density functions, $\int_{-\infty}^\infty f(x) \: dx = 1$. Then,

$Z = \int_{-\infty}^{\infty} \frac{|x+1|}{(x^2+1)^2} = 1.785398$

Great! Now let’s move to R. Unfortunately, R does not have a function for numerical computation of quantiles for arbitrary distribution functions. However, we can build one.

Finding $y$ above is equivalent to the following optimization problem:

$y^* = \arg\min_{y} (F(y) - 0.95)^2$

Then all we need is an optimization routine to find $y$ that minimizes the squared error $F(y) - 0.95)^2$. In R there exists the library nlminb for numerical optimization. The functions are below:

Using this, we find that $y = 0.03161596$. Let’s check this!

Booyah. This is why I’m not a statistics major.