# Solving an integral that involves the quantile function

I'm trying to solve the following integral:

$$\int_{-\infty}^{\mu_1}\exp\{-\rho_{\tau_1}(y-\mu_1)-\rho_{\tau_2}(y-\mu_2)\}\mathbb{I}_{\tau_1<\tau_2}\mathbb{I}_{\mu_1<\mu_2}\,dy$$ without much success. Here $\rho_\tau(u) = \tau u + (\tau-1)u\mathbb{I}_{u<0}$ is the quantile function (see Wiki here), $(\mu_1, \mu_2)\in R^2$, with and $(\tau_1, \tau_2)\in (0,1)^2.$

I've tried with this:

Integrate[
Exp[-(tau1*(y - mu1)*If[y - mu1 >= 0, 1, 0] + (tau1-1)*(y - mu1)*
If[y - mu1 < 0, 1, 0])]*
Exp[-(tau2*(y - mu2)*If[y - mu2 >= 0, 1, 0] + (tau2-1)*(y - mu2)*
If[y - mu2 < 0, 1, 0])], {y, -Infinity, mu1}, Assumptions -> {tau1 < tau2}]


but the output I get is the input its self. Any idea about how to fix it?

Edit: the answer is obviously

$$\frac{e^{(2-\tau_1-\tau_2)\mu_1-\bar\mu}}{2-\tau_1-\tau_2}\,,$$ where $\bar\mu = (1-\tau_1)\mu_1+(1-\tau_2)\mu_2.$

• Try defining the function using Piecewise instead, perhaps. – march Sep 28 '17 at 22:17
• Plots of your integrand (using random values for your taus and mus since none were provided) look pretty explosive for negative values of y. Do you have any reason for supposing that the integral exists? Are there tighter bounds you can put on your parameters? – aardvark2012 Sep 29 '17 at 11:31
• @march tried with "Picewise" and "UitStep" but neither worked. aardvark2012 you are right, there are typos in my Mathematica code and in the definition of the quantile function. Now it must be ok. I can solve it analytically obviously, as such integral is really trivial, but I find it strange that Mathematica is not able to find the solution... – utobi Sep 29 '17 at 15:14

## 3 Answers

You need to include the assumption that mu1 < mu2. Also use Boole rather than If

int = Assuming[{tau1 < tau2, mu1 < mu2},
Integrate[
Exp[-(tau1*(y - mu1)*Boole[y - mu1 >= 0] + (tau1 - 1)*(y - mu1)*
Boole[y - mu1 < 0])]*
Exp[-(tau2*(y - mu2)*Boole[y - mu2 >= 0] + (tau2 - 1)*(y - mu2)*
Boole[y - mu2 < 0])], {y, -Infinity, mu1}] // Simplify]

(* ConditionalExpression[-(E^(-(mu1 - mu2) (-1 + tau2))/(-2 + tau1 + tau2)),
tau1 + tau2 < 2] *)


Comparing to expected result

(int // Normal) ==
Exp[(2 - tau1 - tau2) mu1 - ((1 - tau1) mu1 + (1 - tau2) mu2)]/
(2 - tau1 - tau2) // Simplify

(* True *)


Use PiecewiseExpand on the integrand and include all the assumptions:

Integrate[
Exp[-(tau1*(y - mu1)*If[y - mu1 >= 0, 1, 0] + (tau1 - 1)*(y - mu1)*
If[y - mu1 < 0, 1, 0])]*
Exp[-(tau2*(y - mu2)*If[y - mu2 >= 0, 1, 0] + (tau2 - 1)*(y - mu2)*
If[y - mu2 < 0, 1, 0])] //
PiecewiseExpand,
{y, -Infinity, mu1},
Assumptions -> {0 < tau1 < tau2 < 1, mu1 < mu2}]  (* N.B. *)
(*  -(E^(-(mu1 - mu2) (-1 + tau2))/(-2 + tau1 + tau2))  *)

(* OP's form of the answer *)
mubar = (1 - tau1) mu1 + (1 - tau2) mu2;
Exp[(2 - tau1 - tau2) mu1 - mubar]/(2 - tau1 - tau2) // Simplify
(*  -(E^(-(mu1 - mu2) (-1 + tau2))/(-2 + tau1 + tau2))  *)


I defined the quantile function using HeavisideTheta as

rho[u_, tau_] := tau u + (1 - tau) u HeavisideTheta[-u];


When I tried to integrate your expression using this definition, I was informed that the integral diverges. I tried the expression below instead

Integrate[
Exp[rho[(y - mu1), tau1] + rho[(y - mu2), tau2]] HeavisideTheta[
tau2 - tau1], {y, -Infinity, mu1}]


and got as result

HeavisideTheta[tau2 - tau1]E^(mu1 - mu2)/2

when mu1 <= mu2 and

HeavisideTheta[tau2 - tau1]E^((mu1 - mu2) tau2)/(1 + tau2)

otherwise.

• Thanks for your effort. There was a typo in my definition of the quantile function...sorry for that. So, obviously, your answer is incorrect. – utobi Sep 29 '17 at 15:17