Can't find Inverse Gaussian derivative w.r.t. to its parameter

Question

I was looking for the derivative of the Inverse Gaussian cdf w.r.t to its parameters $\lambda$ (here called $k$) and $\mu$, however I couldn't find a single source on the internet with a full derivation. (I actually need the second order derivatives and partial derivatives, but for now I'm ok with the first oder derivatives). I tried to derive it myself as follows:

$$ P(t) =Pr(X< t)= \Phi\left(\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}-1\right)\right)+e^{\frac{2k}{\mu}}\Phi\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) $$ Where $\Phi$ is the standard normal distribution cdf. $$ \frac{\partial}{\partial k}P(t)=\\ \Phi^{'}\left(\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}-1\right)\right) \color{blue}{\frac{\partial}{\partial k}\left(\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}-1\right)\right)} +\\ \frac{2}{\mu}e^{\frac{2k}{\mu}}\Phi\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) +\\ \Phi^{'}\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) \color{purple}{\frac{\partial}{\partial k}\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right)} e^{\frac{2k}{\mu}} \\= \\ \Phi^{'}\left(\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}-1\right)\right) \color{blue}{\left(\frac{1}{2t\sqrt{\frac{k}{t}}}\left(\frac{t}{\mu}-1\right)\right)} +\\ \frac{2}{\mu}e^{\frac{2k}{\mu}}\Phi\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) +\\ \Phi^{'}\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) \color{purple}{ \left(-\frac{1}{2t \sqrt{\frac{k}{t}}}\left(\frac{t}{\mu}+1\right)\right)} e^{\frac{2k}{\mu}} $$ $$ \frac{\partial}{\partial \mu}P(t)=\\ \Phi^{'}\left(\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}-1\right)\right) \color{blue}{\frac{\partial}{\partial \mu}\left(\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}-1\right)\right)} +\\ \frac{2}{\mu}e^{\frac{2k}{\mu}}\Phi\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) +\\ \Phi^{'}\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) \color{purple}{\frac{\partial}{\partial \mu}\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right)} e^{\frac{2k}{\mu}} \\=\\ \Phi^{'}\left(\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}-1\right)\right) \color{blue}{\left(-\sqrt{\frac{k}{t}}\cdot\frac{t}{\mu^2}\right)} +\\ \frac{2}{\mu}e^{\frac{2k}{\mu}}\Phi\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) +\\ \Phi^{'}\left(-\sqrt{\frac{k}{t}}\left(\frac{t}{\mu}+1\right)\right) \color{purple}{\left(\sqrt{\frac{k}{t}}\frac{t}{\mu^2}\right)} e^{\frac{2k}{\mu}} $$

Does it look right? Is there a smarter/faster way to obtain the second order derivatives?

Answer 1

You asked for a "smarter/faster way": use a program that does symbolic derivatives. Here's an approach with Mathematica. First, define your function.

P[t_] := CDF[NormalDistribution[0, 1], (t/μ - 1) Sqrt[k/t]] + 
  Exp[2 k/μ] CDF[NormalDistribution[0, 1], -(t/μ + 1) Sqrt[k/t]]

$\frac{\partial}{\partial k}P(t)$:

(D[P[t], k] // FullSimplify) //. Erfc[z_] -> 1 - Erf[z] //. 
  Erf[z_] -> 1 - 2 Φ[-Sqrt[2] z] // Together

$$\frac{e^{-\frac{k (t-\mu )^2}{2 \mu ^2 t}} \left(4 \sqrt{\pi } k e^{\frac{k (\mu +t)^2}{2 \mu ^2 t}} \Phi \left(-\frac{\sqrt{\frac{k}{t}} (\mu +t)}{\mu }\right)-\sqrt{2} \mu \sqrt{\frac{k}{t}}\right)}{2 \sqrt{\pi } k \mu }$$

$\frac{\partial^2}{\partial k^2}P(t)$:

D[P[t], {k, 2}] //. Erfc[z_] -> 1 - Erf[z] //. 
  Erf[z_] -> 1 - 2 Φ[-Sqrt[2] z] // FullSimplify

$$\frac{e^{-\frac{k (t-\mu )^2}{2 \mu ^2 t}} \left(-16 \sqrt{\pi } t^3 \left(\frac{k}{t}\right)^{3/2} e^{\frac{k (\mu +t)^2}{2 \mu ^2 t}} \Phi \left(\frac{\sqrt{\frac{k}{t}} (\mu +t)}{\mu }\right)+16 \sqrt{\pi } t^3 \left(\frac{k}{t}\right)^{3/2} e^{\frac{k (\mu +t)^2}{2 \mu ^2 t}}+\sqrt{2} \left(-k t^2+\mu ^2 (k+t)-4 k \mu t\right)\right)}{4 \sqrt{\pi } \mu ^2 t^3 \left(\frac{k}{t}\right)^{3/2}}$$

$\frac{\partial}{\partial \mu}P(t)$:

D[P[t], μ] //. Erfc[z_] -> 1 - Erf[z] //. Erf[z_] -> 1 - 2 Φ[-Sqrt[2] z] // Together

$$\frac{\exp \left(-\frac{k \left(\frac{t}{\mu }-1\right)^2}{2 t}-\frac{k \left(-\frac{t}{\mu }-1\right)^2}{2 t}\right) \left(4 \sqrt{\pi } k \exp \left(\frac{2 k}{\mu }+\frac{k \left(\frac{t}{\mu }-1\right)^2}{2 t}+\frac{k \left(-\frac{t}{\mu }-1\right)^2}{2 t}\right) \Phi \left(-\sqrt{\frac{k}{t}} \left(-\frac{t}{\mu }-1\right)\right)-4 \sqrt{\pi } k \exp \left(\frac{2 k}{\mu }+\frac{k \left(\frac{t}{\mu }-1\right)^2}{2 t}+\frac{k \left(-\frac{t}{\mu }-1\right)^2}{2 t}\right)-\sqrt{2} t \sqrt{\frac{k}{t}} e^{\frac{k \left(-\frac{t}{\mu }-1\right)^2}{2 t}}+\sqrt{2} t \sqrt{\frac{k}{t}} e^{\frac{2 k}{\mu }+\frac{k \left(\frac{t}{\mu }-1\right)^2}{2 t}}\right)}{2 \sqrt{\pi } \mu ^2}$$

$\frac{\partial^2}{\partial \mu^2}P(t)$:

(D[P[t], {μ, 2}] // FullSimplify) //. Erfc[z_] -> 1 - Erf[z] //. Erf[z_] -> 1 - 2 Φ[-Sqrt[2] z] // Together

$$\frac{k e^{-\frac{k (t-\mu )^2}{2 \mu ^2 t}} \left(4 k e^{\frac{k (\mu +t)^2}{2 \mu ^2 t}} \Phi \left(-\frac{\sqrt{\frac{k}{t}} (\mu +t)}{\mu }\right)+4 \mu e^{\frac{k (\mu +t)^2}{2 \mu ^2 t}} \Phi \left(-\frac{\sqrt{\frac{k}{t}} (\mu +t)}{\mu }\right)+\left(-\sqrt{\frac{2}{\pi }}\right) t \sqrt{\frac{k}{t}}\right)}{\mu ^4}$$