<p>The following article explains in detail how to compute quantiles (the inverse CDF) for the inverse Gaussian distribution:</p>

<p>Giner, G, and Smyth, GK (2016). statmod: probability calculations for the inverse Gaussian distribution. R Journal. <a href="http://arxiv.org/abs/1603.06687" rel="nofollow">http://arxiv.org/abs/1603.06687</a></p>

<p>Code for the R language is contained in the R package statmod available from CRAN. For example:</p>

<pre><code>&gt; library(statmod)
&gt; qinvgauss(0.01, lower.tail=FALSE)
[1] 4.98
</code></pre>

<p>computes the 0.01 upper tail quantile of the standard IG distribution.</p>
How do I compute the Inverse gaussian distribution from given CDF?
<p>I want to compute the parameters mu and lambda for the <a href="http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution" rel="nofollow">Inverse Gaussian Distribution</a> given the CDF.</p>

<p>By 'given the CDF' I mean that I have given the data AND the (estimated) quantile for the data I.e.</p>

<pre><code>Quantile - Value

0.01 - 10

0.5 - 12

0.7 - 13
</code></pre>

<p>Now I want to find out the inverse gaussian distribution for this data so that I can e.g. Look up the quantile for value 11 based on my distribution.</p>

<p>How can I find out the values mu and lambda?</p>

<p>The only solution I can think of is using Gradient descent to find the best mu and lambda using RMSE as an error measure.</p>

<p>Isn't there a better solution?</p>

<p>Comment: Matlab's MLE-Algorithm is not an option, since it does not use the quantile data.</p>