# Generalized linear model for Levy distribution with not complete data

I'm trying to build a model that would describe some process of payment and distribution of payments in time. I believe that time of payment has Levy distribution with probability density function:

$f(x,c)=\sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2x}}} {x^{3/2}}$

This distribution depends on parameter c which actually defines the shape of distribution. My task is to build model that explains dependency of this parameter on some explaining variables. I'm trying linear dependency $c = \sum_i \beta_i x_i$

This is example of generalized linear model and is implemented in the VGAM package in R. Problem is that in a sample for building this model I have data only from some period at the beginning and this period is different for different groups of cases. And because of that I can not just to run the model in VGAM package on this data as the result will be incorrect significantly exaggerating probability of early payments.

One possible solution I can think about is to change the likelihood function from which parameter is estimated. If we have information only from time up to t and as cumulative distribution function of Levy distribution is:

$F(x,c)=\textrm{erfc}\left(\sqrt{c/2x}\right)$

the density of distribution up to time t is $f_1(x,c,t)= \frac{f(x,c)}{F(t,c)}$ (where $f(x,c), F(t,c)$ defined as above). This new density functions can be used in estimating regression parameters with maximum likelyhood method. But can it be done in R using methods from VGAM package or usual glm function or some other packages? Or there are some better approaches to my problem? I'm interested in implementation in R.

Thank you in advance for any help!

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