# How to estimate a truncated, user-defined distribution?

Hello,
I want to estimate the parameter of a truncated distribution that is not prebuilt in Stan. In order to understand the process, I follow Annis, Miller, Palmeri (2016) in their approach to user-define the Exponential distribution. The prebuilt code works fine for me:

exp_prebuilt <- stan_model(model_code= '
data {
int N;
vector[N] y;
real L;
}
parameters {
real<lower=0> lambda;
}
model {
real alpha;
real beta;
alpha =1;
beta =1;
lambda ~ gamma(alpha, beta);
for (i in 1:N) {
y[i] ~ exponential(lambda) T[L,];
}
}
')

I then tried to build the exponential distribution by myself, later I want to replace the Exponential with a user-defined function:

exp_user <-stan_model(model_code= '
functions{
real newexp_log(vector x, real lam) {
vector[num_elements(x)] prob;
real lprob;
for(i in 1:num_elements(x)){
prob[i] =  lam*exp(-lam*x[i]);
}
lprob = sum(log(prob));
return lprob;
}
}
data{
int<lower=0> N;
real y[N];
real L;
}
parameters {
real<lower=0> lambda;
}
model {
real alpha;
real beta;
alpha =1;
beta =1;
lambda ~ gamma(alpha, beta);
for (s in 1:N){
y[s] ~ newexp(lambda) T[L,];
}
}
')

I tried to work with this post but am not sure if I implemented it correctly.

I receive two error messages for my code:

1. No matches for: real ~ newexp(real) Available argument signatures for newexp: vector ~ newexp(real)

2. Real return type required for probability function.

Any help in solving this problem, explanations, hints are appreciated!

Stan is quite strict about types. You’ve declared that the first argument to newexp_log is a vector but in the model you visit each element individually, meaning the type passed to newexp is a scalar. The error is trying to say that scalars and vectors are not compatible.
The solution is: change the function argument type to real and remove the loop from the fuction

real newexp_log(real x, real lam) {
real prob;
prob =  lam*exp(-lam*x);
return log(prob);
}

The next problem is that truncation T[L,] requires a cumulative probability distribution function in addition to the probability density function.

real newexp_lcdf(real x, real lam) {
real prob;
prob = exp(-lam*x);
return log(prob);
}

Stan also wants the complementary cumulative distribution function, CCDF.

real newexp_lccdf(real x, real lam) {
real prob;
prob = exp(-lam*x);
return log(1-prob);
}

With these the model compiles.

Niko,
thank you so much for taking the time to help me!
I really appreciate your support and have a great day!