Problem with initialization

Modeling
6

Hi, I have written a vectorized version of a correlated Pareto/NBD:

data{
  
  int<lower=1> N; //  number of customers
  vector<lower=0>[N] x; // number of purchases per customer
  vector<lower=0>[N] t_x; // recency per customer 
  vector<lower=0>[N] T; // observation period per customer
  
}

parameters{
  
  // Individual-level correlated coefficients

  matrix[2, N] eta; // normal errors
  cholesky_factor_corr[2] L_Omega; // cholesky corr.
  row_vector[2] chi; // mean
  vector<lower=0,upper=pi()/2>[2] tau_unif; // scaled variance
    

}

transformed parameters{
  
  vector[N] lambda; // poisson rate (correlated)
  vector[N] mu; // death rate (correlated)
   
  vector<lower=0>[2] tau; // variance of individual-level correlated coefficients
  matrix[N,2] zeta; // individual-level correlated coefficients
 
  for (r in 1:2)
    tau[r] = 2.5 * tan(tau_unif[r]);
   
  zeta = rep_matrix(chi,N) + (diag_pre_multiply(tau,L_Omega) * eta)';  

  lambda=exp(zeta[,1]);
  mu=exp(zeta[,2]);

}


model{
  
  vector[N] lam_mu=lambda+mu;     
  
  to_vector(eta) ~ std_normal();
  L_Omega ~ lkj_corr_cholesky(2);
  to_vector(chi) ~ normal(0,3);

  target += x .* log(lambda) - log(lam_mu) - lam_mu .* t_x + 
        log(mu + lambda .* exp( -(T-t_x) .* (lam_mu) )) ; 

  
}


generated quantities {
  
  matrix[2,2] Omega;
  matrix[2,2] Sigma;
  vector[N] p_alive;
  vector[N] cond_exp_T2;
  int T2=78;
  
  Omega = multiply_lower_tri_self_transpose(L_Omega);
  Sigma = quad_form_diag(Omega, tau);
  
  p_alive = inv( 1 + mu ./ (lambda + mu) .* (exp( (lambda+mu) .* (T-t_x) ) - 1) ) ;
  cond_exp_T2 = lambda ./ mu .* ( 1 - exp(-mu .* (T2-T)) ) .* p_alive; 
}

This model is based on the note from Dr Fader and Dr Hardie. I believe this is also the model by Abe (2009).

What I notice is that Stan is unable to converge when using Gamma distribution for heterogeneity on lambda and mu, and I have tried this original formulation for over a month. But it works really well when replacing it with lognormal distribution.

I hope this help.

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