In this example a nonlinear model is estimated by using the Cauchy distribution. Then a simulation is done for one observation in the data.
The following DATA step creates the data for the model.
/* Generate a Cauchy distributed Y */ data c; format date monyy.; call streaminit(156789); do t=0 to 20 by 0.1; date=intnx('month','01jun90'd,(t*10)1); x=rand('normal'); e=rand('cauchy') + 10 ; y=exp(4*x)+e; output; end; run;
The model to be estimated is






That is, the residuals of the model are distributed as a Cauchy distribution with noncentrality parameter .
The log likelihood for the Cauchy distribution is

The following SAS statements specify the model and the loglikelihood function.
title1 'Cauchy Distribution'; proc model data=c ; dependent y; parm a 2 nc 4; y=exp(a*x); /* Likelihood function for the residuals */ obj = log(constant('pi')*(1+(resid.ync)**2)); errormodel y ~ general(obj) cdf=cauchy(nc); fit y / outsn=s1 method=marquardt; solve y / sdata=s1 data=c(obs=1) random=1000 seed=256789 out=out1; run; title 'Distribution of Y'; proc sgplot data=out1; histogram y; run;
The FIT statement uses the OUTSN= option to output the matrix for residuals from the normal distribution. The matrix is and has value because it is a correlation matrix. The OUTS= matrix is the scalar . Because the distribution is univariate (no covariances), the OUTS= option would produce the same simulation results. The simulation is performed by using the SOLVE statement.
The distribution of is shown in the following output.
Output 19.12.1: Distribution of Y