Simulation the Posterior Distribution for Random Effects

@(@\newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\W}[1]{ \; #1 \; }@)@ This is cppad_mixed--20220519 documentation: Here is a link to its current documentation . Simulation the Posterior Distribution for Random Effects
Syntax
See Also
Prototype
Purpose
manage_gsl_rng
mixed_object
sample
random_ipopt_options
fixed_vec
random_lower
random_upper
random_in
Covariance
error_msg
Example

Syntax

error_msg = mixed_object.sample_random(

     sample,

     fixed_vec,

     random_ipopt_options,

     random_lower,

     random_upper,

     random_in

)

See Also
sample_fixed

Prototype


std::string cppad_mixed::sample_random(
     d_vector&          sample               ,
     const std::string& random_ipopt_options ,
     const d_vector&    fixed_vec            ,
     const d_vector&    random_lower         ,
     const d_vector&    random_upper         ,
     const d_vector&    random_in            )

Purpose
This routine draws samples from the asymptotic posterior distribution for the random effects given the model, the data, and the fixed effects; see sparse observed information .

manage_gsl_rng
It is assumed that get_gsl_rng will return a pointer to a GSL random number generator.

mixed_object
We use mixed_object to denote an object of a class that is derived from the cppad_mixed base class.

sample
This argument has prototype



     CppAD::vector<double>& sample

and its size is a multiple of n_random . The input value of its elements does not matter. We define



     n_sample = sample_size / n_random

If error_msg is empty, upon return for i = 0 , ..., n_sample-1 , j = 0 , ..., n_random-1 ,



     sample[ i * n_random + j ]

is the j-th component of the i-th sample of the optimal random effects. The statistics of these samples is specified under covariance below.

random_ipopt_options
This argument has prototype



     const std::string& random_ipopt_options

and is the ipopt_options for optimizing the random effects.

fixed_vec
This argument specifies the value of the fixed effects vector @(@ \theta @)@.

random_lower
This argument must have size equal to n_random and specifies the lower limits for the optimization of the random effects vector @(@ u @)@. The value minus infinity can be used to specify no lower limit.

random_upper
This argument must have size equal to n_random and specifies the upper limits for the optimization of the random effect. The value plus infinity can be used to specify no lower limit.

random_in
This argument must have size equal to n_random and specifies the initial value used for the optimization of the random effects vector @(@ u @)@. It must hold that



     random_lower[i] <= random_in[i] <= random_upper[i]

for each valid index i .

Covariance
Each sample of the random effects is an independent normal. The mean for this distribution is the optimal random effects @(@ \hat{u} ( \theta ) @)@. The variance of this distribution is the inverse of the observed information matrix; i.e. @[@ f_{uu} [ \theta , \hat{u} ( \theta ) ] ^{-1} @]@ This normal distribution is censored to be within the limits random_lower , random_upper .

error_msg
If error_msg is empty (non-empty), sample values have been calculated (have not been calculated). If error_msg is non-empty, it is a message describing the problem.

Example
The file sample_random.cpp is an example and test of sample_random.

Input File: src/sample_random.cpp