sample_random¶

Simulation the Posterior Distribution for Random Effects¶

Syntax¶

error_msg = mixed_object . sample_random (
        sample ,
        fixed_vec ,
        random_ipopt_options ,
        random_lower ,
        random_upper ,
        random_in ,
        cov_factor
)

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            ,
    double             cov_factor           )

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 .

cov_factor¶

This argument is optional, must be greater than zero, and its default value is one. It can be used to expand or contract the random variation of the random effects; see Covariance below. Note that the factor corresponding to the standard deviations of the samples is the square root of cov_factor .

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 cov_factor times the inverse of the observed information matrix; i.e.

\[\lambda f_{uu} [ \theta , \hat{u} ( \theta ) ] ^{-1}\]

where \(\lambda\) is cov_factor . The samples from this normal distribution are 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 .