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lines 5-49 of file: xrst/model/random_diff.xrst
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{xrst_begin random_diff}
{xrst_spell
  subvector
}

Random Effects Differences Density Function
###########################################

k, K
****
For each :ref:`rate_table@rate_id`
and each :ref:`child<option_table@Parent Node@Children>`
there is a smoothing and corresponding set of model variables.
In addition, for each
:ref:`mulcov_table@mulcov_id`
and each :ref:`subgroup_table@subgroup_id`
corresponding to the
:ref:`mulcov_table@group_id` for this *mulcov_id* ,
there is a smoothing and a corresponding set of model variables.
We use :math:`k` to index these smoothings :math:`s_k` and
:math:`K` for the number of these smoothings.
We use :math:`u^k` to denote the subvector of the random effects
:math:`u` that corresponds to the *k*-th smoothing.

A^u
***
The random effects age difference function is

.. math::

   A^u ( u | \theta ) =  \prod_{k=0}^{K-1} A^s ( s_k , u^k, \theta )

see definition of :ref:`smooth_dage@A^s` .

T^u
***
The random effects time difference function is

.. math::

   T^u ( u | \theta ) =  \prod_{k=0}^{K-1} T^s ( s_k , u^k, \theta )

see definition of :ref:`smooth_dtime@T^s` .

{xrst_end random_diff}