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lines 5-48 of file: xrst/model/fixed_diff.xrst
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{xrst_begin fixed_diff}
{xrst_spell
  subvector
}

Fixed Effects Differences Density Function
##########################################

k, K
****
For each
:ref:`rate_table@rate_id` there is
:ref:`rate_table@parent_smooth_id` .
In addition, for each
:ref:`mulcov_table@mulcov_id`
there is a
:ref:`smooth_id<mulcov_table@group_smooth_id>`
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:`\theta^k` to denote the subvector of the fixed effects
:math:`\theta` that corresponds to the *k*-th smoothing.

A^theta
*******
The fixed effects age difference function is

.. math::

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

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

T^theta
*******
The fixed effects time difference function is

.. math::

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

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

{xrst_end fixed_diff}