Parameter Estimation
====================

See the :mod:`simopt.models.paramesti` module for API details.

Model: Parameter Estimation (PARAMESTI)
---------------------------------------

Description
^^^^^^^^^^^

A model that simulates maximum likelihood estimation for the parameters of
a two-dimensional gamma distribution.

Say a simulation generates output data :math:`{Y_j}`, :math:`Y_j \in [0, \infty] \times [0, \infty]`,
that are i.i.d and known to come from a distribution with the two-dimensional density function

.. math:: f(y1, y2; x^*) = \frac{e^{-y1} y_1^{x^*_1 y_2 - 1}{\Gamma(x^*_1 y_2)} \frac{e^{-y2} y_2^{x^*_2 - 1}{\Gamma(x^*_2)}, y1, y2 > 0,

where :math:`x^* ≡ (x^*_1, x^*_2)`` is the unknown vector of parameters.

Noting that :math:`x_star` maximizes the function

.. math:: g(x) = E [log (f(Y ; x))] = \int_0^\infty \log (f(y; x)) f(y; x^*)dy,

and that

.. math:: G_m(x) = \frac{1}{m} \sum_{j=1}^m \log(f(Y_j ; x))

is a consistent estimator of :math:`g(x)`.
Observations are generated from the distribution specified by a given :math:`x_star`.

Sources of Randomness
^^^^^^^^^^^^^^^^^^^^^

y is a 2-D vector that contributes randomness. Both elements of y are gamma random variables.

Model Factors
^^^^^^^^^^^^^

* x_star: the unknown 2-D parameter that maximizes the expected log likelihood function.
    * Default: [2, 5]
* x: a 2-D variable in the probability density function.
    * Default: [1, 1]

Responses
^^^^^^^^^

* loglik: log likelihood of the pdf.

References
^^^^^^^^^^

This model is designed by Raghu Pasupathy (Virginia Tech) and Shane G. Henderson (Cornell) in 2007.

Optimization Problem: Max Log Likelihood (ParamEstiMaxLogLik)
-------------------------------------------------------------

Decision Variables
^^^^^^^^^^^^^^^^^^

* x

Objectives
^^^^^^^^^^

Minimize the log likelihood of a 2-D gamma random variable.

Constraints
^^^^^^^^^^^

x is in the square (0, 10) × (0, 10).

Problem Factors
^^^^^^^^^^^^^^^

* budget: Maximum number of replications for a solver to take.
    * Default: 1000

Fixed Model Factors
^^^^^^^^^^^^^^^^^^^

N/A

Starting Solution
^^^^^^^^^^^^^^^^^

* x: [1, 1]

Random Solutions
^^^^^^^^^^^^^^^^

Generate :math:`x` i.i.d. uniformly in the square (0, 10) × (0, 10).

Optimal Solution
^^^^^^^^^^^^^^^^

x = [2, 5]

Optimal Objective Function Value
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Known, but not evaluated.