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 \({Y_j}\), \(Y_j \in [0, \infty] \times [0, \infty]\), that are i.i.d and known to come from a distribution with the two-dimensional density function

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

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

Noting that \(x_star\) maximizes the function

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

and that

\[G_m(x) = \frac{1}{m} \sum_{j=1}^m \log(f(Y_j ; x))\]

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

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]

Respones:

  • 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 \(x\) i.i.d. uniformly in the square (0, 10) × (0, 10).

Optimal Solution:

x = [2, 5]

Optimal Objective Function Value:

Known, but not evaluated.