Parameter Estimation

See the 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 \({Y_j}\), \(Y_j \in [0, \infty] \times [0, \infty]\), 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]

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

Optimal Solution

x = [2, 5]

Optimal Objective Function Value

Known, but not evaluated.