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
where \(x^* ≡ (x^*_1, x^*_2)`\) is the unknown vector of parameters.
Noting that \(x_star\) maximizes the function
and that
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.