SSJ
V. labo.

Package umontreal.iro.lecuyer.stat.list.lincv

Provides facilities for computing estimators with linear control variables.

See:
          Description

Class Summary
FunctionOfMultipleMeansTallyWithCV Represents a function of multiple means tally for an estimator with linear control variables.
ListOfTalliesWithCV<E extends Tally> Represents a list of tallies with control variables that inherits the functionalities of a list of tallies, and accepts vectors of length p + q.
 

Package umontreal.iro.lecuyer.stat.list.lincv Description

Provides facilities for computing estimators with linear control variables. Let XRp be a vector of estimators, and CRq be a vector of control variables with known expectation E[C]. The estimator X can be replaced by the linearly controlled estimator

XC = X - βt(C - E[C]),

where β is a constant q×p matrix. The covariance matrix of this estimator is

Cov(XC) = ΣX + βtΣCβ - βtΣCX - ΣCXtβ,

where ΣCX is the q×p covariance matrix of C and X, ΣC is the q×q covariance matrix of C, and ΣX is the p×p covariance matrix of X. The matrix β can be set for | Cov(XC)| <= | Cov(X)|, where | Σ| is the determinant of the matrix Σ. The optimal β matrix, which minimizes the determinant (or the trace) of the covariance matrix, is

β* = ΣC-1ΣCX.

If μ = E[X], ν = E[C], and g(μ) is a continuous function in the neighborhood of μ, let h(μ, ν) = g(μ) - βft(C - ν) be the corresponding function with control variables. Here, βf is a q-dimensional vector being set to reduce the variance of the function of averages with control variables. The optimal βf vector that minimizes the variance is

βf* = β*g(μ) = ΣC-1ΣCXg(μ).

This package provides classes that can help one implement such control variables. During the simulation, or during pilot runs, one adds observations of the estimators and the control variables to a list of tallies specialized for control variables. This list computes the covariances, and can compute an estimate of β* by using matrices of sample covariances. The vector of controlled averages, and the sample controlled covariance matrix can then be obtained. Alternatively, one can reset the list of tallies, keep the β matrix, and add independent vectors of observations.

The package also provides a class for applying control variables on a function of multiple averages. A tally for a function of multiple means with control variables encapsulates a list of tallies with control variables, and an object computing g(μ) and its gradient g(μ). A method is available to estimate the βf vector minimizing the variance of the function of averages.


SSJ
V. labo.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.