imprProbEst (version 1.0.1)

ArgMinDist: A Minimum Distance Estimation

Description

The function calculates a minimum distance estimator for an imprecise probability model. The imprecise probability model consists of upper coherent previsions whose credal sets are given by finite numbers of constraints on the expectations. The parameter set is finite. The estimator chooses that parameter such that the empirical measure lies next to the corresponding credal set with respect to the total variation norm.

Usage

ArgMinDist(x, lbomega, ubomega, epsilon, ImpreciseModel)

Arguments

x

a matrix where each row corresponds to one observation

lbomega

a vector containing the lower bounds of the sample space

ubomega

a vector containing the upper bounds of the sample space

epsilon

a positive real number; step size of the discretization

ImpreciseModel

a list of upper coherent previsions; see 'Details'

Value

ArgMinDist returns a list, e.g. results, containing three components

results[[1]]

the estimation; that is, the number of the minimizing coherent upper prevision in ImpreciseModel

results[[2]]

the total variation distance of the minimizing coherent upper prevision

results[[3]]

the number of linear programms which had to be solved

Details

The matrix x containes independent identically distributed data. Each row corresponds to one observation. The sample space is assumed to be a hyperrectangle in $R^k$. The lower bounds of this hyperrectangle are given by lbomega; the upper bounds of this hyperrectangle are given by ubomega. Accordingly, length(lbomega), length(ubomega) and length(x[,1]) are equal to k. Smaller values of epsilon may lead to more accurate results but increase the calculation time. Too small values of epsilon may cause an error due to RAM limitations. ImpreciseModel containes an imprecise model consisting of upper coherent previsions. ImpreciseModel is a list; each component of ImpreciseModel is again a list which corresponds to an upper coherent prevision. Each upper coherent prevision is given by a list containing a list of functions and a corresponding vector of upper previsions.

For example, the imprecise model ImpreciseModel may consist of three coherent upper previsions ImpreciseModel <- list(CohUpPrev1,CohUpPrev2,CohUpPrev3). CohUpPrev1 may be defined by CohUpPrev1 <- list(ListOfFunctions,UpperPrevisions). Here, ListOfFunctions is a list of functions, e.g., ListOfFunctions <- list(f1,f2,f3,f4). Every function has to accept a single numeric argument and to return a numeric vector of the same length; the infimum of every function has to be 0, the supremum has to be 1. UpperPrevisions is a vector which contains the values of the upper coherent prevision at the functions in ListOfFunctions. That is, e.g., UpperPrevisions[2] is the value of the upper prevision at the function f2. Accordingly, the number of elements of the list ListOfFunctions is equal to length(UpperPrevisions). The estimation is that coherent upper prevision whose credal set has minimal total variation distance to the empirical measure generated by the observations x. Confer Hable (2008) for the definition of this minimum distance estimator; confer Walley (1991) and Hable (2008) for the theory of imprecise probabilities based on coherent upper previsions or coherent lower previsions.

References

Hable, R. (2008) Data-Based Decisions under Complex Uncertainty. Ph.D. thesis, LMU Munich, in preparation.

Walley, P. (1991) Statistical reasoning with imprecise probabilities. Chapman & Hall, London.

Examples

f1 <- function(v){ ifelse( abs(v-1)<1e-10 ,1,0) }
f2 <- function(v){ ifelse( abs(v-2)<1e-10 ,1,0) }
f3 <- function(v){ ifelse( abs(v-3)<1e-10 ,1,0) }
f4 <- function(v){ 1-ifelse( abs(v-3)<1e-10 ,1,0) }

x <- matrix(c(1,2,3,4),nrow=1)

UpperPrevisions1 <- c(1/4-0.03,1/4-0.03,1/4+0.01,1)
ListOfFunctions1 <- list(f1,f2,f3,f4)
CohUpPrev1 <- list(ListOfFunctions1,UpperPrevisions1)

UpperPrevisions2 <- c(1/4-0.04,1/4+0.04,1/4-0.01)
ListOfFunctions2 <- list(f1,f2,f3)
CohUpPrev2 <- list(ListOfFunctions2,UpperPrevisions2)

ImpreciseModel <- list(CohUpPrev1,CohUpPrev2)

lbomega <- 1
ubomega <- 4
epsilon <- 0.01

ArgMinDist(x,lbomega,ubomega,epsilon,ImpreciseModel)