Description Usage Arguments Details Value Note Author(s) References Examples
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.
1 | ArgMinDist(x, lbomega, ubomega, epsilon, ImpreciseModel)
|
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' |
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.
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
|
results[[2]] |
the total variation distance of the minimizing coherent upper prevision |
results[[3]] |
the number of linear programms which had to be solved |
R programming support was given by Matthias Kohl
Robert Hable
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 | 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)
|
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