MonotonicQuantileEstimation {mistral}

Quantile estimation under monotonicity constraints

Description

Estimate a quantile with the constraints that the function is monotone

Usage

  MonotonicQuantileEstimation(f, 
                            inputDimension, 
                            inputDistribution,
                            dir.monot,
                            N.calls,
                            p,
                            method,
                            X.input = NULL,
                            Y.input = NULL)

Arguments

f

a failure fonction

inputDimension

dimension of the inputs

inputDistribution

a list of length ‘inputDimension’ which contains the name of the input distribution and their parameters. For the input "i", inputDistribution[[i]] = list("name_law",c(parameters1,..., parametersN))

dir.monot

vector of size inputDimension which represents the monotonicity of the failure function. dir.monot[i] = -1 (resp. 1) if the failure function f is decreasing (resp. increasing) according with direction i.

N.calls

Number of calls to f allowed

method

there are four methods available. "MonteCarloWB" provides the empirical quantile estimator, "MonteCarloWB" provides the empirical quantile estimator as well as two bounds for the searched quantile, "Bounds" provides two bounds for a quantile from a set of points and "MonteCarloIS" provides an estimate of a quantile based on a sequential framework of simulation.

p

the probability associated to the quantile

X.input

a set of points

Y.input

value of f on X.input

Details

MonotonicQuantileEstimation provides many methods to estimate a quantile under monotonicity constraints.

Value

An object of class list containing the quantile as well as:

qm

A lower bound of the quantile.

qM

A upperer bound of the quantile.

q.hat

An estimate of the quantile.

Um

A lower bounds of the probability obtained from the desing of experiments.

UM

An upper bounds of the probability obtained from the desing of experiments.

XX

Design of experiments

YY

Values of on XX

Note

Inputs X.input and Y.input are useful only for method = "Bounds"

Author(s)

Vincent Moutoussamy

References

Bousquet, N. (2012) Accelerated monte carlo estimation of exceedance probabilities under monotonicity constraints. Annales de la Faculte des Sciences de Toulouse. XXI(3), 557-592.

Examples

## Not run: 
 inputDistribution <- list()
 inputDistribution[[1]] <- list("norm",c(4,1))
 inputDistribution[[2]] <- list("norm",c(0,1))

 inputDimension <- length(inputDistribution)
 dir.monot <- c(1, -1)
 N.calls <- 80

 f <- function(x){
   return(x[1] - x[2])
 }

 probability <- 1e-2

 trueQuantile <- qnorm(probability,
                     inputDistribution[[1]][[2]][1] - inputDistribution[[2]][[2]][1],
                     sqrt(inputDistribution[[1]][[2]][2] + inputDistribution[[1]][[2]][2]))

 resQuantile <- MonotonicQuantileEstimation(f, inputDimension, inputDistribution,
                                      dir.monot, N.calls, p = probability, method = "MonteCarloIS")

 quantileEstimate <- resQuantile[[1]][N.calls, 3]


## End(Not run)