EC_LRT: Compute the Euler characteristic for the generalized...

In TOHM: Testing One Hypothesis Multiple Times

Description

It computes the Euler characteristic (EC) of the generalized Likelihood Ratio test (LRT) field above specified thresholds over a given search area.

Usage

EC_LRT(ck, x, mll, null0, init, lowlim, uplim, THETA)

Arguments

ck	Vector of thresholds defining the excursions sets with respect to which the ECs are computed.
x	A vector or matrix collecting the data on which the LRT is computed.
mll	A function specifying the negative (profile) log-likelihood. See details.
null0	A scalar or vector collecting the values of the free parameters under the null hypothesis. See details.
init	Vector of initial values for the MLE.
lowlim	Vector of lower bounds for the MLE.
uplim	Vector of upper bounds for the MLE.
THETA	A vector or matrix of grid values for the nuisance parameter with respect to which the search is performed.

Details

mll takes as first argument the vector of the parameters for which the MLE is generated. Other arguments of mll are the data vector or matrix (x) and a scalar or vector corresponding to the fixed value for the nuisance parameter with respect to which the profilying is computed (theta, see gLRT). If the latter is a vector it must be of same length of the rows in THETA. If the null model has nuisance parameters, null0 takes as arguments the values of the parameters being tested under the null hypothesis, followed by the estimates of the nuisance parameters obtained assuming that the null hypothesis is true.

Value

Returns a vector of EC values with respect to the thresholds specified in ck.

Author(s)

Sara Algeri

References

S. Algeri and D.A. van Dyk. Testing one hypothesis multiple times: The multidimensional case. arXiv:1803.03858, submitted to the Journal of Computational and Graphical Statistics, 2018.

See Also

EC_T

Examples

#generating data of interest
N<-100
x<-as.matrix(cbind(runif(N*2,172.5,217.5),runif(N*2,-2,58)))
x2<-x[(x[,1]<=217.5)&(x[,1]>=172.5),]
x_sel<-x2[(x2[,2]<=(28+sqrt(30^2-(x2[,1]-195)^2)))&(x2[,2]>=(28-
sqrt(30^2-(x2[,1]-195)^2))),]
data<-x_sel[sample(seq(1:(dim(x_sel)[1])),N),]

#Specifying minus-log-likelihood
kg<-function(theta){integrate(Vectorize(function(x) {
exp(-0.5*((x-theta[1])/0.5)^2)*integrate(function(y) {
exp(-0.5*((y-theta[2])/0.5)^2) }, 28-sqrt(30^2-(x-195)^2),
28+sqrt(30^2-(x-195)^2))$value}) , 172.5, 217.5)$value}
mll<-function(eta,x,theta){
  -sum(log((1-eta)/(pi*(30)^2)+eta*exp(-0.5*((x[,1]-
  theta[1])/0.5)^2-
  0.5*((x[,2]-theta[2])/0.5)^2)/kg(theta)))}

#Specifying search region
theta1<-seq(172.5,217.5,by=15)
theta2<-seq(-2,58,by=10)
THETA<-as.matrix(expand.grid(theta1,theta2))
originalR<-dim(THETA)[1]
rownames(THETA)<-1:(dim(THETA)[1])
THETA2<-THETA[(THETA[,1]<=217.5)&(THETA[,1]>=172.5),]
THETA_sel<-THETA2[(THETA2[,2]<=(28+sqrt(30^2-(THETA2[,1]-
195)^2)))&(THETA2[,2]>=(28-sqrt(30^2-(THETA2[,1]-195)^2))),]

EC_LRT(ck=c(1,8),x=data,mll=mll,null0=0,init=c(0.1),
lowlim=c(0),uplim=c(1), THETA_sel)

TOHM documentation built on March 10, 2021, 1:05 a.m.