gLRT {TOHM}

Compute the generalized Likelihood Ratio Test

Description

Compute the generalized Likelihood Ratio Test (LRT) for a specified value of the nuisance parameter.

Usage

gLRT(theta, mll, x, init, lowlim, uplim, null0)

Arguments

theta

A vector or scalar of the value of the nuisance parameter with respect to which the LRT is computed.

mll

A function specifying the negative (profile) log-likelihood. See details.

x

A vector or matrix collecting the data.

init

A vector or scalar of initial values for the MLE.

lowlim

A vector or scalar of lower bounds for the MLE.

uplim

A vector or scalar of upper bounds for the MLE.

null0

A vector or scalar of the free parameters under the null hypothesis. See details.

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

The value of the generalized LRT for a specified value of theta.

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.

A.C. Davison. Statistical models, volume 11. Cambridge University Press, 2003.

See Also

find_max, TOHM_LRT

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)))}

gLRT(theta=c(200,30),mll=mll,init=0.1,lowlim=0,uplim=1,null0=0,x=data)