find_max: Maximum of Likelihood Ratio Test field

In TOHM: Testing One Hypothesis Multiple Times

Description

It computes the maximum of the generalized Likelihood Ratio Test (LRT) evaluated over a grid of values.

Usage

find_max(x, mll, null0, init, lowlim, uplim, THETA)

Arguments

x	A vector or matrix collecting the data on which the LRT field is computed.
mll	A function specifying the negative (profile) log-likelihood. See details.
null0	A vector or scalar of the free parameters under the null hypothesis. See details.
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.
THETA	A vector or matrix of grid values of 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

max_gLRT	Maximum observed of the LRT field.
theta_max	Value of THETA at which the maximum is observed.

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

gLRT, global_p

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))),]

find_max(x=data,mll=mll,null0=0,init=c(0.1),
lowlim=c(0),uplim=c(1), THETA=THETA_sel)

Example output

Evaluating LRT for each point of the grid...
$max_gLRT
[1] 1.457834

$theta_max
 Var1  Var2 
217.5  18.0 

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