llogisMLE {conf}

Maximum Likelihood Parameter Estimation of a Log Logistic Model with Possibly Censored Data

Description

Estimate log logistic model parameters by the maximum likelihood method using possibly censored data. The corresponding code for this function as well as the manual information included here is attributed to Christophe Pouzat's STAR Package (archived 2022-05-23).

Usage

llogisMLE(yi, ni = numeric(length(yi)) + 1,
          si = numeric(length(yi)) + 1)

Arguments

yi

vector of (possibly binned) observations or a spikeTrain object.

ni

vector of counts for each value of yi; default: numeric(length(yi))+1.

si

vector of counts of uncensored observations for each value of yi; default: numeric(length(yi))+1.

Details

The MLE for the log logistic is not available in closed formed and is therefore obtained numerically obtained by calling optim with the BFGS method.

In order to ensure good behavior of the numerical optimization routines, optimization is performed on the log of parameter scale.

Standard errors are obtained from the inverse of the observed information matrix at the MLE. They are transformed to go from the log scale used by the optimization routine to the requested parameterization.

Value

A list of class durationFit with the following components:

estimate

the estimated parameters, a named vector.

se

the standard errors, a named vector.

logLik

the log likelihood at maximum.

r

a function returning the log of the relative likelihood function.

mll

a function returning the opposite of the log likelihood function using the log of parameter sdlog.

call

the matched call.

Note

The returned standard errors (component se) are valid in the asymptotic limit. You should plot contours using function r in the returned list and check that the contours are reasonably close to ellipses.

Author(s)

Christophe Pouzat [e-mail]

References

Lindsey, J.K. (2004) Introduction to Applied Statistics: A Modelling Approach. OUP.

Lindsey, J.K. (2004) The Statistical Analysis of Stochastic Processes in Time. CUP.

See Also

dllogis, invgaussMLE, gammaMLE.

Examples

## Not run: 
## Simulate sample of size 100 from a log logisitic
## distribution
set.seed(1102006,"Mersenne-Twister")
sampleSize <- 100
location.true <- -2.7
scale.true <- 0.025
sampLL <- rllogis(sampleSize,location=location.true,scale=scale.true)
sampLLmleLL <- llogisMLE(sampLL)
rbind(est = sampLLmleLL$estimate,se = sampLLmleLL$se,true = c(location.true,scale.true))

## Estimate the log relative likelihood on a grid to plot contours
Loc <- seq(sampLLmleLL$estimate[1]-4*sampLLmleLL$se[1],
               sampLLmleLL$estimate[1]+4*sampLLmleLL$se[1],
               sampLLmleLL$se[1]/10)
Scale <- seq(sampLLmleLL$estimate[2]-4*sampLLmleLL$se[2],
             sampLLmleLL$estimate[2]+4*sampLLmleLL$se[2],
             sampLLmleLL$se[2]/10)
sampLLmleLLcontour <- sapply(Loc, function(m) sapply(Scale, function(s) sampLLmleLL$r(m,s)))
## plot contours using a linear scale for the parameters
## draw four contours corresponding to the following likelihood ratios:
##  0.5, 0.1, Chi2 with 2 df and p values of 0.95 and 0.99
X11(width=12,height=6)
layout(matrix(1:2,ncol=2))
contour(Loc,Scale,t(sampLLmleLLcontour),
        levels=c(log(c(0.5,0.1)),-0.5*qchisq(c(0.95,0.99),df=2)),
        labels=c("log(0.5)",
          "log(0.1)",
          "-1/2*P(Chi2=0.95)",
          "-1/2*P(Chi2=0.99)"),
        xlab="Location",ylab="Scale",
        main="Log Relative Likelihood Contours"
        )
points(sampLLmleLL$estimate[1],sampLLmleLL$estimate[2],pch=3)
points(location.true,scale.true,pch=16,col=2)
## The contours are not really symmetrical about the MLE we can try to
## replot them using a log scale for the parameters to see if that improves
## the situation
contour(Loc,log(Scale),t(sampLLmleLLcontour),
        levels=c(log(c(0.5,0.1)),-0.5*qchisq(c(0.95,0.99),df=2)),
        labels="",
        xlab="log(Location)",ylab="log(Scale)",
        main="Log Relative Likelihood Contours",
        sub="log scale for parameter: scale")
points(sampLLmleLL$estimate[1],log(sampLLmleLL$estimate[2]),pch=3)
points(location.true,log(scale.true),pch=16,col=2)

## make a parametric boostrap to check the distribution of the deviance
nbReplicate <- 10000
sampleSize <- 100
system.time(
            devianceLL100 <- replicate(nbReplicate,{
              sampLL <- rllogis(sampleSize,location=location.true,scale=scale.true)
              sampLLmleLL <- llogisMLE(sampLL)
              -2*sampLLmleLL$r(location.true,scale.true)
            }
                                       )
            )[3]

## Get 95 and 99
ci <- sapply(1:nbReplicate,
                 function(idx) qchisq(qbeta(c(0.005,0.025,0.975,0.995),
                                            idx,
                                            nbReplicate-idx+1),
                                      df=2)
             )
## make QQ plot
X <- qchisq(ppoints(nbReplicate),df=2)
Y <- sort(devianceLL100)
X11()
plot(X,Y,type="n",
     xlab=expression(paste(chi[2]^2," quantiles")),
     ylab="MC quantiles",
     main="Deviance with true parameters after ML fit of log logistic data",
     sub=paste("sample size:", sampleSize,"MC replicates:", nbReplicate)
     )
abline(a=0,b=1)
lines(X,ci[1,],lty=2)
lines(X,ci[2,],lty=2)
lines(X,ci[3,],lty=2)
lines(X,ci[4,],lty=2)
lines(X,Y,col=2)

## End(Not run)