Maximum Likelihood Parameter Estimation of a Weibull Model with Possibly Censored Data

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Description

Estimate Weibull model parameters by the maximum likelihood method using possibly censored data.

Usage

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weibullMLE(yi, ni = numeric(length(yi)) + 1,
           si = numeric(length(yi)) + 1, shape.min = 0.05, shape.max = 5)

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.

shape.min

numeric, the inital guess of the minimal possible value of the shape parameter, used by optimise.

shape.max

numeric, the inital guess of the maximal possible value of the shape parameter, used by optimise.

Details

There is no closed form expression for the MLE of a Weibull distribution. The numerical method implemented here uses the profile likelihood described by Kalbfleisch (1985) pp 56-58.

In order to ensure good behavior of the numerical optimization routines, optimization is performed on the log of the parameters (shape and 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 parameterization requested.

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 the parameters.

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 christophe.pouzat@gmail.com

References

Kalbfleisch, J. G. (1985) Probability and Statistical Inference. Volume 2: Statistical Inference. Springer-Verlag.

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

See Also

Weibull, invgaussMLE, lnormMLE, gammaMLE

Examples

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## Not run: 
## Simulate sample of size 100 from a weibull distribution
set.seed(1102006,"Mersenne-Twister")
sampleSize <- 100
shape.true <- 2.5
scale.true <- 0.085
sampWB <- rweibull(sampleSize,shape=shape.true,scale=scale.true)
sampWBmleWB <- weibullMLE(sampWB)
rbind(est = sampWBmleWB$estimate,se = sampWBmleWB$se,true = c(shape.true,scale.true))

## Estimate the log relative likelihood on a grid to plot contours
Shape <- seq(sampWBmleWB$estimate[1]-4*sampWBmleWB$se[1],
               sampWBmleWB$estimate[1]+4*sampWBmleWB$se[1],
               sampWBmleWB$se[1]/10)
Scale <- seq(sampWBmleWB$estimate[2]-4*sampWBmleWB$se[2],
             sampWBmleWB$estimate[2]+4*sampWBmleWB$se[2],
             sampWBmleWB$se[2]/10)
sampWBmleWBcontour <- sapply(Shape, function(sh) sapply(Scale, function(sc) sampWBmleWB$r(sh,sc)))
## plot contours using a linear scale for the parameters
## draw four contours corresponding to the following likelihood ratios:
##  0.5, 0.1, Chi2 with 2 df and p values of 0.95 and 0.99
X11(width=12,height=6)
layout(matrix(1:2,ncol=2))
contour(Shape,Scale,t(sampWBmleWBcontour),
        levels=c(log(c(0.5,0.1)),-0.5*qchisq(c(0.95,0.99),df=2)),
        labels=c("log(0.5)",
          "log(0.1)",
          "-1/2*P(Chi2=0.95)",
          "-1/2*P(Chi2=0.99)"),
        xlab="shape",ylab="scale",
        main="Log Relative Likelihood Contours"
        )
points(sampWBmleWB$estimate[1],sampWBmleWB$estimate[2],pch=3)
points(shape.true,scale.true,pch=16,col=2)
## The contours are not really symmetrical about the MLE we can try to
## replot them using a log scale for the parameters to see if that improves
## the situation
contour(log(Shape),log(Scale),t(sampWBmleWBcontour),
        levels=c(log(c(0.5,0.1)),-0.5*qchisq(c(0.95,0.99),df=2)),
        labels="",
        xlab="log(shape)",ylab="log(scale)",
        main="Log Relative Likelihood Contours",
        sub="log scale for the parameters")
points(log(sampWBmleWB$estimate[1]),log(sampWBmleWB$estimate[2]),pch=3)
points(log(shape.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(
            devianceWB100 <- replicate(nbReplicate,{
              sampWB <- rweibull(sampleSize,shape=shape.true,scale=scale.true)
              sampWBmleWB <- weibullMLE(sampWB)
              -2*sampWBmleWB$r(shape.true,scale.true)
            }
                                       )
            )[3]

## Get 95 and 99% confidence intervals for the QQ plot
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(devianceWB100)
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 gamma 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)

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