# gammaMLE: Maximum Likelihood Parameter Estimation of a Gamma Model with... In STAR: Spike Train Analysis with R

## Description

Estimate Gamma model parameters by the maximum likelihood method using possibly censored data. Two different parameterizations of the Gamma distribution can be used.

## Usage

 ```1 2``` ```gammaMLE(yi, ni = numeric(length(yi)) + 1, si = numeric(length(yi)) + 1, scale = TRUE) ```

## 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`. `scale` logical should the scale (`TRUE`) or the rate parameterization (`FALSE`) be used?

## Details

There is no closed form expression for the MLE of a Gamma distribution. The numerical method implemented here uses the profile likelihood described by Monahan (2001) pp 210-216.

In order to ensure good behavior of the numerical optimization routines, optimization is performed on the log of the parameters (`shape` and `scale` or `rate`).

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 [email protected]

## References

Monahan, J. F. (2001) Numerical Methods of Statistics. CUP.

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

`GammaDist`, `invgaussMLE`, `lnormMLE`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83``` ```## Not run: ## Simulate sample of size 100 from a gamma distribution set.seed(1102006,"Mersenne-Twister") sampleSize <- 100 shape.true <- 6 scale.true <- 0.012 sampGA <- rgamma(sampleSize,shape=shape.true,scale=scale.true) sampGAmleGA <- gammaMLE(sampGA) rbind(est = sampGAmleGA\$estimate,se = sampGAmleGA\$se,true = c(shape.true,scale.true)) ## Estimate the log relative likelihood on a grid to plot contours Shape <- seq(sampGAmleGA\$estimate[1]-4*sampGAmleGA\$se[1], sampGAmleGA\$estimate[1]+4*sampGAmleGA\$se[1], sampGAmleGA\$se[1]/10) Scale <- seq(sampGAmleGA\$estimate[2]-4*sampGAmleGA\$se[2], sampGAmleGA\$estimate[2]+4*sampGAmleGA\$se[2], sampGAmleGA\$se[2]/10) sampGAmleGAcontour <- sapply(Shape, function(sh) sapply(Scale, function(sc) sampGAmleGA\$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(sampGAmleGAcontour), 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(sampGAmleGA\$estimate[1],sampGAmleGA\$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(sampGAmleGAcontour), 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(sampGAmleGA\$estimate[1]),log(sampGAmleGA\$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( devianceGA100 <- replicate(nbReplicate,{ sampGA <- rgamma(sampleSize,shape=shape.true,scale=scale.true) sampGAmleGA <- gammaMLE(sampGA) -2*sampGAmleGA\$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(devianceGA100) 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) ```