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I have two question about profile log-likelihood. One is of theoretical nature and one is about a plot in R. I illustrate the two questions in a application of profile-likelihood in EVT. I use the package evd.

> data(lisbon)
> fgev(lisbon)

Call: fgev(x = lisbon) 
Deviance: 241.2459 

Estimates
    loc    scale    shape  
96.0319  12.8526  -0.1988  

Standard Errors
   loc   scale   shape  
2.6171  1.8346  0.1284  

Optimization Information
  Convergence: successful 
  Function Evaluations: 24 
  Gradient Evaluations: 13 

> plot(profile(fgev(lisbon)),ci=c(0.95,0.99))

The result is the following plot:

enter image description here

My two question are about this plot.

theoretical one: For log-likelihood profile you can write one of the parameter as a function of the other, i.e. your log-likelihood function $l(\alpha,\beta)=l(\alpha,f(\alpha))$ for some function $f$.
Here I have a log-likelihood function $l(\mu,\sigma,\gamma)$ of three parameters location $\mu$, scale $\sigma$ and shape $\gamma$. In the profile log-likelihood of the location parameter I fix $\sigma$ and $\gamma$ and just vary $\mu$. But which values are used for $\sigma$ and $\gamma$? I don't think it is possible to write $\sigma=f_1(\mu)$ and $\gamma=f_2(\mu)$ for two functions $f_1$ and$ f_2$?

about the plot: In the plot what is indicated by the dashed lines?

Many thanks for your help

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1 Answer 1

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For each of the three plots, the horizontal dashed lines give a hint for a confidence interval on the parameter of interest for the two selected confidence levels $0.95$ and $0.99$. The lower and upper bounds of a such interval are the abscissas of the points where the profile log-likelihood intersect the horizontal line. For instance, the $95\%$ interval for the shape parameter ranges from about $-0.44$ to $0.08$. You can find these values from the third plot by using test values e.g. using abline(v = 0.08) or the interactive function locator.

The horizontal line position is found by subtracting to the maximal log-likelihood $\ell_{\text{max}} \approx -120.62$ half the corresponding upper quantile of the chi-square distribution with $1$ degree of freedom (number of parameters of interest). For instance, the $95\%$ level ordinate is

logLik(fgev(lisbon))[1] -0.5 * qchisq(p = 0.95, df = 1)

The lines are at the same ordinate for the three plots (which use different scales).

For each plot/parameter of interest, the profile is obtained by using a grid of values. For each of these grid values, a numerical optimisation over the two other parameters is performed, so no explicit formula is required. For instance, on the first plot the value corresponding to an abscissa $\mu$ is $$ \ell_{\text{p}}(\mu) := \max_{\sigma,\,\gamma} \, \ell(\mu,\,\sigma,\,\gamma) = \ell(\mu,\,\widehat{\sigma}_\mu, \,\widehat{\gamma}_\mu) $$ where the vector $[\widehat{\sigma}_\mu, \,\widehat{\gamma}_\mu]^\top$ is found by numerical optimisation of $\ell$ with $\mu$ fixed. The confidence interval is defined by $\ell_{\text{p}}(\mu) > \ell_{\text{max}} - q/2$ where $q$ is the quantile as returned by qchisq.

To see how this can be achieved in R, you can have a look at the simpler code of the gev.profxi function in the ismev package by J. Heffernann and A. Stephenson, in the gev.R file of the source package.

Profile likelihood can be used in some cases to simplify a ML estimation, and in a fairly more general context to infer on the parameters. In the first case, it is essential to have an explicit formula, since computation time is the primary concern. In the second case, time may not be a problem. A special case is when the parameter of interest is scalar, since then we want a confidence interval rather than a confidence region and we can proceed as in your example.

Note that the derivation of the confidence limits as used here is based on asymptotic theory, see e.g the book by A.C. Davison Statistical Models. My opinion is that for EVT distributions (GEV and Generalised Pareto), a fairly large number of observations is required to reach an accurate approximation of the confidence levels. With so few as $30$ observations, a poor approximation of the confidence level is to be feared.

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