Profilikelihood

Profilikelihood is a method used in statistical inference, particularly in the context of maximum likelihood estimation. It is employed when dealing with models that have multiple parameters, and the interest lies in understanding the uncertainty or estimating the value of a single parameter (or a subset of parameters) while accounting for the influence of the other parameters.

The core idea of profilikelihood is to fix a specific value for the parameter of interest and then maximize the likelihood function with respect to all other nuisance parameters. This process is repeated for a range of possible values of the parameter of interest. The resulting profile likelihood function is a function of the parameter of interest only.

The profile likelihood function can then be used to construct confidence intervals or regions for the parameter of interest. A common approach is to use a likelihood ratio test statistic based on the profile likelihood. For a given significance level, the set of parameter values for which the likelihood ratio test statistic is below a certain threshold defines the confidence interval.

Compared to standard maximum likelihood estimation, profilikelihood offers advantages when the number of nuisance parameters is large, as it can be more computationally efficient and provide more accurate confidence intervals. It is widely used in various fields, including biostatistics, econometrics, and engineering, for parameter estimation and uncertainty quantification.