Profiellikelihood

Profiellikelihood is a term sometimes encountered as a variant spelling of profile likelihood, though the standard term in statistics is profile likelihood. It refers to a method for making inferences about a parameter of interest when nuisance parameters are present.

Definition and construction

In a parametric model with data y and parameter vector θ = (φ, ψ), where φ represents the parameter(s) of interest and ψ the nuisance parameters, the profile likelihood for φ is obtained by maximizing the full likelihood over ψ for each fixed φ. Formally, Lp(φ) = supψ L(φ, ψ; y). The maximum of L over all φ gives the maximum likelihood estimate, while Lp(φ) provides a reduced likelihood function that reflects the information about φ after accounting for nuisance parameters.

Inference

Profile likelihoods are used to construct confidence intervals and perform hypothesis tests for φ. A common approach is the profile likelihood ratio statistic: LR(φ) = -2 log [Lp(φ) / Lp(φ̂)], where φ̂ maximizes the full likelihood. Under regularity conditions and large samples, LR(φ) has an approximate chi-square distribution, enabling likelihood-based confidence intervals and tests (Wilks’ theorem).

Applications and considerations

Profile likelihoods are widely used in nonlinear models, mixed-effects models, survival analysis, and econometrics, especially when nuisance parameters complicate direct inference. Computation can be intensive, requiring repeated optimization over ψ for many φ values. Advantages include favorable invariance properties and reliance on likelihood theory; limitations include potential instability when nuisance dimensions are large and sensitivity to model specification.

See also: likelihood ratio test, Wilks’ theorem, maximum likelihood estimation, nuisance parameter.