likelihoodlike

Likelihoodlike is a term used in statistics to describe a function that behaves similarly to a likelihood function but does not necessarily satisfy all formal criteria that define a likelihood. It is often used to denote surrogate, approximate, or non-probabilistic constructs that can be used for parameter inference in a manner akin to likelihoods.

Definition and scope: For data x and parameter θ in a parameter space Θ, a function L(θ; x) is likelihoodlike if: (1) L(θ; x) ≥ 0 for all θ; (2) larger values indicate greater agreement with the data; (3) it is defined on Θ (or a subset) and can be used to form comparisons, such as ratios L(θ1; x) / L(θ2; x); (4) it need not be proportional to P(x|θ) or integrate to 1 over Θ. In this sense, a likelihoodlike function may serve as a surrogate for a true likelihood in estimation and inference.

Relation to other concepts: Likelihoodlike functions often overlap with pseudo-likelihood, composite likelihood, or quasi-likelihood, which are built to simplify or stabilize inference when a full likelihood is intractable or undefined. They may also arise from estimating equations or other non-probabilistic criteria that still yield useful comparative information about θ.

Usage and caveats: Inference using likelihoodlike functions relies on the relative magnitudes of L(θ; x) rather than on a formal probabilistic interpretation. Standard results such as likelihood-ratio tests and confidence intervals may require additional justification or resampling methods to ensure correct frequentist properties.

See also: likelihood, pseudo-likelihood, composite likelihood, quasi-likelihood, estimating equations, likelihood ratio test.