Threeparameter

Threeparameter is a term used to describe models or distributions that are specified by three free parameters. It does not refer to a single canonical model, but rather to a class of three-parameter families encountered across statistics, psychometrics, and reliability engineering. The extra parameter relative to two-parameter families provides additional flexibility to capture features such as shift, asymmetry, or guessing behavior.

In distribution theory, three-parameter families extend two-parameter forms by including a location or threshold parameter in addition to shape and scale. Examples include the three-parameter lognormal distribution, which adds a shift parameter to the standard lognormal, and the three-parameter Weibull distribution, which includes a threshold or location term. These models can better fit data with skewness or a shifted support than their two-parameter counterparts.

In psychometrics, the three-parameter logistic model (3PL) is a prominent application in item response theory. It models the probability that a person with latent ability θ answers an item correctly as P(θ) = c + (1 − c) / [1 + exp(−D a (θ − b))]. Here a is the item discrimination parameter, b is the difficulty parameter, and c is the guessing parameter that accounts for the chance of a correct answer by guessing on difficult items. The 3PL adds the guessing parameter relative to simpler two-parameter models.

Estimation of three-parameter models typically uses maximum likelihood or Bayesian methods. Challenges include identifiability and interpretability of parameters, especially with limited data or strong parameter correlations. Model selection often relies on information criteria and diagnostic checks.