twoparameter

Twoparameter, often written as two-parameter, is an adjective describing any system, model, or family of objects described by two independent parameters. It contrasts with one-parameter models, which are specified by a single scalar, and with higher-parameter models that use more parameters. The two parameters may control location, scale, shape, orientation, amplitude, or other properties, depending on the context.

In mathematics, a two-parameter family of curves consists of all curves obtained by varying two parameters in a functional form F(x, y; a, b) = 0. A simple example is y = a x + b, where a and b determine slope and intercept. More generally, the family can be represented implicitly by equations or explicit forms such as y = a sin(bx) or y = a e^{b x}. The parameters can be estimated from data or chosen to meet constraints.

In statistics and probability, many distributions and models are two-parameter. Common examples include the two-parameter Weibull distribution with shape and scale parameters, and the two-parameter logistic model (2PL) in item response theory, which includes a discrimination parameter a and a difficulty parameter b for each item. This is often contrasted with one-parameter versions that share a common parameter across items, such as the Rasch model.

Parameter estimation for two-parameter models typically uses maximum likelihood, least squares, or Bayesian methods. Issues of identifiability, collinearity, and interpretability are common: with two parameters there can be trade-offs in how they affect model outputs, and care is needed to ensure unique fitting.