Enkeltparametre

Enkeltparametre refers to a model or statistical distribution that is determined by a single unknown parameter, while all other aspects of the model are fixed or known. In this sense, the parameter serves as the sole degree of freedom that governs the behavior or shape of the probability distribution or function. The concept is often contrasted with multi-parameter models where several parameters must be estimated simultaneously.

Common examples in statistics include one-parameter families such as the Poisson distribution with rate λ, the exponential distribution with rate λ, and the normal distribution with known variance σ² where the mean μ is the only unknown parameter. In addition, location families where a single location parameter θ shifts the distribution, e.g., F(x−θ), are often treated as one-parameter models. These examples illustrate how a single parameter can capture the essential variability in the data under a specified structure.

Estimation and inference in one-parameter models are typically straightforward. Maximum likelihood estimation, method of moments, and Bayesian methods are commonly applied, often yielding closed-form estimators. For example, the MLEs include: λ̂ = x̄ for Poisson, λ̂ = n/∑x_i for Exponential, and μ̂ = x̄ for Normal with known σ². Fisher information and standard errors are readily computed, enabling confidence intervals and hypothesis tests.

Limitations of enkeltparametre models arise when data exhibit more complex patterns that require additional flexibility, such as multiple sources of variation or nonlinear effects. In such cases, models with more parameters or hierarchical structures may be more appropriate. The concept remains a useful baseline for understanding how a single parameter shapes a model's behavior.