parameterestimation

Parameter estimation is the process of using observed data to infer the values of parameters in a mathematical model. Parameters determine the model’s behavior, while data provide evidence about their values. The goal is to select parameter values that best explain the data under a probabilistic framework and to quantify uncertainty.

Common methods include maximum likelihood estimation (MLE), least squares, method of moments, and Bayesian estimation. MLE chooses parameters that maximize the likelihood of the observed data. Least squares minimizes squared residuals. Method of moments matches sample moments to theoretical ones. Bayesian estimation combines a prior with data to obtain a posterior distribution.

Estimator properties include unbiasedness, consistency, and efficiency. An unbiased estimator’s expected value equals the true parameter; a consistent estimator converges to the true value as sample size grows; an efficient estimator has minimal variance among unbiased estimators, often characterized by Fisher information via the Cramér–Rao bound. Identifiability and model misspecification affect accuracy and interpretability of estimates.

Practical considerations include numerical optimization or sampling, and reporting uncertainty with standard errors, confidence intervals, or credible intervals. In time series and dynamic models, techniques such as Kalman filtering or state-space methods are common. Regularization can reduce overfitting in high-dimensional models and improve generalization.

Example: in linear regression y = Xβ + ε with ε ~ N(0, σ^2 I), ordinary least squares yields β̂ = (X′X)^{-1}X′y, which coincides with the MLE under Gaussian errors. Parameter estimation thus encompasses a range of methods and theory for inferring model parameters from data.