gammahat

Gammahat, typically written as gamma_hat, is a common notation in statistics and related fields used to denote an estimator of a parameter gamma. The exact meaning of gamma_hat depends on the context and model, and it is not a single fixed object. In general, gamma_hat represents the value obtained from data that aims to approximate the true parameter gamma.

In parametric models, gamma_hat is often the estimated shape or other parameter gamma of a distribution or model. For example, in a gamma distribution with shape parameter gamma (often denoted k or alpha) and a scale parameter, gamma_hat may refer to the estimated shape parameter obtained by maximum likelihood or by the method of moments. The maximum likelihood estimator for the shape parameter typically requires numerical optimization and can involve the digamma function, while the method of moments provides a closed-form option such as gamma_hat = m^2 / v, where m is the sample mean and v the sample variance.

In hidden Markov models and related time-series methods, gamma_hat commonly denotes the posterior probability of being in a given state at a specific time, computed by procedures like the forward-backward algorithm. These gamma_hat values serve as inputs to re-estimation steps in algorithms such as the expectation-maximization (EM) framework.

In regression and econometric contexts, gamma_hat may designate the estimated value of a parameter named gamma, a coefficient, or a functional parameter depending on the model specification. Across contexts, gamma_hat is subject to sampling variability and, under standard regularity conditions, is consistent and asymptotically normal; standard errors are typically obtained from Fisher information, bootstrap methods, or other resampling techniques.

See also: gamma distribution, maximum likelihood estimation, hidden Markov model.