gammahatk

gammahatk is the common notation for the k-th estimated component of a parameter vector gamma. In statistical models where gamma governs scale or shape parameters, gammahatk represents the estimate of gamma_k obtained from data under a specified estimation procedure, such as maximum likelihood, method of moments, or Bayesian posterior mean.

Notation and computation: When gamma is estimated jointly, gammahatk is derived from solving the likelihood equations with respect to gamma or by sampling from the posterior distribution p(gamma | data). In frequentist settings, asymptotic theory often yields that sqrt(n)(gammahat - gamma) converges to a multivariate normal distribution with covariance given by the inverse Fisher information. In Bayesian contexts, gammahatk can denote the posterior mean or median of gamma_k.

Applications: gammahatk appears in generalized linear models with gamma-family responses, survival models with gamma frailty, and hierarchical models where gamma controls dispersion, scale, or rate parameters. It is also used in time series notation for the k-th component of a gamma-related parameter and, in some texts, to denote the sample autocovariance gamma_hat(k) for lag k in stationary processes.

Interpretation and properties: The interpretation depends on the model. As with other estimators, gammahatk is subject to sampling error, and bias correction or shrinkage may be applied in small samples. Under standard regularity conditions, gammahatk is consistent for gamma_k and asymptotically normal.

See also: gamma distribution, parameter estimation, Bayesian inference, Fisher information, autoregressive models, time series.