statostata

Statostata is a term used in theoretical statistics to describe a class of statistics that explicitly incorporate an auxiliary state variable into their calculation. The idea is to produce statistics whose behavior adapts to the current state of the system, enabling state-dependent summaries of data.

Definition and formulation: Let X denote observed data and S denote a state variable or regime. A statostata is any function T(X, S) that outputs a statistic, often a scalar, whose value depends on both X and S. When S is fixed, T reduces to a conventional statistic on X. Common examples include conditional means E[X|S=s], conditional variances Var(X|S=s), or more general functions T(X, S) that summarize X with reference to the state S.

Construction and properties: Statostata are typically constructed from a model that links X and S, such as X having a distribution conditional on S. They are evaluated per state, which can yield a vector of statistics across states. The statistical properties, such as bias and variance, depend on the estimation method and the correctness of the X–S model. They relate to conditional statistics and to state-space approaches, but are distinguished by their explicit dependence on the state variable.

Applications and limitations: The concept appears in time-series with regimes, econometrics with regime-switching models, and environmental science where summaries should reflect changing conditions. Limitations include the need for reliable state information, potential misclassification, and increased complexity or variance when data within states are sparse.

See also: state space model, conditional expectation, regime-switching, adaptive statistics.