standard_est: Computes the Standard Estimator of the Autocovariance...

In CovEsts: Nonparametric Estimators for Covariance Functions

Computes the Standard Estimator of the Autocovariance Function.

Description

This function computes the following two estimates of the autocovariance function depending on the parameter pd.

Usage

standard_est(
  X,
  pd = TRUE,
  maxLag = length(X) - 1,
  type = "autocovariance",
  meanX = mean(X)
)

Arguments

X	A vector representing observed values of the time series.
pd	Whether a positive-definite estimate should be used. Defaults to TRUE.
maxLag	An optional parameter that determines the maximum lag to compute the estimated autocovariance function at. Defaults to length(X) - 1.
type	Compute either the 'autocovariance' or 'autocorrelation'. Defaults to 'autocovariance'.
meanX	The average value of X. Defaults to mean(X).

Details

For pd = TRUE:

\widehat{C}(h) = \frac{1}{N} \sum_{j=1}^{N-h} ( X(j) - \bar{X} ) ( X(j + h) - \bar{X} ) .

For pd = FALSE:

\widehat{C}(h) = \frac{1}{N - h} \sum_{j=1}^{N-h} ( X(j) - \bar{X} ) ( X(j + h) - \bar{X} ) .

This function will generate autocovariance values for lags h from the set \{0, \dots, \mbox{maxLag}\}.

The positive-definite estimator must be used cautiously when estimating over all lags as the sum of all values of the autocorrelation function equals to -1/2. For the nonpositive-definite estimator a similar constant summation property holds.

Value

A vector whose values are the autocovariance estimates.

References

Bilchouris, A. & Olenko, A (2025). On Nonparametric Estimation of Covariogram. Austrian Statistical Society 54(1), 112-137. https://doi.org/10.17713/ajs.v54i1.1975

Examples

X <- c(1, 2, 3)
standard_est(X, pd = FALSE, maxLag = 2, meanX = mean(X))

CovEsts documentation built on Sept. 10, 2025, 10:39 a.m.