adjusted_est: Compute the Kernel Regression Estimator.

In CovEsts: Nonparametric Estimators for Covariance Functions

Compute the Kernel Regression Estimator.

Description

This function computes the kernel regression estimator of the autocovariance function.

Usage

adjusted_est(
  X,
  x,
  t,
  b,
  kernel_name = "gaussian",
  kernel_params = c(),
  pd = TRUE,
  type = "autocovariance",
  meanX = mean(X),
  custom_kernel = FALSE
)

Arguments

X	A vector representing observed values of the time series.
x	A vector of lags.
t	The arguments at which the autocovariance function is calculated at.
b	Bandwidth parameter, greater than 0.
kernel_name	The name of the symmetric kernel (see kernel_symm) function to be used. Possible values are: gaussian, wave, rational_quadratic, and bessel_j. Alternatively, a custom kernel function can be provided, see the examples.
kernel_params	A vector of parameters of the kernel function. See kernel_symm for parameters.
pd	Whether a positive-definite estimate should be used. Defaults to TRUE.
type	Compute either the 'autocovariance' or 'autocorrelation'. Defaults to 'autocovariance'.
meanX	The average value of X. Defaults to mean(X).
custom_kernel	If a custom kernel is to be used or not. Defaults to FALSE.

Details

The kernel regression estimator of an autocovariance function is defined as

\hat{\rho}(t) = \left( \sum_{i=1}^{N} \sum_{j=1}^{N} \check{X}_{ij} K((t - (t_{i} - t_{j})) / b) \right) \left( \sum_{i=1}^{N} \sum_{j=1}^{N} K((t - (t_{i} - t_{j})) / b) \right)^{-1},

where \check{X}_{ij} = (X(t_{i}) - \bar{X}) (X(t_{j}) - \bar{X}).

If pd is TRUE, the estimator will be made positive-definite through the following procedure

1. Take the discrete Fourier cosine transform, \widehat{\mathcal{F}}(\theta), of the estimated autocovariance function

2. Compute a modified spectrum \widetilde{\mathcal{F}}(\theta) = \max(\widehat{\mathcal{F}}(\theta), 0) for all sample frequencies.

3. Perform the Fourier inversion to obtain a new estimator.

Value

A vector whose values are the kernel regression estimates.

References

Hall, P. & Patil, P. (1994). Properties of nonparametric estimators of autocovariance for stationary random fields. Probability Theory and Related Fields 99(3), 399-424. https://doi.org/10.1007/bf01199899

Hall, P., Fisher, N. I., & Hoffmann, B. (1994). On the nonparametric estimation of covariance functions. The Annals of Statistics 22(4), 2115-2134. https://doi.org/10.1214/aos/1176325774

Examples

X <- c(1, 2, 3, 4)
adjusted_est(X, 1:4, 1:3, 0.1, "gaussian")
my_kernel <- function(x, theta, params) {
  stopifnot(theta > 0, length(x) >= 1)
  return(exp(-((abs(x) / theta)^params[1])) * (2 * theta  * gamma(1 + 1/params[1])))
}
adjusted_est(X, 1:4, 1:3, 0.1, my_kernel, c(0.25), custom_kernel = TRUE)

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