truncated_est: Compute the Truncated Kernel Regression Estimator.

In CovEsts: Nonparametric Estimators for Covariance Functions

Compute the Truncated Kernel Regression Estimator.

Description

This function computes the truncated kernel regression estimator, based on the kernel regression estimator \hat{\rho}(\cdot), see adjusted_est.

Usage

truncated_est(
  X,
  x,
  t,
  T1,
  T2,
  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.
T1	The first truncation point, T_{1} > 0.
T2	The second truncation point, T_{2} > T_{1} > 0.
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

This function computes the truncated kernel regression estimator,

\hat{\rho}_{1}(t) = \left\{ \begin{array}{ll} \hat{\rho}(t) & 0 \leq t \leq T_{1} \\ \hat{\rho}(T_{1}) (T_{2} - t)(T_{2} - T_{1})^{-1} & T_{1} < t \leq T_{2} \\ 0 & t > T_{2} \end{array} \right.

where \hat{\rho}(\cdot) is the kernel regression estimator, see adjusted_est.

Compared to adjusted_est, this function brings down the estimate to zero linearly between T_{1} and T_{2}. In the case of short-range dependence, this may be beneficial as it can remove estimation artefacts at large lags.

To make this estimator positive-definite, the following procedure is used:

1. Take the discrete Fourier cosine transform \widehat{\mathcal{F}}(\theta).

2. Find the smallest frequency where its associated value in the spectral domain is negative

   \hat{\theta} = \inf\{ \theta > 0 : \widehat{\mathcal{F}}(\theta)) < 0\}.

3. Set all values starting at the frequency to zero.

4. Perform the Fourier inversion.

If \hat{\theta} is a small frequency, most of the spectrum equals zero, resulting in an inaccurate estimate of the autocovariance function, see Bilchouris and Olenko (2025).

Value

A vector whose values are the truncated 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

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, 4)
truncated_est(X, 1:4, 1:3, 1, 2, 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])))
}
truncated_est(X, 1:4, 1:3, 1, 2, 0.1, my_kernel, c(0.25), custom_kernel = TRUE)

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