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 = c("gaussian", "wave", "rational_quadratic", "bessel_j"),
  kernel_params = c(),
  pd = TRUE,
  type = c("autocovariance", "autocorrelation"),
  meanX = mean(X),
  custom_kernel = FALSE,
  parallel = FALSE,
  ncores = parallel::detectCores() - 1,
  cl_export = NULL,
  cl = NULL
)

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_ec) 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_ec 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.
parallel	Whether or not the computations should be done in parallel or not. Defaults to FALSE.
ncores	The number of cores to be used in the parallel computations. Defaults to the number cores - 1 (threads if hyperthreading is available), calculated from parallel::detectCores() - 1.
cl_export	A vector of any additional functions or variables to export for parallel computations. This may be required if estimator is not within the package. Defaults to NULL.
cl	An optional cluster object created by parallel::makeCluster. Defaults to NULL, which creates a temporary PSOCK cluster.

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 CovEsts S3 object (list) with the following values

acf

A numeric vector containing the autocovariance/autocorrelation estimates.

lags

A numeric vector containing the lag indices used to compute the estimates on.

est_type

The type of estimate, namely 'autocorrelation' or 'autocovariance', this depends on the type parameter.

est_used

The estimator function used, in this case, 'truncated_est'.

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, params) {
  stopifnot(params[1] > 0, length(x) >= 1)
  return(exp(-((abs(x) / params[1])^params[2])) * (2 * params[1] * gamma(1 + 1/params[2])))
}

truncated_est(X, 1:4, 1:3, 1, 2, 0.1, my_kernel, c(0.25), custom_kernel = TRUE)
## Not run: 
library(parallel)
X <- c(1, 2, 3, 4)
my_cl <- makePSOCKcluster(2)
truncated_est(X, 1:4, 1:3, 1, 2, 0.1, "gaussian", parallel = TRUE, cl = my_cl)
stopCluster(my_cl)

## End(Not run)

CovEsts documentation built on April 19, 2026, 5:06 p.m.