The Path to the Data Folder

Description

This variable provides the path to the data folder within the package.

Value

The path to the package's internal data folder as a character string.

The Path to the External Data Folder for Non-R Data Files

Description

This variable provides the path to the extdata folder within the package, where non-standard R data files are stored.

Value

The path to the package's external data folder (for non-standard R data files) as a character string.

Define the inverse Fourier transform function of W

Description

Define the inverse Fourier transform function of W

Usage

W_kernel(u, L = 10)

Arguments

Value

A numerical value or vector representing the inverse Fourier transform of the infinite order kernel at the given time or space point(s).

Define the Fourier transform of a infinite kernel proposed in Schennach 2004

Description

Define the Fourier transform of a infinite kernel proposed in Schennach 2004

Usage

W_kernel_ft(xi, xi_lb = 0.5, xi_ub = 1.5)

Arguments

Value

A numerical value or vector representing the Fourier transform of the infinite order kernel at the given frequency/frequencies.

Bias bound approach for conditional expectation estimation

Description

Estimates the density at a given point or across a range, and provides visualization options for density, bias, and confidence intervals.

Usage

biasBound_condExpectation(
  Y,
  X,
  x = NULL,
  h = NULL,
  h_method = "cv",
  alpha = 0.05,
  est_Ar = NULL,
  resol = 100,
  xi_lb = NULL,
  xi_ub = NULL,
  methods_get_xi = "Schennach",
  if_plot_ft = FALSE,
  ora_Ar = NULL,
  if_plot_conditional_mean = TRUE,
  kernel.fun = "Schennach2004",
  if_approx_kernel = TRUE,
  kernel.resol = 1000
)

Arguments

Value

A list containing various outputs including estimated values, plots, and intervals.

Examples

# Example 1: point estimation of conditional expectation of Y on X
biasBound_condExpectation(
 Y = sample_data$Y,
 X = sample_data$X,
 x = 1,
 h = 0.09,
 kernel.fun = "Schennach2004"
)

# Example 2: conditional expectation with automatic bandwidth selection using cross-validation
# biasBound_condExpectation(
# Y = sample_data$Y,
#  X = sample_data$X,
#  h = NULL,
#  h_method = "cv",
#  xi_lb = 1,
#  xi_ub = 12,
#  kernel.fun = "Schennach2004"
# )

# Example 3: conditional expectation with automatic bandwidth selection using Silverman's rule
# biasBound_condExpectation(
# Y = sample_data$Y,
#  X = sample_data$X,
#  h = NULL,
#  h_method = "silverman",
#  methods_get_xi = "Schennach",
#  if_plot_ft = TRUE,
#  kernel.fun = "Schennach2004"
# )

Bias bound approach for density estimation

Description

Estimates the density at a given point or across a range, and provides visualization options for density, bias, and confidence intervals.

Usage

biasBound_density(
  X,
  x = NULL,
  h = NULL,
  h_method = "cv",
  alpha = 0.05,
  resol = 100,
  xi_lb = NULL,
  xi_ub = NULL,
  methods_get_xi = "Schennach",
  if_plot_density = TRUE,
  if_plot_ft = FALSE,
  ora_Ar = NULL,
  kernel.fun = "Schennach2004",
  if_approx_kernel = TRUE,
  kernel.resol = 1000
)

Arguments

Value

A list containing various outputs including estimated values, plots, and intervals.

Examples

# Example 1: Specifying x for point estimation with manually selected xi range
# from a fixed bandwidth
biasBound_density(
  X = sample_data$X,
  x = 1,
  h = 0.09,
  xi_lb = 1,
  xi_ub = 12,
  if_plot_ft = TRUE,
  kernel.fun = "Schennach2004"
)

# Example 2: Density estimation with automatic bandwidth selection using cross-validation
# biasBound_density(
#   X = sample_data$X,
#   h = NULL,
#   h_method = "cv",
#   xi_lb = 1,
#   xi_ub = 12,
#   if_plot_ft = FALSE,
#   kernel.fun = "Schennach2004"
# )

# Example 3: Density estimation with automatic bandwidth selection using Silverman's rule
#  biasBound_density(
#   X = sample_data$X,
#   h = NULL,
#   h_method = "silverman",
#   methods_get_xi = "Schennach",
#   if_plot_ft = TRUE,
#   kernel.fun = "Schennach2004"
# )

Create a configuration object for bias bound estimations

Description

Create a configuration object for bias bound estimations

Usage

create_biasBound_config(
  X,
  Y = NULL,
  h = NULL,
  h_method = "cv",
  use_fft = TRUE,
  alpha = 0.05,
  resol = 100,
  xi_lb = NULL,
  xi_ub = NULL,
  methods_get_xi = "Schennach",
  kernel.fun = "Schennach2004",
  if_approx_kernel = TRUE,
  kernel.resol = 1000
)

Arguments

Value

A configuration object (list) with all parameters

Create kernel functions based on configuration

Description

Create kernel functions based on configuration

Usage

create_kernel_functions(
  kernel.fun = "Schennach2004",
  if_approx_kernel = TRUE,
  kernel.resol = 1000
)

Arguments

Value

A list containing kernel function, its Fourier transform, and the kernel type

Cross-Validation for Bandwidth Selection

Description

Implements least-squares cross-validation for bandwidth selection with any kernel function. Uses the self-convolution approach for accurate estimation of the integral term.

Usage

cv_bandwidth(
  X,
  h_grid = NULL,
  kernel_func,
  kernel_type = "normal",
  grid_size = 512
)

Arguments

Value

A scalar representing the optimal bandwidth that minimizes the cross-validation score.

Examples

# Generate sample data
X <- rnorm(100)
# Get optimal bandwidth using cross-validation with a normal kernel
kernel_functions <- create_kernel_functions("normal")
h_opt <- cv_bandwidth(X, kernel_func = kernel_functions$kernel,
                     kernel_type = kernel_functions$kernel_type)

Epanechnikov Kernel

Description

Epanechnikov Kernel

Usage

epanechnikov_kernel(u)

Arguments

Value

Returns the value of the Epanechnikov kernel function at the given input.

Fourier Transform Epanechnikov Kernel

Description

Fourier Transform Epanechnikov Kernel

Usage

epanechnikov_kernel_ft(xi)

Arguments

Value

Returns the value of the Fourier transform of the Epanechnikov kernel at the given frequency/frequencies.

Approximation Function for Intensive Calculations

Description

This function provides a lookup-based approximation for calculations that are computationally intensive. Once computed, it stores the results in an environment and uses linear interpolation for new data points to speed up subsequent computations.

Usage

fun_approx(u, u_lb = -100, u_ub = 100, resol = 10000, fun = W_kernel)

Arguments

Details

The fun_approx function works by initially creating a lookup table of function values based on the range specified by u_lb and u_ub and the resolution resol. This precomputation only happens once for a given set of parameters (u_lb, u_ub, resol, and fun). Subsequent calls to fun_approx with the same parameters use the lookup table to find the closest precomputed points to the requested u values and then return an interpolated result.

Linear interpolation is used between the two closest precomputed points in the lookup table. This ensures a smooth approximation for values in between sample points.

This function is especially useful for computationally intensive functions where recalculating function values is expensive or time-consuming. By using a combination of precomputation and interpolation, fun_approx provides a balance between accuracy and speed.

Value

A vector of approximated function values corresponding to u.

Generate Sample Data

Description

This function used for generate some sample data for experiment

Usage

gen_sample_data(size, dgp, seed = NULL)

Arguments

Value

A numeric vector of length size. The elements of the vector are generated according to the specified dgp:

normal

Normally distributed values with mean 0 and standard deviation 2.

chisq

Chi-squared distributed values with df = 10.

mixed

Half normally distributed (mean 0, sd = 2) and half chi-squared distributed (df = 10) values.

poly

Values from a polynomial cumulative distribution function on [0,1].

2_fold_uniform

Sum of two uniformly distributed random numbers.

Kernel point estimation

Description

Computes the point estimate using the specified kernel function.

Usage

get_avg_f1x(X, x, h, inf_k)

Arguments

Value

A scalar representing the kernel density estimate at point x.

Kernel point estimation

Description

Computes the point estimate using the specified kernel function.

Usage

get_avg_fyx(Y, X, x, h, inf_k)

Arguments

Value

A scalar representing the kernel density estimate at point x.

Compute Sample Average of Fourier Transform Magnitude

Description

Compute Sample Average of Fourier Transform Magnitude

Usage

get_avg_phi(Y = 1, X, xi)

Arguments

Value

Returns the sample estimation of expected Fourier transform at frequency xi.

Compute log sample average of fourier transform and get mod

Description

Compute log sample average of fourier transform and get mod

Usage

get_avg_phi_log(Y = 1, X, ln_xi)

Arguments

Value

Returns the log sample estimation of expected Fourier transform at frequency xi.

get the conditional variance of Y on X for given x

Description

get the conditional variance of Y on X for given x

Usage

get_conditional_var(X, Y, x, h, kernel_func)

Arguments

Value

Returns a non-negative scalar representing the estimated conditional variance of Y given X at the point x. Returns 0 if the computed variance is negative.

get the estimation of A and r

Description

This function estimates the parameters A and r by optimizing an objective function over a specified range of frequency values and r values.

Usage

get_est_Ar(Y = 1, X, xi_interval, r_stepsize = 150)

Arguments

Details

The function internally defines a range for the natural logarithm of frequency values (ln_xi_range) and a range for the parameter r (r_range). It then defines an optimization function optim_ln_A to minimize the integral of a given function over the ln_xi_range. The actual estimation is done by finding the r and A value that minimizes the the area of the line \ln A - r \ln \xi under the constraint that the line should not go below the Fourier transform curve.

Value

A named vector with elements est_A and est_r representing the estimated values of A and r, respectively.

get the estimation of B

Description

get the estimation of B

Usage

get_est_B(Y)

Arguments

Value

The mean of the absolute values of the elements in Y, representing the estimated value of B.

Estimation of bias b1x

Description

Computes the bias estimate for given parameters.

Usage

get_est_b1x(X, h, est_Ar, inf_k_ft, ...)

Arguments

Value

A scalar representing the bias b1x estimate.

Estimation of bias byx

Description

Estimation of bias byx

Usage

get_est_byx(Y, X, ...)

Arguments

Value

A scalar representing the bias byx estimate.

get the estimation of Vy

Description

get the estimation of Vy

Usage

get_est_vy(Y)

Arguments

Estimation of sigma

Description

Computes the sigma estimate for given parameters.

Usage

get_sigma(X, x, h, inf_k)

Arguments

Value

A scalar representing the sigma estimate at point x. Returns 0 if the density estimate is negative.

Estimation of sigma_yx

Description

Estimation of sigma_yx

Usage

get_sigma_yx(Y, X, x, h, inf_k)

Arguments

Value

Returns a scalar representing the estimated value of sigma_yx at the point x. Returns 0 if either fyx or conditional variance is negative.

get xi interval

Description

get xi interval

Usage

get_xi_interval(Y = 1, X, methods = "Schennach")

Arguments

Details

The "Schennach" method computes the xi interval by performing a test based on the Schennach's theorem, adjusting the upper bound xi_ub if the test condition is met. The "Schennach_loose" method provides a looser calculation of the xi interval without performing the Schennach's test.

Value

A list containing the lower (xi_lb) and upper (xi_ub) bounds of the xi interval.

Kernel Regression function

Description

Kernel Regression function

Usage

kernel_reg(X, Y, x, h, kernel_func)

Arguments

Value

Returns a scalar representing the estimated value of the regression function at the point x.

Normal Kernel Function

Description

Normal Kernel Function

Usage

normal_kernel(u)

Arguments

Value

Returns the value of the Normal kernel function at the given input.

Fourier Transform of Normal Kernel

Description

Fourier Transform of Normal Kernel

Usage

normal_kernel_ft(xi)

Arguments

Value

Returns the value of the Fourier transform of the Normal kernel at the given frequency/frequencies.

Plot the Fourier Transform

Description

Plot the Fourier Transform of the

Usage

plot_ft(X, xi_interval, ft_plot.resol = 500)

Arguments

Details

C = 1, the parameter in O(1/n^{0.25}), see more details in in Schennach (2020).

Value

A ggplot object representing the plot of the Fourier transform.

Examples

plot_ft(
  sample_data$X,
  xi_interval = list(xi_lb = 1, xi_ub = 50),
  ft_plot.resol = 1000
)

Generate n samples from the distribution

Description

Generate n samples from the distribution

Usage

rpoly01(n, k = 5)

Arguments

Value

A vector of n samples from the specified polynomial distribution.

CDF: f(x) = (x-1)^k + 1

Sample Data

Description

Sample Data

Usage

sample_data

Format

A data frame with 1000 rows and 2 variables:

X

Numeric vector, generated from 2 fold uniform distribution.

Y

Numeric vector, Y = -X^2 + 3*X + rnorm(1000)*X.

Select Optimal Bandwidth

Description

Selects an optimal bandwidth using the specified method.

Usage

select_bandwidth(
  X,
  Y = NULL,
  method = "cv",
  kernel.fun = "normal",
  if_approx_kernel = TRUE,
  kernel.resol = 1000
)

Arguments

Value

A scalar representing the optimal bandwidth.

Examples

# Generate sample data
X <- rnorm(100)
# Get optimal bandwidth using cross-validation with normal kernel
h_opt <- select_bandwidth(X, method = "cv", kernel.fun = "normal")
# Get optimal bandwidth using Silverman's rule with Schennach kernel
h_opt <- select_bandwidth(X, method = "silverman", kernel.fun = "Schennach2004")

Silverman's Rule of Thumb for Bandwidth Selection

Description

Implements Silverman's rule of thumb for selecting an optimal bandwidth in kernel density estimation.

Usage

silverman_bandwidth(X, kernel_type = "normal")

Arguments

Value

A scalar representing the optimal bandwidth.

Examples

# Generate sample data
X <- rnorm(100)
# Get optimal bandwidth using Silverman's rule
h_opt <- silverman_bandwidth(X, kernel_type = "normal")

Infinite Kernel Function

Description

Infinite Kernel Function

Usage

sinc(u)

Arguments

Value

The value of the sinc function at each point in u.

Define the closed form FT of the infinite order kernel sin(x)/(pi*x)

Description

Define the closed form FT of the infinite order kernel sin(x)/(pi*x)

Usage

sinc_ft(x)

Arguments

Value

The value of the Fourier Transform of the sinc function at each point in x.

True density of 2-fold uniform distribution

Description

True density of 2-fold uniform distribution

Usage

true_density_2fold(x)

Arguments

Value

The value of the true density of the 2-fold uniform distribution at each point in x.