Bias bound approach for conditional expectation estimation — biasBound

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 = "snr",
  noise_floor = "auto",
  envelope_use_Y = TRUE,
  integer_r = TRUE,
  ora_Ar = NULL,
  kernel.fun = "Schennach2004",
  if_approx_kernel = TRUE,
  kernel.resol = 1000
)

Arguments

Y: A numerical vector of sample data.
X: A numerical vector of sample data.
x: Optional. A scalar or range of points where the density is estimated. If NULL, a range is automatically generated.
h: A scalar bandwidth parameter. If NULL, the bandwidth is automatically selected using the method specified in 'h_method'.
h_method: Method for automatic bandwidth selection when h is NULL. Options are "cv" (cross-validation) and "silverman" (Silverman's rule of thumb). Default is "cv".
alpha: Confidence level for intervals. Default is 0.05.
est_Ar: Optional list of estimates for A and r. If NULL, they are computed using get_est_Ar().
resol: Resolution for the estimation range. Default is 100.
xi_lb: Optional. Lower bound for the interval of Fourier Transform frequency xi. Used for determining the range over which A and r is estimated. If NULL, it is automatically determined based on the methods_get_xi.
xi_ub: Optional. Upper bound for the interval of Fourier Transform frequency xi. Similar to xi_lb, it defines the upper range for A and r estimation. If NULL, the upper bound is determined based on the methods_get_xi.
methods_get_xi: A string selecting the frequency-window rule used when xi_lb/xi_ub are NULL: "snr" (default; a signal-to-noise cutoff that selects a valid window at realistic sample sizes), "Schennach" (the data-driven rule of Schennach 2020, Theorem 2), or "Schennach_loose" (the initial, un-refined interval).
noise_floor: Noise-floor form for the Schennach test: "auto" (default), "compact", or "general".
envelope_use_Y: If TRUE (default), fit the regression envelope to the cross-spectrum |phi_YX|; if FALSE, fit it to the marginal spectrum |phi_X|.
integer_r: If TRUE (default), clamp the fitted envelope slope up to r = 2 when it falls below the minimum smoothness assumed by Schennach (2020, Definition 2), i.e. r < 2, and refit A; this keeps the bias-bound integral finite. Slopes >= 2 are left unchanged.
ora_Ar: Optional list of oracle values for A and r (for research/comparison purposes).
kernel.fun: A string specifying the kernel function to be used. Options are "Schennach2004", "sinc", "normal", "epanechnikov".
if_approx_kernel: Logical. If TRUE, uses approximations for the kernel function.
kernel.resol: The resolution for kernel function approximation. See fun_approx.

Value

An object of class bbnp_regression with components:

fitted_values: \(E[Y|X=x]\) estimates (for range estimation)
x: Evaluation points
estimate: Point estimate (for single x)
conf_int: List containing lower, upper bounds and conf_level. Note that the confidence interval can be unbounded (i.e., contain -Inf or Inf) in regions where the estimated marginal density \(\hat f(x)\) is very close to zero, because the estimator is formed as a ratio involving \(1/\hat f(x)\).
bias_bound: List containing b1x, byx, est_A, est_r, est_B, xi_interval
std_error: Standard errors
marginal_density: f(x) estimates
joint_density: f_YX estimates
call: The function call
bandwidth: Bandwidth used
n: Sample size
kernel: Kernel type
data: Original data (X, Y)

Use plot(), summary(), coef(), fitted(), and confint() methods to work with the result.

Examples

# \donttest{
# Example 1: Point estimation at x = 1
X <- gen_sample_data(size = 500, dgp = "2_fold_uniform", seed = 1)
Y <- 2 * X - X^2 + rnorm(length(X), sd = 0.3)
fit <- biasBound_condExpectation(Y = Y, X = X, x = 1, h = 0.09)
print(fit)
#> Bias-Bounded Conditional Expectation Estimation
#> 
#> Call:
#> biasBound_condExpectation(Y = Y, X = X, x = 1, h = 0.09)
#> 
#> Sample size: n = 500
#> Bandwidth:   h = 0.0900 (user-specified) 
#> Kernel:      Schennach2004 
#> 
#> Bias bound parameters:
#>   A = 4.4348, r = 2.0000, B = 0.8231
#>   bias bounds: b1x = 0.0000, byx = 0.1307
#> 
#> Point estimate at x = 1.0000: E[Y|X=x] = 0.9987
#> Confidence level: 95%
#> 
#> Use summary() for detailed statistics
#> Use plot() to visualize results
#> Use fitted() to extract fitted values
fitted(fit)
#> [1] 0.998705

# Example 2: Range estimation with plots
fit2 <- biasBound_condExpectation(Y = Y, X = X, h = NULL, h_method = "cv")
plot(fit2)              # Regression plot

plot(fit2, type = "ft") # Fourier transform plot

summary(fit2)
#> Summary: Bias-Bounded Conditional Expectation Estimation
#> ============================================================
#> 
#> Call:
#> biasBound_condExpectation(Y = Y, X = X, h = NULL, h_method = "cv")
#> 
#> Sample Information:
#>   Sample size (n):  500
#>   Bandwidth (h):    0.2603
#>   Kernel function:  Schennach2004
#> 
#> Bias Bound Parameters:
#>   A (amplitude):    4.4348
#>   r (decay rate):   2.0000
#>   B (Y bound):      0.8231
#>   b1x (bias f(x)):  0.0000
#>   byx (bias f_YX):  0.3780
#>   Xi interval:      [2.3941, 5.3114]
#> 
#> Range Estimation:
#>   Fitted values (E[Y|X]):
#>    min Q1.25% median   mean Q3.75%    max 
#> 0.4481 0.7389 0.8435 0.7993 0.8935 0.9087 
#> 
#>   Marginal density f(x):
#>    min   mean    max 
#> 0.1418 0.5338 0.8132 
#> 
#>   Standard errors:
#>    min   mean    max 
#> 0.0067 0.0114 0.0136 
#> 
# }