Density Estimation with rbbnp

This article provides a comprehensive guide to density estimation using the biasBound_density() function.

The biasBound_density() Function

The main function for density estimation takes the following key arguments:

biasBound_density(
  X,                    # Data vector
  x = NULL,             # Evaluation points (auto if NULL)
  h = NULL,             # Bandwidth (auto-selected if NULL)
  h_method = "cv",      # "cv" or "silverman"
  alpha = 0.05,         # Confidence level
  kernel.fun = "Schennach2004",
  xi_lb = NULL,         # Frequency range lower bound
  xi_ub = NULL          # Frequency range upper bound
)

Basic Example

# Generate sample data from 2-fold uniform distribution
X <- gen_sample_data(size = 1000, dgp = "2_fold_uniform", seed = 42)

# Estimate density
fit <- biasBound_density(X, h = 0.08, kernel.fun = "Schennach2004")

# Display results
fit
#> Bias-Bounded Density Estimation
#> 
#> Call:
#> biasBound_density(X = X, h = 0.08, kernel.fun = "Schennach2004")
#> 
#> Sample size: n = 1000
#> Bandwidth:   h = 0.0800 (user-specified) 
#> Kernel:      Schennach2004 
#> 
#> Bias bound parameters:
#>   A = 3.4851, r = 1.9202
#>   bias bound b1x = 0.1209
#> 
#> Evaluation points: 100 (range: [-0.1746, 2.1780])
#> Confidence level: 95%
#> 
#> Use summary() for detailed statistics
#> Use plot() to visualize results

Understanding the Output

The result is an S3 object of class bbnp_density containing:

# Key parameters
coef(fit)
#>        A        r        h 
#> 3.485103 1.920169 0.080000

# Detailed summary
summary(fit)
#> Summary: Bias-Bounded Density Estimation
#> ============================================================
#> 
#> Call:
#> biasBound_density(X = X, h = 0.08, kernel.fun = "Schennach2004")
#> 
#> Sample Information:
#>   Sample size (n):  1000
#>   Bandwidth (h):    0.0800
#>   Kernel function:  Schennach2004
#> 
#> Bias Bound Parameters:
#>   A (amplitude):    3.4851
#>   r (decay rate):   1.9202
#>   b1x (bias bound): 0.1209
#>   Xi interval:      [2.4074, 13.5376]
#> 
#> Range Estimation:
#>   Density estimates:
#>    min Q1.25% median   mean Q3.75%    max 
#> 0.0000 0.1111 0.4273 0.4217 0.7125 0.8798 
#> 
#>   Standard errors:
#>    min   mean    max 
#> 0.0000 0.0341 0.0563

Visualizing Results

Density Plot

plot(fit)

Interpreting the plot:

Element	Description
Blue line	Kernel density estimate $\hat{f}(x)$
Orange band	Bias range: $[\hat{f} - \bar{b}, \hat{f} + \bar{b}]$
Green band	95% CI: $[\hat{f} - \bar{b} - z\hat{\sigma}, \hat{f} + \bar{b} + z\hat{\sigma}]$

Fourier Transform Analysis

The package automatically detects smoothness via Fourier analysis:

plot(fit, type = "ft")

Interpreting the Fourier plot:

Black curve: Empirical Fourier transform $|\hat{\phi}(\xi)|$
Grey lines: Frequency bounds $[\underline{\xi}, \bar{\xi}_n]$
Red line: Estimated envelope $A|\xi|^{-r}$

The slope $r$ indicates smoothness: larger $r$ = smoother function.

Bandwidth Selection

Cross-Validation

fit_cv <- biasBound_density(X, h = NULL, h_method = "cv", kernel.fun = "normal")
cat("CV bandwidth:", coef(fit_cv)["h"], "\n")
#> CV bandwidth: 0.1878156

plot(fit_cv) + ggtitle("Cross-Validation Bandwidth")

Silverman’s Rule of Thumb

fit_silv <- biasBound_density(X, h = NULL, h_method = "silverman", kernel.fun = "normal")
cat("Silverman bandwidth:", coef(fit_silv)["h"], "\n")
#> Silverman bandwidth: 0.09390781

plot(fit_silv) + ggtitle("Silverman's Rule Bandwidth")

Direct Bandwidth Selection

# Select bandwidth without full estimation
h_cv <- select_bandwidth(X, method = "cv", kernel.fun = "normal")
h_silv <- select_bandwidth(X, method = "silverman", kernel.fun = "normal")

cat("CV:", h_cv, "\nSilverman:", h_silv)
#> CV: 0.1878156 
#> Silverman: 0.09390781

Kernel Functions

Available Kernels

Kernel	Order	Best For
Schennach2004	Infinite	High smoothness (recommended)
sinc	Infinite	Any smoothness level
normal	2	Moderate smoothness
epanechnikov	2	Limited smoothness

Comparing Kernels

fit_sch <- biasBound_density(X, kernel.fun = "Schennach2004")
fit_sinc <- biasBound_density(X, kernel.fun = "sinc")
fit_norm <- biasBound_density(X, kernel.fun = "normal")
fit_epan <- biasBound_density(X, kernel.fun = "epanechnikov")

grid.arrange(
  plot(fit_sch) + ggtitle("Schennach2004 (infinite order)"),
  plot(fit_sinc) + ggtitle("Sinc (infinite order)"),
  plot(fit_norm) + ggtitle("Normal (2nd order)"),
  plot(fit_epan) + ggtitle("Epanechnikov (2nd order)"),
  ncol = 2
)

Recommendation: Use "Schennach2004" for most applications as it adapts to any smoothness level.

Custom Frequency Range

You can manually specify the frequency range for Fourier analysis:

# Default (automatic)
fit_auto <- biasBound_density(X, h = 0.08)

# Custom range
fit_custom <- biasBound_density(X, h = 0.08, xi_lb = 2, xi_ub = 8)

grid.arrange(
  plot(fit_auto, type = "ft") + ggtitle("Automatic Frequency Range"),
  plot(fit_custom, type = "ft") + ggtitle("Custom Range [2, 8]"),
  ncol = 1
)

Extracting Results

# Confidence intervals as matrix
ci <- confint(fit)
head(ci, 10)
#>       lower     upper
#>  [1,]     0 0.1209374
#>  [2,]     0 0.1209374
#>  [3,]     0 0.1209374
#>  [4,]     0 0.1209374
#>  [5,]     0 0.1209374
#>  [6,]     0 0.1302624
#>  [7,]     0 0.1464837
#>  [8,]     0 0.1637745
#>  [9,]     0 0.1833616
#> [10,]     0 0.2053695

# Evaluation points
x_points <- fit$x
head(x_points)
#> [1] -0.1746134 -0.1508492 -0.1270850 -0.1033208 -0.0795566 -0.0557924

# Density estimates
f_hat <- fit$f_hat
head(f_hat)
#> NULL

Comparison with Undersmoothing

The bias-bound approach yields narrower CIs than undersmoothing:

h_opt <- select_bandwidth(X, method = "cv", kernel.fun = "sinc")

# Bias-bound with optimal bandwidth
result_opt <- biasBound_density(X, h = h_opt, kernel.fun = "sinc")

# Undersmoothing (half optimal)
result_under <- biasBound_density(X, h = h_opt * 0.5, kernel.fun = "sinc")

grid.arrange(
  plot(result_opt) + ggtitle(paste0("Bias Bound (h = ", round(h_opt, 3), ")")),
  plot(result_under) + ggtitle(paste0("Undersmoothing (h = ", round(h_opt/2, 3), ")")),
  ncol = 1
)

The bias-bound approach provides valid coverage while maintaining efficient estimation.

Xinyu Dai and Susanne M. Schennach

2026-02-14