trawl_est: Bias-corrected nonparametric estimation of the trawl function

Description

This function implements bias correction for the nonparametric trawl estimation proposed in Sauri and Veraart (2022).

Usage

trawl_est(data, Delta)

Value

trawlfct Returns the original nonparametric trawl function estimate obtained from nonpar_trawlest.

trawlfct_biascor Returns the bias-corrected trawl function estimate, computed as the original estimate minus the bias correction term \(0.5 \times \Delta \times \hat{a}'(t)\).

Arguments

data: Data to be used in the trawl function estimation.
Delta: Width of the grid on which we observe the data

Details

This function performs bias correction on the nonparametric trawl function estimates obtained using the methodology of Sauri and Veraart (2022). The bias correction is implemented by estimating the derivative of the trawl function and applying the correction \(0.5 \times \Delta \times \hat{a}'(t)\), where \(\hat{a}'(t)\) is the estimated derivative at lag \(t\).

Suppose the data is observed on the grid 0, Delta, 2Delta, ..., (n-1)Delta. Given the path contained in data, the function returns both the original and bias-corrected estimates of the trawl function.

The bias correction is based on the theoretical result that the leading bias term in nonparametric trawl function estimation is proportional to Delta times the derivative of the true trawl function.

Examples

Run this code

# \donttest{
  #Simulation of a Gaussian trawl process with exponential trawl function
  n<-2000
  Delta<-0.5

  trawlfct_par <-0.5
  distr<-"Gauss"
  distr_par<-c(0,1) #mean 0, std 1
  set.seed(233)

  a <- function(x){exp(-trawlfct_par*x)}
  path <- sim_weighted_trawl_gen(n, Delta, a,
                                 distr, distr_par)$path
  #Plot the path
  library(ggplot2)
  df <- data.frame(time = seq(0,n,1), value=path)
  p <- ggplot(df, aes(x=time, y=path))+
    geom_line()+
    xlab("l")+
    ylab("Trawl process")
  p



  result <- trawl_est(path, Delta)

  original_estimate <- result$trawlfct[1:30]
  bias_corrected <- result$trawlfct_biascor[1:30]



  # True exponential values for the same lags
  lags <- 0:(length(original_estimate)-1)
  true_vals <- a(lags * Delta)

  # Plot all three
  plot(lags, original_estimate,
       type = "l", col = "blue", lwd = 2,
       xlab = "Lag index", ylab = "Trawl function estimate",
       main = "Trawl function: Original vs bias-corrected vs true",
       ylim = range(c(original_estimate, bias_corrected, true_vals), na.rm = TRUE))
  lines(lags, bias_corrected,
        col = "red", lwd = 2, lty = 2)
  lines(lags, true_vals,
        col = "black", lwd = 2, lty = 3)

  legend("topright",
         legend = c("Original", "Bias-corrected", "True exponential"),
         col = c("blue", "red", "black"),
         lty = c(1, 2, 3), lwd = 2)
# }