# Nonparametric estimation of the trawl function This function implements the nonparametric trawl estimation proposed in Sauri and Veraart (2022). ```r nonpar_trawlest(data, Delta, lag = 100) ``` ## Arguments - `data`: Data to be used in the trawl function estimation. - `Delta`: Width of the grid on which we observe the data - `lag`: The lag until which the trawl function should be estimated ## Returns ahat Returns the lag-dimensional vector $(\hat a(0), \hat a(\Delta), \ldots, \hat a((lag-1) \Delta)).$ Here, $\hat a(0)$ is estimated based on the realised variance estimator. a0_alt Returns the alternative estimator of a(0) using the same methodology as the one used for t>0. Note that this is not the recommended estimator to use, but can be used for comparison purposes. ## Details Estimation of the trawl function using the methodology proposed in Sauri and Veraart (2022). Suppose the data is observed on the grid 0, Delta, 2Delta, ..., (n-1)Delta. Given the path contained in data, the function returns the lag-dimensional vector $$ (\hat a(0), \hat a(\Delta), \ldots, \hat a((lag-1) \Delta)). $$ In the case when lag=n, the n-1 dimensional vector $$ (\hat a(0), \hat a(\Delta), \ldots, \hat a((n-2) \Delta)) $$ is returned. ## Examples ```r ##Simulate a trawl process ##Determine the sampling grid my_n <- 5000 my_delta <- 0.1 my_t <- my_n*my_delta ###Choose the model parameter #Exponential trawl function: my_lambda <- 2 #Poisson marginal distribution trawl my_v <- 1 #Set the seed set.seed(1726) #Simulate the trawl process Poi_data<-ambit::sim_weighted_trawl(my_n, my_delta, "Exp", my_lambda, "Poi", my_v)$path #Estimate the trawl function my_lag <- 100+1 PoiEx_trawl <- nonpar_trawlest(Poi_data, my_delta, lag=my_lag)$a_hat #Plot the estimated trawl function and superimpose the true one l_seq <- seq(from = 0,to = (my_lag-1), by = 1) esttrawlfct.data <- base::data.frame(l=l_seq[1:31], value=PoiEx_trawl[1:31]) p1 <- ggplot2::ggplot(esttrawlfct.data, ggplot2::aes(x=l,y=value))+ ggplot2::geom_point(size=3)+ ggplot2::geom_function(fun = function(x) acf_Exp(x*my_delta,my_lambda), colour="red", size=1.5)+ ggplot2::xlab("l")+ ggplot2::ylab(latex2exp::TeX("$\\hat{a}(\\cdot)$ for Poisson trawl process")) p1 ```

nonpar_trawlest function