# Estimate distribution of the fACF under the iid. hypothesis using Imhof's method Estimate the distribution of the autocorrelation function under the hypothesis of strong functional white noise. This function uses Imhof's method to estimate the distribution. ```r estimate_iid_distr_Imhof(Y, v, autocovSurface, matindex, figure = FALSE, ...) ``` ## Arguments - `Y`: Matrix containing the discretized values of the functional time series. The dimension of the matrix is $(n x m)$, where $n$ is the number of curves and $m$ is the number of points observed in each curve. - `v`: Discretization points of the curves, by default `seq(from = 0, to = 1, length.out = 100)`. - `autocovSurface`: An $(m x m)$ matrix with the discretized values of the autocovariance operator $\hat{C}_{0}$, obtained by calling the function `obtain_autocovariance`. The value $m$ indicates the number of points observed in each curve. - `matindex`: A vector containing the L2 norm of the autocovariance function. It can be obtained by calling function `obtain_suface_L2_norm`. - `figure`: Logical. If `TRUE`, plots the estimated distribution. - `...`: Further arguments passed to the `plot` function. ## Returns Return a list with: * `ex`: Knots where the distribution has been estimated * `ef`: Discretized values of the estimated distribution. ## Examples ```r # Example 1 N <- 100 v <- seq(from = 0, to = 1, length.out = 10) sig <- 2 Y <- simulate_iid_brownian_bridge(N, v, sig) nlags <- 1 autocovSurface <- obtain_autocovariance(Y,nlags) matindex <- obtain_suface_L2_norm (v,autocovSurface) # Remove lag 0 matindex <- matindex[-1] Imhof_dist <- estimate_iid_distr_Imhof(Y,v,autocovSurface,matindex) plot(Imhof_dist$ex,Imhof_dist$ef,type = "l",main = "ecdf obtained by Imhof's method") grid() # Example 2 N <- 400 v <- seq(from = 0, to = 1, length.out = 50) sig <- 2 Y <- simulate_iid_brownian_bridge(N, v, sig) autocovSurface <- obtain_autocovariance(Y,nlags) matindex <- obtain_suface_L2_norm (v,autocovSurface) # Remove lag 0 matindex <- matindex[-1] Imhof_dist <- estimate_iid_distr_Imhof(Y,v,autocovSurface,matindex) plot(Imhof_dist$ex,Imhof_dist$ef,type = "l",main = "ecdf obtained by Imhof's method") grid() ```

estimate_iid_distr_Imhof function