# Numerical properties of statistical estimators operating on the linear fractional stable motion. The function is useful, for instance, when one needs to compute standard deviation of $\widehat \alpha_{high}$ estimator given a fixed set of parameters. ```r MCestimLFSM(Nmc, s, m, M, alpha, H, sigma, fr, Inference, ...) ``` ## Arguments - `Nmc`: Number of Monte Carlo repetitions - `s`: sequence of path lengths - `m`: discretization. A number of points between two nearby motion points - `M`: truncation parameter. A number of points at which the integral representing the definition of lfsm is calculated. So, after M points back we consider the rest of the integral to be 0. - `alpha`: self-similarity parameter of alpha stable random motion. - `H`: Hurst parameter - `sigma`: Scale parameter of lfsm - `fr`: frequency. Either "H" or "L" - `Inference`: statistical function to apply to sample paths - `...`: parameters to pass to Inference ## Returns It returns a list containing the following components: - **data**: a data frame, values of the estimates depending on path length s - **data_nor**: a data frame, normalized values of the estimates depending on path length s - **means, biases, sds**: data frames: means, biases and standard deviations of the estimators depending on s - **Inference**: a function used to obtain estimates - **alpha, H, sigma**: the parameters for which MCestimLFSM performs path generation - **freq**: frequency, either 'L' for low- or 'H' for high frequency ## Details MCestimLFSM performs Monte-Carlo experiments to compute parameters according to procedure Inference. More specifically, for each element of s it generates Nmc lfsm sample paths with length equal to s[i], performs the statistical inference on each, obtaining the estimates, and then returns their different statistics. It is vital that the estimator returns a list of named parameters (one or several of 'sigma', 'alpha' and 'H'). MCestimLFSM uses the names to lookup the true parameter value and compute its bias. For sample path generation MCestimLFSM uses a light-weight version of path, path_fast. In order to be applied, function Inference must accept argument 'path' as a sample path. ## Examples ```r #### Set of global parameters #### m<-25; M<-60 p<-.4; p_prime<-.2; k<-2 t1<-1; t2<-2 NmonteC<-5e1 S<-c(1e2,3e2) alpha<-1.8; H<-0.8; sigma<-0.3 # How to plot empirical density theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC, m=m,M=M,alpha=alpha,H=H, sigma=sigma,ContinEstim, t1=t1,t2=t2,p=p,k=k) l_plot<-Plot_dens(par_vec=c('sigma','alpha','H'), MC_data=theor_3_1_H_clt, Nnorm=1e7) # For MCestimLFSM() it is vital that the estimator returns a list of named parameters H_hat_f <- function(p,k,path) {hh<-H_hat(p,k,path); list(H=hh)} theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC, m=m,M=M,alpha=alpha,H=H, sigma=sigma,H_hat_f, p=p,k=k) # The estimator can return one, two or three of the parameters. est_1 <- function(path) list(H=1) theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC, m=m,M=M,alpha=alpha,H=H, sigma=sigma,est_1) est_2 <- function(path) list(H=0.8, alpha=1.5) theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC, m=m,M=M,alpha=alpha,H=H, sigma=sigma,est_2) est_3 <- function(path) list(sigma=5, H=0.8, alpha=1.5) theor_3_1_H_clt<-MCestimLFSM(s=S,fr='H',Nmc=NmonteC, m=m,M=M,alpha=alpha,H=H, sigma=sigma,est_3) ```

MCestimLFSM function