# Minimum distance estimation in the autoregression model of the known order. Estimates the autoressive coefficients in the $X_t = \rho' Z_t + \xi_t $ where $Z_t$ is the vector of $q$ observations at times $t-1,...,t-q$. ```r KoulArMde(X, AR_Order, IntMeasure, TuningConst = 1.345) ``` ## Arguments - `X`: - Vector of `n` observed values. - `AR_Order`: - Order of the autoregression model. - `IntMeasure`: - Symmetric and $\sigma$-finite measure: Lebesgue, Degenerate, and Robust - `TuningConst`: - Used only for Robust measure. ## Returns rhohat - Minimum distance estimator of $\rho$. residual - Residuals after minimum distance estimation. ObjVal - Value of the objective function at minimum distance estimator. ## Examples ```r ##### Generate stationary AR(2) process with 10 observations n <- 10 q <- 2 rho <- c(-0.2, 0.8) ##### Generate true parameters rho = (-0.2, 0.8)' eps <- rnorm(n, 0,1) ##### Generate innovations from N(0,1) X <- rep(0, times=n) for (i in 1:n){ tempCol <- rep(0, times=q) for (j in 1:q){ if(i-j<=0){ tempCol[j] <- 0 }else{ tempCol[j] <- X[i-j] } } X[i] <- t(tempCol)%*% rho + eps[i] } IntMeasure <- "Lebesgue" ##### Define Lebesgue measure MDEResult <- KoulArMde(X, q, IntMeasure, TuningConst=1.345) rhohat <- MDEResult$rhohat ##### Obtain minimum distance estimator resid <- MDEResult$residual ##### Obtain residual objVal <- MDEResult$ObjVal ##### Obtain the value of the objective function IntMeasure <- "Degenerate" ##### Define degenerate measure at 0 MDEResult <- KoulArMde(X, q, IntMeasure, TuningConst=1.345) rhohat <- MDEResult$rhohat ##### Obtain minimum distance estimator resid <- MDEResult$residual ##### Obtain residual objVal <- MDEResult$ObjVal ##### Obtain the value of the objective function IntMeasure <- "Robust" ##### Define "Robust" measure at 0 TuningConst <- 3 ##### Define the tuning constant MDEResult <- KoulArMde(X, q, IntMeasure, TuningConst) resid <- MDEResult$residual ##### Obtain residual objVal <- MDEResult$ObjVal ##### Obtain the value of the objective function ``` ## References [1] Kim, J. (2018). A fast algorithm for the coordinate-wise minimum distance estimation. J. Stat. Comput. Simul., 3: 482 - 497 [2] Kim, J. (2020). Minimum distance estimation in linear regression model with strong mixing errors. Commun. Stat. - Theory Methods., 49(6): 1475 - 1494 [3] Koul, H. L (1985). Minimum distance estimation in linear regression with unknown error distributions. Statist. Probab. Lett., 3: 1-8. [4] Koul, H. L (1986). Minimum distance estimation and goodness-of-fit tests in first-order autoregression. Ann. Statist., 14 1194-1213. [5] Koul, H. L (2002). Weighted empirical process in nonlinear dynamic models. Springer, Berlin, Vol. 166 ## See Also KoulLrMde() and Koul2StageMde()

KoulArMde function