# KoulArMde: Minimum distance estimation in the autoregression model of... In KoulMde: Koul's Minimum Distance Estimation in Regression and Image Segmentation Problems

## Description

Estimates the autoressive coefficients in the X_t = ρ' Z_t + ξ_t where Z_t is the vector of q observations at times t-1,...,t-q.

## Usage

 `1` ```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 σ-finite measure: Lebesgue, Degenerate, and Robust `TuningConst` - Used only for Robust measure.

## Value

rhohat - Minimum distance estimator of ρ.

residual - Residuals after minimum distance estimation.

ObjVal - Value of the objective function at minimum distance estimator.

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39``` ```##### 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 ```