# Minimum distance estimation in linear regression model. Estimates the regression coefficients in the model $Y=X\beta + \epsilon$. ```r KoulLrMde(Y, X, D, b0, IntMeasure, TuningConst = 1.345) ``` ## Arguments - `Y`: - Vector of response variables in linear regression model. - `X`: - Design matrix of explanatory variables in linear regression model. - `D`: - Weight Matrix. Dimension of D should match that of X. Default value is XA where A=(X'X)^(-1/2). - `b0`: - Initial value for beta. - `IntMeasure`: - Symmetric and $\sigma$-finite measure: Lebesgue, Degenerate, and Robust - `TuningConst`: - Used only for Robust measure. ## Returns betahat - Minimum distance estimator of $\beta$. residual - Residuals after minimum distance estimation. ObjVal - Value of the objective function at minimum distance estimator. ## Examples ```r #################### n <- 10 p <- 3 X <- matrix(runif(n*p, 0,50), nrow=n, ncol=p) #### Generate n-by-p design matrix X beta <- c(-2, 0.3, 1.5) #### Generate true beta = (-2, 0.3, 1.5)' eps <- rnorm(n, 0,1) #### Generate errors from N(0,1) Y <- X%*%beta + eps D <- "default" #### Use the default weight matrix b0 <- solve(t(X)%*%X)%*%(t(X)%*%Y) #### Set initial value for beta IntMeasure <- "Lebesgue" ##### Define Lebesgue measure MDEResult <- KoulLrMde(Y,X,D, b0, IntMeasure, TuningConst=1.345) betahat <- MDEResult$betahat ##### Obtain minimum distance estimator resid <- MDEResult$residual ##### Obtain residual objVal <- MDEResult$ObjVal ##### Obtain the value of the objective function IntMeasure <- "Degenerate" ##### Define degenerate measure at 0 MDEResult <- KoulLrMde(Y,X,D, b0, IntMeasure, TuningConst=1.345) betahat <- MDEResult$betahat ##### Obtain minimum distance estimator resid <- MDEResult$residual ##### Obtain residual objVal <- MDEResult$ObjVal ##### Obtain the value of the objective function IntMeasure <- "Robust" ##### Define "Robust" measure TuningConst <- 3 ##### Define the tuning constant MDEResult <- KoulLrMde(Y,X,D, b0, IntMeasure, TuningConst) betahat <- MDEResult$betahat ##### Obtain minimum distance estimator resid <- MDEResult$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 KoulArMde() and Koul2StageMde()

KoulLrMde function