KoulLrMde: Minimum distance estimation in linear regression model. In KoulMde: Koul's Minimum Distance Estimation in Regression and Image Segmentation Problems

Description

Estimates the regression coefficients in the model Y=Xβ + ε.

Usage

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

Value

betahat - Minimum distance estimator of β.

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``` ```#################### 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 ```