meanreg {lpme}

Local Polynomial Estimators for Nonparametric Mean Regression

Description

This function provides both the DFC (Delaigle, Fan, and Carroll, 2009) and HZ (Huang and Zhou, 2017) local linear estimators for solving the errors-in-variables problem. The local linear estimator in the absence of measurement error is also provided.

Usage

meanreg(Y, W, bw, xgrid=NULL, method="HZ", sig=NULL, error="laplace", FT_fu)

Arguments

Y

an n by 1 response vector.

W

an n by 1 predictor vector.

bw

bandwidth.

xgrid

the grid values to estimate the responses.

method

the method to be used; method="HZ" returns the estimator proposed by Huang and Zhou (2017); method="DFC" returns the estimator proposed by Delaigle, Fan, and Carroll (2009); method="naive" returns the local linear estimator ignoring measurement error.

sig

standard deviation of the measurement error; sig=NULL returns the naive estimators ignoring measurement error.

error

the distribution assumed for the measurement error; error="laplace" is for Laplace distribution; error="normal" is for Gaussian distribution; error="user" is for user-assumed distribution.

FT_fu

a function for the Fourier transform of density of error, which is required only when error="user".

Value

The results include the grid points xgrid for predictor and corresponding fitted responses yhat.

Author(s)

Haiming Zhou and Xianzheng Huang

References

Huang, X. and Zhou, H. (2017). An alternative local polynomial estimator for the error-in-variables problem. Journal of Nonparametric Statistics, 29: 301-325.

Delaigle, A., Fan, J., and Carroll, R. (2009). A design-adaptive local polynomial estimator for the errors-in-variables problem. Journal of the American Statistical Association, 104: 348-359.

See Also

meanregbwSIMEX

Examples

#############################################
## X - True covariates
## W - Observed covariates
## Y - individual response
library(lpme)

## sample size:
n =100;
## Function gofx(x) to estimate
gofx  = function(x){ 1/4*x + x^2/4 }
xgrid = seq(-2, 2, 0.02)

## Generate data
sigma_e  = 0.5;
sigma_x = 1; X = rnorm(n, 0, sigma_x); 
## Sample Y
Y  = gofx(X) + rnorm(n, 0, sigma_e);
##------------------  method Based on X ---------------------------
ghat_X= meanreg(Y, X, 0.1, method="naive", xgrid=xgrid);

## reliability ratio
lambda=0.85;
sigma_u  = sqrt(1/lambda-1)*sigma_x;
print( sigma_x^2/(sigma_x^2 + sigma_u^2) );
W=X+rnorm(n,0,sigma_u);
#W=X+rlaplace(n,0,sigma_u/sqrt(2));
  
##------------------  method Based on W ---------------------------
ghat_W=meanreg(Y, W, 0.1, method="naive", xgrid=xgrid);

##------------------ JASA method -------------------------------------
h = 0.13; 
ghat_JASA=meanreg(Y, W, h, method="DFC", sig=sigma_u, 
                  error="laplace", xgrid=xgrid);

##------------------ Our method -------------------------
ghat_NEW=meanreg(Y, W, h, method="HZ", sig=sigma_u, 
                 error="laplace", xgrid=xgrid);

## plots
plot(xgrid, gofx(xgrid), "l", main="Individual", lwd="2")
lines(xgrid, ghat_NEW$yhat, lty="dashed", col="2",lwd="3")
lines(xgrid, ghat_JASA$yhat, lty="dotted", col="3",lwd="3")
lines(xgrid, ghat_X$yhat, lty="dashed", col="4",lwd="2")
lines(xgrid, ghat_W$yhat, lty="dashed", col="5",lwd="3")