Local Polynomial Estimators for the Errors-in-Variables Problem

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Description

This function provides both the DFC (Delaigle, Fan, and Carroll, 2009) and HZ (Huang and Zhou, 2014+) local polynomial estimators for solving the errors-in-variables problem.

Usage

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lpme(Y, W, bw, method="HZ", sig=NULL, error="laplace", xgridmin=-2, xgridmax=2,
     data=sys.frame(sys.parent()), na.action=na.fail, work.dir=NULL)

Arguments

Y

an n by 1 response vector.

W

an n by 1 predictor vector.

bw

bandwidth.

method

the method to be used; method="HZ" returns the estimator proposed by Huang and Zhou (2015); method="DFC" returns the estimator proposed by Delaigle, Fan, and Carroll (2009); method="naive" returns the local polynomial estimator ignoring measurement error.

sig

standard deviation of the measurement error.

error

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

xgridmin

the minimum value where the estimated responsed is evaluated at; default is -2.

xgridmax

the maximum value where the estimated responsed is evaluated at; defualt is 2.

data

data frame.

na.action

a function that indicates what should happen when the data contain NAs. The default action (na.fail) causes lpme to print an error message and terminate if there are any incomplete observations.

work.dir

working directory.

Details

This function provides both the DFC (Delaigle, Fan, and Carroll, 2009) and HZ (Huang and Zhou, 2014+) local polynomial estimators for solving the errors-in-variables problem.

Value

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

Author(s)

Haiming Zhou <zhouh@email.sc.edu> and Xianzheng Huang <huang@stat.sc.edu>

References

Delaigle, A. and Hall, P. (2008). Using SIMEX for smoothing-parameter choice in errors-in-variables problems. Journal of the American Statistical Association, 103, 280-287.

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.

Huang, X. and Zhou, H. (2014+). An alternative local polynomial estimator for the errors-in-variable problem. Submitted.

Examples

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#############################################
## X - True covariates
## W - Observed covariates
## Y - individual response
rm(list=ls())
library(lpme)

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

## 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= lpme(Y, X, 0.1, method="naive");

## 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=lpme(Y, W, 0.1, method="naive");

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

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

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

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