DLL {DLL}

DLL

Description

It constructs the Decorrelated Local Linear estimator and estimates its standard error. It further constructs the confidence interval for the derivative of the function of interest.

Usage

DLL(
  X,
  y,
  D.ind,
  d0,
  h = NULL,
  lam.seq = NULL,
  treatment.SAM = FALSE,
  data.swap = FALSE,
  quant.trans = FALSE,
  alpha = 0.05
)

Arguments

X

the covariates matrix, of dimension n \times p

y

the outcome vector, of length n

D.ind

the column index(es) of X, indicating the index(es) of the variable(s) of interest. It can be a scalar or a vector. If vector, then do inference for each index of the sequence.

d0

evaluation points for derivative estimation. It can be scalar or vector.

h

bandwidth, computed by Rule of Thumb from the package “locpol” if not provided.

lam.seq

a sequence of tuning parameters considered in fitting the sparse additive model. Cross validation is used to choose the best one. If not provided(default), the default sequence ranges from 5e-3 to 1 with the length of 100. If provided, the sequence needs to be in a decreasing order for the reason of computation efficiency.

treatment.SAM

Whether a sparse additive model is used for fitting the treatment model? If 'False'(default), Lasso with cross validation is used to fit the treatment model. Default is 'FALSE'

data.swap

Whether data swapping is conducted or not? Default is 'FALSE'

quant.trans

Whether quantile transformation is conducted or not? Default is 'FALSE'

alpha

the significance level. Default is 0.05

Value

est

point estimates of the function derivative

est.se

estimated standard errors of est

CI

list of lower and upper bounds of confidence intervals

d0

evaluation points

bw.save

selected bandwidth at each element of d0

sigma1.sq

estimated variance of the error term in the outcome model

Examples

# evaluation points
d0 = c(-0.5,0.25)
f = function(x) 1.5*sin(x)
f.deriv = function(x) 1.5*cos(x)
g1 = function(x) 2*exp(-x/2)
g2 = function(x) (x-1)^2 - 25/12
g3 = function(x) x - 1/3
g4 = function(x) 0.75*x
g5 = function(x) 0.5*x
# sample size and dimension of X
n = 200
p = 100
# covariance structure of D and X
Cov_Matrix = toeplitz(c(1, 0.7, 0.5, 0.3, seq(0.1, 0, length.out = p-3)))
set.seed(123)
# X represents the (D,X) here
X = MASS::mvrnorm(n,rep(-0.25,p+1),Sigma = Cov_Matrix)
e = rnorm(n,sd=1)
# generating response
y = f(X[,1]) + g1(X[,2]) + g2(X[,3]) + g3(X[,4]) + g4(X[,5]) + g5(X[,6]) + e
### DLL inference
DLL.model = DLL(X=X, y=y, D.ind = 1, d0 = d0)
# true values
f.deriv(d0)
# point estimates
DLL.model$est
# standard errors
DLL.model$est.se
# confidence interval
DLL.model$CI