# DLL

The goal of DLL is to implement the Decorrelated Local Linear estimator proposed in <arxiv:1907.12732>. It constructs the confidence interval for the derivative of the function of interest under the high-dimensional sparse additive model.

## Installation

You can install the released version of DLL from CRAN with:

``install.packages("DLL")``

## Example

This is a basic example which shows you how to solve a common problem:

``````library(DLL)
library(MASS)
# 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 = 500
p = 500

# 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 = 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)
#>  1.316374 1.453369``````

point estimates

``````DLL.model\$est
#>            f1
#> -0.5 1.258581
#> 0.25 1.659544``````

standard errors

``````DLL.model\$est.se
#>             f1
#> -0.5 0.3911074
#> 0.25 0.4301377``````

confidence interval

``````DLL.model\$CI
#> \$f1
#>          lower    upper
#> -0.5 0.4920249 2.025138
#> 0.25 0.8164900 2.502599``````