R packge for the inference under the density ratio model with semicontinous data

R package ZeroDRM

This package is developed based on Yuan et al. (2021a) and Yuan et al. (2021b). This package provides the estimators and confidence intervals for the functionals and Gini indices of two semicontibous populations under the density ratio models (DRMs).

Table of Contents

Installation
Functions
Usage
References

Installation

Install this package from Github with

# If the package "devtool" has not been installed, install it fisrt with 
#install.packages("devtools")
library(devtools)
devtools::install_github("MengYuan-UW/ZeroDRM")
library(ZeroDRM)

Functions

This package contains the following functions:

• The Density Ratio Model (DRM): a function to fit the DRM.
• The inference on the functional (ZeroFunc): a function to estimate the functionals and construct the confidence interval under the DRM.
• The inference on the mean ratio (MeanRatio): a function to estimate the mean ratio of two populations and construct the confidence interval under the DRM.
• The inference on the Gini indices (ZeroGini): a function to estimate the Gini indices and construct the confidence intervals under the DRM.
• The inference on the difference of two Gini indices (DiffGini): a function to estimate the difference of Gini indices of two populations and construct the confidence intervals under the DRM.

Usage

We provide two examples.

• Example 1: inference on population means and mean ratio

#Data generation function
DataGen = function(nu,para,size){
  sample = rbinom(size,1,nu)
  #since nu is the probability that x = 0
  n0 = sum(sample==1)
  n1 = sum(sample==0)
  data = c(rep(0,n0),
           rlnorm(n1,meanlog = para[1],sdlog = para[2]))
  data
}
# zero proportion
nu = c(0.3,0.5)
# mean for lognormal distribution in log scale
a = c(1/3,2/3)

set.seed(123)
x = DataGen(nu[1],c(a[1],1),50)
y = DataGen(nu[2],c(a[2],1),50)

# Now move to the inference part

# define the arguments in the 'ZeroFunc'
# (1) define the basis function q(x) = log(x)
basis = function(x){log(x)}
# (2) u function used to define the psi = c(mu_0, mu_1), where mu_0 and mu_1 are population means for x and y, respectively
Ufun = function(x,nu,theta){
  u1 = (1-nu[1])*x
  u2 = (1-nu[2])*x*exp(theta%*%c(1,log(x)))
  c(u1,u2)
}
# (3) the first derative of u functin with respect to parametres nu
Pnu = function(x,nu,theta){
  pu1 = c(-x,0)
  pu2 = c(0,-x*exp(theta%*%c(1,log(x))))
  rbind(pu1,pu2)
}
# (4)the first derative of u functin with respect to parametres theta
Ptheta = function(x,nu,theta){
  pu1 = c(0,0)
  pu2 = (1-nu[2])*x*c(exp(theta%*%c(1,log(x))))*c(1,log(x))
  rbind(pu1,pu2)
}

# get the estimate of psi and asymptotic variance-covaraince matrix
ZeroFunc(x,y,basis,method="optimal",Ufun,Pnu,Ptheta)

# otain the estimate of the mean ratio mu_1/mu_0 and asymptotic standard deviation
# construct the 95% confidence intervals using difference methods under the density ratio model
MeanRatio(x,y,basis,CItype = "None")
MeanRatio(x,y,basis,CItype = "NA-DRM")
MeanRatio(x,y,basis,CItype = "log-DRM")

• Example 2: inference on the Gini indices of two populations and their difference

# get the data 
data(Ilocos,package = "ineq")
x = Ilocos$income[Ilocos$urban == "urban"&Ilocos$pro == "Pangasinan"]
y = Ilocos$income[Ilocos$urban == "rural"&Ilocos$pro == "Pangasinan"]

# define the basis function q(x)
basis = function(x){log(x)}

# get the estimate of Gini indices and 95% Wald-type confidence intervals
ZeroGini(x,y,basis,CItype = "NA-DRM")

# get the estimate of the difference of Gini indices and 95% Wald-type confidence intervals
DiffGini(x,y,basis,CItype = "NA-DRM")

References

Yuan, M., Wang, C., Lin, B., and Li, P. (2021a). Semiparametric inference on general functionals of two semicontinuous populations. Annals of the Institute of Statistical Mathematics. In press.

Yuan, M., Li, P., and Wu, C. (2021b). Semiparametric inference on Gini indices of two semicontinuous populations under density ratio models. arXiv:2106.02741.