Package eel. September 1, 2015

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1 Type Package Title Etended Empirical Likelihood Version 1.1 Date Author Fan Wu and Yu Zhang Maintainer Yu Zhang Package eel September 1, 2015 Compute the etended empirical log likelihood ratio (Tsao & Wu, 2014) for the mean and parameters defined by estimating equations. Depends emplik, rootsolve License GPL (>= 2) NeedsCompilation no Repository CRAN Date/Publication :31:11 R topics documented: eel-package EEL EEL_est EMLogLR ep_factor ep_factor_est prime_image prime_image_est print.eel summary.eel Inde 14 1

2 2 eel-package eel-package Etended Empirical Likelihood Compute the etended empirical log likelihood ratio (Tsao & Wu, 2014) for the mean and parameters defined by estimating equations. Details Inde of help topics: EEL EEL_est EMLogLR eel-package ep_factor ep_factor_est prime_image prime_image_est print.eel summary.eel Etended empirical log likelihood ratio for the mean Etended empirical log likelihood ratio for parameters defined by estimating equaitons Original empirical log likelihood ratio Etended Empirical Likelihood Calculating epansion factor for EEL for the mean Calculating epansion factor for EEL for parameters defined by estimating equations Calculating prime-image based on similarity mapping for the mean Calculating prime-image based on similarity mapping for parameters defined by estimating equations Printing EEL objects Summarizing EEL objects The etended empirical log likelihood ratio for the mean is computed by calling the function EEL(), and that for the parameter defined estimating equations is computed by calling the function EEL_est(). This package requires pre-installation of two packages "emplik" and "rootsolve". These are needed for computing the prime image of a point as well as the final etended empirical log likelihood ratio value as described in Tsao and Wu (2013, 2014). Only the first-order EEL discussed Tsao and Wu (2013, 2014) is included in this package. Fan Wu and Yu Zhang Maintainer: Yu Zhang <yuz@uvic.ca> References Tsao, M. (2013). Etending the empirical likelihood by domain epansion. The Canadian Journal of Statistics, 41 (2),

3 EEL 3 Tsao, M., & Wu, F. (2013). Empirical likelihood on the full parameter space. Annals of Statistics, 0 (00), doi: /13-AOS1143 Tsao, M., & Wu, F. (2014). Etended empirical likelihood for estimating equations.biometrika, 1-8. doi: /biomet/asu014 EMLogLR, EEL, EEL_est, ep_factor, prime_image, prime_image_est, ep_factor_est, # EXAMPLE: computing the EEL for the mean of a bivariate random variable # Generating a sample of n=40 bivariate observations. # For this eample, we do this through a univariate normal random number generator. uninorm<- rnorm(40*2,5,1) multnorm<-matri(uninorm,ncol=2) # To calculate the EEL for a point =c(5,2), use EEL(=multnorm,=c(5,2)) # an eample to use the EEL_est in the case of estimating equation # generate regression dataset # random variable dm2<-runif(100,min=0,ma=100) dm<-matri(0,100,2) dm[,1]=1 dm[,2]=dm2 # set the initial beta value beta0<-c(1,2) # generate random errors and calculate the response variable errdata<-rnorm(100,0,1) ydata<-dm%*%beta0+errdata # calculate the maimum empirical likelihood estimates beta_lse<-solve(t(dm)%*%dm)%*%(t(dm)%*%ydata) num=eel_est(=dm,=c(1,2),_tilda=beta_lse, "g<-matri(0,nrow=100,ncol=2) for(i in 1:2){g[,i]<-dm[,i]*(ydata-dm%*%as.matri())} g") summary(num) EEL Etended empirical log likelihood ratio for the mean

4 4 EEL Calculate the etended empirical log likelihood ratio for a multi-dimensional mean EEL(, ) ## Default S3 method: EEL(,) Data matri. The value at which the etended empirical likelihood is to be evaluated. An object of class EEL, basically a list including elements the value at which the etended empirical likelihood is to be evaluated; prime the prime-image inside the conve hull for the point ; estimating equation the estimating equation here is "-"; epansion oel_log eel_log the value of the epansion factor gamma; the original empirical log likelihood ratio value; the etended empirical log likelihood ratio value. Yu Zhang & Fan Wu EMLogLR, ep_factor, prime_image, print.eel, summary.eel, EEL_est # EXAMPLE: computing the EEL for the mean of a bivariate random variable # Generating a sample of n=40 bivariate observations. # For this eample, we do this through a univariate normal random number generator. uninorm<- rnorm(40*2,5,1) multnorm<-matri(uninorm,ncol=2) # To calculate the EEL for a point =c(5,3), use EEL(=multnorm,=c(5,3))

5 EEL_est 5 EEL_est Etended empirical log likelihood ratio for parameters defined by estimating equaitons Calculate the etended empirical log likelihood ratio for parameters defined by estimating equations EEL_est(,, _tilda, equation) ## Default S3 method: EEL_est(,, _tilda, equation) _tilda equation Data matri. at which the EEL for the parameters defined by estimating equations will be evaluated. The maimum empirical likelihood estimator of the unknown parameter. The estimating equation, must be put inside quotation marks and has to be a function of. An object of class EEL, basically a list including elements prime value at which the EEL for the parameters defined by estimating equations will be evaluated; the prime-image inside the conve hull for the point ; estimating equation the estimating equation; epansion oel_log eel_log the value of the epansion factor gamma; the original empirical log likelihood ratio value; the etended empirical log likelihood ratio value. Yu Zhang EMLogLR,ep_factor_est,prime_image_est,print.EEL,summary.EEL,eel-package, EEL

6 6 EMLogLR # EXAMPLE: computing the EEL for the mean of a bivariate random variable # Generating a sample of n=40 bivariate observations. # For this eample, we do this through a univariate normal random number generator. uninorm<- rnorm(40*2,5,1) multnorm<-matri(uninorm,ncol=2) # To calculate the EEL for a point =c(5,3), use _tilda=colmeans(multnorm-as.vector(c(5,3))) EEL_est(=multnorm,=c(5,3),_tilda, "-") EMLogLR Original empirical log likelihood ratio The function etracts the empirical log likelihood ratio value produced by el.test() function from package "emplik". EMLogLR(, mean) mean Data matri. The mean value to be evaluated. The function will return a numerical value representing the original empirical log likelihood ratio. Yu Zhang & Fan Wu EEL =rnorm(50,0,1) # find the empirical log likelihood ratio at point 0 EMLogLR(,0)

7 ep_factor 7 ep_factor Calculating epansion factor for EEL for the mean The function calculates the first order epansion factor of EEL for the mean. ep_factor(, ) ## Default S3 method: ep_factor(,) Data matri of interest. The value to be evaluated. Details The first order epansion factor for calculating EEL is defined as γ(n, l(θ)) = 1 + l(θ) 2n. The function will return a numerical value representing the value of the epansion factor calculated. Fan Wu & Yu Zhang References Tsao, M. (2013). Etending the empirical likelihood by domain epansion. The Canadian Journal of Statistics, 41 (2), EEL, prime_image, ep_factor_est =rnorm(400,0,3) ep_factor(,0)

8 8 ep_factor_est ep_factor_est Calculating epansion factor for EEL for parameters defined by estimating equations The function calculates the first order epansion factor of EEL for the parameters defined by estimating equations. ep_factor_est(,, equation) ## Default S3 method: ep_factor_est(,,equation) equation Data Matri. The value to be evaluated. The estimating equation by which the parameters are defined. Details The first order epansion factor for calculating EEL is defined as γ(n, l(θ)) = 1 + l(θ) 2n. The estimating equation input has to be a function of. The function will return a numerical value representing the value of the epansion factor calculated. Yu Zhang and Fan Wu References Tsao, M. (2013). Etending the empirical likelihood by domain epansion. The Canadian Journal of Statistics, 41 (2), eel-package, ep_factor, EEL_est, prime_image_est

9 prime_image 9 #generate data with theoretical mean 2 and standard deviation 1 =rnorm(100,2,1) ep_factor_est(=, =2, equation="-") prime_image Calculating prime-image based on similarity mapping for the mean The function calculates the prime image of the given point based on the similarity mapping defined in the EEL calculaiton. prime_image(_tilda,, ) ## Default S3 method: prime_image(_tilda,,) _tilda Sample mean or the maimum empirical likelihood estimate of the estimating equaitons. to be evaluated. Data matri. Details The prime image was found by solving the equation f(ζ ) = ζ. See the reference paper for details. The function returns a vector, with the same length as the mean, representing the prime image of the point, based on the similarity mapping defind in EEL calculation. Fan Wu and Yu Zhang References Tsao, M. (2013). Etending the empirical likelihood by domain epansion. The Canadian Journal of Statistics, 41 (2), Tsao, M., & Wu, F. (2013). Empirical likelihood on the full parameter space. Annals of Statistics, 0 (00), doi: /13-AOS1143

10 10 prime_image_est EEL,ep_factor,eel-package,prime_image_est =rnorm(50,6,2) # find the prime image of true mean prime_image(_tilda=mean(),=6,=) prime_image_est Calculating prime-image based on similarity mapping for parameters defined by estimating equations The function calculates the prime image of a given point based on the similarity mapping defined in the EEL calculaiton. prime_image_est(_tilda,,, equation) ## Default S3 method: prime_image_est(_tilda,,, equation) _tilda equation The maimum empirical likelihood estimates for parameters defined by estimating equations. The value to be evaluated. Data Matri. The estimating equation by which the parameters are defined, must be put as a function of. Details The prime image was found by solving the equation f(ζ ) = ζ. See the reference paper for details. The function returns a vector, with the same dimension as the mean, representing the prime image of the point, based on the similarity mapping defind in EEL calculation.

11 print.eel 11 Yu Zhang References Tsao, M. (2013). Etending the empirical likelihood by domain epansion. The Canadian Journal of Statistics, 41 (2), Tsao, M., & Wu, F. (2013). Empirical likelihood on the full parameter space. Annals of Statistics, 0 (00), doi: /13-AOS1143 EEL,ep_factor_est,EEL_est,prime_image =rnorm(50,6,2) # find the prime image of true mean prime_image_est(_tilda=mean(),=6,=,"-") print.eel Printing EEL objects The function prints the etended empirical log likelihood ratio. ## S3 method for class EEL print(,...) EEL object.... Further arguments passed to or from other methods. The function prints the etended empirical log likelihood ratio value of the EEL object. Yu Zhang

12 12 summary.eel EEL # EXAMPLE: computing the EEL for the mean of a bivariate random variable # Generating a sample of n=40 bivariate observations. # For this eample, we do this through a univariate normal random number generator. uninorm<- rnorm(40*2,5,1) multnorm<-matri(uninorm,ncol=2) # To calculate the EEL for a point =c(5,3), use obj=eel(=multnorm,=c(5,3)) print(obj) summary.eel Summarizing EEL objects The function prints a summary of EEL objects. ## S3 method for class EEL summary(object,...) object An EEL object.... Additional arguments affecting the summary produced. The function prints a summary of EEL objects including the point at which the EEL is to be evaluated; estimating equation the estimating equation at which the paremeters are defined; log oel ratio empirical log likelihood ratio for the point ; prime image prime-image of defined by the similarity mapping in EEL calculation; epansion factor value of the epansion factor γ(, θ) log eel ratio ; value of the etended empirical log likelihood ratio.

13 summary.eel 13 Yu Zhang EEL,print.EEL, EEL_est # EXAMPLE: computing the EEL for the mean of a bivariate random variable # Generating a sample of n=40 bivariate observations. # For this eample, we do this through a univariate normal random number generator. uninorm<- rnorm(40*2,5,1) multnorm<-matri(uninorm,ncol=2) # To calculate the EEL for a point =c(5,3), use obj=eel(=multnorm,=c(5,3)) summary(obj)

14 Inde Topic EEL EEL_est, 5 ep_factor_est, 8 prime_image_est, 10 print.eel, 11 summary.eel, 12 Topic Empirical EEL, 3 eel-package, 2 Topic Estimating EEL_est, 5 Topic Likelihood eel-package, 2 Topic epansion ep_factor, 7 Topic image prime_image, 9 prime_image_est, 10 Topic mapping ep_factor, 7 prime_image, 9 Topic nonparametric EEL, 3 Topic printing print.eel, 11 Topic summary summary.eel, 12 EEL, 3, 3, 5 7, eel-package, 2 EEL_est, 3, 4, 5, 8, 11, 13 EMLogLR, 3 5, 6 ep_factor, 3, 4, 7, 8, 10 ep_factor_est, 3, 5, 7, 8, 11 prime_image, 3, 4, 7, 9, 11 prime_image_est, 3, 5, 8, 10, 10 print.eel, 4, 5, 11, 13 summary.eel, 4, 5, 12 14