cvmgof-package: cvmgof
In cvmgof: Cramer-von Mises Goodness-of-Fit Tests
Description
Details
Author(s)
References
See Also
Examples
Description
It implements three goodness-of-fit tests to test the validity of the regression function in the regression model. Two of them (Alcala et al. '99 and Van Keilegom et al. '12) are “directional” in that they detect departures from mainly the regression function assumption of the model or “global” (Ducharme and Ferrigno '12) with the conditional distribution function. The establishment of such statistical tests requires nonparametric estimators and the use of wild bootstrap methods for the simulations.
Details
Package: cvmgof
Type: Package
Version: 1.0.3
Date: 2021-01-11
License: Cecill
Author(s)
Romain Azais, Sandie Ferrigno and Marie-Jose Martinez
References
J. T. Alcala, J. A. Cristobal, and W. Gonzalez Manteiga. Goodness-of-fit test for linear
models based on local polynomials. Statistics & Probability Letters, 42(1), 39:46, 1999.
G. R. Ducharme and S. Ferrigno. An omnibus test of goodness-of-fit for conditional distributions
with applications to regression models. Journal of Statistical Planning and
Inference, 142, 2748:2761, 2012.
I. Van Keilegom, W. Gonzalez Manteiga, and C. Sanchez Sellero. Goodness-of-fit tests in
parametric regression based on the estimation of the error distribution. Test, 17, 401:415,
2008.
R. Azais, S. Ferrigno and M-J Martinez. cvmgof: An R package for Cramer-von Mises goodness-of-fit tests in regression models. Submitted. January 2021.hal-03101612
See Also
Examples
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# require(lattice) # Only for plotting conditional CDF
########################################################################
# Simulation
set.seed(1)
# The following example tests are computed from only 25 data points
# The seed is fixed to avoid NA in estimates
# Data simulation
n = 25 # Dataset size
data.X = runif(n,min=0,max=5) # X
data.Y = 0.2*data.X^2-data.X+2+rnorm(n,mean=0,sd=0.3) # Y
# plot(data.X,data.Y,xlab='X',ylab='Y',pch='+')
########################################################################
# Estimation of the link function (uncomment the following code block)
# bandwidth = 0.75 # Here, the bandwidth is arbitrarily fixed
#
# xgrid = seq(0,5,by=0.1)
# ygrid_df = df.linkfunction.estim(xgrid,data.X,data.Y,bandwidth)
# ygrid_acgm = acgm.linkfunction.estim(xgrid,data.X,data.Y,bandwidth)
# ygrid_vkgmss = vkgmss.linkfunction.estim(xgrid,data.X,data.Y,bandwidth)
#
# plot(xgrid,ygrid_df,type='l',col='blue',lty=1,lwd=2,xlab='X',ylab='Y',ylim=c(0.25,2.5))
# lines(xgrid,ygrid_acgm,type='l',col='red',lty=2,lwd=2)
# lines(xgrid,ygrid_vkgmss,type='l',col='dark green',lty=3,lwd=2)
# lines(xgrid,0.2*xgrid^2-xgrid+2,lwd=0.5,col='gray')
# # Ducharme and Ferrigno: blue
# # Alcala et al.: red
# # Van Keilegom et al.: dark green
# # true link function: gray
#
# # Estimation of the conditional CDF (only Ducharme and Ferrigno estimator)
# xgrid = seq(0.5,4.5,by=0.1)
# ygrid = seq(-1,3,by=0.1)
# cdf_df = df.cdf.estim(xgrid,ygrid,data.X,data.Y,bandwidth)
#
# wireframe(cdf_df, drape=TRUE,
# col.regions=rainbow(100),
# zlab='CDF(y|x)',
# xlab='x',ylab='y',zlim=c(0,1.01))
#
# # Estimation of residuals cdf (only Van Keilegom et al. estimator)
#
# egrid = seq(-5,5,by=0.1)
# res.cdf_vkgmss = vkgmss.residuals.cdf.estim(egrid,data.X,data.Y,0.5)
#
# plot(egrid,res.cdf_vkgmss,type='l',xlab='e',ylab='CDF(e)')
#
# # Estimation of residuals standard deviation (only Van Keilegom et al. estimator)
#
# sd_vkgmss = vkgmss.sd.estim(xgrid,data.X,data.Y,bandwidth)
#
# plot(xgrid,sd_vkgmss, type='l',xlab='X',ylab='SD(X)')
# abline(h=0.3)
########################################################################
# Bandwidth selection under H0 (uncomment the following code block)
# # We want to test if the link function is f(x)=0.2*x^2-x+2
# # The answer is yes (see the definition of data.Y above)
# # We generate a dataset under H0 to estimate the optimal bandwidth under H0
#
# linkfunction.H0 = function(x){0.2*x^2-x+2}
#
# data.X.H0 = runif(n,min=0,max=5)
# data.Y.H0 = linkfunction.H0(data.X.H0)+rnorm(n,mean=0,sd=0.3)
#
# h.opt.df = df.bandwidth.selection.linkfunction(data.X.H0, data.Y.H0,linkfunction.H0)
# h.opt.acgm = acgm.bandwidth.selection.linkfunction(data.X.H0, data.Y.H0,linkfunction.H0)
# h.opt.vkgmss = vkgmss.bandwidth.selection.linkfunction(data.X.H0, data.Y.H0,linkfunction.H0)
# # Ducharme and Ferrigno: 1.184604
# # Alcala et al.: 0.7716453
# # Van Keilegom et al.: 0.6780543
########################################################################
# Test statistics under H0 (uncomment the following code block)
# # Remainder:
# # Ducharme and Ferrigno test is on the conditional CDF and not on the link function
# # Thus we need to define the conditional CDF associated
# # with the link function under H0 to evaluate this test
# # Alcala et al. and Van Keilegom et al. tests are on the link function
#
# # Optimal bandwidths estimated at the previous step
# h.opt.df = 1.184604
# h.opt.acgm = 0.7716453
# h.opt.vkgmss = 0.6780543
#
# cond_cdf.H0 = function(x,y)
# {
# out=matrix(0,nrow=length(x),ncol=length(y))
# for (i in 1:length(x)){
# x0=x[i]
# out[i,]=pnorm(y-linkfunction.H0(x0),0,0.3)
# }
# out
# }
# # cond_cdf.H0 is the conditional CDF associated with linkfunction.H0
# # with additive Gaussian noise (standard deviation=0.3)
#
# df.statistics(data.X,data.Y,cond_cdf.H0,h.opt.df)
# acgm.statistics(data.X,data.Y,linkfunction.H0,h.opt.acgm)
# vkgmss.statistics(data.X,data.Y,linkfunction.H0,h.opt.vkgmss)
########################################################################
# Test (bootstrap) under H0
h.opt.df = 1.184604 # Optimal bandwidth estimated above
linkfunction.H0 = function(x){0.2*x^2-x+2}
cond_cdf.H0 = function(x,y)
{
out=matrix(0,nrow=length(x),ncol=length(y))
for (i in 1:length(x)){
x0=x[i]
out[i,]=pnorm(y-linkfunction.H0(x0),0,0.3)
}
out
}
test_df.H0 = df.test.bootstrap(data.X,data.Y,cond_cdf.H0,
0.05,h.opt.df,bootstrap=c(20,'Mammen'),
integration.step = 0.1)
test_acgm.H0 = acgm.test.bootstrap(data.X,data.Y,linkfunction.H0,
0.05,bandwidth='optimal',bootstrap=c(20,'Mammen'),
verbose=FALSE,integration.step = 0.1)
test_vkgmss.H0 = vkgmss.test.bootstrap(data.X,data.Y,linkfunction.H0,
0.05,bandwidth='optimal',bootstrap=c(20,'Mammen'),
verbose=FALSE)
# test_acgm$decision is a string: 'accept H0' or 'reject H0'
# test_acgm$bandwidth is a float (optimal bandwidth under H0
# (only for Alcala and Van Keilegom tests) if bandwidth = 'optimal')
# test_acgm$pvalue is a float but it could be a string
# ('< 0.02' for instance or 'None' if the test can not be evaluated)
# test_acgm$test_statistics is a float but it could be a string
# ('None' if the test can not be evaluated)
# The 3 tests accept H0
########################################################################
# Test (bootstrap) under H1 (uncomment the following code block)
# # We want to test if the link function is f(x)=0.5*cos(x)+1
# # The answer is no (see the definition of data.Y above)
#
# linkfunction.H1=function(x){0.8*cos(x)+1}
#
# plot(xgrid,linkfunction.H0(xgrid),type='l',ylim=c(-1,2))
# lines(xgrid,linkfunction.H1(xgrid),type='l')
#
# data.X.H1 = data.X.H0
# data.Y.H1 = linkfunction.H1(data.X.H1)+rnorm(n,mean=0,sd=0.3)
# h.opt.df = df.bandwidth.selection.linkfunction(data.X.H1, data.Y.H1,linkfunction.H1)
#
# cond_cdf.H1=function(x,y)
# {
# out=matrix(0,nrow=length(x),ncol=length(y))
# for (i in 1:length(x)){
# x0=x[i]
# out[i,]=pnorm(y-linkfunction.H1(x0),0,0.3)
# }
# out
# }
#
# test_df.H1 = df.test.bootstrap(data.X,data.Y,cond_cdf.H1,
# 0.05,h.opt.df,bootstrap=c(20,'Mammen'),
# integration.step = 0.1)
# test_acgm.H1 = acgm.test.bootstrap(data.X,data.Y,linkfunction.H1,
# 0.05,bandwidth='optimal',bootstrap=c(20,'Mammen'),
# integration.step = 0.1,verbose=FALSE)
# test_vkgmss.H1 = vkgmss.test.bootstrap(data.X,data.Y,linkfunction.H1,
# 0.05,bandwidth='optimal',bootstrap=c(20,'Mammen'),
# verbose=FALSE)
#
# # From only 25 points, only Van Keilegom et al. test rejects H0
# # while Ducharme and Ferrigno and Alcala et al. tests accept H0
########################################################################
cvmgof documentation built on Jan. 16, 2021, 5:40 p.m.
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Description Details Author(s) References See Also Examples
Description
It implements three goodness-of-fit tests to test the validity of the regression function in the regression model. Two of them (Alcala et al. '99 and Van Keilegom et al. '12) are “directional” in that they detect departures from mainly the regression function assumption of the model or “global” (Ducharme and Ferrigno '12) with the conditional distribution function. The establishment of such statistical tests requires nonparametric estimators and the use of wild bootstrap methods for the simulations.
Details
| Package: | cvmgof |
| Type: | Package |
| Version: | 1.0.3 |
| Date: | 2021-01-11 |
| License: | Cecill |
Author(s)
Romain Azais, Sandie Ferrigno and Marie-Jose Martinez
References
J. T. Alcala, J. A. Cristobal, and W. Gonzalez Manteiga. Goodness-of-fit test for linear models based on local polynomials. Statistics & Probability Letters, 42(1), 39:46, 1999.
G. R. Ducharme and S. Ferrigno. An omnibus test of goodness-of-fit for conditional distributions with applications to regression models. Journal of Statistical Planning and Inference, 142, 2748:2761, 2012.
I. Van Keilegom, W. Gonzalez Manteiga, and C. Sanchez Sellero. Goodness-of-fit tests in parametric regression based on the estimation of the error distribution. Test, 17, 401:415, 2008.
R. Azais, S. Ferrigno and M-J Martinez. cvmgof: An R package for Cramer-von Mises goodness-of-fit tests in regression models. Submitted. January 2021.hal-03101612
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 | # require(lattice) # Only for plotting conditional CDF
########################################################################
# Simulation
set.seed(1)
# The following example tests are computed from only 25 data points
# The seed is fixed to avoid NA in estimates
# Data simulation
n = 25 # Dataset size
data.X = runif(n,min=0,max=5) # X
data.Y = 0.2*data.X^2-data.X+2+rnorm(n,mean=0,sd=0.3) # Y
# plot(data.X,data.Y,xlab='X',ylab='Y',pch='+')
########################################################################
# Estimation of the link function (uncomment the following code block)
# bandwidth = 0.75 # Here, the bandwidth is arbitrarily fixed
#
# xgrid = seq(0,5,by=0.1)
# ygrid_df = df.linkfunction.estim(xgrid,data.X,data.Y,bandwidth)
# ygrid_acgm = acgm.linkfunction.estim(xgrid,data.X,data.Y,bandwidth)
# ygrid_vkgmss = vkgmss.linkfunction.estim(xgrid,data.X,data.Y,bandwidth)
#
# plot(xgrid,ygrid_df,type='l',col='blue',lty=1,lwd=2,xlab='X',ylab='Y',ylim=c(0.25,2.5))
# lines(xgrid,ygrid_acgm,type='l',col='red',lty=2,lwd=2)
# lines(xgrid,ygrid_vkgmss,type='l',col='dark green',lty=3,lwd=2)
# lines(xgrid,0.2*xgrid^2-xgrid+2,lwd=0.5,col='gray')
# # Ducharme and Ferrigno: blue
# # Alcala et al.: red
# # Van Keilegom et al.: dark green
# # true link function: gray
#
# # Estimation of the conditional CDF (only Ducharme and Ferrigno estimator)
# xgrid = seq(0.5,4.5,by=0.1)
# ygrid = seq(-1,3,by=0.1)
# cdf_df = df.cdf.estim(xgrid,ygrid,data.X,data.Y,bandwidth)
#
# wireframe(cdf_df, drape=TRUE,
# col.regions=rainbow(100),
# zlab='CDF(y|x)',
# xlab='x',ylab='y',zlim=c(0,1.01))
#
# # Estimation of residuals cdf (only Van Keilegom et al. estimator)
#
# egrid = seq(-5,5,by=0.1)
# res.cdf_vkgmss = vkgmss.residuals.cdf.estim(egrid,data.X,data.Y,0.5)
#
# plot(egrid,res.cdf_vkgmss,type='l',xlab='e',ylab='CDF(e)')
#
# # Estimation of residuals standard deviation (only Van Keilegom et al. estimator)
#
# sd_vkgmss = vkgmss.sd.estim(xgrid,data.X,data.Y,bandwidth)
#
# plot(xgrid,sd_vkgmss, type='l',xlab='X',ylab='SD(X)')
# abline(h=0.3)
########################################################################
# Bandwidth selection under H0 (uncomment the following code block)
# # We want to test if the link function is f(x)=0.2*x^2-x+2
# # The answer is yes (see the definition of data.Y above)
# # We generate a dataset under H0 to estimate the optimal bandwidth under H0
#
# linkfunction.H0 = function(x){0.2*x^2-x+2}
#
# data.X.H0 = runif(n,min=0,max=5)
# data.Y.H0 = linkfunction.H0(data.X.H0)+rnorm(n,mean=0,sd=0.3)
#
# h.opt.df = df.bandwidth.selection.linkfunction(data.X.H0, data.Y.H0,linkfunction.H0)
# h.opt.acgm = acgm.bandwidth.selection.linkfunction(data.X.H0, data.Y.H0,linkfunction.H0)
# h.opt.vkgmss = vkgmss.bandwidth.selection.linkfunction(data.X.H0, data.Y.H0,linkfunction.H0)
# # Ducharme and Ferrigno: 1.184604
# # Alcala et al.: 0.7716453
# # Van Keilegom et al.: 0.6780543
########################################################################
# Test statistics under H0 (uncomment the following code block)
# # Remainder:
# # Ducharme and Ferrigno test is on the conditional CDF and not on the link function
# # Thus we need to define the conditional CDF associated
# # with the link function under H0 to evaluate this test
# # Alcala et al. and Van Keilegom et al. tests are on the link function
#
# # Optimal bandwidths estimated at the previous step
# h.opt.df = 1.184604
# h.opt.acgm = 0.7716453
# h.opt.vkgmss = 0.6780543
#
# cond_cdf.H0 = function(x,y)
# {
# out=matrix(0,nrow=length(x),ncol=length(y))
# for (i in 1:length(x)){
# x0=x[i]
# out[i,]=pnorm(y-linkfunction.H0(x0),0,0.3)
# }
# out
# }
# # cond_cdf.H0 is the conditional CDF associated with linkfunction.H0
# # with additive Gaussian noise (standard deviation=0.3)
#
# df.statistics(data.X,data.Y,cond_cdf.H0,h.opt.df)
# acgm.statistics(data.X,data.Y,linkfunction.H0,h.opt.acgm)
# vkgmss.statistics(data.X,data.Y,linkfunction.H0,h.opt.vkgmss)
########################################################################
# Test (bootstrap) under H0
h.opt.df = 1.184604 # Optimal bandwidth estimated above
linkfunction.H0 = function(x){0.2*x^2-x+2}
cond_cdf.H0 = function(x,y)
{
out=matrix(0,nrow=length(x),ncol=length(y))
for (i in 1:length(x)){
x0=x[i]
out[i,]=pnorm(y-linkfunction.H0(x0),0,0.3)
}
out
}
test_df.H0 = df.test.bootstrap(data.X,data.Y,cond_cdf.H0,
0.05,h.opt.df,bootstrap=c(20,'Mammen'),
integration.step = 0.1)
test_acgm.H0 = acgm.test.bootstrap(data.X,data.Y,linkfunction.H0,
0.05,bandwidth='optimal',bootstrap=c(20,'Mammen'),
verbose=FALSE,integration.step = 0.1)
test_vkgmss.H0 = vkgmss.test.bootstrap(data.X,data.Y,linkfunction.H0,
0.05,bandwidth='optimal',bootstrap=c(20,'Mammen'),
verbose=FALSE)
# test_acgm$decision is a string: 'accept H0' or 'reject H0'
# test_acgm$bandwidth is a float (optimal bandwidth under H0
# (only for Alcala and Van Keilegom tests) if bandwidth = 'optimal')
# test_acgm$pvalue is a float but it could be a string
# ('< 0.02' for instance or 'None' if the test can not be evaluated)
# test_acgm$test_statistics is a float but it could be a string
# ('None' if the test can not be evaluated)
# The 3 tests accept H0
########################################################################
# Test (bootstrap) under H1 (uncomment the following code block)
# # We want to test if the link function is f(x)=0.5*cos(x)+1
# # The answer is no (see the definition of data.Y above)
#
# linkfunction.H1=function(x){0.8*cos(x)+1}
#
# plot(xgrid,linkfunction.H0(xgrid),type='l',ylim=c(-1,2))
# lines(xgrid,linkfunction.H1(xgrid),type='l')
#
# data.X.H1 = data.X.H0
# data.Y.H1 = linkfunction.H1(data.X.H1)+rnorm(n,mean=0,sd=0.3)
# h.opt.df = df.bandwidth.selection.linkfunction(data.X.H1, data.Y.H1,linkfunction.H1)
#
# cond_cdf.H1=function(x,y)
# {
# out=matrix(0,nrow=length(x),ncol=length(y))
# for (i in 1:length(x)){
# x0=x[i]
# out[i,]=pnorm(y-linkfunction.H1(x0),0,0.3)
# }
# out
# }
#
# test_df.H1 = df.test.bootstrap(data.X,data.Y,cond_cdf.H1,
# 0.05,h.opt.df,bootstrap=c(20,'Mammen'),
# integration.step = 0.1)
# test_acgm.H1 = acgm.test.bootstrap(data.X,data.Y,linkfunction.H1,
# 0.05,bandwidth='optimal',bootstrap=c(20,'Mammen'),
# integration.step = 0.1,verbose=FALSE)
# test_vkgmss.H1 = vkgmss.test.bootstrap(data.X,data.Y,linkfunction.H1,
# 0.05,bandwidth='optimal',bootstrap=c(20,'Mammen'),
# verbose=FALSE)
#
# # From only 25 points, only Van Keilegom et al. test rejects H0
# # while Ducharme and Ferrigno and Alcala et al. tests accept H0
########################################################################
|
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