L2DensityGoFtest (version 0.6.0)

S.n: Density goodness-of-fit test statistic based on discretized L2 distance

Description

Implements the density goodness of fit test statistic \(\hat{S}_n(h)\) of Bagkavos, Patil and Wood (2021), based on aggregation of local discrepancies between the fitted parametric density and a nonparametric empirical density estimator.

Usage

S.n(xin, h,  dist, p1, p2)

Value

A vector with the value of the test statistic as well as the Delta value used for its calculation

Arguments

xin

A vector of data points - the available sample size.

h

The bandwidth to use, typically the output of hopt.edgeworth.

dist

The null distribution.

p1

Parameter 1 (vector or object) for the null distribution.

p2

Parameter 2 (vector or object) for the null distribution.

Author

R implementation and documentation: Dimitrios Bagkavos <[e-mail]>

Details

Implements the test statistic used for testing the hypothesis $$H_0: f(x) = f_0(x, p1, p2) \;\; vs \;\; H_a: f(x) \neq f_0(x, p1, p2).$$ This density goodness-of-fit test is based on a discretized approximation of the L2 distance. Assuming that \(n\) is the number of observations and \(g = (max(xin)-min(xin))/n^{-drate}\) is the number of bins in which the range of the data is split, the test statistic is: $$ S_n(h) = n \Delta^2 h^{-1/2} {\sum\sum}_{i \neq j} K \{ (X_i-X_j)h^{-1}\} \{Y_i -f_0(X_i) \}\{Y_j -f_0(X_j) \} $$ where \(K\) is the Epanechnikov kernel implemented in this package with the Epanechnikov function. The null model \(f_0\) is specified through the dist argument with parameters passed through the p1 and p2 arguments. The test is implemented either with bandwidth hopt.edgeworth or with bandwidth hopt.be which provide the value of \(h\) needed for calculation of \( S_n(h)\) and the critical value used to determine acceptance or rejection of the null hypothesis. See the example below for an application to a real world dataset.

References

Bagkavos, Patil and Wood: Nonparametric goodness-of-fit testing for a continuous multivariate parametric model, (2021), under review.

See Also

S.n.Boot

Examples

library(fGarch)
library(boot)
 if (FALSE) data(EuStockMarkets)
DAX <- as.ts(EuStockMarkets[,"DAX"])
dax <-  diff(log(DAX))#[,"DAX"]

# Fit a GARCH(1,1) model to dax returns:
lll<-garchFit(~ garch(1,1), data = as.ts(dax), trace = FALSE, cond.dist ="std")
# define the model innovations, to be used as input to the test statistic
xin<-lll@residuals /lll@sigma.t
# exclude smallest value - only for uniform presentation of results
#(this step can be excluded):
xin = xin[xin!= min(xin)]

#inputs for the test statistic:
#kernel function to use in implementing the statistic
#and functional estimates for optimal h:
kfun<-"epanechnikov"
a.sig<-0.05 #define the significance level
#null hypothesis is that the innovations are normaly distributed:
Nulldist<-"normal"

p1<-mean(xin)
p2<- sd(xin)
#Power optimal bandwidth:
h<-hopt.edgeworth(xin,   Nulldist, kfun, p1, p2, a.sig )
h.be <- hopt.be(xin)
# Edgeworth cutoff point:
cutoff<-cutoff.edgeworth(xin,   Nulldist, kfun, p1, p2, a.sig )
# Bootstrap cutoff point:
cutoff.boot<-cutoff.bootstrap(xin, 100,  "permutation", Nulldist, h.be, kfun, p1, p2, a.sig)
# Asympt. Norm. cutoff point:
cutoff.asympt<-cutoff.asymptotic( Nulldist,   p1, p2, a.sig )

TestStatistic<-S.n(xin, h, Nulldist, p1, p2)
TestStatistic.be<-S.n(xin, h.be, Nulldist, p1, p2)

cat("L2 test statistic value with power opt. band:", TestStatistic[1],
"\nL2 test statistic value Barry-Essen bandwidth:", TestStatistic.be[1],
"\ncritical value asymptotic:", round(cutoff.asympt,3), "critical value bootstrap:",
round(cutoff.boot,3),  "critical value Edgeworth:", round(cutoff,3), "\n")
#L2 test statistic value Edgeworth: 7.257444
#L2 test statistic value Berry-Esseen bandwidth: 10.97069
# critical value Asymptotically Norm.:  1.801847
# critical value Edgeworth: 2.140446
# critical value bootstrap: 6.040048
# L2 test statistic >  critical value on all occasions, hence normality is rejected