View source: R/cutoff.bootstrap.R
cutoff.bootstrap | R Documentation |
Implements a bootstrap critical value for testing the goodness-of-fit of a parametrically estimated density with the test statistic S.n
.
cutoff.bootstrap(xin, M, sim, dist, h.use, kfun, p1, p2, sig.lev)
xin |
A vector of data points - the available sample. |
M |
Number of bootstrap replications. |
sim |
A character string indicating the type of simulation required: "ordinary" (the default), "parametric", "balanced", "permutation", or "antithetic". |
dist |
The null distribution. |
h.use |
The test statistic bandwidth, best implemented with |
kfun |
The kernel to use in the density estimates used in the bandwidth expression. |
p1 |
Parameter 1 (vector or object) for the null distribution. |
p2 |
Parameter 2 (vector or object) for the null distribution. |
sig.lev |
Significance level of the hypothesis test. |
Implements the bootstrap based finite sample critical value defined in Section 2.6, Bagkavos, Patil and Wood (2021), and calculated as follows:
1. Resample the observations \mathcal{X}=\{X_1, …, X_n\} to obtain M bootstrap samples, denoted by \mathcal{X}_m^\ast=\{ X_{1m}^\ast, …, X_{nm}^\ast\}, where for each m=1,… , M, \mathcal{X}_m^\ast is sampled randomly, with replacement, from \mathcal{X}. Write \hat{θ}=θ(\mathcal{X}) for the estimator of θ based on the original sample \mathcal{X} and, for each m, define the bootstrap estimator of θ by \hat{θ}_m^\ast = θ(\mathcal{X}_m^\ast), where θ(\cdot) is the relevant functional for the parameter θ.
2. For m=1, … , M, use \mathcal{X}_m^\ast =\{X_{1m}^\ast, …, X_{nm}^\ast\} and \hat θ_m^\ast from the previous step to calculate n Δ^{2d} h^{-d/2} \hat S_{n,m}^\ast(hρ),m=1, …, M.
3. Calculate \ell_α^\ast as the 1-α empirical quantile of the values n Δ^{2d} h^{-d/2} \hat S_{n,m}^\ast(hρ), m=1, …, M. Then \ell_α^\ast approximately satisfies P^\ast [ n Δ^{2d} h^{-d/2}\hat S_{n,m}^\ast(hρ)> \ell_α^\ast ]=1-α, where P^\ast indicates the bootstrap probability measure conditional on \mathcal{X}.
A scalar, the estimate of the bootstrap critical value at the given significance level.
Dimitrios Bagkavos
R implementation and documentation: Dimitrios Bagkavos <dimitrios.bagkavos@gmail.com>
Bagkavos, Patil and Wood: Nonparametric goodness-of-fit testing for a continuous multivariate parametric model, (2021), under review.
Gao and Gijbels, Bandwidth selection in nonparametric kernel testing, pp. 1584-1594, JASA (2008)
cutoff.asymptotic, cutoff.edgeworth
library(nor1mix) library(boot) SampleSize<-80 M<-1000 dist<- "normixt" kfun<- Epanechnikov p1 <-MW.nm2 p2 <-1 sig.lev <- 0.05 sim<-"ordinary" ## Not run: #Run the following to compare the asymptotic and bootstrap cut-off points on 4 occasions: for(i in 15:18) { set.seed(i) xin<-rnorMix(SampleSize, p1) h.use <- hopt.be(xin) l.a.a<-cutoff.asymptotic( dist, p1, p2, sig.lev ) l.a.b<- cutoff.bootstrap(xin, M, sim, dist, h.use, kfun, p1, p2, sig.lev) #print the result of each iteration: cat("Asympt. cut.off= ", l.a.a, "Boot. cut.off= ", l.a.b, "\n") } ## End(Not run)
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