S.nd | R Documentation |
Implements the multivariate (d >=2) 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.
S.nd(xin, h, dist, p1, p2)
xin |
A matrix (n x d) of data points - the available sample with n rows and d columns, each column corresponds to a different coordinate axis. |
h |
The bandwidth vector to use, typically the output of |
dist |
The null distribution. |
p1 |
Parameter 1 (vector or object) for the null distribution. |
p2 |
Parameter 2 (vector or object) for the null distribution. |
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 Δ^2 {∑∑}_{i \neq j} K \{ (X_{i1}-X_{j1})h_1^{-1}, …, (X_{id}-X_{jd})h_d^{-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.
A vector with the value of the test statistic as well as the Delta value used for its calculation
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.
S.n
library(mvtnorm) sigma <- matrix(c(4,2,2,3), ncol=2) x <- rmvnorm(n=100, mean=c(1,2), sigma=sigma) h.be1 <- hopt.be(x[,1]) h.be2 <- hopt.be(x[,2]) h<-c(h.be1, h.be2) Nulldist<-"normal" S.nd(x, h, Nulldist, c(1,2), sigma)
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