Probabilities of high excursions of Gaussian fields

Multivariate goodness-of-fit tests based on kernel density estimators

“…Further, we discuss the problem of analytical approximation of the distribution of statistic 3, which could be used as a test statistic with a fixed choice of the smoothing matrix W . The problem is investigated using the theory of high excursions of Gaussian (and, in some sense, close to Gaussian) random fields introduced in [25,27]. In addition, the obtained results lead to the analytical approximations of functions µ(•), γ(•) and quintile c α (•) used in the definitions of the statistics (5) and 7, respectively.…”

“…In practice, the critical region of the test is found by means of Monte Carlo simulations. The problem of analytical approximation of the distribution of the test statistic under the null hypothesis is discussed, using the theory of high excursions of Gaussian (and, in some sense, close to Gaussian) random fields developed in [25,27].…”

Goodness of Fit Tests Based on Kernel Density Estimators

“…Further consider an empirical field ξ h (x) with respect x ∈ I and h ∈ J . The required approximation for P (u) could be obtained by applying the results of Rudzkis and Bakshaev (2012). It has been shown that if a differentiable (in the mean square sense) Gaussian random field {η(t), t ∈ T } with Eη(t) ≡ 0, Dη(t) ≡ 1 and continuous trajectories defined on the mdimensional interval T ⊂ R m satisfies certain smoothness and regularity conditions, then P{−v(t) < η(t) < u(t), t ∈ T } ∼ = e −Q , as ∀t ∈ T u(t), v(t) > χ, χ → ∞, where v(•) and u(•) are smooth enough functions and Q is a certain constructive functional depending on u, v, T and the matrix function R(t) = cov(η ′ (t), η ′ (t)).…”

“…In practice the critical region of the test is established by means of Monte Carlo simulations. The problem of analytical approximation of the distribution of the test statistic under the null hypothesis is discussed, using the theory of high excursions of Gaussian (and, in some sense, close to Gaussian) random processes and fields developed by Rudzkis ( , 2012. Besides some of the already mentioned researchers, the asymptotic distributions of deviations of kernel density estimators in uniform metric were also considered by Piterbarg and Konakov (1982Konakov ( , 1983, Muminov (2011Muminov ( , 2012.…”

“…Simulation results show that the proposed test is a powerful competitor to the existing classical criteria testing goodness of fit against a specific type of alternative hypothesis. An analytical way for establishing the asymptotic distribution of the test statistic is proposed, using the theory of high excursions of close to Gaussian random processes and fields introduced by Rudzkis ( , 2012.…”