On the Power of Some Nonparametric Isotropy Tests

Krzysztof Szymoniak-Książek

doi:10.18778/0208-6018.350.03

Abstract

In this paper, properties of nonparametric significance tests verifying the random field isotropy hypothesis are discussed. In particular, the subject of the conducted analysis is the probability of rejecting the null hypothesis when it is true. A potential significant difference of empirical rejection probability from the assumed significance level could distort the results of statistical inference. The tests proposed by Guan, Sherman, Calvin (2004) and Lu, Zimmerman (2005) are considered. A simulation study has been carried out through generating samples from a given theoretical distribution and repeatedly testing the null hypothesis. Isotropic distributions are considered, among others, those based on a multidimensional normal distribution. The main aim of the paper is to compare both considered nonparametric significance tests verifying the random field isotropy hypothesis. For this purpose, the empirical rejection probabilities for both tests have been calculated and compared with the assumed significance level.

Highlights

In spatial statistics, observations of the study variables are treated as realisations of the spatial stochastic process, understood as a collection of random variablesX ( ) = Xt t∈T indexed by a coordinate vector T ⊂ n
Empirical rejection probabilities depending on the significance level for Guan, Sherman, Calvin and Lu, Zimmerman tests were calculated (10,000 realisations were used)
Empirical rejection probability values greater than the significance level mean that the test rejects the null hypothesis more often than the user is will‐ ing to accept

Summary

Introduction

Observations of the study variables are treated as realisations of the spatial stochastic process, understood as a collection of random variables. These are processes for which the expected value is constant and the covariance function depends only on the shift vector, i.e. One of the basic tools used to study the variability structure of studied phenomena is the variogram. It is a measure defined in locations shifted by the vector h as 2γ (h) = Var ( X s – Xt ) , where h = s – t. An important assumption used in the estimation of variograms is random field isotropy (Sherman, 2010)

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