Abstract
Nonparametric estimation of probability density functions, both marginal and joint densities, is a very useful tool in statistics. The kernel method is popular and applicable to dependent data, including time series and spatial data. But at least for the joint density, one has had to assume that data are observed at regular time intervals or on a regular grid in space. Though this is not very restrictive in the time series case, it often is in the spatial case. In fact, to a large degree it has precluded applications of nonparametric methods to spatial data because such data often are irregularly positioned over space. In this article, we propose nonparametric kernel estimators for both the marginal and in particular the joint probability density functions for nongridded spatial data. Large sample distributions of the proposed estimators are established under mild conditions, and a new framework of expanding-domain infill asymptotics is suggested to overcome the shortcomings of spatial asymptotics in the existing literature. A practical, reasonable selection of the bandwidths on the basis of cross-validation is also proposed. We demonstrate by both simulations and real data examples of moderate sample size that the proposed methodology is effective and useful in uncovering nonlinear spatial dependence for general, including non-Gaussian, distributions. Supplementary materials for this article are available online.
Highlights
Mation since this in turn can be used in nonlinear modeling in Nonparametric estimation methods are well established for time series and have found extensive practical applications; see, for example, Fan and Yao (2003) and Terasvirta, Tjøstheim, and Granger (2010)
If the observations were taken at irregular time intervals, there would not be enough observation points to estimate the joint density for a specified time difference
It may be worth pointing out that our defined domainexpanding infill asymptotic by (2.3) and (2.4) is different from an alternative version of mixed asymptotic adopted by Hall and Patil (1994); see Matsuda and Yajima (2009), who assumed in the notation of this article, that si = (A1ui,1, A2ui,2), where ui =, i = 1, 2, . . . , N, are independently and identically distributed random vectors with a probability density function fU(u) of a compact support in [0, 1]2, and A1 = A1,N → ∞, A2 = A2,N → ∞ as N → ∞
Summary
Mation since this in turn can be used in nonlinear modeling in Nonparametric estimation methods are well established for time series and have found extensive practical applications; see, for example, Fan and Yao (2003) and Terasvirta, Tjøstheim, and Granger (2010). Because of physical constraints measurement stations cannot usually be put on a regular grid in space This means that spatial analysis has been almost completely dominated by parametric models; for example, parametric models for covariance functions (or variogram) in kriging (Cressie 1993; Stein 1999). One of the main motivations for nonparametric analysis in time series is density estimation, in particular joint density estiregression and additive modeling In all of this the regular grid assumption is vital. Is a brief overview of the article: In Section 2, the methodology for the estimation of the marginal and joint probability density functions together with a new framework of expanding-domain infill asymptotics for irregularly positioned spatial data is proposed.
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