Page 157 - Special Topic Session (STS) - Volume 2

STS474 Yoshihiro Y.

On Gaussian semiparametric estimation for
two-dimensional intrinsic stationary random
fields
Yoshihiro Yajima
Tohoku University, Sendai, Japan

Abstract
We propose a Gaussian semiparametric estimator for semiparametric models
of two-dimensional intrinsic stationary random fields (ISRFs) observed on a
regular grid and derive its asymptotic properties. Originally this estimator is
an approximate likelihood estimator in a frequency domain for long memory
models of stationary and nonstationary time series (Robinson (1995); Velasco
(1999)). We apply it to two dimensional ISRFs. These ISRFs include a fractional
Brownian field, which is a Gaussian random field and is used to model many
physical processes in space. The estimator is consistent and has the limiting
normal distribution as the sample size goes to infinity. We conduct a
computational simulation to compare the performance of it with those of
different estimators.

Keywords
spatio-temporal models; local Whittle estimator; fractional Brownian field

1. Introduction

Let {() ∶ ∈ } be a random field. Throughout this paper, we
specialize to the two-dimensional random field, = 2 . Whereas for the
moment, for ease of description, we consider a general d-dimensional setting.

If {X(s)} satisfies that for any fixed (∈ ), the increment () = ( +

) − () is a stationary random field, {()} is called an ISRF. Then {X (s)} is
characterized by
(( + ) − ()) = 0,
(( + ) − ()) = 2 ();

where 2 () is the variogram function (see Chilés & Delfiner (2012); Cressie

(1993)). Hereafter we also assume that () = 0.
For and , let (, ) be the inner product and ∥ ∥ be the norm.
Then if 2 () is a continuous function on satisfying () = 0, it has the

spectral representation
1−cos((,))

2 () = ∫ (2) () + (), (1)

where ()(≥ 0) is a quadratic form and () is a positive, symmetric
measure such that ∥ ∥ () is continuous at the origin and
2

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