Page 158 - Special Topic Session (STS) - Volume 2

STS474 Yoshihiro Y.
∥ ∥ 2 (2)
∫ 2 () < ∞
1 +∥ ∥
(See (see Chilés & Delfiner (2012); Cressie (1993); Solo (1992); Yaglom (1957).)
Hereafter we assume that () = 0 and () is absolutely continuous with
density (). Then (1) and (2) reduce to

1−cos((,))

2 () = ∫ (2) (),

and
∥ ∥ 2
∫ 2 () < ∞,
1 +∥ ∥
respectively.
An interesting special class of ISRF's that is often applied to empirical data
analysis in space is a fractional Brownian field (FBF)(see Adler (1981);
Mandelbrot& Van Ness (1968); Zhu & Stein (2002) and the references therein).
A FBF is a Gaussian ISRF and has 2 () = ∥ ∥ 2 , which is equivalent to

() = ,
∥ ∥ +2
where
⁄
= 2 2+ Γ( + 2 2 Γ(1 − )⁄ , 0 < < 1.
⁄
2

(Yaglom (1957)). is a scale parameter and is a smoothness parameter
with larger values corresponding to smoother surfaces.
Hereafter we assume that = 2 and {()} is a two-dimensional
Gaussian ISRF. Then we also denote λ by ( , ) and the spectral density
2
1
function () by ( , ) respectively.
2
1
In this paper we consider the following class of the spectral density
function, which includes a FBF.

Assumption 1 ( , ) is expressed by
2
1
( , ) =∥ ∥ −2ℎ−2 ( , ), 0 < < 1,
2
1

1
2
where ( , ),is a nonnegative with (0,0) > 0, symmetric, ( , ) =

2
1
1

2
(− , − ),twice continuously differentiable function for −∞ < , < ∞

1
1
2
2
and is bounded with bounded first and second order partial derivatives.

2. Methodology
We denote the sampling sites and observations by = (, ) and
( )(, = 1, … , )respectively. Then the sample size is .
2

Definition 2.1(Data Tapers of order ). {ℎ : = 1, … , } is called a sequence

of data tapers of order if it satisfies the following conditions.
ℎ = 1.
(1) ℎ is positive and symmetric around = /2 with max 1≤≤

(2) For any > 0, there exists a constant , 0 < < ∞, which may depend on
so that ∑ ℎ = holds.
=1
=1
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