Reference Work Entry

International Encyclopedia of Statistical Science

pp 459-460


Estimation Problems for Random Fields

  • Mikhail P. MoklyachukAffiliated withKyiv National Taras Shevchenko University

Estimation problems for random fields X (t),   t ∈ ℝ n (estimation of the unknown mathematical expectation, estimation of the correlationfunction, estimation of regression parameters, extrapolation, interpolation, filtering,etc) are similar to the corresponding problems for stochastic processes (random fields of dimension 1). Complications usually are caused by the form of domain of points {tj} = D ⊂ ℝ n, where observations {X (tj) } are given, and by the dimension of the field. The complications can be overcome by considering specific domains of observations and particular classes of random fields.

Say in the domain D ⊂ ℝ n there are given observations of the random field
$$X\left (t\right ) = \sum \limits_{i=1}^{q}\theta,{g}_{ i}(t) + Y (t),$$
where gi (t), i = 1, , q, are known non-random functions, θi, i = 1, ,   q, are unknown parameters, and Y (t) is a random field with EY (t) = 0. The problem is to estimate the regression parameters θi, i = 1, ,   q. This problem includes as a ...
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