Asymptotic properties of rank estimators in a simple
spatial linear regression model under spatial sampling
Received: 28 November 2017 / Accepted: 21 May 2018
Ó Japanese Federation of Statistical Science Associations 2018
Abstract In this study, we derive the asymptotic normality of a class of rank
estimators in a simple spatial linear regression model, when errors form a strongly
mixing random ﬁeld and when the spatial data are both on the lattice and on the
irregularly spaced spatial sites. This result in turn is used to investigate the
asymptotic relative efﬁciency (ARE) of these estimators relative to the LSE. In
addition, we conduct numerical experiments under both the lattice and the irregu-
larly spaced sampling, which lends support to the robustness of these estimators
compared to the LSE.
Keywords Spatial regression Strong-mixing random ﬁeld Spatial
sampling designs Rank estimator Robustness Asymptotic normality
The least-squares estimator (LSE) is a popular estimator because it has desirable
properties under regularity conditions and is always easily available. However, it is
well known that it is not robust against heavy-tail error distributions.
On the other hand, rank estimators are known to have such a robustness property.
Hodges and Lehmann (1963) originally introduced their estimator to ensure that the
robustness of rank tests is inherited from estimation problems. In an attempt to
extend the Hodges–Lehmann estimator for location problems to regression
(1969, 1971) then introduced the rank estimators in the i.i.d.
case for simple and multiple regression. When errors in the simple linear regression
& Eika Yamada
Graduate School of Economics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku,
Tokyo 113-0033, Japan
Jpn J Stat Data Sci