Semiparametric nonhomogeneity analysis
Carey E. Priebe, David J. Marchette, George W. Rogers · Journal of Statistical Planning and Inference · 1997
Let ξ(x, ω) be a ‘piecewise stationary’ random field, defined as an embedding of stationary random fields ξi(x, ω) via the polytomous field m(x, ω). The domain of definition is partitioned into disjoint regions Ri. Denote the marginals for each ξi(x, ω) by xi(ξ) so that ξ(x, ω) ∼ αi(ξ) for x ∈ Ri. Define homogeneity as the situation in which all the αi are identical versus nonhomogeneity in which there exist at least two regions with differing marginals. To perform a test of these hypotheses without assuming parametric structure for the αi or choosing a specific type of nonhomogeneity in the alternative requires estimates \̂gai for each region. However, the competing requirements of estimation without restrictive assumptions versus small-area investigation to determine the unknown locations of potential nonhomogeneities lead to an impasse which cannot easily be overcome and has led to a dichotomy of approaches — parametric versus nonparametric. This paper develops a borrowed strength methodology which can be used to improve upon the local estimates which are obtainable by either fully nonparametric methods or by simple parametric procedures. The approach involves estimating the marginals as a generalized mixture model, and the improvement derives from using all the observed data, borrowing strength from potentially dissimilar regions, to impose constraints on the local estimation problems.