Optimal robust estimation for discrete time stochastic processes
Let \(Y_ 1,\ldots,Y_ n\) be a sample of n observations from a discrete time stochastic process whose distribution depends on \(\theta\), a real valued parameter taking values in an open subset of the \(p\)-dimensional Euclidean space. If \(T_ n=T_ n(Y_ 1,\ldots,Y_ n)\) denotes a \(p\)-dimensional vector estimating function for \(\theta\) then the estimate of \(\theta\) will be the solution of the system of estimating equations \(T_ n=0\). The class \({\mathcal T}\) of zero mean square integrable vector estimating functions which are a.s. differentiable with respect to \(\theta\) and such that \(E \dot T_ n=(E \partial T_{n,i}/\partial \theta_ j)\) and \(E T_ nT_ n'\) are nonsingular, is considered. \(T_ n^*\in {\mathcal T}_ 1\subset {\mathcal T}\) is defined as \(O_ F\)-optimal within \({\mathcal T}_ 1\) if \[ (E \dot T_ n)^{-1}(E T_ nT_ n')((E \dot T_ n)^{-1})'-(E \dot T_ n^*)^{-1}(E T_ n^*T_ n^{*'})((E \dot T^*_ n)^{-1})' \] is non-negative definite for all \(T_ n\in {\mathcal T}_ 1\). Attention is restricted to the subset \({\mathcal S}\) of \({\mathcal T}\) having elements \[ S_ n=\sum^{n}_{i=1}(\sum^{i}_{j=1}b_{j,i} k'_{i-j})h_ i \] where \(b_{j,i}\) are constant vectors of dimension p and \(h_ i\) and \(k_{i-j}\) are specified vectors of dimension q satisfying \(E(h_ i| {\mathcal F}_{i-1})=0\) a.s., \(i=1,...,n\) and \(E(k_{i-j}| {\mathcal F}_{i-j-1})=0\), \(i>j\), where \({\mathcal F}_ k\) denotes the past history \(\sigma\)-field. The \(O_ F\)-optimal \(S_ n^*\) within the class \({\mathcal S}\) is given. Examples of a regression model with autoregressive errors and a general contamination model are presented to illustrate applications of the \(O_ F\)-optimal vector estimating function. The loss of efficiency in using \(S_ n^*\) in the illustrations is discussed.
constructing robust quasi-likelihood estimating functions
zero mean square integrable vector estimating functions
scientific article; zbMATH DE number 4070103
scientific article