On the effect of substituting parameter estimators in limiting \(\chi ^ 2U\) and V statistics
In many statistical problems, we need to know the effect of substituting an estimator of a parameter into a statistic. Let \(X_ 1,...,X_ n\) be i.i.d. random vectors. Let \({\hat \lambda}\) be the consistent estimator of \(\lambda\) such that \({\hat \lambda}=\lambda +n^{- 1}\sum^{n}_{i=1}\alpha (X_ i)=o_ p(n^{-1/2})\) where \(\alpha\) is a function. For some function g and for a finite positive measure M, let \[ V_ n({\hat \lambda})=n^{-2}\int [\sum^{n}_{i=1}g(X_ i,\quad t: {\hat \lambda})]^ 2dM(t) \] and \[ V_ n=n^{-2}\int [\sum^{n}_{i=1}g(X_ i,\quad t: \lambda)+d_ 1\mu (t: \lambda)'\alpha (x)]^ 2dM(t) \] where \(d_ 1\mu (t:\lambda)\) represents the vector of partial derivatives of \(E_{\lambda}[g(X_ 1\), \(t: \gamma)\)] at \(\gamma =\lambda.\) In this paper, it is shown that under some conditions \(n(V_ n({\hat \lambda})-V_ n)\) converges in probability to zero and the distribution of \(nV_ n({\hat \lambda})\) converges weakly to a distribution of a weighted sum of (possibly infinitely many) independent \(\chi^ 2_ 1\) random variables. The result shows that the limiting distribution of \(nV_ n({\hat \lambda})\) is affected by substitution of \({\hat \lambda}\) when \(\lambda\) is unknown. In addition, the analogous result for U-statistics with estimators in their kernel is obtained. Examples are also given.
scientific article; zbMATH DE number 4001184