Bandwidth selection for local polynomial smoothing of multinomial data
The paper deals with the development of methodology for selecting local and global bandwidths for local polynomial smoothing of multinomial data. Let \(P=(P_{1}, \ldots,P_{k})^{T}\) denote the probabilities of a \(k\)-cell multinomial distribution and let \(\bar P=(\bar P_{1},\ldots,\bar P_{k})^{T}\) be cell proportions. The performance of the estimator \(\hat P^{A}(h)\) for estimating \(P^{A}=I_{A}P\) is measured by \[ MSSE_{A}(h,P)=\sum_{i\in A} E[\hat P_{i}(h)-P_{i}]^{2}, \] where \(A\) is a subset of the set of indexes \(\{1,\ldots,k\}\), \(h>0\) is the bandwidth. The optimal bandwidth is \(h_{M}^{A}= {\text{arg min}}_{h>0}(MSSE_{A}(h,P))\). A double smoothing bandwidth selection strategy involves replacing the unknown \(P\) in \(MSSE_{A}(h,P)\) by an estimate \(S_{g^{A} \bar P}\), where \(g^{A}\) is another bandwidth, \(S_{h}\) is a smoother matrix. The authors propose an algorithm for searching the optimal bandwidth. An application of the obtained bandwidth selection strategy to mine data is presented.
scientific article; zbMATH DE number 1336637