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Estimator selection for regression functions in exponential families with application to changepoint detection

Juntong Chen
Dec 2022
摘要
We observe $n$ independent pairs of random variables $(W_{i}, Y_{i})$ forwhich the conditional distribution of $Y_{i}$ given $W_{i}=w_{i}$ belongs to aone-parameter exponential family with parameter${\mathbf{\gamma}}^{*}(w_{i})\in{\mathbb{R}}$ and our aim is to estimate theregression function ${\mathbf{\gamma}}^{*}$. Our estimation strategy is asfollows. We start with an arbitrary collection of piecewise constant candidateestimators based on our observations and by means of the same observations, weselect an estimator among the collection. Our approach is agnostic to thedependencies of the candidate estimators with respect to the data and cantherefore be unknown. From this point of view, our procedure contrasts withother alternative selection methods based on data splitting, cross validation,hold-out etc. To illustrate its theoretical performance, we establish anon-asymptotic risk bound for the selected estimator. We then explain how toapply our procedure to the changepoint detection problem in exponentialfamilies. The practical performance of the proposed algorithm is illustrated bya comparative simulation study under different scenarios and on two realdatasets from the copy numbers of DNA and British coal disasters records.
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