Cramér-Rao Bound for Estimation after Model Selection and Its Application to Sparse Vector Estimation

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6 Scopus citations


In many practical parameter estimation problems, such as coefficient estimation of polynomial regression, the true model is unknown and thus, a model selection step is performed prior to estimation. The data-based model selection step affects the subsequent estimation. In particular, the oracle Cramér-Rao bound (CRB), which is based on knowledge of the true model, is inappropriate for post-model-selection performance analysis and system design outside the asymptotic region. In this paper, we investigate post-model-selection parameter estimation of a vector with an unknown support set, where this support set represents the model. We analyze the estimation performance of coherent estimators that force unselected parameters to zero. We use the mean-squared-selected-error (MSSE) criterion and introduce the concept of selective unbiasedness in the sense of Lehmann unbiasedness. We derive a non-Bayesian Cramér-Rao-type bound on the MSSE and on the mean-squared-error (MSE) of any coherent estimator with a specific selective-bias function in the Lehmann sense. We implement the selective CRB for the special case of sparse vector estimation with an unknown support set. Finally, we demonstrate in simulations that the proposed selective CRB is an informative lower bound on the performance of the maximum selected likelihood estimator for a general linear model with the generalized information criterion and for sparse vector estimation with one step thresholding. It is shown that for these cases the selective CRB outperforms the oracle CRB and Sando-Mitra-Stoica CRB (SMS-CRB) [1].

Original languageEnglish
Article number9385866
Pages (from-to)2284-2301
Number of pages18
JournalIEEE Transactions on Signal Processing
StatePublished - 1 Jan 2021


  • Non-Bayesian selective estimation
  • coherence estimation
  • estimation after model selection
  • selective Cramér-Rao bound
  • sparse vector estimation

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering


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