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Title: On the sparsity of signals in a random sample
Keywords: Large covariance matrix, Method-of-moments, Signal sequence, Sparsity
Issue Date: 2-Nov-2011
Citation: JIANG BINYAN (2011-11-02). On the sparsity of signals in a random sample. ScholarBank@NUS Repository.
Abstract: The ?large p small n? data sets are frequently encountered by various researchers during the past decades. One of the commonly used assumptions for these data sets is that the data set is sparse. Various methods have been developed in dealing with model selection, signal detection or large covariance matrix estimation. However, as far as we know, the problem of estimating the ?sparsity? has not been addressed thoroughly yet. Here loosely speaking, sparsity is interpreted as the proportion of parameters taking the value 0. Our work in this thesis contains two parts. The first part deals with estimating the sparsity of a sparse random sequence. An estimator is constructed from a sample analog of certain Hermitian trigonometric matrices. To evaluate our estimator, upper and lower bounds for the minimax convergence rate are derived. The second part deals with estimating the sparsity of a large covariance matrix or correlation matrix. This to some degree is related to the problem of finding a universal data-dependent threshold for the elements of a sample correlation matrix. We propose two estimators based on different methods. Consistency of these two estimators is proved under mild conditions. Simulation studies for both parts show that the proposed estimators can have significantly smaller mean absolute errors than their main competitors.
Appears in Collections:Ph.D Theses (Open)

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