Abstract
In this paper we consider covariance structural models with which we associate semidefinite programming problems. We discuss statistical properties of estimates of the respective optimal value and optimal solutions when the ‘true’ covariance matrix is estimated by its sample counterpart. The analysis is based on perturbation theory of semidefinite programming. As an example we consider asymptotics of the so-called minimum trace factor analysis. We also discuss the minimum rank matrix completion problem and its SDP counterparts.
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Notes
As it was pointed before, the constraints in (2.20) are invariant with respect to replacing matrix E by matrix ET for an arbitrary nonsingular \((p-r)\times (p-r)\) matrix T. Therefore, unless stated otherwise, for the sake of computational convenience we assume that matrix E has orthonormal columns.
The index set \(\tau \subset \{1,\ldots ,p\}\times \{1,\ldots ,p\}\) is symmetric in the sense that if \((i,j)\in \tau \), then \((j,i)\in \tau \), and is formed by such (i, j) that \((i,j-n_1)\in \iota \) for \(1\le i\le n_1\) and \(n_1+1\le j\le n_1+n_2\), and the respective (j, i) otherwise.
In case of normally distributed population the covariance matrix \(\varGamma \) is completely defined by the respective matrix \(\varSigma _0\) (see (3.1)). In general, components of \(\varGamma \) can be estimated by the respective estimates of fourth order moments.
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The author is indebted to anonymous referees for constructive comments which helped to improve the manuscript.
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This research was partly supported by NSF grant 1633196 and DARPA EQUiPS program grant SNL 014150709.
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Shapiro, A. Statistical inference of semidefinite programming. Math. Program. 174, 77–97 (2019). https://doi.org/10.1007/s10107-018-1250-z
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DOI: https://doi.org/10.1007/s10107-018-1250-z
Keywords
- Semidefinite programming
- Minimum trace factor analysis
- Matrix completion problem
- Minimum rank
- Nondegeneracy
- Statistical inference
- Asymptotics