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Statistical analysis of sparse approximate factor models. (English) Zbl 07246819
The authors consider a sequence of $$n$$ i.i.d. observations of a $$p$$-dimensional random vector $$(X_i)$$, having the factor structure $$X_i=\Lambda\,F_i+\epsilon_i$$, where $$\Lambda$$ is the loading $$p\times m$$ matrix, $$F_i$$ is the vector of centred factor variables and $$\epsilon_i$$ are the errors – the idiosyncratic variables. The dimension $$m>0$$ is known. The variance is var$$(X_i)=\Lambda\,\Lambda'+\Psi$$. Factors and idiosyncratic variables are assumed to be uniformly sub-Gaussian. The authors provide $$\ell_1$$-, $$\ell_2$$- and $$\ell_{\infty}$$-error bounds. Their approach is based on a two-step estimation: first the matrices $$\Lambda$$ and $$\Psi$$ are obtained through a Gaussian quasi-maximum likelihood (QML) estimation (in this step $$\Psi$$ is assumed to be diagonal). Conditionally on this first step estimation, the diagonality assumption on $$\Psi$$ is relaxed, and by means of various regularisers, both Gaussian QML and least squares loss function are used to obtain a sparse error covariance matrix. The support recovery property is also established. The results are supported by simulations.
##### MSC:
 62H25 Factor analysis and principal components; correspondence analysis 62J07 Ridge regression; shrinkage estimators (Lasso) 62F12 Asymptotic properties of parametric estimators
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