7,140 research outputs found
The Ordered Qualitative Model For Credit Rating Transitions
Information on the expected changes in credit quality of obligors is contained in credit migration matrices which trace out the movements of firms across ratings categories in a given period of time and in a given group of bond issuers. The rating matrices provided by Moody’s, Standard &Poor’s and Fitch became crucial inputs to many applications, including the assessment of risk on corporate credit portfolios (CreditVar) and credit derivatives pricing. We propose a factor probit model for modeling and prediction of credit rating matrices that are assumed to be stochastic and driven by a latent factor. The filtered latent factor path reveals the effect of the economic cycle on corporate credit ratings, and provides evidence in support of the PIT (point-in-time) rating philosophy. The factor probit model also yields the estimates of cross-sectional correlations in rating transitions that are documented empirically but not fully accounted for in the literature and in the regulatory rules established by the Basle Committee.Credit Rating, Migration, Migration Correlation, Credit Risk, Probit Model, Latent Factor, Business Cycle.
Nonlinear shrinkage estimation of large-dimensional covariance matrices
Many statistical applications require an estimate of a covariance matrix
and/or its inverse. When the matrix dimension is large compared to the sample
size, which happens frequently, the sample covariance matrix is known to
perform poorly and may suffer from ill-conditioning. There already exists an
extensive literature concerning improved estimators in such situations. In the
absence of further knowledge about the structure of the true covariance matrix,
the most successful approach so far, arguably, has been shrinkage estimation.
Shrinking the sample covariance matrix to a multiple of the identity, by taking
a weighted average of the two, turns out to be equivalent to linearly shrinking
the sample eigenvalues to their grand mean, while retaining the sample
eigenvectors. Our paper extends this approach by considering nonlinear
transformations of the sample eigenvalues. We show how to construct an
estimator that is asymptotically equivalent to an oracle estimator suggested in
previous work. As demonstrated in extensive Monte Carlo simulations, the
resulting bona fide estimator can result in sizeable improvements over the
sample covariance matrix and also over linear shrinkage.Comment: Published in at http://dx.doi.org/10.1214/12-AOS989 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Tyler's Covariance Matrix Estimator in Elliptical Models with Convex Structure
We address structured covariance estimation in elliptical distributions by
assuming that the covariance is a priori known to belong to a given convex set,
e.g., the set of Toeplitz or banded matrices. We consider the General Method of
Moments (GMM) optimization applied to robust Tyler's scatter M-estimator
subject to these convex constraints. Unfortunately, GMM turns out to be
non-convex due to the objective. Instead, we propose a new COCA estimator - a
convex relaxation which can be efficiently solved. We prove that the relaxation
is tight in the unconstrained case for a finite number of samples, and in the
constrained case asymptotically. We then illustrate the advantages of COCA in
synthetic simulations with structured compound Gaussian distributions. In these
examples, COCA outperforms competing methods such as Tyler's estimator and its
projection onto the structure set.Comment: arXiv admin note: text overlap with arXiv:1311.059
A data driven equivariant approach to constrained Gaussian mixture modeling
Maximum likelihood estimation of Gaussian mixture models with different
class-specific covariance matrices is known to be problematic. This is due to
the unboundedness of the likelihood, together with the presence of spurious
maximizers. Existing methods to bypass this obstacle are based on the fact that
unboundedness is avoided if the eigenvalues of the covariance matrices are
bounded away from zero. This can be done imposing some constraints on the
covariance matrices, i.e. by incorporating a priori information on the
covariance structure of the mixture components. The present work introduces a
constrained equivariant approach, where the class conditional covariance
matrices are shrunk towards a pre-specified matrix Psi. Data-driven choices of
the matrix Psi, when a priori information is not available, and the optimal
amount of shrinkage are investigated. The effectiveness of the proposal is
evaluated on the basis of a simulation study and an empirical example
Robust Orthogonal Complement Principal Component Analysis
Recently, the robustification of principal component analysis has attracted
lots of attention from statisticians, engineers and computer scientists. In
this work we study the type of outliers that are not necessarily apparent in
the original observation space but can seriously affect the principal subspace
estimation. Based on a mathematical formulation of such transformed outliers, a
novel robust orthogonal complement principal component analysis (ROC-PCA) is
proposed. The framework combines the popular sparsity-enforcing and low rank
regularization techniques to deal with row-wise outliers as well as
element-wise outliers. A non-asymptotic oracle inequality guarantees the
accuracy and high breakdown performance of ROC-PCA in finite samples. To tackle
the computational challenges, an efficient algorithm is developed on the basis
of Stiefel manifold optimization and iterative thresholding. Furthermore, a
batch variant is proposed to significantly reduce the cost in ultra high
dimensions. The paper also points out a pitfall of a common practice of SVD
reduction in robust PCA. Experiments show the effectiveness and efficiency of
ROC-PCA in both synthetic and real data
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