Semiparametric posterior limits

We review the Bayesian theory of semiparametric inference following Bickel and Kleijn (2012) and Kleijn and Knapik (2013). After an overview of efficiency in parametric and semiparametric estimation problems, we consider the Bernstein-von Mises theorem (see, e.g., Le Cam and Yang (1990)) and generalize it to (LAN) regular and (LAE) irregular semiparametric estimation problems. We formulate a version of the semiparametric Bernstein-von Mises theorem that does not depend on least-favourable submodels, thus bypassing the most restrictive condition in the presentation of Bickel and Kleijn (2012). The results are applied to the (regular) estimation of the linear coefficient in partial linear regression (with a Gaussian nuisance prior) and of the kernel bandwidth in a model of normal location mixtures (with a Dirichlet nuisance prior), as well as the (irregular) estimation of the boundary of the support of a monotone family of densities (with a Gaussian nuisance prior).Comment: 47 pp., 1 figure, submitted for publication. arXiv admin note: substantial text overlap with arXiv:1007.017

Superconformal hypermultiplets

We present theories of N=2 hypermultiplets in four spacetime dimensions that are invariant under rigid or local superconformal symmetries. The target spaces of theories with rigid superconformal invariance are (4n)-dimensional {\it special} hyper-K\"ahler manifolds. Such manifolds can be described as cones over tri-Sasakian metrics and are locally the product of a flat four-dimensional space and a quaternionic manifold. The latter manifolds appear in the coupling of hypermultiplets to N=2 supergravity. We employ local sections of an Sp$(n)\times{\rm Sp}(1)$ bundle in the formulation of the Lagrangian and transformation rules, thus allowing for arbitrary coordinatizations of the hyper-K\"ahler and quaternionic manifolds

Recovery, detection and confidence sets of communities in a sparse stochastic block model

Posterior distributions for community assignment in the planted bi-section model are shown to achieve frequentist exact recovery and detection under sharp lower bounds on sparsity. Assuming posterior recovery (or detection), one may interpret credible sets (or enlarged credible sets) as consistent confidence sets. If credible levels grow to one quickly enough, credible sets can be interpreted as frequentist confidence sets without conditions on the parameters. In the regime where within-class and between-class edge-probabilities are very close, credible sets may be enlarged to achieve frequentist asymptotic coverage. The diameters of credible sets are controlled and match rates of posterior convergence.Comment: 22 pp., 2 fi

Clearing price distributions in call auctions

We propose a model for price formation in financial markets based on clearing of a standard call auction with random orders, and verify its validity for prediction of the daily closing price distribution statistically. The model considers random buy and sell orders, placed following demand- and supply-side valuation distributions; an equilibrium equation then leads to a distribution for clearing price and transacted volume. Bid and ask volumes are left as free parameters, permitting possibly heavy-tailed or very skewed order flow conditions. In highly liquid auctions, the clearing price distribution converges to an asymptotically normal central limit, with mean and variance in terms of supply/demand-valuation distributions and order flow imbalance. By means of simulations, we illustrate the influence of variations in order flow and valuation distributions on price/volume, noting a distinction between high- and low-volume auction price variance. To verify the validity of the model statistically, we predict a year's worth of daily closing price distributions for 5 constituents of the Eurostoxx 50 index; Kolmogorov-Smirnov statistics and QQ-plots demonstrate with ample statistical significance that the model predicts closing price distributions accurately, and compares favourably with alternative methods of prediction

The semiparametric Bernstein-von Mises theorem

In a smooth semiparametric estimation problem, the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior. It is shown that, under certain straightforward and interpretable conditions, the assertion of Le Cam's acclaimed, but strictly parametric, Bernstein-von Mises theorem [Univ. California Publ. Statist. 1 (1953) 277-329] holds in the semiparametric situation as well. As a consequence, Bayesian point-estimators achieve efficiency, for example, in the sense of H\'{a}jek's convolution theorem [Z. Wahrsch. Verw. Gebiete 14 (1970) 323-330]. The model is required to satisfy differentiability and metric entropy conditions, while the nuisance prior must assign nonzero mass to certain Kullback-Leibler neighborhoods [Ghosal, Ghosh and van der Vaart Ann. Statist. 28 (2000) 500-531]. In addition, the marginal posterior is required to converge at parametric rate, which appears to be the most stringent condition in examples. The results are applied to estimation of the linear coefficient in partial linear regression, with a Gaussian prior on a smoothness class for the nuisance.Comment: Published in at http://dx.doi.org/10.1214/11-AOS921 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Analysis and improvement of the vector quantization in SELP (Stochastically Excited Linear Prediction)

The Stochastically Excited Linear Prediction (SELP) algorithm is described as a speech coding method employing a two-stage vector quantization. The first stage uses an adaptive codebook which efficiently encodes the periodicity of voiced speech, and the second stage uses a stochastic codebook to encode the remainder of the excitation signal. The adaptive codebook performs well when the pitch period of the speech signal is larger than the frame size. An extension is introduced, which increases its performance for the case that the frame size is longer than the pitch period. The performance of the stochastic stage, which improves with frame length, is shown to be best in those sections of the speech signal where a high level of short-term correlations is present. It can be concluded that the SELP algorithm performs best during voiced speech where the pitch period is longer than the frame length