8,015 research outputs found
Spatial aggregation of local likelihood estimates with applications to classification
This paper presents a new method for spatially adaptive local (constant)
likelihood estimation which applies to a broad class of nonparametric models,
including the Gaussian, Poisson and binary response models. The main idea of
the method is, given a sequence of local likelihood estimates (``weak''
estimates), to construct a new aggregated estimate whose pointwise risk is of
order of the smallest risk among all ``weak'' estimates. We also propose a new
approach toward selecting the parameters of the procedure by providing the
prescribed behavior of the resulting estimate in the simple parametric
situation. We establish a number of important theoretical results concerning
the optimality of the aggregated estimate. In particular, our ``oracle'' result
claims that its risk is, up to some logarithmic multiplier, equal to the
smallest risk for the given family of estimates. The performance of the
procedure is illustrated by application to the classification problem. A
numerical study demonstrates its reasonable performance in simulated and
real-life examples.Comment: Published in at http://dx.doi.org/10.1214/009053607000000271 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Intersection Bounds: Estimation and Inference
We develop a practical and novel method for inference on intersection bounds,
namely bounds defined by either the infimum or supremum of a parametric or
nonparametric function, or equivalently, the value of a linear programming
problem with a potentially infinite constraint set. We show that many bounds
characterizations in econometrics, for instance bounds on parameters under
conditional moment inequalities, can be formulated as intersection bounds. Our
approach is especially convenient for models comprised of a continuum of
inequalities that are separable in parameters, and also applies to models with
inequalities that are non-separable in parameters. Since analog estimators for
intersection bounds can be severely biased in finite samples, routinely
underestimating the size of the identified set, we also offer a
median-bias-corrected estimator of such bounds as a by-product of our
inferential procedures. We develop theory for large sample inference based on
the strong approximation of a sequence of series or kernel-based empirical
processes by a sequence of "penultimate" Gaussian processes. These penultimate
processes are generally not weakly convergent, and thus non-Donsker. Our
theoretical results establish that we can nonetheless perform asymptotically
valid inference based on these processes. Our construction also provides new
adaptive inequality/moment selection methods. We provide conditions for the use
of nonparametric kernel and series estimators, including a novel result that
establishes strong approximation for any general series estimator admitting
linearization, which may be of independent interest
Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates
We establish optimal convergence rates for a decomposition-based scalable
approach to kernel ridge regression. The method is simple to describe: it
randomly partitions a dataset of size N into m subsets of equal size, computes
an independent kernel ridge regression estimator for each subset, then averages
the local solutions into a global predictor. This partitioning leads to a
substantial reduction in computation time versus the standard approach of
performing kernel ridge regression on all N samples. Our two main theorems
establish that despite the computational speed-up, statistical optimality is
retained: as long as m is not too large, the partition-based estimator achieves
the statistical minimax rate over all estimators using the set of N samples. As
concrete examples, our theory guarantees that the number of processors m may
grow nearly linearly for finite-rank kernels and Gaussian kernels and
polynomially in N for Sobolev spaces, which in turn allows for substantial
reductions in computational cost. We conclude with experiments on both
simulated data and a music-prediction task that complement our theoretical
results, exhibiting the computational and statistical benefits of our approach
Accurate Short-Term Yield Curve Forecasting using Functional Gradient Descent
We propose a multivariate nonparametric technique for generating reliable shortterm historical yield curve scenarios and confidence intervals. The approach is based on a Functional Gradient Descent (FGD) estimation of the conditional mean vector and covariance matrix of a multivariate interest rate series. It is computationally feasible in large dimensions and it can account for non-linearities in the dependence of interest rates at all available maturities. Based on FGD we apply filtered historical simulation to compute reliable out-of-sample yield curve scenarios and confidence intervals. We back-test our methodology on daily USD bond data for forecasting horizons from 1 to 10 days. Based on several statistical performance measures we find significant evidence of a higher predictive power of our method when compared to scenarios generating techniques based on (i) factor analysis, (ii) a multivariate CCC-GARCH model, or (iii) an exponential smoothing covariances estimator as in the RiskMetricsTM approach.Conditional mean and variance estimation, Filtered Historical Simulation, Functional Gradient Descent, Term structure; Multivariate CCC-GARCH models
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