Monte Carlo Estimation of the Density of the Sum of Dependent Random Variables

We study an unbiased estimator for the density of a sum of random variables that are simulated from a computer model. A numerical study on examples with copula dependence is conducted where the proposed estimator performs favourably in terms of variance compared to other unbiased estimators. We provide applications and extensions to the estimation of marginal densities in Bayesian statistics and to the estimation of the density of sums of random variables under Gaussian copula dependence

A new kernel-based approach to system identification with quantized output data

In this paper we introduce a novel method for linear system identification with quantized output data. We model the impulse response as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. This serves as a starting point to cast our system identification problem into a Bayesian framework. We employ Markov Chain Monte Carlo methods to provide an estimate of the system. In particular, we design two methods based on the so-called Gibbs sampler that allow also to estimate the kernel hyperparameters by marginal likelihood maximization via the expectation-maximization method. Numerical simulations show the effectiveness of the proposed scheme, as compared to the state-of-the-art kernel-based methods when these are employed in system identification with quantized data.Comment: 10 pages, 4 figure

Iterative procedures for identification of nonlinear interconnected systems

This work addresses the identification problem of a discrete-time nonlinear system composed by linear and nonlinear subsystems. Systems in this class will be represented by Linear Fractional Transformations. Iterative identification procedures are examined, that alternate between the estimation of the linear and the nonlinear components. The burden of identification falls naturally on the nonlinear subsystem, as techniques for identification of linear systems have long been established. Two approaches are examined. A point-wise identification of the nonlinearity, recently proposed in the literature, is applied and its advantages and drawbacks are outlined. An alternative procedure that employs piecewise affine approximation techniques is proposed. Numerical examples demonstrate the efficiency of the proposed algorithm

Stochastic approximation of score functions for Gaussian processes

We discuss the statistical properties of a recently introduced unbiased stochastic approximation to the score equations for maximum likelihood calculation for Gaussian processes. Under certain conditions, including bounded condition number of the covariance matrix, the approach achieves $O(n)$ storage and nearly $O(n)$ computational effort per optimization step, where $n$ is the number of data sites. Here, we prove that if the condition number of the covariance matrix is bounded, then the approximate score equations are nearly optimal in a well-defined sense. Therefore, not only is the approximation efficient to compute, but it also has comparable statistical properties to the exact maximum likelihood estimates. We discuss a modification of the stochastic approximation in which design elements of the stochastic terms mimic patterns from a $2^n$ factorial design. We prove these designs are always at least as good as the unstructured design, and we demonstrate through simulation that they can produce a substantial improvement over random designs. Our findings are validated by numerical experiments on simulated data sets of up to 1 million observations. We apply the approach to fit a space-time model to over 80,000 observations of total column ozone contained in the latitude band $40^{\circ}$-$50^{\circ}$N during April 2012.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS627 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org