On Bayes Sequential Estimation of the Location Parameter with Nonsmooth Density

We consider the problem of sequential estimation of the location parameter for a density with irregular behavior at some points (discontinuity, infinite values of the derivative, and so on). Thus, for our problem we have no finite Fisher information. In this situation, sequential estimation is usually more preferable compared to estimation based on samples of fixed size. In this paper, we establish the asymptotic efficiency of the Bayes sequential estimation plans and find their limit distribution. Bibliography: 14 titles.     

Estimating the support of multivariate densities under measurement error

We consider the problem of estimating the support of a multivariate density based on contaminated data. We introduce an estimator, which achieves consistency under weak conditions on the target density and its support, respecting the assumption of a known error density. Especially, no smoothness or sharpness assumptions are needed for the target density. Furthermore, we derive an iterative and easily computable modification of our estimation and study its rates of convergence in a special case; a numerical simulation is given.     

Discrete–time nonlinear filtering with marked point process observations: ii. risk–sensitive filters

We discuss in this article the risk–sensitive filtering problem of estimating a nonlinear signal process, with nonadditive non–Gaussian noise, via a marked point process observation. This extends to the risk sensitive case all the risk–neutral results studied in Dufour and Kannan [2].By going into a change of measure, we derive the unnormalized conditional density of the signal conditioned on the observation history. We also derive the unnormalized prediction density. Using these, we present two separate expressions for the optimal estimate of the signal. A similar analysis of the smoothing density of the signal is also studied under both the risk–sensitive and risk–neutral cases. We specialize the above optimal estimation to the linear signal dynamics and marked point process observation under some Gaussian assumptions. We obtain a Kalman type risk-sensitive filter. Due to the special nature of the observation process, the conditional mean and covariance estimates directly depend now on the point process     

On a boundary value problem for delay differential equations of population dynamics and Chebyshev approximation

In this paper, a boundary value problem for delay differential equations of population dynamics is considered. We obtain approximate solutions by using Chebyshev polynomial series and Newton–Raphson's procedure and give the error estimation. The method of the error estimation has been obtained in an existence theorem proved by a part of the authors. We carry out some numerical experiments by a computer language MATLAB.     

The problem of estimation in continuous/discrete dynamic systems

We consider the problem of optimal nonlinear estimation in a continuous/discrete dynamic system whose state vector is a piecewise-continuous function and the observations are represented by a collection of continuous and discrete processes. We obtain equations that determine the piecewise-continuous conditional probability density function of the process being estimated, on the basis of which we form optimal estimates, as well as an exact representation of the solution of these equations and corresponding estimation algorithms for the problem of linear optimal continuous/discrete estimation. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Asymptotic bounds on the quality of the nonparametric regression estimation in

We consider the problem of the estimation of the regression function from the observations where the selection is in the available statistic, while the noises, are conditionally independent for fixed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 88–101, 1980.In conclusion, we note that the method of obtaining a lower bound, applied in Sec. 3, is similar to the method of [9], applied for the estimation of the density. This method can be used also for other problems of nonparametric estimation.     

Spectral estimation of the Lévy density in partially observed affine models

The problem of estimating the Lévy density of a partially observed multidimensional affine process from low-frequency and mixed-frequency data is considered. The estimation methodology is based on the log-affine representation of the conditional characteristic function of an affine process and local linear smoothing in time. We derive almost sure uniform rates of convergence for the estimated Lévy density both in mixed-frequency and low-frequency setups and prove that these rates are optimal in the minimax sense. Finally, the performance of the estimation algorithms is illustrated in the case of the Bates stochastic volatility model.     

Estimation of Kullback–Leibler Divergence by Local Likelihood

Motivated from the bandwidth selection problem in local likelihood density estimation and from the problem of assessing a final model chosen by a certain model selection procedure, we consider estimation of the Kullback–Leibler divergence. It is known that the best bandwidth choice for the local likelihood density estimator depends on the distance between the true density and the ‘vehicle’ parametric model. Also, the Kullback–Leibler divergence may be a useful measure based on which one judges how far the true density is away from a parametric family. We propose two estimators of the Kullback-Leibler divergence. We derive their asymptotic distributions and compare finite sample properties. Research of Young Kyung Lee was supported by the Brain Korea 21 Projects in 2004. Byeong U. Park’s research was supported by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.     

Nonparametric estimation for irregularly sampled Lévy processes

We consider nonparametric statistical inference for Lévy processes sampled irregularly, at low frequency. The estimation of the jump dynamics as well as the estimation of the distributional density are investigated. Non-asymptotic risk bounds are derived and the corresponding rates of convergence are discussed under global as well as local regularity assumptions. Moreover, minimax optimality is proved for the estimator of the jump measure. Some numerical examples are given to illustrate the practical performance of the estimation procedure.     

On nonlinear TAR processes and threshold estimation

We consider the problem of threshold estimation for autoregressive time series with a ??space switching?? in the situation when the regression is nonlinear and the innovations have a smooth, possibly non-Gaussian, probability density. Assuming that the unknown threshold parameter is sampled from a continuous positive prior density, we find the asymptotic distribution of the Bayes estimator. As is usual in the singular estimation problems, the sequence of Bayes estimators is asymptotically efficient, attaining the minimax risk lower bound.     

The normal approximation rate for the drift estimator of multidimensional diffusions

We consider the problem of the density and drift estimation by the observation of a trajectory of an \mathbbR^d{\mathbb{R}^{d}}-dimensional homogeneous diffusion process with a unique invariant density. We construct estimators of the kernel type based on discretely sampled observations and study their asymptotic distribution. An estimate of the rate of normal approximation is given.     

Nonparametric estimation of the anisotropic probability density of mixed variables

The problem of nonparametric estimation of the joint probability density of a vector of continuous and ordinal/nominal categorical random variables with bounded support is considered. There are numerous publications devoted to the cases of either continuous or categorical variables, and the curse of dimensionality and strong regularity assumptions are the two familiar issues in the literature. Mixed variables occur in practically all applications of the statistical science and, nonetheless, the literature devoted to the joint density estimation is practically next to none. This paper develops the theory of estimation of the density of mixed variables which is on par with results known for simpler settings. Specifically, a data-driven estimator is developed that adapts to unknown anisotropic smoothness of the joint density and, whenever the density depends on a smaller number of variables, performs a dimension reduction that implies the corresponding optimal rate of the mean integrated squared error (MISE) convergence. The results hold without traditional, in the density estimation literature, minimal regularity assumptions like differentiability or continuity of the density. The procedure of estimation is based on mimicking an oracle-estimator that knows the underlying density, and the main theoretical result is the oracle inequality which relates the MISEs of the estimator and the oracle-estimator. The proof is based on a new exponential inequality for Sobolev statistics which is of interest on its own merits.     

Multivariate density estimation using dimension reducing information and tail flattening transformations

We propose a nonparametric multiplicative bias corrected transformation estimator designed for heavy tailed data. The multiplicative correction is based on prior knowledge and has a dimension reducing effect at the same time as the original dimension of the estimation problem is retained. Adding a tail flattening transformation improves the estimation significantly-particularly in the tail-and provides significant graphical advantages by allowing the density estimation to be visualized in a simple way. The combined method is demonstrated on a fire insurance data set and in a data-driven simulation study.     

Nonparametric inference for the spectral measure of a bivariate pure-jump semimartingale

We develop a nonparametric estimator for the spectral density of a bivariate pure-jump Itô semimartingale from high-frequency observations of the process on a fixed time interval with asymptotically shrinking mesh of the observation grid. The process of interest is locally stable, i.e., its Lévy measure around zero is like that of a time-changed stable process. The spectral density function captures the dependence between the small jumps of the process and is time invariant. The estimation is based on the fact that the characteristic exponent of the high-frequency increments, up to a time-varying scale, is approximately a convolution of the spectral density and a known function depending on the jump activity. We solve the deconvolution problem in Fourier transform using the empirical characteristic function of locally studentized high-frequency increments and a jump activity estimator.     

The root–unroot algorithm for density estimation as implemented via wavelet block thresholding

We propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem. This regression problem is created by binning the original observations into many small size bins, and by then applying a suitable form of root transformation to the binned data counts. In principle many common nonparametric regression estimators could then be applied to the transformed data. We propose use of a wavelet block thresholding estimator in this paper. Finally, the estimated regression function is un-rooted by squaring and normalizing. The density estimation procedure achieves simultaneously three objectives: computational efficiency, adaptivity, and spatial adaptivity. A numerical example and a practical data example are discussed to illustrate and explain the use of this procedure. Theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. There are three key steps in the technical argument: Poissonization, quantile coupling, and oracle risk bound for block thresholding in the non-Gaussian setting. Some of the technical results may be of independent interest.     

Thresholding methods to estimate copula density

This paper deals with the problem of multivariate copula density estimation. Using wavelet methods we provide two shrinkage procedures based on thresholding rules for which knowledge of the regularity of the copula density to be estimated is not necessary. These methods, said to be adaptive, have proved to be very effective when adopting the minimax and the maxiset approaches. Moreover we show that these procedures can be discriminated in the maxiset sense. We provide an estimation algorithm and evaluate its properties using simulation. Finally, we propose a real life application for financial data.     

Asymptotic Efficiency of Maximum Entropy Estimates

Doklady Mathematics - The problem of entropy estimation of probability density functions with allowance for real data is posed (the maximum entropy estimation (MEE) problem). Global existence...     

Stieltjes representation of the 3D Bruggeman effective medium and Padé approximation

The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Padé approximation method introduced in [39] to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies. The problem of reconstruction of the Stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite, is used to characterize its geometric structure. We present two different proofs of the Stieltjes analytical representation for the effective complex permittivity in the 3D Bruggeman effective medium model: one proof is based on direct calculation, the other one is the derivation of the representation using Stieltjes inversion formula. We show that the continuous spectral density in the integral representation for the Bruggeman EMA model can be efficiently approximated by a rational function. A rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition. We show results of numerical rational approximation of Bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium. The volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values.