Nonparametric density estimation for multivariate bounded data

We propose a new nonparametric estimator for the density function of multivariate bounded data. As frequently observed in practice, the variables may be partially bounded (e.g., nonnegative) or completely bounded (e.g., in the unit interval). In addition, the variables may have a point mass. We reduce the conditions on the underlying density to a minimum by proposing anonparametric approach. By using a gamma, a beta, or a local linear kernel (also called boundary kernels), in a product kernel, the suggested estimator becomes simple in implementation and robust to the well known boundary bias problem. We investigate the mean integrated squared error properties, including the rate of convergence, uniform strong consistency and asymptoticnormality. We establish consistency of the least squares cross-validation method to select optimal bandwidth parameters. A detailed simulation study investigates the performance of the estimators. Applications using lottery and corporate finance data are provided.asymmetric kernels, multivariate boundary bias, nonparametric multivariate density estimation, asymptotic properties, bandwidth selection, least squares cross- validation

Density and Hazard Rate Estimation for Censored and ?-mixing Data Using Gamma Kernels

In this paper we consider the nonparametric estimation for a density and hazard rate function for right censored ?-mixing survival time data using kernel smoothing techniques. Since survival times are positive with potentially a high concentration at zero, one has to take into account the bias problems when the functions are estimated in the boundary region. In this paper, gamma kernel estimators of the density and the hazard rate function are proposed. The estimators use adaptive weights depending on the point in which we estimate the function, and they are robust to the boundary bias problem. For both estimators, the mean squared error properties, including the rate of convergence, the almost sure consistency and the asymptotic normality are investigated. The results of a simulation demonstrate the excellent performance of the proposed estimators.Gamma kernel, Kaplan Meier, density and hazard function, mean integrated squared error, consistency, asymptotic normality.

Nonparametric Density Estimation for Multivariate Bounded Data

We propose a new nonparametric estimator for the density function of multivariate bounded data. As frequently observed in practice, the variables may be partially bounded (e.g., nonnegative) or completely bounded (e.g., in the unit interval). In addition, the variables may have a point mass. We reduce the conditions on the underlying density to a minimum by proposing a nonparametric approach. By using a gamma, a beta, or a local linear kernel (also called boundary kernels), in a product kernel, the suggested estimator becomes simple in implementation and robust to the well-known boundary bias problem. We investigate the mean integrated squared error properties, including the rate of convergence, uniform strong consistency and asymptotic normality. We establish consistency of the least squares cross-validation method to select optimal bandwidth parameters. A detailed simulation study investigates the performance of the estimators. Applications using lottery and corporate finance data are provided.Asymmetric kernels, multivariate boudnary bias, nonparametric multivariate density estimation, asymptotic properties, bandwidth selection, least squares cross-validation

Nonparametric density estimation for multivariate bounded data.

We propose a new nonparametric estimator for the density function of multivariate bounded data. As frequently observed in practice, the variables may be partially bounded (e.g., nonnegative) or completely bounded (e.g., in the unit interval). In addition, the variables may have a point mass. We reduce the conditions on the underlying density to a minimum by proposing a nonparametric approach. By using a gamma, a beta, or a local linear kernel (also called boundary kernels), in a product kernel, the suggested estimator becomes simple in implementation and robust to the well known boundary bias problem. We investigate the mean integrated squared error properties, including the rate of convergence, uniform strong consistency and asymptotic normality. We establish consistency of the least squares cross-validation method to select optimal bandwidth parameters. A detailed simulation study investigates the performance of the estimators. Applications using lottery and corporate finance data are provided.Asymmetric kernels, multivariate boundary bias, nonparametric multivariate density estimation, asymptotic properties, bandwidth selection, least squares cross-validation.

Nonparametric Density Estimation for Positive Time Series

The Gaussian kernel density estimator is known to have substantial problems for bounded random variables with high density at the boundaries. For i.i.d. data several solutions have been put forward to solve this boundary problem. In this paper we propose the gamma kernel estimator as density estimator for positive data from a stationary ?-mixing process. We derive the mean integrated squared error, almost sure convergence and asymptotic normality. In a Monte Carlo study, where we generate data from an autoregressive conditional duration model and a stochastic volatility model, we find that the gamma kernel outperforms the local linear density estimator. An application to data from financial transaction durations, realized volatility and electricity price data is provided.Gamma kernel, nonparametric density estimation, mixing process, transaction durations, realised volatility.

Asymptotic properties of the Bernstein density copula for dependent data

Copulas are extensively used for dependence modeling. In many cases the data does not reveal how the dependence can be modeled using a particular parametric copula. Nonparametric copulas do not share this problem since they are entirely data based. This paper proposes nonparametric estimation of the density copula for α-mixing data using Bernstein polynomials. We study the asymptotic properties of the Bernstein density copula, i.e., we provide the exact asymptotic bias and variance, we establish the uniform strong consistency and the asymptotic normality

Semiparametric Multivariate Density Estimation for Positive Data Using Copulas.

In this paper we estimate density functions for positive multivariate data. We propose a semiparametric approach. The estimator combines gamma kernels or local linear kernels, also called boundary kernels, for the estimation of the marginal densities with semiparametric copulas to model the dependence. This semiparametric approach is robust both to the well known boundary bias problem and the curse of dimensionality problem. We derive the mean integrated squared error properties, including the rate of convergence, the uniform strong consistency and the asymptotic normality. A simulation study investigates the finite sample performance of the estimator. We find that univariate least squares cross validation, to choose the bandwidth for the estimation of the marginal densities, works well and that the estimator we propose performs very well also for data with unbounded support. Applications in the field of finance are provided.Asymptotic properties, asymmetric kernels, boundary bias, copula, curse of dimension, least squares cross validation.

Semiparametric multivariate density estimation for positive data using copulas

In this paper we estimate density functions for positive multivariate data. We propose a semiparametric approach. The estimator combines gamma kernels or local linear kernels, also called boundary kernels, for the estimation of the marginal densities with semiparametric copulas to model the dependence. This semiparametric approach is robust both to the well known boundary bias problem and the curse of dimensionality problem. We derive the mean integrated squared error properties, including the rate of convergence, the uniform strong consistency and the asymptotic normality. A simulation study investigates the finite sample performance of the estimator. We find that univariate least squares cross validation, to choose the bandwidth for the estimation of the marginal densities, works well and that the estimator we propose performs very well also for data with unbounded support. Applications in the field of finance are provided.asymptotic properties, asymmetric kernels, boundary bias, copula, curse of dimension, least squares cross validation

Nonparametric Beta Kernel Estimator for Long Memory Time Series

The paper introduces a new nonparametric estimator of the spectral density that is given in smoothing the periodogram by the probability density of Beta random variable (Beta kernel). The estimator is proved to be bounded for short memory data, and diverges at the origin for long memory data. The convergence in probability of the relative error and Monte Carlo simulations suggest that the estimator automaticaly adapts to the long- or the short-range dependency of the process. A cross-validation procedure is also studied in order to select the nuisance parameter of the estimator. Illustrations on historical as well as most recent returns and absolute returns of the S&P500 index show the reasonable performance of the estimation, and show that the data-driven estimator is a valuable tool for the detection of long-memory as well as hidden periodicities in stock returns.spectral density, long rage dependence, nonparametric estimation

A nonparametric copula based test for conditional independence with applications to Granger causality

nonparametric tests, conditional independence, Granger non-causality, Bernstein density copula, bootstrap, finance, volatility asymmetry, leverage effect, volatility feedback effect, macroeconomics