Simple estimation of the mode of a multivariate density

The authors consider an estimate of the mode of a multivariate probability density using a kernel estimate drawn from a random sample. The estimate is defined by maximizing the kernel estimate over the set of sample values. The authors show that this estimate is strongly consistent and give an almost sure rate of convergence. This rate depends on the sharpness of the density near the true mode, which is measured by a peak index.     

The stochastic approximation method for the estimation of a multivariate probability density

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil [1994. On the efficiency of on-line density estimators. IEEE Trans. Inform. Theory 40, 1504–1512]. We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator.     

Weak and strong uniform consistency rates of kernel density estimates for randomly censored data

Let X₁,., X_n, be i.i.d. random variables with distribution function F, and let Y₁,.,.,Y_n be i.i.d. with distribution function G. For i = 1, 2,.,., n set δ_i, = 1 if X_i ≤ Y_i, and 0 otherwise, and X_i, = min{X_i, K_i}. A kernel-type density estimate of f, the density function of F w.r.t. Lebesgue measure on the Borel o-field, based on the censored data (δ_i, X_i), i = 1,.,.,n, is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher-order differentiability assumption on f. A procedure for relaxing such assumptions is also proposed.     

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.     

An efficient method for the estimation of multivariate moving averge models

Durbin's (1959) efficient method for the estimation of univariate moving average models is generalized to the vector case. Strong consistency and asymptotic normality of the estimator is proved. A simulation experiment is performed to illustrate the behaviour of the method in finite samples.     

Modified cross-validation in density estimation

We propose a new modified (biased) cross-validation method for adaptively determining the bandwidth in a nonparametric density estimation setup. It is shown that the method provides consistent minimizers. Some simulation results are reported on which compare the small sample behavior of the new and the classical cross-validation selectors.     

On multivariate variable-kernel density estimates for time series

Let X₁ be a strictly stationary multiple time series with values in R^d and with a common density f. Let X₁,.,.,X_n, be n consecutive observations of X₁. Let k = k_n, be a sequence of positive integers, and let H_ni be the distance from X_i to its kth nearest neighbour among X_j, j i. The multivariate variable-kernel estimate f_n, of f is defined by where K is a given density. The complete convergence of f_n, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.     

Robust recursive estimation in nonlinear time series

Aase (1983) has dealt with recursive estimation in nonlinear time series of autoregressive type including its asymptotic properties. This contribution modifies the results for the case of nonlinear time series with outliers using the principle of M-estimation from robust statistics. Strong consistency of the robust recursive estimates is preserved under corresponding assumptions. Several types of such estimates are compared by means of a numerical simulation.     

Parametric estimation under a class of multivariate distributions

In this paper the parameters of some members of a class of multivariate distributions, which was constructed by AL-Hussaini and Ateya (2003), are estimated by using the maximum likelihood and Bayes methods.     

Uniform strong consistency of histogram density estimation for dependent process

Histogram density estimator is very intuitive and easy to compute and has been widely adopted. Especially in today's big data environment, people pay more attention to the computational cost and are more willing to choose estimators with less to compute. And so, many scholars have been interested in the various estimates based on the histogram technique. Under strong mixing process, this article studies the uniform strong consistency of histogram density estimator and the convergence rate. Our conditions on the mixing coefficient and the bin width are very mild.     

Robust estimation in the multivariate normal model with variance components

Fisher consistent and Fréchet differentiable statistical functionals have been already used by Bednarski and Zontek [Robust estimation of parameters in a mixed unbalanced model. Ann Statist. 1996;24(4):1493–1510] to get a robust estimator of parameters in a two-way crossed classification mixed model. This way of robust estimation appears also in the variance components model with a commutative covariance matrix [Zmy?lony, Zontek. Robust M-estimator of parameters in variance components model. Discuss Math Probab Stat. 2002;22:61–71]. In this paper it is shown that a modification of this method does not involve any assumptions about commutation of covariance matrix. The theoretical results have been completed with computer simulation studies. Robustness of considered estimator and possibility of approximation of the estimator's distribution with some multivariate normal distribution for both model and contaminated data have been confirmed there.     

The strong uniform convergence of multivariate variable kernel estimates

We show that sup, completely as, where f is a uniformly continuous density on are independent random vectors with common density f, and f_n is the variable kernel estimate Here H_ni is the distance between X_i and its kth nearest neighbour, K is a given density satisfying some regularity conditions, and k is a sequence of integers with the property that log asn     

Consistent deconvolution in density estimation

Suppose we have n observations from X = Y + Z, where Z is a noise component with known distribution, and Y has an unknown density f. When the characteristic function of Z is nonzero almost everywhere, we show that it is possible to construct a density estimate f_n such that for all f, Iim_n| |=0.     

On the estimation of extreme directional multivariate quantiles

Abstract

In multivariate extreme value theory (MEVT), the focus is on analysis outside of the observable sampling zone, which implies that the region of interest is associated to high risk levels. This work provides tools to include directional notions into the MEVT, giving the opportunity to characterize the recently introduced directional multivariate quantiles (DMQ) at high levels. Then, an out-sample estimation method for these quantiles is given. A bootstrap procedure carries out the estimation of the tuning parameter in this multivariate framework and helps with the estimation of the DMQ. Asymptotic normality for the proposed estimator is provided and the methodology is illustrated with simulated data-sets. Finally, a real-life application to a financial case is also performed.     

On nonparametric density estimation for multivariate linear long-memory processes

We consider nonparametric estimation of the density function and its derivatives for multivariate linear processes with long-range dependence. In a first step, the asymptotic distribution of the multivariate empirical process is derived. In a second step, the asymptotic distribution of kernel density estimators and their derivatives is obtained.     

Parameter estimation for the skew Ornstein-Uhlenbeck processes based on discrete observations

Abstract

In this paper, the drift parameter estimation for the one-dimensional skew Ornstein-Uhlenbeck process is considered. We derived the moment estimator in terms of the sample moments and invariant density. Then, we proved the strong consistency and asymptotic normality. Finally, some numerical experiments are presented to show the effect of the moment estimator.     

Nonparametric density estimation from censored data

Nonparametric estimation of the probability density function f° of a lifetime distribution based on arbitrarily right-censor-ed observations from f° has been studied extensively in recent years. In this paper the density estimators from censored data that have been obtained to date are outlined. Histogram, kernel-type, maximum likelihood, series-type, and Bayesian nonparametric estimators are included. Since estimation of the hazard rate function can be considered as giving a density estimate, all known results concerning nonparametric hazard rate estimation from censored samples are also briefly mentioned.