Local maxima of the sample functions of the N-parameter Bessel process

AbstractIn this paper we show that almost every sample function of the N-parameter Bessel process associated with the N-parameter Wiener process has a local maximum. In addition some properties related to the local maxima are investigated

Kernel density estimation for linear processes

Let X1,...,Xn be n consecutive observations of a linear process , where [mu] is a constant and {Zt} is an innovation process consisting of independent and identically distributed random variables with mean zero and finite variance. Assume that X1 has a probability density [latin small letter f with hook]. Uniform strong consistency of kernel density estimators of [latin small letter f with hook] is established, and their rates of convergence are obtained. The estimators can achieve the rate of convergence (n-1 log n)1/3 in L[infinity] norm restricted to compacts under weak conditions.kernel density estimator bandwidth linear process uniform convergence

Hazard rate estimation on random fields

AbstractConsider observations (representing lifelengths) taken on a random field indexed by lattice points. Our purpose is to estimate the hazard rate r(x), which is the rate of failure at time x for the survivors up to time x. We estimate r(x) by the nonparametric estimator constructed in terms of a kernel-type estimator for f(x) and the natural estimator for F¯(x). Under some general mixing assumptions, the limiting distribution of the estimator at multiple points is shown to be multivariate normal. The result is useful in establishing confidence bands for r(x) with x in an interval

On the First-Order Autoregressive Process with Infinite Variance

For a first-order autoregressive process Y = β Yt−1 + null where the null null 'S are i.i.d. and belong to the domain of attraction of a stable law, the strong consistency of the ordinary least-squares estimator b of β is obtained for β = 1, and the limiting distribution of b is established as a functional of a Lévy process. Generalizations to seasonal difference models are also considered. These results are useful in testing for the presence of unit roots when the null null 'S are heavy-tailed.

On the Best Unbiased Estimate for the Mean of a Short Autoregressive Time Series

A simple formula for computing the best linear unbiased estimate of the mean of an autoregressive process as well as its variance is given. Numerical results show that the estimate can have much lower variance than that of the usual sample mean.

Asymptotic normality of frequency polygons for random fields

International audienceThe purpose of this paper is to investigate asymptotic normality of the frequency polygon estimator of a stationary mixing random field indexed by multidimensional lattice points space Z(N). Appropriate choices of the bandwidths are found

Asymptotic normality of frequency polygons for random fields

International audienc

Frequency polygons for weakly dependent processes

The purpose of this paper is to investigate the frequency polygon as a density estimator for stationary strong mixing processes. Optimal bin widths which asymptotically minimize integrated mean square errors (IMSE) are derived. Under weak conditions, frequency polygons achieve the same rate of convergence to zero of the IMSE as kernel estimators. They can also attain the optimal uniform rate of convergence ((n-1logn)1/3 under general conditions. Frequency polygons thus appear to be very good density estimators with respect to both criteria of IMSE and uniform convergence.Density estimation Mixing process Bin width Frequency polygons

Kernel density estimation for random fields (density estimation for random fields)

Kernel-type estimators of the multivariate density of stationary random fields indexed by multidimensional lattice points space are investigated. Sufficient conditions for kernel estimators to converge uniformly are obtained. The estimators can attain the optimal rates L[infinity] of convergence. The results apply to a large class of spatial processes.Random field Kernel Bandwidth