Showing papers on "Multivariate kernel density estimation published in 1975"
TL;DR: Applications of gradient estimation to pattern recognition are presented using clustering and intrinsic dimensionality problems, with the ultimate goal of providing further understanding of these problems in terms of density gradients.
Abstract: Nonparametric density gradient estimation using a generalized kernel approach is investigated. Conditions on the kernel functions are derived to guarantee asymptotic unbiasedness, consistency, and uniform consistency of the estimates. The results are generalized to obtain a simple mcan-shift estimate that can be extended in a k -nearest-neighbor approach. Applications of gradient estimation to pattern recognition are presented using clustering and intrinsic dimensionality problems, with the ultimate goal of providing further understanding of these problems in terms of density gradients.
3,487 citations
TL;DR: A modification of thekernel estimate is proposed that allows the data to play a role in this smoothing and, at the same time, retains the desirable features of the kernel estimate.
Abstract: A disadvantage of the kernel estimate of a probability density is that the degree of smoothing about the observation points is chosen without regard to the data. A modification of the kernel estimate is proposed that allows the data to play a role in this smoothing and, at the same time, retains the desirable features of the kernel estimate.
70 citations
TL;DR: Using kernel estimates of the Parzen type, a naive sequential nonparametric density estimation procedure is developed and the asymptotic distribution structure of the stopping variable is examined.
Abstract: Using kernel estimates of the Parzen type, a naive sequential nonparametric density estimation procedure is developed. The asymptotic distribution structure of the stopping variable is examined. The stopping variable is shown to have finite moments of ail order and is shown to be dosed. The stopping variable N depends on some preassigned error \varepsilon , and it is shown that N diverges strongly to \infty as \varepsilon converges to zero. Finally, with \hat{f}_n(x) being a kernel-type estimator, it is shown that \hat{f}_N(X) converges to f(x) , the true density at x , with probability one as \varepsilon converges to zero.
28 citations
Abstract: K. S. Lii and M. Rosenblatt. Asymptotic behavior of a spline estimate of a density function. Comput. Math. Appl., 1(2):223–235, 1975. Reprinted with permission of Elsevier Inc.
17 citations
20 May 1975
TL;DR: The approach adopted is interval analysis, oriented so that the end computations can be easily and economically performed on modern computers and the ideas involved are unified by using the Hilbert spaces.
Abstract: In many real problems, such as in physical, biological, socio-economic sciences, medicine and certain natural sciences, one is faced with probabilistic models To completely specify these models one must know the form of either probability distribution or density function In practice one does not know the exact forms, rather one has to approximate these functions from given data A great number of researchers have investigated different aspects of this problem In this paper we concentrate on two broad classes of density estimators, namely kernel and spline function based estimators The approach adopted is interval analysis, oriented so that the end computations can be easily and economically performed on modern computers In order to understand clearly and simply, the ideas involved are unified by using the Hilbert spaces
6 citations
TL;DR: The rate at which the mean square error decreases as sample size increases is evaluated for general $L 1$ kernel estimates and for the Fourier integral estimate for a probability density function as discussed by the authors.
Abstract: The rate at which the mean square error decreases as sample size increases is evaluated for general $L^1$ kernel estimates and for the Fourier integral estimate for a probability density function. The estimates are then compared on the basis of these rates.
TL;DR: In this paper, the Galton-Watson branching process is considered and estimators for the offspring probabilities and probability generating functions and for the extinction probability are defined for the non-subcritical case and the subcritical case.
Abstract: Estimators for the offspring probabilities and probability generating functions and for the extinction probability are defined for the Galton-Watson branching process. These estimators are shown to be conditionally consistent and asymptotically normal in the non-subcritical case and to lack these properties in the subcritical case. Corresponding questions are considered when immigration is permitted.