共查询到20条相似文献,搜索用时 31 毫秒
1.
Probability Density Function Estimation Using Gamma Kernels 总被引:6,自引:0,他引:6
Song Xi Chen 《Annals of the Institute of Statistical Mathematics》2000,52(3):471-480
We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n
–4/5
x
–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators. 相似文献
2.
In this paper, we discuss the estimation of a density function based on censored data by the kernel smoothing method when the survival and the censoring times form a stationary α-mixing sequence. A Berry-Esseen type bound is derived for the kernel density estimator at a fixed point x. For practical purposes, a randomly weighted estimator of the density function is also constructed and investigated. 相似文献
3.
《Journal of computational and graphical statistics》2013,22(3):750-769
One of the main objectives of this article is to derive efficient nonparametric estimators for an unknown density fX. It is well known that the ordinary kernel density estimator has, despite several good properties, some serious drawbacks. For example, it suffers from boundary bias and it also exhibits spurious bumps in the tails. We propose a semiparametric transformation kernel density estimator to overcome these defects. It is based on a new semiparametric transformation function that transforms data to normality. A generalized bandwidth adaptation procedure is also developed. It is found that the newly proposed semiparametric transformation kernel density estimator performs well for unimodal, low, and high kurtosis densities. Moreover, it detects and estimates densities with excessive curvature (e.g., modes and valleys) more effectively than existing procedures. In conclusion, practical examples based on real-life data are presented. 相似文献
4.
Abhishek Bhattacharya David B. Dunson 《Annals of the Institute of Statistical Mathematics》2012,64(4):687-714
This article considers a broad class of kernel mixture density models on compact metric spaces and manifolds. Following a Bayesian approach with a nonparametric prior on the location mixing distribution, sufficient conditions are obtained on the kernel, prior and the underlying space for strong posterior consistency at any continuous density. The prior is also allowed to depend on the sample size n and sufficient conditions are obtained for weak and strong consistency. These conditions are verified on compact Euclidean spaces using multivariate Gaussian kernels, on the hypersphere using a von Mises-Fisher kernel and on the planar shape space using complex Watson kernels. 相似文献
5.
Toshio Honda 《Annals of the Institute of Statistical Mathematics》2009,61(2):413-439
We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators.
We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of
the kernel density estimators with the order of bandwidths fixed as h = cn
−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior
of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations.
In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study
is also carried out to examine small sample properties. 相似文献
6.
Ayanendranath Basu Bruce G. Lindsay 《Annals of the Institute of Statistical Mathematics》1994,46(4):683-705
A general class of minimum distance estimators for continuous models called minimum disparity estimators are introduced. The conventional technique is to minimize a distance between a kernel density estimator and the model density. A new approach is introduced here in which the model and the data are smoothed with the same kernel. This makes the methods consistent and asymptotically normal independently of the value of the smoothing parameter; convergence properties of the kernel density estimate are no longer necessary. All the minimum distance estimators considered are shown to be first order efficient provided the kernel is chosen appropriately. Different minimum disparity estimators are compared based on their characterizing residual adjustment function (RAF); this function shows that the robustness features of the estimators can be explained by the shrinkage of certain residuals towards zero. The value of the second derivative of theRAF at zero,A
2, provides the trade-off between efficiency and robustness. The above properties are demonstrated both by theorems and by simulations. 相似文献
7.
A. Meister 《Mathematical Methods of Statistics》2007,16(1):63-76
This paper addresses the statistical problem of density deconvolution under the condition that the density to be estimated
has compact support. We introduce a new estimation procedure, which establishes faster rates of convergence for smooth densities
as compared to the optimal rates for smooth densities with unbounded support. This framework also allows us to relax the usual
condition of known error density with non-vanishing Fourier transform, so that a nonparametric class of densities is valid;
therefore, even the shape of the noise density need not be assumed. These results can also be generalized for fast decaying
densities with unbounded support. We prove optimality of the rates in the underlying experiment and study the practical performance
of our estimator by numerical simulations.
相似文献
8.
Direct importance estimation for covariate shift adaptation 总被引:2,自引:0,他引:2
Masashi Sugiyama Taiji Suzuki Shinichi Nakajima Hisashi Kashima Paul von Bünau Motoaki Kawanabe 《Annals of the Institute of Statistical Mathematics》2008,60(4):699-746
A situation where training and test samples follow different input distributions is called covariate shift. Under covariate shift, standard learning methods such as maximum likelihood estimation are no longer consistent—weighted
variants according to the ratio of test and training input densities are consistent. Therefore, accurately estimating the
density ratio, called the importance, is one of the key issues in covariate shift adaptation. A naive approach to this task is to first estimate training and
test input densities separately and then estimate the importance by taking the ratio of the estimated densities. However,
this naive approach tends to perform poorly since density estimation is a hard task particularly in high dimensional cases.
In this paper, we propose a direct importance estimation method that does not involve density estimation. Our method is equipped
with a natural cross validation procedure and hence tuning parameters such as the kernel width can be objectively optimized.
Furthermore, we give rigorous mathematical proofs for the convergence of the proposed algorithm. Simulations illustrate the
usefulness of our approach. 相似文献
9.
Michael L. Wenocur 《Queueing Systems》1989,4(2):115-135
A variety of methods for approximating probability density functions on the positive half-line are presented and discussed. In particular, the method of moments and orthogonal expansion methods are studied. We give a new, computational proof that continuous probability densities vanishing at can be uniformly approximated by generalized hyper-exponential densities. The same denseness property is also shown to hold for families of densities expressible as sums of Erlang densitieswith common fixed rate parameter.This research was supported in part by Air Force Office of Scientific Research Contract F49620-86-C-0022. 相似文献
10.
Jeffrey D. Hart 《Statistics & probability letters》1984,2(6):363-369
The ability of a kernel density estimator to resolve modes of the underlying density is investigated. For various bimodal densities and three different kernels, the smallest sample size required for the expectation of an optimally smoothed kernel estimator to be bimodal is determined. The optimality criterion employed is equivalent to asymptotic mean integrated squared error for sufficiently smooth densities. 相似文献
11.
《Journal of multivariate analysis》1987,21(1):53-66
Analytical and numerical results are given for determining the location of the mode of a class of bivariate gamma densities as a function of the parameters. The model location for a class of bivariate gammas as considered by Kibble (1941, Sankhya A 5 137–150) is shown to satisfy a nonlinear differential equation in ϱ, the correlation coefficient for fixed shape parameter. Qualitative and asymptotic properties of the modal location are also given. Whenever the shape parameters are unequal, analytical and numerical results are used to provide a conjecture for the modal location in the general case. 相似文献
12.
In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered. 相似文献
13.
Wen-Jun Shen Hau-San Wong Quan-Wu Xiao Xin Guo Stephen Smale 《Foundations of Computational Mathematics》2014,14(5):951-984
We attempt to establish geometrical methods for amino acid sequences. To measure the similarities of these sequences, a kernel on strings is defined using only the sequence structure and a good amino acid substitution matrix (e.g. BLOSUM62). The kernel is used in learning machines to predict binding affinities of peptides to human leukocyte antigen DR (HLA-DR) molecules. On both fixed allele (Nielsen and Lund in BMC Bioinform. 10:296, 2009) and pan-allele (Nielsen et al. in Immunome Res. 6(1):9, 2010) benchmark databases, our algorithm achieves the state-of-the-art performance. The kernel is also used to define a distance on an HLA-DR allele set based on which a clustering analysis precisely recovers the serotype classifications assigned by WHO (Holdsworth et al. in Tissue Antigens 73(2):95–170, 2009; Marsh et al. in Tissue Antigens 75(4):291–455, 2010). These results suggest that our kernel relates well the sequence structure of both peptides and HLA-DR molecules to their biological functions, and that it offers a simple, powerful and promising methodology to immunology and amino acid sequence studies. 相似文献
14.
Majid Mojirsheibani 《Statistical Inference for Stochastic Processes》2006,9(1):97-107
A strong approximation of the smoothed empirical process of strictly stationary α-mixing random variables by a sequence of iid Gaussian processes will be studied.
Here, the smoothing is done via kernel density estimators. No assumptions are made on the support of the kernel; in fact,
our main results are stated for kernels with possibly an infinite support.
Received June 2003; Accepted February 2004. 相似文献
15.
Jonathan C. Marshall 《Journal of multivariate analysis》2010,101(4):949-963
In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets. 相似文献
16.
We study the problem of finding the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle
inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lower bounds showing that
these procedures attain optimal rates of aggregation. As an example, we consider aggregation of multivariate kernel density
estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically
exact sense. We prove that, for Pinsker’s kernel, the proposed aggregates are sharp asymptotically minimax simultaneously
over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex
aggregation procedure.
相似文献
17.
Evarist Gin Armelle Guillou 《Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques》2002,38(6):907
Let fn denote the usual kernel density estimator in several dimensions. It is shown that if {an} is a regular band sequence, K is a bounded square integrable kernel of several variables, satisfying some additional mild conditions ((K1) below), and if the data consist of an i.i.d. sample from a distribution possessing a bounded density f with respect to Lebesgue measure on Rd, then for some absolute constant C that depends only on d. With some additional but still weak conditions, it is proved that the above sequence of normalized suprema converges a.s. to
. Convergence of the moment generating functions is also proved. Neither of these results require f to be strictly positive. These results improve upon, and extend to several dimensions, results by Silverman [13] for univariate densities. 相似文献
18.
A.A. Balkema 《Journal of multivariate analysis》2010,101(7):1738-1754
This paper compares the shape of the level sets for two multivariate densities. The densities are positive and continuous, and have the same dependence structure. The density f is heavy-tailed. It decreases at the same rate-up to a positive constant-along all rays. The level sets {f>c} for c↓0, have a limit shape, a bounded convex set. We transform each of the coordinates to obtain a new density g with Gaussian marginals. We shall also consider densities g with Laplace, or symmetric Weibull marginal densities. It will be shown that the level sets of the new light-tailed density g also have a limit shape, a bounded star-shaped set. The boundary of this set may be written down explicitly as the solution of a simple equation depending on two positive parameters. The limit shape is of interest in the study of extremes and in risk theory, since it determines how the extreme observations in different directions relate. Although the densities f and g have the same copula-by construction-the shapes of the level sets are not related. Knowledge of the limit shape of the level sets for one density gives no information about the limit shape for the other density. 相似文献
19.
Given a sample of n observations from a density ƒ on
d, a natural estimator of ƒ(x) is formed by counting the number of points in some region
surrounding x and dividing this count by the d dimensional volume of
. This paper presents an asymptotically optimal choice for
. The optimal shape turns out to be an ellipsoid, with shape depending on x. An extension of the idea that uses a kernel function to put greater weight on points nearer x is given. Among nonnegative kernels, the familiar Bartlett-Epanechnikov kernel used with an ellipsoidal region is optimal. When using higher order kernels, the optimal region shapes are related to Lp balls for even positive integers p. 相似文献