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局部自适应密度估计研究AI 解读

Theoretical and practical aspects of the sample-point adaptive positive kernel density estimator are examined. A closed-form expression for the mean integrated squared error is obtained through the device of preprocessing the data by binning. With this expression, the exact behavior of the optimally adaptive smoothing parameter function is studied for the first time. The approach differs from most earlier techniques in that bias of the adaptive estimator remains O(h 2) and…

多元密度估计的交叉验证AI 解读

In recent years, the focus of study in smoothing parameter selection for kernel density estimation has been on the univariate case, while multivariate kernel density estimation has been largely neglected. In part, this may be due to the perception that calibrating multivariate densities is substantially more difficult. In this article, we explicitly derive and compare multivariate versions of the bootstrap method of Taylor, the least-squares cross-validation method developed…

稳健非参数回归的 L1 方法AI 解读

Consider the problem of estimating the regression function underlying a set of data that is contaminated by a heavy-tailed error distribution. The two standard approaches to such a problem are each flawed. Robust parametric least squares is appropriate only if there is good reason to believe that the underlying function has some particular form, whereas most nonparametric regression methods are asymptotically equivalent to kernel regression methods, which are not resistant…

多元密度估计：理论、实践与可视化AI 解读

Representation and Geometry of Multivariate Data. Nonparametric Estimation Criteria. Histograms: Theory and Practice. Frequency Polygons. Averaged Shifted Histograms. Kernel Density Estimators. The Curse of Dimensionality and Dimension Reduction. Nonparametric Regression and Additive Models. Special Topics. Appendices. Indexes.

摘要暂缺

可变带宽核密度估计AI 解读

We investigate some of the possibilities for improvement of univariate and multivariate kernel density estimates by varying the window over the domain of estimation, pointwise and globally. Two general approaches are to vary the window width by the point of estimation and by point of the sample observation. The first possibility is shown to be of little efficacy in one variable. In particular, nearest-neighbor estimators in all versions perform poorly in one and two…

摘要暂缺

密度估计中的有偏与无偏交叉验证AI 解读

Nonparametric density estimation requires the specification of smoothing parameters. The demands of statistical objectivity make it highly desirable to base the choice on properties of the data set. In this article we introduce some biased cross-validation criteria for selection of smoothing parameters for kernel and histogram density estimators, closely related to one investigated in Scott and Factor (1981). These criteria are obtained by estimating L 2 norms of derivatives…

平均平移直方图：有效的多维非参数密度估计量AI 解读

We introduce two nonparametric multivariate density estimators that are particularly suitable for application in interactive computing environments. These estimators are statistically comparable to kernel methods and computationally comparable to histogram methods. Asymptotic theory of the estimators is presented and examples with univariate and simulated trivariate Gaussian data are illustrated.

频率多边形：理论与应用AI 解读

In this article I investigate the theoretical properties and applications of the frequency polygon, which is constructed by connecting with straight lines the mid-bin values of a histogram. For estimating an unknown probability density function using a random sample, the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error, achieving the same rate of convergence to zero of the integrated mean squared error as…

过度平滑的非参数密度估计AI 解读

The optimal histogram for a sample of size n from a density defined on the entire real line requires at least (2n)1/3 bins, under mild smoothness conditions. Similar bounds exist for the frequency polygon and kernel estimators. Values near these bounds give nearly optimal results for a variety of smooth densities, providing good, quick density estimates.

三种数据驱动非参数概率密度估计方法的蒙特卡洛研究AI 解读

Although the theoretical properties of modern nonparametric probability density estimators have been studied for 25 years, there remains the practical problem of how to specify the amount of bias or smoothing in a density estimate based on a random sample. In this paper we review and evaluate three recently developed data-based algorithms that completely specify a density estimate from a random sample. Using Monte Carlo techniques, we compare the statistical accuracy of…

改进非负核密度估计量的收敛速度AI 解读

To improve the rate of decrease of integrated mean square error for nonparametric kernel density estimators beyond $0(n^{-\frac{4}{5}}),$ we must relax the constraint that the density estimate be a bonafide density function, that is, be nonnegative and integrate to one. All current methods for kernel (and orthogonal series) estimators relax the nonnegativity constraint. In this paper we show how to achieve similar improvement by relaxing the integral constraint only. This is…

最优直方图与数据驱动直方图AI 解读

In this paper the formula for the optimal histogram bin width is derived which asymptotically minimizes the integrated mean squared error. Monte Carlo methods are used to verify the usefulness of this formula for small samples. A data-based procedure for choosing the bin width parameter is proposed, which assumes a Gaussian reference standard and requires only the sample size and an estimate of the standard deviation. The sensitivity of the procedure is investigated using…