Bayesian density estimation using bernstein polynomials

We propose a Bayesian nonparametric procedure for density estimation, for data in a closed, bounded interval, say [0,1]. To this aim, we use a prior based on Bemstein polynomials. This corresponds to expressing the density of the data as a mixture of given beta densities, with random weights and a random number of components. The density estimate is then obtained as the corresponding predictive density function. Comparison with classical and Bayesian kernel estimates is provided. The proposed procedure is illustrated in an example; an MCMC algorithm for approximating the estimate is also discussed.     

A class of random polynomials with an invariance property

A class of polynomials generalizing the polynomial score for a location parameter and possessing a certain invariance property is studied. It is shown that the reciprocal of the variance of such polynomials is super-additive, a result similar to Stam's inequality.     

On the grouped selection and model complexity of the adaptive elastic net

Lasso proved to be an extremely successful technique for simultaneous estimation and variable selection. However lasso has two major drawbacks. First, it does not enforce any grouping effect and secondly in some situation lasso solutions are inconsistent for variable selection. To overcome this inconsistency adaptive lasso is proposed where adaptive weights are used for penalizing different coefficients. Recently a doubly regularized technique namely elastic net is proposed which encourages grouping effect i.e. either selection or omission of the correlated variables together. However elastic net is also inconsistent. In this paper we study adaptive elastic net which does not have this drawback. In this article we specially focus on the grouped selection property of adaptive elastic net along with its model selection complexity. We also shed some light on the bias-variance tradeoff of different regularization methods including adaptive elastic net. An efficient algorithm was proposed in the line of LARS-EN, which is then illustrated with simulated as well as real life data examples.     

An algebraic and Bayesian technology for object recognition

This paper presents a new robust, low computational cost technology for recognizing free-form objects in three-dimensional (3D) range data, or, in two dimensional (2D) curve data in the image plane. Objects are represented by implicit polynomials (i.e. 3D algebraic surfaces or 2D algebraic curves) of degree greater than two, and are recognized by computing and matching vectors of their algebraic invariants (which are functions of their coefficients that are invariant to translations, rotations and general linear transformations). Such polynomials of the fourth degree can represent objects considerably more complicated than quadrics and super-quadrics, and can realize object recognition at significantly lower computational cost. Unfortunately, the coefficients of high degree implicit polynomials are highly sensitive to small changes in the data to which the polynomials are fit, thus often making recognition based on these polynomial coefficients or their invariants unreliable. We take two approaches to the problem: one involves restricting the polynomials to those which represent bounded curves and surfaces, and the other approach is to use Bayesian recognizers. The Bayesian recognizers are amazingly stable and reliable, even when the polynomials have unbounded zero sets and very large coefficient variability. The Bayesian recognizers are a unique interplay of algebraic functions and statistical methods. In this paper, we present these recognizers and show that they work effectively, even when data are missing along a large portion of an object boundary due, for example, to partial occlusion.     

An algebraic and Bayesian technology for object recognition

This paper presents a new robust, low computational cost technology for recognizing free-form objects in three-dimensional (3D) range data, or, in two dimensional (2D) curve data in the image plane. Objects are represented by implicit polynomials (i.e. 3D algebraic surfaces or 2D algebraic curves) of degree greater than two, and are recognized by computing and matching vectors of their algebraic invariants (which are functions of their coefficients that are invariant to translations, rotations and general linear transformations). Such polynomials of the fourth degree can represent objects considerably more complicated than quadrics and super-quadrics, and can realize object recognition at significantly lower computational cost. Unfortunately, the coefficients of high degree implicit polynomials are highly sensitive to small changes in the data to which the polynomials are fit, thus often making recognition based on these polynomial coefficients or their invariants unreliable. We take two approaches to the problem: one involves restricting the polynomials to those which represent bounded curves and surfaces, and the other approach is to use Bayesian recognizers. The Bayesian recognizers are amazingly stable and reliable, even when the polynomials have unbounded zero sets and very large coefficient variability. The Bayesian recognizers are a unique interplay of algebraic functions and statistical methods. In this paper, we present these recognizers and show that they work effectively, even when data are missing along a large portion of an object boundary due, for example, to partial occlusion.     

Minimax weights for generalised M‐estimation in biased regression models

The authors consider the construction of weights for Generalised M‐estimation. Such weights, when combined with appropriate score functions, afford protection from biases arising through incorrectly specified response functions, as well as from natural variation. The authors obtain minimax fixed weights of the Mallows type under the assumption that the density of the independent variables is correctly specified, and they obtain adaptive weights when this assumption is relaxed. A simulation study indicates that one can expect appreciable gains in precision when the latter weights are used and the various sources of model uncertainty are present.     

EXPECTATIONS OF RATIOS OF QUADRATIC FORMS IN NORMAL VARIABLES: EVALUATING SOME TOP-ORDER INVARIANT POLYNOMIALS

Chikuse's (1987) algorithm constructs top-order invariant polynomials with multiple matrix arguments. Underlying it is a set of simultaneous equations for which all integer solutions must be found. Each solution represents a component of the sum of terms which comprise the polynomial. The system of equations has a specialised structure which may be exploited to obtain a polynomial with r matrix arguments in terms of a polynomial with r-1 matrix arguments. This is demonstrated for two particular polynomials that have two matrix arguments. These results are applied to problems involving expectations of ratios of quadratic forme in normal variables; analytic as well as computable formulae are derived.     

Stability of the roots of random algebraic polynomials

In this paper we consider the behavior of the roots of random algebraic polynomials. A code was developed which generates a sample of random algebraic polynomials, calculates the roots of each sample polynomial, and then calculates the averages of the roots. Finally, the roots of the deterministic algebraic polynomial whose coefficients are the averages of the sample coefficients are calculated. These data are then tabulated and graphically displayed. The relationship between the averages of the roots of the sample polynomials and the roots of the average polynomial is discussed.     

Weighted nonparametric regression

In the fixed design regression model, additional weights are considered for the Nad a ray a-Watson and Gasser-Miiller kernel estimators. We study their asymptotic behavior and the relationships between new and classical estimators. For a simple family of weights, and considering the AIMSEAS global loss criterion, we show some possible theoretical advantages. An empirical study illustrates the performance of the weighted kernel estimators in theoretical ideal situations and in simulated data sets. Also some results concerning the use of weights for local polynomial estimators are given.     

Bayesian fractional polynomials

This paper sets out to implement the Bayesian paradigm for fractional polynomial models under the assumption of normally distributed error terms. Fractional polynomials widen the class of ordinary polynomials and offer an additive and transportable modelling approach. The methodology is based on a Bayesian linear model with a quasi-default hyper-g prior and combines variable selection with parametric modelling of additive effects. A Markov chain Monte Carlo algorithm for the exploration of the model space is presented. This theoretically well-founded stochastic search constitutes a substantial improvement to ad hoc stepwise procedures for the fitting of fractional polynomial models. The method is applied to a data set on the relationship between ozone levels and meteorological parameters, previously analysed in the literature.     

Bayesian curve estimation by polynomial of random order

A Bayesian method of estimating an unknown regression curve by a polynomial of random order is proposed. A joint distribution is assigned over both the set of possible orders of the polynomial and the polynomial coefficients. Reversible jumps Markov chain Monte Carlo (MCMC) (Green, Biometrika 82 (1995) 711-32), are used to compute required posteriors. The methodology is extended to polynomials of random order with discontinuities and to piecewise polynomials of random order to handle wiggly curves. The effectiveness of the methodology is illustrated with a number of examples.     

Tailoring the Gaussian Law for Excess Kurtosis and Skewness by Hermite Polynomials

This article aims at reshaping the normal law to account for tail-thickness and asymmetry, of which there is plenty of evidence in financial data. The inspiration to address the issue was provided by the orthogonality of Hermite polynomials with the Gaussian density as a weight function, with the Gram–Charlier expansion as background. A solution is then devised accordingly, by embodying skewness and excess-kurtosis in a normal kernel, via third- and forth-degree polynomial tune-up. Features of the densities so obtained are established in the main theorem of this article. In addition, a glance is cast at the issue of embodying between-squares correlation, and a solution is outlined.     

Variable selection of the quantile varying coefficient regression models

As a useful supplement to mean regression, quantile regression is a completely distribution-free approach and is more robust to heavy-tailed random errors. In this paper, a variable selection procedure for quantile varying coefficient models is proposed by combining local polynomial smoothing with adaptive group LASSO. With an appropriate selection of tuning parameters by the BIC criterion, the theoretical properties of the new procedure, including consistency in variable selection and the oracle property in estimation, are established. The finite sample performance of the newly proposed method is investigated through simulation studies and the analysis of Boston house price data. Numerical studies confirm that the newly proposed procedure (QKLASSO) has both robustness and efficiency for varying coefficient models irrespective of error distribution, which is a good alternative and necessary supplement to the KLASSO method.     

Combination-Based Permutation Tests: Equipower Property and Power Behavior in Presence of Correlation

Multivariate combination-based permutation tests have been widely used in many complex problems. In this paper we focus on the equipower property, derived directly from the finite-sample consistency property, and we analyze the impact of the dependency structure on the combined tests. At first, we consider the finite-sample consistency property which assumes that sample sizes are fixed (and possibly small) and considers on each subject a large number of informative variables. Moreover, since permutation test statistics do not require to be standardized, we need not assume that data are homoscedastic in the alternative. The equipower property is then derived from these two notions: consider the unconditional permutation power of a test statistic T for fixed sample sizes, with V ? 2 independent and identically distributed variables and fixed effect δ, calculated in two ways: (i) by considering two V-dimensional samples sized m₁ and m₂, respectively; (ii) by considering two unidimensional samples sized n₁ = Vm₁ and n₂ = Vm₂, respectively. Since the unconditional power essentially depends on the non centrality induced by T, and two ways are provided with exactly the same likelihood and the same non centrality, we show that they are provided with the same power function, at least approximately. As regards both investigating the equipower property and the power behavior in presence of correlation we performed an extensive simulation study.     

Why High-Order Polynomials Should Not Be Used in Regression Discontinuity Designs

It is common in regression discontinuity analysis to control for third, fourth, or higher-degree polynomials of the forcing variable. There appears to be a perception that such methods are theoretically justified, even though they can lead to evidently nonsensical results. We argue that controlling for global high-order polynomials in regression discontinuity analysis is a flawed approach with three major problems: it leads to noisy estimates, sensitivity to the degree of the polynomial, and poor coverage of confidence intervals. We recommend researchers instead use estimators based on local linear or quadratic polynomials or other smooth functions.     

Generalized likelihood ratio tests for the structure of semiparametric additive models

Semiparametric additive models (SAMs) are very useful in multivariate nonparametric regression. In this paper, the authors study nonparametric testing problems for the nonparametric components of SAMs. Using the backfitting algorithm and the local polynomial smoothing technique, they extend to SAMs the generalized likelihood ratio tests of Fan &Jiang (2005). The authors show that the proposed tests possess the Wilks‐type property and that they can detect alternatives nearing the null hypothesis with a rate arbitrarily close to root‐n while error distributions are unspecified. They report simulations which demonstrate the Wilks phenomenon and the powers of their tests. They illustrate the performance of their approach by simulation and using the Boston housing data set.     

Extended Analysis of At Least Partially Ordered Multi‐factor ANOVA

For multifactor experimental designs in which the levels of at least one of the factors are ordered we show how to construct components that provide a deep nonparametric scrutiny of the data. The components assess generalised correlations and the resulting tests include and extend the Page and umbrella tests. Application of the tests described is straightforward. Orthonormal polynomials on the ANOVA responses and the factors are required and the formulae needed are given subsequently. These depend on the moments of the responses and of each factor and are easily calculated. Products of at least two of these orthonormal polynomials are then used as inputs into standard ANOVA routines. For example, using the first order orthonormal polynomial on factor A and the second order orthonormal polynomial on the ANOVA response will assess if, with increasing levels of factor A there is an umbrella response with either an increase and then a decrease or a decrease and then an increase.     

Addendum to fitting polynomials to main effects in multi factor experiments: A note concerning polynomial coefficients when all factors are quantitative

When all factors are quantitative, cell means may be expressed as a polynomial function of products of powers of the associated quantitative classification variables. Existence and uniqueness of this polynomial is established for unbalanced data with unequal spacing for no missing cells. The relationship between the coefficients of this polynomial and the main effects polynomials are derived for main effects which are arbitrary weighted averages of the cell means.     

A unified treatment for the asymptotic normality of the coefficients of polynomials related to orthogonal ones

Suitable transformations on the orthogonal polynomials lead to polynomials with nonnegative coefficients. In this work, the asymptotic normality for the nonnegative coefficients of these polynomials is derived based on the nature of the weight function of the orthogonal polynomials. In particular, orthogonal polynomial cases from both classical and semi-classical systems are included as well as the singular behaved Pollaczek polynomial case.     

基于自适应迭代更新的函数型数据聚类方法研究

函数型数据的稀疏性和无穷维特性使得传统聚类分析失效。针对此问题,本文在界定函数型数据概念与内涵的基础上提出了一种自适应迭代更新聚类分析。首先,基于数据参数信息实现无穷维函数空间向有限维多元空间的过渡;在此基础上,依据变量信息含量的差异构建了自适应赋权聚类统计量,并依此为函数型数据的相似性测度进行初始类别划分;进一步地,在给定阈值限制下,对所有函数的初始类别归属进行自适应迭代更新,将收敛的优化结果作为最终的类别划分。随机模拟和实证检验表明,与现有的同类函数型聚类分析相比,文中方法的分类正确率显著提高,体现了新方法的相对优良性和实际问题应用中的有效性。