On Rates of Convergence for Bayesian Density Estimation

Abstract.  We consider the problem of estimating a compactly supported density taking a Bayesian nonparametric approach. We define a Dirichlet mixture prior that, while selecting piecewise constant densities, has full support on the Hellinger metric space of all commonly dominated probability measures on a known bounded interval. We derive pointwise rates of convergence for the posterior expected density by studying the speed at which the posterior mass accumulates on shrinking Hellinger neighbourhoods of the sampling density. If the data are sampled from a strictly positive, α -Hölderian density, with α  ∈ ( 0,1] , then the optimal convergence rate n^{− α / (2 α +1)} is obtained up to a logarithmic factor. Smoothing histograms by polygons, a continuous piecewise linear estimator is obtained that for twice continuously differentiable, strictly positive densities satisfying boundary conditions attains a rate comparable up to a logarithmic factor to the convergence rate n ^−4/5 for integrated mean squared error of kernel type density estimators.     

On an adaptive version of the Metropolis–Hastings algorithm with independent proposal distribution

In this paper, we present a general formulation of an algorithm, the adaptive independent chain (AIC), that was introduced in a special context in Gåsemyr et al . [ Methodol. Comput. Appl. Probab. 3 (2001)]. The algorithm aims at producing samples from a specific target distribution Π, and is an adaptive, non-Markovian version of the Metropolis–Hastings independent chain. A certain parametric class of possible proposal distributions is fixed, and the parameters of the proposal distribution are updated periodically on the basis of the recent history of the chain, thereby obtaining proposals that get ever closer to Π. We show that under certain conditions, the algorithm produces an exact sample from Π in a finite number of iterations, and hence that it converges to Π. We also present another adaptive algorithm, the componentwise adaptive independent chain (CAIC), which may be an alternative in particular in high dimensions. The CAIC may be regarded as an adaptive approximation to the Gibbs sampler updating parametric approximations to the conditionals of Π.     

Estimation of global temperature fields from scattered observations by a spherical-wavelet-based spatially adaptive method

Summary.  The paper considers the problem of estimating the entire temperature field for every location on the globe from scattered surface air temperatures observed by a network of weather-stations. Classical methods such as spherical harmonics and spherical smoothing splines are not efficient in representing data that have inherent multiscale structures. The paper presents an estimation method that can adapt to the multiscale characteristics of the data. The method is based on a spherical wavelet approach that has recently been developed for a multiscale representation and analysis of scattered data. Spatially adaptive estimators are obtained by coupling the spherical wavelets with different thresholding (selective reconstruction) techniques. These estimators are compared for their spatial adaptability and extrapolation performance by using the surface air temperature data.     

Modeling for Risk Assessment of Neurotoxic Effects

The regulation of noncancer toxicants, including neurotoxicants, has usually been based upon a reference dose (allowable daily intake). A reference dose is obtained by dividing a no-observed-effect level by uncertainty (safety) factors to account for intraspecies and interspecies sensitivities to a chemical. It is assumed that the risk at the reference dose is negligible, but no attempt generally is made to estimate the risk at the reference dose. A procedure is outlined that provides estimates of risk as a function of dose. The first step is to establish a mathematical relationship between a biological effect and the dose of a chemical. Knowledge of biological mechanisms and/or pharmacokinetics can assist in the choice of plausible mathematical models. The mathematical model provides estimates of average responses as a function of dose. Secondly, estimates of risk require selection of a distribution of individual responses about the average response given by the mathematical model. In the case of a normal or lognormal distribution, only an estimate of the standard deviation is needed. The third step is to define an adverse level for a response so that the probability (risk) of exceeding that level can be estimated as a function of dose. Because a firm response level often cannot be established at which adverse biological effects occur, it may be necessary to at least establish an abnormal response level that only a small proportion of individuals would exceed in an unexposed group. That is, if a normal range of responses can be established, then the probability (risk) of abnormal responses can be estimated. In order to illustrate this process, measures of the neurotransmitter serotonin and its metabolite 5-hydroxyindoleacetic acid in specific areas of the brain of rats and monkeys are analyzed after exposure to the neurotoxicant methylene-dioxymethamphetamine. These risk estimates are compared with risk estimates from the quantal approach in which animals are classified as either abnormal or not depending upon abnormal serotonin levels.     

Improved Monte Carlo inference for models with additive error

Some statistical models defined in terms of a generating stochastic mechanism have intractable distribution theory, which renders parameter estimation difficult. However, a Monte Carlo estimate of the log-likelihood surface for such a model can be obtained via computation of nonparametric density estimates from simulated realizations of the model. Unfortunately, the bias inherent in density estimation can cause bias in the resulting log-likelihood estimate that alters the location of its maximizer. In this paper a methodology for radically reducing this bias is developed for models with an additive error component. An illustrative example involving a stochastic model of molecular fragmentation and measurement is given.     

Confidence Intervals for Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

投资者信心和中国宏观经济波动——基于MIDAS混频模型

将投资者信心指数引入MIDAS混频模型之中,可以考察新冠疫情背景下投资者信心对于中国宏观经济波动的影响。基于投资者信心指数和“三驾马车”对我国季度GDP增长率进行预测,通过实证发现:引入投资者信心指数的MIDAS混频模型在预测精准度方面和基准模型进行比较,预测精准度高于未加入投资者信心的模型,均方根残差比值更小;在多元MIDAS混频模型之中,加入投资者信心指数的回归模型对我国GDP的实时预报和短期预测结果更加稳定,可为决策者提供更加精确的参考区间;宏观经济的波动对投资者信心指数变化反应,和投资、消费和进出口相比是最灵敏的。这为定量追踪宏观经济波动提供了一个新的视角。     

Approximation of density functions by orthogonal series with grouped data

Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕ_j} be a complete orthonormal sequence of functions onL ₂(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the firstm of the ϑ_j’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. Research partially supported by: CNR grant, n. 93. 00837. CT10.     

A simple,automatic and adaptive bivariate density estimator based on conditional densities

The standard approach to non-parametric bivariate density estimation is to use a kernel density estimator. Practical performance of this estimator is hindered by the fact that the estimator is not adaptive (in the sense that the level of smoothing is not sensitive to local properties of the density). In this paper a simple, automatic and adaptive bivariate density estimator is proposed based on the estimation of marginal and conditional densities. Asymptotic properties of the estimator are examined, and guidance to practical application of the method is given. Application to two examples illustrates the usefulness of the estimator as an exploratory tool, particularly in situations where the local behaviour of the density varies widely. The proposed estimator is also appropriate for use as a pilot estimate for an adaptive kernel estimate, since it is relatively inexpensive to calculate.     

多元正态分布广义方差区间估计的改进

讨论了多元正态分布广义方差的区间估计问题,给出了在覆盖率及长度上均优于最优仿射同变区间估计的改进估计．