Likelihood Factorizations for Mixed Discrete and Continuous Variables

Some general remarks are made about likelihood factorizations, distinguishing parameter-based factorizations and concentration-graph factorizations. Two parametric families of distributions for mixed discrete and continuous variables are discussed. Conditions on graphs are given for the circumstances under which their joint analysis can be split into separate analyses, each involving a reduced set of component variables and parameters. The result shows marked differences between the two families although both involve the same necessary condition on prime graphs. This condition is both necessary and sufficient for simplified estimation in Gaussian and for discrete log linear models.     

Weighted Approximations to Continuous Time Martingales with Applications

A weighted approximation to a sequence of continuous time martingales by a time transformed Wiener process is established. The basic tool of proof is the Skorohod imbedding for martingale difference sequences. As an application of the main result a useful weighted approximation to the randomly weighted uniform empirical process is derived. A number of other applications are also discussed.     

Random Bernstein Polynomials

Random Bernstein polynomials which are also probability distribution functions on the closed unit interval are studied. The probability law of a Bernstein polynomial so defined provides a novel prior on the space of distribution functions on [0, 1] which has full support and can easily select absolutely continuous distribution functions with a continuous and smooth derivative. In particular, the Bernstein polynomial which approximates a Dirichlet process is studied. This may be of interest in Bayesian non-parametric inference. In the second part of the paper, we study the posterior from a Bernstein–Dirichlet prior and suggest a hybrid Monte Carlo approximation of it. The proposed algorithm has some aspects of novelty since the problem under examination has a changing dimension parameter space.     

Beta-Bernstein Smoothing for Regression Curves with Compact Support

ABSTRACT. The problem of boundary bias is associated with kernel estimation for regression curves with compact support. This paper proposes a simple and uni(r)ed approach for remedying boundary bias in non-parametric regression, without dividing the compact support into interior and boundary areas and without applying explicitly different smoothing treatments separately. The approach uses the beta family of density functions as kernels. The shapes of the kernels vary according to the position where the curve estimate is made. Theyare symmetric at the middle of the support interval, and become more and more asymmetric nearer the boundary points. The kernels never put any weight outside the data support interval, and thus avoid boundary bias. The method is a generalization of classical Bernstein polynomials, one of the earliest methods of statistical smoothing. The proposed estimator has optimal mean integrated squared error at an order of magnitude n ^−4/5, equivalent to that of standard kernel estimators when the curve has an unbounded support.     

Understanding exponential smoothing via kernel regression

Exponential smoothing is the most common model-free means of forecasting a future realization of a time series. It requires the specification of a smoothing factor which is usually chosen from the data to minimize the average squared residual of previous one-step-ahead forecasts. In this paper we show that exponential smoothing can be put into a nonparametric regression framework and gain some interesting insights into its performance through this interpretation. We also use theoretical developments from the kernel regression field to derive, for the first time, asymptotic properties of exponential smoothing forecasters.     

Biased Bootstrap Methods for Reducing the Effects of Contamination

Contamination of a sampled distribution, for example by a heavy-tailed distribution, can degrade the performance of a statistical estimator. We suggest a general approach to alleviating this problem, using a version of the weighted bootstrap. The idea is to 'tilt' away from the contaminated distribution by a given (but arbitrary) amount, in a direction that minimizes a measure of the new distribution's dispersion. This theoretical proposal has a simple empirical version, which results in each data value being assigned a weight according to an assessment of its influence on dispersion. Importantly, distance can be measured directly in terms of the likely level of contamination, without reference to an empirical measure of scale. This makes the procedure particularly attractive for use in multivariate problems. It has several forms, depending on the definitions taken for dispersion and for distance between distributions. Examples of dispersion measures include variance and generalizations based on high order moments. Practicable measures of the distance between distributions may be based on power divergence, which includes Hellinger and Kullback–Leibler distances. The resulting location estimator has a smooth, redescending influence curve and appears to avoid computational difficulties that are typically associated with redescending estimators. Its breakdown point can be located at any desired value ε∈ (0, ½) simply by 'trimming' to a known distance (depending only on ε and the choice of distance measure) from the empirical distribution. The estimator has an affine equivariant multivariate form. Further, the general method is applicable to a range of statistical problems, including regression.     

A Goodness-of-fit Test for Exponentiality Based on the Memoryless Property

In this investigation a test of goodness of fit for exponentiality is proposed. This procedure applies equally whether the scale and/or the location parameters of the distribution are known or not. The limiting null and non-null distributions of the test statistic are normal under minimal conditions. Monte Carlo critical values for small sample sizes are given and the power of the test is calculated for various alternatives showing that it compares favourably relatively to other more complicated published procedures.     

A modified Kolmogorov-Smirnov test for a rectangular distribution with unknown parameters: Computation of the distribution of the test statistic

A recursive scheme for the calculation of the distribution of the test statistic of a modified Kolmogorov-Smirnov-test for a rectangular distribution with unknown parameters is given.     

Bayesian estimation for the Pareto income distribution

The Bayes estimators of the Gini index, the mean income and the proportion of the population living below a prescribed income level are obtained in this paper on the basis of censored income data from a pareto income distribution. The said estimators are obtained under the assumptions of a two-parameter exponential prior distribution and the usual squared error loss function. This work is also extended to the case when the income data are grouped and the exact incomes for the individuals in the population are not available. The method for the assessment of the hyperparameters is also outlined. Finally, the results are generalized for the doubly truncated gamma prior distribution. Now deceased.     

伽马分布的形状参数的一种估计

本文利用服从伽马分布的随机变量 X 与其逆 X~(-1)的协方差只与形状参数有关这一性质,给出伽马分布形状参数的所谓自逆协方差估计,进而构造了相应的无偏估计,并证明了这类估计的大样本性质:强相合性以及渐近正态性。