Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).     

Estimating Mixing Densities in Exponential Family Models for Discrete Variables

This paper is concerned with estimating a mixing density g using a random sample from the mixture distribution f(x)=∫f x | θ)g(θ)dθ where f(· | θ) is a known discrete exponen tial family of density functions. Recently two techniques for estimating g have been proposed. The first uses Fourier analysis and the method of kernels and the second uses orthogonal polynomials. It is known that the first technique is capable of yielding estimators that achieve (or almost achieve) the minimax convergence rate. We show that this is true for the technique based on orthogonal polynomials as well. The practical implementation of these estimators is also addressed. Computer experiments indicate that the kernel estimators give somewhat disappoint ing finite sample results. However, the orthogonal polynomial estimators appear to do much better. To improve on the finite sample performance of the orthogonal polynomial estimators, a way of estimating the optimal truncation parameter is proposed. The resultant estimators retain the convergence rates of the previous estimators and a Monte Carlo finite sample study reveals that they perform well relative to the ones based on the optimal truncation parameter.     

All Invariant Moments of the Wishart Distribution

Abstract.  In this paper, we compute moments of a Wishart matrix variate U of the form E ( Q ( U )) where Q ( u ) is a polynomial with respect to the entries of the symmetric matrix u , invariant in the sense that it depends only on the eigenvalues of the matrix u . This gives us in particular the expected value of any power of the Wishart matrix U or its inverse U ^{− 1}. For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U . This means that all moments can be obtained through the multiplication of three matrices with known entries. Practically, the moments are obtained by computer with an extremely simple Maple program.     

A Dimensional CLT for Non-Central Wilks' Lambda in Multivariate Analysis

Abstract.  We consider the non-central distribution of the classical Wilks' lambda statistic for testing the general linear hypothesis in MANOVA. We prove that as the dimension of the observation vector goes to infinity, Wilks' lambda obeys a central limit theorem under simple growth conditions on the non-centrality matrix. In one case we also prove a stronger result: the saddlepoint cumulative distribution function (CDF) approximation for the standardized version of Wilks' lambda converges uniformly on compact sets to the standard normal CDF. These theoretical results go some way towards explaining why saddlepoint approximations to the distribution of Wilks' lambda retain excellent accuracy in high-dimensional cases.     

A note on nonparametric estimation of circular conditional densities

ABSTRACT

The conditional density offers the most informative summary of the relationship between explanatory and response variables. We need to estimate it in place of the simple conditional mean when its shape is not well-behaved. A motivation for estimating conditional densities, specific to the circular setting, lies in the fact that a natural alternative of it, like quantile regression, could be considered problematic because circular quantiles are not rotationally equivariant. We treat conditional density estimation as a local polynomial fitting problem as proposed by Fan et al. [Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika. 1996;83:189–206] in the Euclidean setting, and discuss a class of estimators in the cases when the conditioning variable is either circular or linear. Asymptotic properties for some members of the proposed class are derived. The effectiveness of the methods for finite sample sizes is illustrated by simulation experiments and an example using real data.     

Practical performance of local likelihood for circular density estimation

ABSTRACT

Local likelihood has been mainly developed from an asymptotic point of view, with little attention to finite sample size issues. The present paper provides simulation evidence of how likelihood density estimation practically performs from two points of view. First, we explore the impact of the normalization step of the final estimate, second we show the effectiveness of higher order fits in identifying modes present in the population when small sample sizes are available. We refer to circular data, nevertheless it is easily seen that our findings straightforwardly extend to the Euclidean setting, where they appear to be somehow new.     

Splines as a Useful and Convenient Statistical Tool

The framework for a unified statistical theory of spline regression assuming fixed knots using the truncated polynomial or “+” function representation is presented. In particular, a partial ordering of some spline models is introduced to clarify their relationship and to indicate the hypotheses that can be tested by using either standard multiple regression procedures or a little-used conditional test developed by Hotelling (1940). The construction of spline models with polynomial pieces of different degrees is illustrated. A numerical example from a chemical experiment is given by using the GLM procedure of the statistical software package SAS (Barr et al. 1976).     

Statistical theory of shape under elliptical models via QR decomposition

The statistical shape theory via QR decomposition and based on Gaussian and isotropic models is extended in this paper to the families of non-isotropic elliptical distributions. The new shape distributions are easily computable and then the inference procedure can be studied with the resulting exact densities. An application in Biology is studied under two Kotz models, the best distribution (non-Gaussian) is selected by using a modified Bayesian information criterion (BIC)*.     

Lévy Processes,Phase-Type Distributions,and Martingales

Lévy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance, etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance gamma processes, CGMY Lévy processes (tempered stable processes), NIG (normal inverse Gaussian) Lévy processes, and hyperbolic Lévy processes. We consider here a dense class of Lévy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of Lévy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a Lévy process with reflection in 0. The solutions are in terms of roots to polynomials, and the basic equations are derived by purely probabilistic arguments using martingale optional stopping; a particularly useful martingale is the so-called Kella-Whitt martingale. Also, the relation to fluid models with a Brownian component is discussed.