Comparison between the estimated of nonparametric methods by using the methodology of quantile regression models

This paper study two stratified quantile regression models of the marginal and the conditional varieties. We estimate the quantile functions of these models by using two nonparametric methods of smoothing spline (B-spline) and kernel regression (Nadaraya-Watson). The estimates can be obtained by solve nonparametric quantile regression problem which means minimizing the quantile regression objective functions and using the approach of varying coefficient models. The main goal is discussing the comparison between the estimators of the two nonparametric methods and adopting the best one between them

Nonparametric regression analysis

textNonparametric regression uses nonparametric and flexible methods in analyzing complex data with unknown regression relationships by imposing minimum assumptions on the regression function. The theory and applications of nonparametric regression methods with an emphasis on kernel regression, smoothing spines and Gaussian process regression are reviewed in this report. Two datasets are analyzed to demonstrate and compare the three nonparametric regression models in R.Statistic

One and Two-Step Estimation of Time Variant Parameters and Nonparametric Quantiles

This dissertation develops and discusses several one-step and two-step smoothing methods of time variant nonparametric quantiles and time variant parameters from probability models. First, we investigate and develop nonparametric techniques for measuring extreme quantiles. The method involves aggregating data by an explanatory variable such as time and smoothing the resulting data with a nonparametric method like kernel, local polynomial or spline smoothing. We demonstrate both in application and simulation that this two-step procedure of quantile estimation is superior to the parametric quantile regression. We then develop a one-step method which combines the strength of maximum likelihood estimation with a local kernel function. This local maximum likelihood estimation is applied in both a discrete and continuous case of distribution, and we consider polynomial expansions of the unknown parameter in each case. In the continuous case, we choose a distribution with two parameters and iteratively solve for each to smooth the data. Results indicate that the one-step procedure can yield improvement over the corresponding two-step methods mentioned previously in both application cases and simulation exercises. We also explore nonparametric techniques for estimating volatility of financial data. We develop a residual based method for estimating the conditional variance function using local composite quantile regression, and compare this to using local least squares regression. These methods are applied on the asset returns for many individual firms, with promising results in favor of local composite quantile regression. Comparisons of these nonparametric techniques in forecasting also indicate some improvement over using a traditional autoregressive model for heteroscedastic data

Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field

Meta-analysis of functional neuroimaging data using Bayesian nonparametric binary regression

In this work we perform a meta-analysis of neuroimaging data, consisting of locations of peak activations identified in 162 separate studies on emotion. Neuroimaging meta-analyses are typically performed using kernel-based methods. However, these methods require the width of the kernel to be set a priori and to be constant across the brain. To address these issues, we propose a fully Bayesian nonparametric binary regression method to perform neuroimaging meta-analyses. In our method, each location (or voxel) has a probability of being a peak activation, and the corresponding probability function is based on a spatially adaptive Gaussian Markov random field (GMRF). We also include parameters in the model to robustify the procedure against miscoding of the voxel response. Posterior inference is implemented using efficient MCMC algorithms extended from those introduced in Holmes and Held [Bayesian Anal. 1 (2006) 145--168]. Our method allows the probability function to be locally adaptive with respect to the covariates, that is, to be smooth in one region of the covariate space and wiggly or even discontinuous in another. Posterior miscoding probabilities for each of the identified voxels can also be obtained, identifying voxels that may have been falsely classified as being activated. Simulation studies and application to the emotion neuroimaging data indicate that our method is superior to standard kernel-based methods.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS523 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org