Generalized least squares cross‐validation in kernel density estimation

The kernel density estimation is a popular method in density estimation. The main issue is bandwidth selection, which is a well‐known topic and is still frustrating statisticians. A robust least squares cross‐validation bandwidth is proposed, which significantly improves the classical least squares cross‐validation bandwidth for its variability and undersmoothing, adapts to different kinds of densities, and outperforms the existing bandwidths in statistical literature and software.     

A number‐of‐modes reference rule for density estimation under multimodality

We consider kernel density estimation for univariate distributions. The question of interest is as follows: given that the data analyst has some background knowledge on the modality of the data (for instance, ‘data of this type are usually bimodal’), what is the adequate bandwidth to choose? We answer this question by extending Silverman's idea of ‘normal‐reference’ to that of ‘reference to a Gaussian mixture’. The concept is illustrated in the light of real data examples.     

A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation

This paper presents a Bayesian approach to bandwidth selection for multivariate kernel regression. A Monte Carlo study shows that under the average squared error criterion, the Bayesian bandwidth selector is comparable to the cross-validation method and clearly outperforms the bootstrapping and rule-of-thumb bandwidth selectors. The Bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state-price density of Arrow–Debreu securities with the S&P 500 index options data and the DAX index options data. The proposed Bayesian bandwidth selector represents a data-driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state-price density pioneered by Aït-Sahalia and Lo [Aït-Sahalia, Y., Lo, A.W., 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499, 547.]     

非线性计量经济模型的非参数估计方法研究

本文提出使用核估计的方法构造平滑转移模型(STR)的非参数模拟最大似然估计(NPSML),给出了NPSML估计量的构造方法、渐近性质以及相应的核函数和窗宽的选择准则,并利用滑动窗宽算法对估计量的构造过程进行了改进。通过Monte Carlo实验证明,该方法是可靠的,并且当误差项存在序列相关时,此种估计量是稳健的。     

A simple approach to the parametric estimation of potentially nonstationary diffusions

A simple and robust approach is proposed for the parametric estimation of scalar homogeneous stochastic differential equations. We specify a parametric class of diffusions and estimate the parameters of interest by minimizing criteria based on the integrated squared difference between kernel estimates of the drift and diffusion functions and their parametric counterparts. The procedure does not require simulations or approximations to the true transition density and has the simplicity of standard nonlinear least-squares methods in discrete time. A complete asymptotic theory for the parametric estimates is developed. The limit theory relies on infill and long span asymptotics and is robust to deviations from stationarity, requiring only recurrence.     

Fast Kernel Smoothing of Point Patterns on a Large Network using Two‐dimensional Convolution

We propose a computationally efficient and statistically principled method for kernel smoothing of point pattern data on a linear network. The point locations, and the network itself, are convolved with a two‐dimensional kernel and then combined into an intensity function on the network. This can be computed rapidly using the fast Fourier transform, even on large networks and for large bandwidths, and is robust against errors in network geometry. The estimator is consistent, and its statistical efficiency is only slightly suboptimal. We discuss bias, variance, asymptotics, bandwidth selection, variance estimation, relative risk estimation and adaptive smoothing. The methods are used to analyse spatially varying frequency of traffic accidents in Western Australia and the relative risk of different types of traffic accidents in Medellín, Colombia.     

An Improved Test for Earnings Management Using Kernel Density Estimation

Abstract

This paper describes improvements on methods developed by Burgstahler and Dichev (1997, Earnings management to avoid earnings decreases and losses, Journal of Accounting and Economics, 24(1), pp. 99–126) and Bollen and Pool (2009, Do hedge fund managers misreport returns? Evidence from the pooled distribution, Journal of Finance, 64(5), pp. 2257–2288) to test for earnings management by identifying discontinuities in distributions of scaled earnings or earnings forecast errors. While existing methods use preselected bandwidths for kernel density estimation and histogram construction, the proposed test procedure addresses the key problem of bandwidth selection by using a bootstrap test to endogenise the selection step. The main advantage offered by the bootstrap procedure over prior methods is that it provides a reference distribution that cannot be globally distinguished from the empirical distribution rather than assuming a correct reference distribution. This procedure limits the researcher's degrees of freedom and offers a simple procedure to find and test a local discontinuity. I apply the bootstrap density estimation to earnings, earnings changes, and earnings forecast errors in US firms over the period 1976–2010. Significance levels found in earlier studies are greatly reduced, often to insignificant values. Discontinuities cannot be detected in analysts’ forecast errors, while such findings of discontinuities in earlier research can be explained by a simple rounding mechanism. Earnings data show a large drop in loss aversion after 2003 that cannot be detected in changes of earnings.     

Generalized runs tests for the IID hypothesis

We provide a family of tests for the IID hypothesis based on generalized runs, powerful against unspecified alternatives, providing a useful complement to tests designed for specific alternatives, such as serial correlation, GARCH, or structural breaks. Our tests have appealing computational simplicity in that they do not require kernel density estimation, with the associated challenge of bandwidth selection. Simulations show levels close to nominal asymptotic levels. Our tests have power against both dependent and heterogeneous alternatives, as both theory and simulations demonstrate.     

Extrapolation‐based Bandwidth Selectors: A Review and Comparative Study with Discussion on Bivariate Applications

Cross‐validation is a widely used tool in selecting the smoothing parameter in a non‐parametric procedure. However, it suffers from large sampling variation and tends to overfit the data set. Many attempts have been made to reduce the variance of cross‐validation. This paper focuses on two recent proposals of extrapolation‐based cross‐validation bandwidth selectors: indirect cross‐validation and subsampling‐extrapolation technique. In univariate case, we notice that using a fixed value parameter surrogate for indirect cross‐validation works poorly when the true density is hard to estimate, while the subsampling‐extrapolation technique is more robust to non‐normality. We investigate whether a hybrid bandwidth selector could benefit from the advantages of both approaches and compare the performance of different extrapolation‐based bandwidth selectors through simulation studies, real data analyses and large sample theory. A discussion on their extension to bivariate case is also presented.     

Least absolute deviations estimation for the censored regression model

This paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression (or censored ‘Tobit’) model. The proposed estimator is a generalization of least absolute deviations estimation for the standard linear model, and, unlike estimation methods based on the assumption of normally distributed error terms, the estimator is consistent and asymptotically normal for a wide class of error distributions, and is also robust to heteroscedasticity. The paper gives the regularity conditions and proofs of these large-sample results, and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances.