协整参数的自举推断

 针对完全修正最小二乘(full-modified ordinary least square,简称FMOLS）估计方法,给出一种协整参数的自举推断程序,证明零假设下自举统计量与检验统计量具有相同的渐近分布。关于检验功效的研究表明,虽然有约束自举的实际检验水平表现良好,但如果零假设不成立,自举统计量的分布是不确定的,因而其经验分布不能作为检验统计量精确分布的有效估计。实际应用中建议使用无约束自举,因为无论观测数据是否满足零假设,其自举统计量与零假设下检验统计量都具有相同的渐近分布。最后,利用蒙特卡洛模拟对自举推断和渐近推断的有限样本表现进行比较研究。     

Monte Carlo Approximations of the Quantiles of a Sample Statistic

We consider in this article the problem of numerically approximating the quantiles of a sample statistic for a given population, a problem of interest in many applications, such as bootstrap confidence intervals. The proposed Monte Carlo method can be routinely applied to handle complex problems that lack analytical results. Furthermore, the method yields estimates of the quantiles of a sample statistic of any sample size though Monte Carlo simulations for only two optimally selected sample sizes are needed. An analysis of the Monte Carlo design is performed to obtain the optimal choices of these two sample sizes and the number of simulated samples required for each sample size. Theoretical results are presented for the bias and variance of the numerical method proposed. The results developed are illustrated via simulation studies for the classical problem of estimating a bivariate linear structural relationship. It is seen that the size of the simulated samples used in the Monte Carlo method does not have to be very large and the method provides a better approximation to quantiles than those based on an asymptotic normal theory for skewed sampling distributions.     

Bootstrap Testing for Changes in Persistence with Heavy-Tailed Innovations

This article considers the problem of detection for changes in persistence with heavy-tailed innovations. We adopt a ratio type test and derive its null asymptotic distribution which is dependent on the stable index. Then a residual-based bootstrap is proposed when the stable index is unknown. Our procedure requires drawing bootstrap samples of size m < T, T being the size of original sample. We establish the convergence in probability of the bootstrap distribution function assuming that m → ∞ and m/T → 0. A Monte Carlo study has shown that the bootstrap improve the finite sample size and power compared to the asymptotic test, especially for small stable index.     

Bootstrapping Spatial Median for Location Problems

In multivariate location problems, the sample mean is most widely used, having various advantages. It is, however, very sensitive to outlying observations and inefficient for data from heavy tailed distributions. In this situation, the spatial median is more robust than the sample mean and could be a reasonable alternative. We reviewed several spatial median based testing methods for multivariate location and compared their significance level and power through Monte Carlo simulations. The results show that bootstrap method is efficient for the estimation of the covariance matrix of the sample spatial median. We also proposed bootstrap simultaneous confidence intervals based on the spatial median for multiple comparisons in the multi-sample case.     

Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm

Two new implementations of the EM algorithm are proposed for maximum likelihood fitting of generalized linear mixed models. Both methods use random (independent and identically distributed) sampling to construct Monte Carlo approximations at the E-step. One approach involves generating random samples from the exact conditional distribution of the random effects (given the data) by rejection sampling, using the marginal distribution as a candidate. The second method uses a multivariate t importance sampling approximation. In many applications the two methods are complementary. Rejection sampling is more efficient when sample sizes are small, whereas importance sampling is better with larger sample sizes. Monte Carlo approximation using random samples allows the Monte Carlo error at each iteration to be assessed by using standard central limit theory combined with Taylor series methods. Specifically, we construct a sandwich variance estimate for the maximizer at each approximate E-step. This suggests a rule for automatically increasing the Monte Carlo sample size after iterations in which the true EM step is swamped by Monte Carlo error. In contrast, techniques for assessing Monte Carlo error have not been developed for use with alternative implementations of Monte Carlo EM algorithms utilizing Markov chain Monte Carlo E-step approximations. Three different data sets, including the infamous salamander data of McCullagh and Nelder, are used to illustrate the techniques and to compare them with the alternatives. The results show that the methods proposed can be considerably more efficient than those based on Markov chain Monte Carlo algorithms. However, the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension.     

Asymptotically refined score and GOF tests for inverse Gaussian models

ABSTRACT

The score test and the GOF test for the inverse Gaussian distribution, in particular the latter, are known to have large size distortion and hence unreliable power when referring to the asymptotic critical values. We show in this paper that with the appropriately bootstrapped critical values, these tests become second-order accurate, with size distortion being essentially eliminated and power more reliable. Two major generalizations of the score test are made: one is to allow the data to be right-censored, and the other is to allow the existence of covariate effects. A data mapping method is introduced for the bootstrap to be able to produce censored data that are conformable with the null model. Monte Carlo results clearly favour the proposed bootstrap tests. Real data illustrations are given.     

部分函数型线性变系数模型的序列相关检验

部分函数型线性变系数模型(PFLVCM)是近几年出现的一个比较灵活、应用广泛的新模型。在实际应用中,搜集到的经济和金融数据往往存在序列相关性。如果不考虑数据间的相关性直接对其进行建模,会影响模型中参数估计的精度和有效性。本文主要研究了PFLVCM中误差的序列相关性的检验问题,基于经验似然,把标量时间序列数据相关性检验的方法拓展到函数型数据中,提出了经验对数似然比检验统计量,并在零假设下得到了检验统计量的近似分布。通过蒙特卡洛数值模拟说明该统计量在有限样本下有良好的水平和功效。最后,把该方法用于检验美国商业用电消费数据是否有序列相关性,证明该统计量的有效性和实用性。     

Assessment of individual bioequivalence using sufficient bootstrap procedure

This paper proposes a sufficient bootstrap method, which uses only the unique observations in the resamples, to assess the individual bioequivalence under 2 × 4 randomized crossover design. The finite sample performance of the proposed method is illustrated by extensive Monte Carlo simulations as well as a real‐experimental data set, and the results are compared with those obtained by the traditional bootstrap technique. Our records reveal that the proposed method is a good competitor or even better than the classical percentile bootstrap confidence limits.     

Ascent-based Monte Carlo expectation– maximization

Summary.  The expectation–maximization (EM) algorithm is a popular tool for maximizing likelihood functions in the presence of missing data. Unfortunately, EM often requires the evaluation of analytically intractable and high dimensional integrals. The Monte Carlo EM (MCEM) algorithm is the natural extension of EM that employs Monte Carlo methods to estimate the relevant integrals. Typically, a very large Monte Carlo sample size is required to estimate these integrals within an acceptable tolerance when the algorithm is near convergence. Even if this sample size were known at the onset of implementation of MCEM, its use throughout all iterations is wasteful, especially when accurate starting values are not available. We propose a data-driven strategy for controlling Monte Carlo resources in MCEM. The algorithm proposed improves on similar existing methods by recovering EM's ascent (i.e. likelihood increasing) property with high probability, being more robust to the effect of user-defined inputs and handling classical Monte Carlo and Markov chain Monte Carlo methods within a common framework. Because of the first of these properties we refer to the algorithm as 'ascent-based MCEM'. We apply ascent-based MCEM to a variety of examples, including one where it is used to accelerate the convergence of deterministic EM dramatically.     

An automated (Markov chain) Monte Carlo EM algorithm

We present an automated Monte Carlo EM (MCEM) algorithm which efficiently assesses Monte Carlo error in the presence of dependent Monte Carlo, particularly Markov chain Monte Carlo, E-step samples and chooses an appropriate Monte Carlo sample size to minimize this Monte Carlo error with respect to progressive EM step estimates. Monte Carlo error is gauged though an application of the central limit theorem during renewal periods of the MCMC sampler used in the E-step. The resulting normal approximation allows us to construct a rigorous and adaptive rule for updating the Monte Carlo sample size each iteration of the MCEM algorithm. We illustrate our automated routine and compare the performance with competing MCEM algorithms in an analysis of a data set fit by a generalized linear mixed model.     

Bootstrapping p Values and Power in the First-Order Autoregression: A Monte Carlo Investigation

The small-sample behavior of the bootstrap is investigated as a method for estimating p values and power in the stationary first-order autoregressive model. Monte Carlo methods are used to examine the bootstrap and Student-t approximations to the true distribution of the test statistic frequently used for testing hypotheses on the underlying slope parameter. In contrast to Student's t, the results suggest that the bootstrap can accurately estimate p values and power in this model in sample sizes as small as 5–10.     

A weighted bootstrap approach to bootstrap iteration

The operation of resampling from a bootstrap resample, encountered in applications of the double bootstrap, maybe viewed as resampling directly from the sample but using probability weights that are proportional to the numbers of times that sample values appear in the resample. This suggests an approximate approach to double-bootstrap Monte Carlo simulation, where weighted bootstrap methods are used to circumvent much of the labour involved in compounded Monte Carlo approximation. In the case of distribution estimation or, equivalently, confidence interval calibration, the new method may be used to reduce the computational labour. Moreover, the method produces the same order of magnitude of coverage error for confidence intervals, or level error for hypothesis tests, as a full application of the double bootstrap.     

基于带跳时变系数模型的PPI与CPI相关性研究

2008年以来,我国PPI与CPI走势出现了多次背离与分化,从整体上看,两者相关性很弱。但从动态视角来看,由于相关关系可能会因时而变,整体相关性有可能被关系本身的方向和强弱变化所削弱甚至掩盖。为准确反映两者相关性的动态变化,本文放宽时变系数函数的光滑性假设,提出了带跳时变系数模型,并给出一种非参数三步估计方法：首先,估计系数函数中跳点的位置和个数;然后,基于估计的跳点和Bootstrap方法选择的窗宽给出系数函数的最终估计;最后,利用蒙特卡洛模拟评价本文提出的非参数估计和窗宽选择方法的有限样本性质。通过对2008年1月至2017年12月我国PPI和CPI月度同比数据的实证分析,我们发现该模型能较好地刻画PPI与CPI相关性的时变和带跳特征,进而也验证了该模型的应用价值。     

Permutation Tests for Comparing Inequality Measures

ABSTRACT

Asymptotic and bootstrap tests for inequality measures are known to perform poorly in finite samples when the underlying distribution is heavy-tailed. We propose Monte Carlo permutation and bootstrap methods for the problem of testing the equality of inequality measures between two samples. Results cover the Generalized Entropy class, which includes Theil’s index, the Atkinson class of indices, and the Gini index. We analyze finite-sample and asymptotic conditions for the validity of the proposed methods, and we introduce a convenient rescaling to improve finite-sample performance. Simulation results show that size correct inference can be obtained with our proposed methods despite heavy tails if the underlying distributions are sufficiently close in the upper tails. Substantial reduction in size distortion is achieved more generally. Studentized rescaled Monte Carlo permutation tests outperform the competing methods we consider in terms of power.     

Coverage plots for assessing the variability of estimated contours of a density

Methods for assessing the variability of an estimated contour of a density are discussed. A new method called the coverage plot is proposed. Techniques including sectioning and bootstrap techniques are compared for a particular problem which arises in Monte Carlo simulation approaches to estimating the spatial distribution of risk in the operation of weapons firing ranges. It is found that, for computational reasons, the sectioning procedure outperforms the bootstrap for this problem. The roles of bias and sample size are also seen in the examples shown.     

空间经济计量滞后模型Bootstrap Moran检验功效的模拟分析

 当误差项不服从独立同分布时,利用Moran’s I统计量的渐近检验,无法有效判断空间经济计量滞后模型2SLS估计残差间存在空间关系与否。本文采用两种基于残差的Bootstrap方法,诊断空间经济计量滞后模型残差中的空间相关关系。大量Monte Carlo模拟结果显示,从功效角度看,无论误差项服从独立同分布与否,与渐近检验相比,Bootstrap Moran检验都具有更好的有限样本性质,能够更有效地进行空间相关性检验。尤其是,在样本量较小和空间衔接密度较高情况下,Bootstrap Moran检验的功效显著大于渐近检验。     

Introducing model uncertainty by moving blocks bootstrap

It is common in parametric bootstrap to select the model from the data, and then treat as if it were the true model. Chatfield (1993, 1996) has shown that ignoring the model uncertainty may seriously undermine the coverage accuracy of prediction intervals. In this paper, we propose a method based on moving block bootstrap for introducing the model selection step in the resampling algorithm. We present a Monte Carlo study comparing the finite sample properties of the proposel method with those of alternative methods in the case of prediction intervas.     

Bootstrap Bartlett correction in inflated beta regression

The inflated beta regression model aims to enable the modeling of responses in the intervals (0, 1], [0, 1), or [0, 1]. In this model, hypothesis testing is often performed based on the likelihood ratio statistic. The critical values are obtained from asymptotic approximations, which may lead to distortions of size in small samples. In this sense, this article proposes the bootstrap Bartlett correction to the statistic of likelihood ratio in the inflated beta regression model. The proposed adjustment only requires a simple Monte Carlo simulation. Through extensive Monte Carlo simulations the finite sample performance (size and power) of the proposed corrected test is compared to the usual likelihood ratio test and the Skovgaard adjustment already proposed in the literature. The numerical results evidence that inference based on the proposed correction is much more reliable than that based on the usual likelihood ratio statistics and the Skovgaard adjustment. At the end of the work, an application to real data is also presented.     

Bayesian and Classical Approaches for Hypotheses Testing in Trisomies

We introduce classical approaches for testing hypotheses on the meiosis I non disjunction fraction in trisomies, such as the likelihood-ratio, bootstrap, and Monte Carlo procedures. To calculate the p-values for the bootstrap and Monte Carlo procedures, different transformations in the data are considered. Bootstrap confidence intervals are also used as a tool to perform hypotheses tests. A Monte Carlo study is carried out to compare the proposed test procedures with two Bayesian ones: Jeffreys and Pereira-Stern tests. The results show that the likelihood-ratio and the Bayesian tests present the best performance. Down syndrome data are analyzed to illustrate the procedures.     

Minimum variance unbiased estimation based on bootstrap iterations

Practical computation of the minimum variance unbiased estimator (MVUE) is often a difficult, if not impossible, task, even though general theory assures its existence under regularity conditions. We propose a new approach based on iterative bootstrap bias correction of the maximum likelihood estimator to accurately approximate the MVUE. Viewing bootstrap iteration as a Markov process, we develop a computational algorithm for bias correction based on arbitrarily many bootstrap iterations. The algorithm, when applied parametrically to finite sample spaces, does not involve Monte Carlo simulation. For infinite sample spaces, a nonparametric version of the algorithm is combined with a preliminary round of Monte Carlo simulation to yield an approximate MVUE. Both algorithms are computationally more efficient and stable than conventional simulation-based bootstrap iterations. Examples are given of both finite and infinite sample spaces to illustrate the effectiveness of our new approach. Supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7026/97P).