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带移入的临界分枝过程中 Bootstrap 的无效性AI 解读

This article considers a case of parametric bootstrap when the observations consist of generation sizes of the branching process with immigration together with the immigration component of each generation. Suppose we estimate the offspring mean $m$ by the maximum likelihood estimator (m.l.e.). It is then shown that the bootstrap version of the standardized m.l.e. does not have the same limiting distribution as the standardized m.l.e., under the assumption that $m = 1$ (critic…

平稳序列经验过程的分块 BootstrapAI 解读

We apply the bootstrap for general stationary observations, proposed by Kunsch, to the empirical process for $p$-dimensional random vectors. It is known that the empirical process in the multivariate case converges weakly to a certain Gaussian process. We show that the bootstrapped empirical process converges weakly to the same Gaussian process almost surely, assuming that the block length $l$ for constructing bootstrap replicates satisfies $l(n) = O(n^{1/2-\varepsilon}), 0 < \varepsilon < \frac{1}{2}$…

平稳观测经验过程的分块自助法有效性AI 解读

We show that the empirical process of the block-based bootstrap observations from a stationary sequence converges weakly to an appropriate Gaussian process, conditionally in probability and almost surely, depending upon the block length. This bootstrap was introduced by Kunsch and later by Liu and Singh. Applications in estimation of the sampling distribution of a compactly differentiable functional are indicated.

缺失数据、插补与自助法AI 解读

Missing data refers to a class of problems made difficult by the absence of some portions of a familiar data structure. For example, a regression problem might have some missing values in the predictor vectors. This article concerns nonparametric approaches to assessing the accuracy of an estimator in a missing data situation. Three main topics are discussed: bootstrap methods for missing data, these methods' relationship to the theory of multiple imputation, and…

皮特曼接近性度量：统计估计量的比较AI 解读

Preface Part I. Introduction 1. Evolution of Estimation Theory Least Squares Method of Moments Maximum Likelihood Uniformly Minimum Variance Unbiased Estimation Biased Estimation Bayes and Empirical Bayes Influence Functions and Resampling Techniques Future Directions 2. PMC Comes of Age PMC: A Product of Controversy PMC as an Intuitive Criterion 3. The Scope of the Book History, Motivation, and Controversy of PMC A Unified Development of PMC Part II. Development of Pitman's…

基于枢轴估计函数的重抽样方法AI 解读

Suppose that, under a semiparametric model setting, one is interested in drawing inferences about a finite-dimensional parameter vector β based on an estimating function. Generally a consistent point estimator β^ for βo, the true value for β, can be easily obtained by finding a root of the corresponding estimating equation. To estimate the variance of β^, however, may involve complicated and subjective nonparametric functional estimates. In this paper, a general and simple…

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自助法鞍点近似AI 解读

We compare saddlepoint approximations to the exact distributions of a studentized mean and to its bootstrap approximation. We show that, on bounded sets, these empirical saddlepoint approximations achieve second order relative errors uniformly. We also consider the relative errors for larger deviations. It follows that the studentized-t bootstrap p-value and the coverage of the bootstrap confidence interval have second order relative errors.

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自助法鞍点近似AI 解读

We compare saddlepoint approximations to the exact distributions of a studentized mean and to its bootstrap approximation. We show that, on bounded sets, these empirical saddlepoint approximations achieve second order relative errors uniformly. We also consider the relative errors for larger deviations. It follows that the studentized-t bootstrap p-value and the coverage of the bootstrap confidence interval have second order relative errors.

使用加权似然 Bootstrap 的近似贝叶斯推断AI 解读

SUMMARY We introduce the weighted likelihood bootstrap (WLB) as a way to simulate approximately from a posterior distribution. This method is often easy to implement, requiring only an algorithm for calculating the maximum likelihood estimator, such as iteratively reweighted least squares. In the generic weighting scheme, the WLB is first order correct under quite general conditions. Inaccuracies can be removed by using the WLB as a source of samples in the…

指数族的稳健预测分布AI 解读

The bootstrap predictive distribution considered by Harris (1989) is modified by replacing the maximum likelihood estimator of the unknown parameter by its minimum Hellinger distance estimator. The predictive distribution thus obtained is asymptotically superior to the corresponding estimative distribution in terms of the average Kullback-Leibler divergence, when the true distribution is in the natural exponential family. Simulation results are provided for the binomial and…

基于枢轴估计函数的重抽样方法AI 解读

SUMMARY Suppose that, under a semiparametric model setting, one is interested in drawing inferences about a finite-dimensi onal parameter vector /? based on an estimating function. Generally a consistent point estimator /J for /?0, the true value for /J, can be easily obtained by finding a root of the corresponding estimating equation. To estimate the variance of ft, however, may involve complicated and subjective nonparametric functional estimates. In this paper, a general…

Monte Carlo 近似与迭代自助法AI 解读

We discuss optimal choice of the numbers of resamples in the two stages of the iterated bootstrap, when that technique is used to calibrate bootstrap confidence intervals. If the numbers of resamples in the first and second stages are denoted by B and C respectively, we show that it is optimal to take C to be approximately a constant multiple of B½. The value of the constant is derived, and shown to depend only on the nominal coverage level of a confidence interval. However,…

加权抽样的贝叶斯方法AI 解读

Bayesian statistical inference for sampling from weighted distribution models is studied. Small-sample Bayesian bootstrap clone (BBC) approximations to the posterior distribution are discussed. A second-order property for the BBC in unweighted i.i.d. sampling is given. A consequence is that BBC approximations to a posterior distribution of the mean and to the sampling distribution of the sample average, can be made asymptotically accurate by a proper choice of the random…

响应自适应治疗分配设计下的渐近推断AI 解读

A response-adaptive treatment allocation design for a clinical trial attempts to place the majority of patients on the treatment that appears more successful, based on the responses of patients already treated. One example of such a design is the randomized play-the-winner rule developed by Wei and Durham, which randomizes the treatment assignment probabilities according to the outcomes of treatments previously assigned. For a trial with dichotomous treatment responses and a…

退化 U 统计量广义自助法的一致性AI 解读

A generalized bootstrap version is defined for degenerate $U$-statistics. Our main result shows that the (conditional) distribution function of the bootstrapped degenerate $U$-statistic provides a consistent estimator for the unknown distribution function of the degenerate $U$-statistic under consideration. For the proof we rely on a rank statistic approach.

测量误差模型中线性函数关系的 Bootstrap 置信域AI 解读

We suggest bootstrap methods for constructing confidence bands (and intervals) for an unknown linear functional relationship in an errors-invariables model. It is assumed that the ratio of error variances is known to lie within an interval $\Lambda = \lbrack\lambda_1, \lambda_2\rbrack$. A confidence band is constructed for the range of possible linear relationships when $\lambda \in \Lambda$. Meaningful results are obtained even in the extreme case $\Lambda = \lbrack 0, \infty\rbrack$…

非参数与参数回归拟合的比较AI 解读

In general, there will be visible differences between a parametric and a nonparametric curve estimate. It is therefore quite natural to compare these in order to decide whether the parametric model could be justified. An asymptotic quantification is the distribution of the integrated squared difference between these curves. We show that the standard way of bootstrapping this statistic fails. We use and analyse a different form of bootstrapping for this task. We call this…

L1回归估计量分布的解析与自助法近似AI 解读

Edgeworth and bootstrap approximations to estimator distributions in L1 regression are described. Analytic approximations based on Edgeworth expansions that mix lattice and nonlattice components and allow for an intercept term in the regression are developed under mild conditions, which do not even require a density for the error distribution. Under stronger assumptions on the error distribution, the Edgeworth expansion assumes a simpler form. Bootstrap approximations are…

Cox模型中随协变量变化的中位生存时间置信带AI 解读

Let ξp (x) be the pth quantile of the distribution of the life length of an individual with covariate vector x in the Cox model. We introduce an estimator ξp (x) of ξp (x) and develop several families of confidence bands for ξ(x) as a function of x. To construct one type of band, we proceed as follows. We show that as n → ∞, where n is the number of individuals in the study, ) converges weakly to a Gaussian process W(x) with a complicated covariance structure. We then…

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