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插补调查数据的 Bootstrap 方法AI 解读

Most surveys use imputation to compensate for missing data. However, treating the imputed data set as the complete data set and directly applying existing methods (e.g., the linearization, the jackknife, and the bootstrap) for variance estimation and/or statistical inference does not produce valid results, because these methods do not account for the effect of missing data and/or imputation. In this article we show that correct bootstrap estimates can be obtained by…

插补调查数据的自助法AI 解读

Most surveys use imputation to compensate for missing data. However, treating the imputed data set as the complete data set and directly applying existing methods (e.g., the linearization, the jackknife, and the bootstrap) for variance estimation and/or statistical inference does not produce valid results, because these methods do not account for the effect of missing data and/or imputation. In this article we show that correct bootstrap estimates can be obtained by…

Bootstrap 模型选择AI 解读

In a regression problem, typically there are p explanatory variables possibly related to a response variable, and we wish to select a subset of the p explanatory variables to fit a model between these variables and the response. A bootstrap variable/model selection procedure is to select the subset of variables by minimizing bootstrap estimates of the prediction error, where the bootstrap estimates are constructed based on a data set of size n. Although the bootstrap…

复杂调查抽样中的 L 统计量AI 解读

We study linear combinations of order statistics ($L$-statistics) in survey problems with a stratified multistage sampling design. Two general types of $L$-statistics are considered: smooth $L$-statistics with weights generated by a smooth function and nonsmooth $L$-statistics (sample quantiles). The trimmed sample mean, the decile mean and variance, the sample Lorenz curve and the sample Gini's family parameters are examples of smooth $L$-statistics or functions of smooth $L$…

迭代加权最小二乘估计量AI 解读

In a heteroscedastic linear model, we establish the asymptotic normality of the iterative weighted least squares estimators with weights constructed by using the within-group residuals obtained from the previous model fitting. An adaptive procedure is proposed which ensures that the iterative process stops after a finite number of iterations and produces an estimator asymptotically equivalent to the best estimator that can be obtained by using the iterative procedure.…

通过交叉验证选择线性模型AI 解读

We consider the problem of selecting a model having the best predictive ability among a class of linear models. The popular leave-one-out cross-validation method, which is asymptotically equivalent to many other model selection methods such as the Akaike information criterion (AIC), the C p , and the bootstrap, is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of…

统计泛函的可微性与 Jackknife 一致性AI 解读

In statistical applications the unknown parameter of interest can frequently be defined as a functional $\theta=T(F)$, where F is an unknown population. Statistical inferences about $\theta$ are usually made based on the statistic $T(F_n)$, where $F_n$ is the empirical distribution. Assessing $T(F_n)$ (as an estimator of $\theta$) or making large sample inferences usually requires a consistent estimator of the asymptotic variance of $T(F_n)$. Asymptotic behavior of the…

样本分位数的平衡重复复制法之渐近性质AI 解读

Inference, including variance estimation, can be made from stratified samples by selecting a balanced set of subsamples. This balanced subsampling method is generically called the balanced repeated replication method in survey data analysis, which includes McCarthy's balanced half-samples method and its extensions for more general stratified designs. We establish the asymptotic consistency of the balanced repeated replication variance estimators when the parameter of…

SUMMARY In a heteroscedastic linear regression model, the ordinary jackknife estimator of the asymptotic covariance matrix of the weighted least squares estimator is proved to be inconsistent. A modified jackknife procedure is proposed and shown to produce consistent estimators of the asymptotic covariance matrix. Finite sample performances of the jackknife, the modified jackknife and the customary S method are discussed. Some empirical results are also presented.

刀切法方差估计的一般理论AI 解读

The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with $d$, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded $d$…

Jackknife 方差估计近似的效率与一致性AI 解读

The problem considered is the computation reduction for general delete-d jackknife variance estimators. The delete-d jackknife estimator was proved consistent (Shao and Wu 1986), and in this article its mean squared error is shown to have order o(n –2), where n is the sample size. These properties are not shared by the traditional delete-1 jackknife in some situations. Use of the delete-d jackknife, however, requires ( n d ) recomputations of a point estimate θ, which…

线性模型中方差与偏差估计的重抽样方法AI 解读

Let $g$ be a nonlinear function of the regression parameters $\beta$ in a heteroscedastic linear model and $\hat{\beta}$ be the least squares estimator of $\beta.$ We consider the estimation of the variance and bias of $g(\hat{\beta})$ [as an estimator of $g(\beta)$] by using three resampling methods: the weighted jackknife, the unweighted jackknife and the bootstrap. The asymptotic orders of the mean squared errors and biases of the resampling variance and bias estimators…

线性模型中 Jackknife 方差估计量的异方差稳健性AI 解读

The asymptotic unbiasedness and consistency of three types of jackknife variance estimators in the presence of error variance heteroscedasticity in linear models are studied. The results are given in terms of the number of observations deleted and measures of imbalance of the model. The consistency of a class of Wu's weighted jackknife variance estimators for nonlinear parameters is also studied. A necessary and sufficient condition is given for the asymptotic unbiasedness…