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利用相互删失方法联合估计系统与部件可靠性AI 解读

Let $F$ denote the life distribution of a coherent structure of independent components. Suppose that we have a sample of independent systems, all having the same structure. Each system is continuously observed until it fails. For every component in each system, either a failure time or a censoring time is recorded. A failure time is recorded if the component fails before or at the time of system failure; otherwise a censoring time is recorded. We introduce a method for…

平滑与自助法AI 解读

Recent attention has focussed on possible improvements in performance of estimators which might flow from using the smoothed bootstrap. We point out that in a great many problems, such as those involving functions of vector means, any such improvements will be only second-order effects. However, we argue that substantial and significant improvements can occur in problems where local properties of underlying distributions play a decisive role. This situation often occurs in…

随机效应模型的正态性评估AI 解读

When one uses the unbalanced, mixed linear model $\mathbf{y}_i = \mathbf{X}_i\mathbf{\alpha} + \mathbf{Z}_i\mathbf{\beta}_i + \varepsilon_i, i = 1, \cdots, n$ to analyze data from longitudinal experiments with continuous outcomes, it is customary to assume $\varepsilon_i \sim_{\operatorname{ind}} \mathscr{N}(\mathbf{0}, \sigma^2\mathbf{I}_i)$ independent of $\mathbf{\beta}_i \sim_{\operatorname{iid}} \mathscr{N}(\mathbf{0,\Delta})$, where $\sigma^2$ and the elements of an…

基于局部中位数的核估计量的渐近性质AI 解读

The desire to make nonparametric regression robust leads to the problem of conditional median function estimation. Under appropriate regularity conditions, a sequence of local median estimators can be chosen to achieve the optimal rate of convergence $n^{-1/(2+d)}$ both pointwise and in the $L^q (1 \leq q < \infty)$ norm restricted to a compact. It can also be chosen to achieve the optimal rate of convergence $(n^{-1} \log n)^{1/(2+d)}$ in the $L^\infty$ norm restricted to a…

Slowly decaying serial correlations have, if ignored, disastrous effects on classical tests and confidence intervals. We propose a modified t test for the location parameter which takes into account such correlations. An approximate distribution function of the test statistic is given under the assumption of fractional Gaussian noise. The results are generalized to prediction intervals for a future average. An example is given to illustrate the methods.

线性模型的自适应 L 估计AI 解读

Asymptotically efficient (adaptive) estimators for the slope parameters of the linear regression model are constructed based upon the "regression quantile" statistics suggested by Koenker and Bassett. The estimators are natural analogues of the adaptive $L$-estimators of location of Sacks, but employ kernel-density type estimators of the optimal $L$-estimator weight function.

M过程与M估计AI 解读

We relate the asymptotic behavior of $M$-estimators of the regression parameter in a linear model in which the dimension of the regression parameter may increase with the sample size to the stochastic equicontinuity of an associated $M$-process. The approach synthesises a number of results for the dimensionally fixed regression model and then extends these results in a direct unified way. The resulting theorems require only mild conditions on the $\psi$-function and the…

Bootstrap与经验Edgeworth展开的二阶及Lᵖ比较AI 解读

The bootstrap estimate of distribution functions of studentized statistics is shown to be more accurate than even the two-term empirical Edgeworth expansion, thus strengthening the claim of superiority of the bootstrap over the normal approximation method. The two methods are compared not only with respect to bounded bowl-shaped loss functions but also with respect to squared error loss and, more generally, in $L^p$-norms.

使用 Bootstrap 构造分布函数置信带AI 解读

We first discuss the construction of bootstrap confidence bands for the distribution function F of a population for simple random sampling but do not assume that F is continuous. The known alternative approach is to use the quantiles from the tabled Kolmogorov distribution. This approach is known to be conservative. It has already been shown in Bickel and Freedman (1981) that the bootstrap confidence band has the correct coverage probability asymptotically. We show by…

利用辅助信息估计有限总体中位数AI 解读

SUMMARY This paper deals with the estimation, under simple random sampling, of a finite population median in the presence of an auxiliary variable. An estimator which is a simple modification of the ratio estimator for the mean is studied along with two other proposed estimators derived from a different approach. These estimators are applicable in situations where only the population median or a grouped frequency distribution of the auxiliary variable is known. On the basis…

Bootstrap的对偶重抽样AI 解读

An antithetic variates method for the bootstrap is proposed and discussed. It is applicable quite generally, to bias estimation, distribution function estimation and quantile estimation, for example. It is based on an ‘antithetic permutation’ of the sample, which amounts to ranking values of a certain function of the data. Once this has been done B uniform resampling operations may be immediately converted into 2B ‘effective’ resampling operations, yielding greater…

序列相关缓慢衰减数据的位置参数检验AI 解读

Slowly decaying serial correlations have, if ignored, disastrous effects on classical tests and confidence intervals. We propose a modified t test for the location parameter which takes into account such correlations. An approximate distribution function of the test statistic is given under the assumption of fractional Gaussian noise. The results are generalized to prediction intervals for a future average. An example is given to illustrate the methods.

离散、连续不变非参数估计及多项分布对应问题的可容许性AI 解读

Discrete and multinomial analogs are defined for classical (continuous) invariant nonparametric problems of estimating the sample cumulative distribution function (sample c.d.f.) and the sample median. Admissibility of classical estimators and their analogs is investigated. In discrete (including multinomial) settings the sample c.d.f. is shown to be an admissible estimator of the population c.d.f. under the invariant weighted Cramer-von Mises loss function $L_1(F, \hat{F}) = \int \big\lbrack (F(t) - \hat{F}(t))^2/(F(t)(1 - F(t))) \big\rbrack dF(t).$…

多维空间中拟合优度检验的新息方法AI 解读

We present a solution to the goodness-of-fit problem for multivariate observations, using the innovation process for the (sequential) empirical distribution function with respect to a conveniently chosen linear ordering or scanning system in $R^m$.

条件泛函估计量的强一致收敛速率AI 解读

Strong uniform consistency rates are established for kernel type estimators of functionals of the conditional distribution function, under general conditions. The present treatment unifies a number of specific problems previously studied separately in the literature. Some of these applications we treat in detail, including regression curve estimation, density estimation, estimation of conditional df's, $L$-smoothing and $M$-smoothing. Various previous results in the…

Anderson–Darling检验统计量分布函数的近似AI 解读

A theoretical approximation to the upper tail area of the Anderson-Darling test statistic is derived by applying a result of Zolotarev (1961) to the characteristic function. This approximation is particularly good in the upper tail. An empirical distribution is given that is accurate over the complete domain of the distribution.

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Kolmogorov–Smirnov损失下分布函数的最优不变估计AI 解读

Given a random sample of size $n$ from an unknown continuous distribution function $F$, we consider the problem of estimating $F$ nonparametrically from a decision theoretic approach. In our treatment, we assume the Kolmogorov-Smirnov loss function and the group of all one-to-one monotone transformations of real numbers onto themselves which leave the sample values invariant. Under this setup, we obtain a best invariant estimator of $F$ which is shown to be unique. This…

In this paper the asymptotic behaviour of the product limit estimator $F_n$ of an unknown distribution is investigated. We give an approximation of the difference $F_n(x) - F(x)$ by a Gaussian process and also by the average of i.i.d. processes. We get almost as good an approximation of the stochastic process $F_n(x) - F(x)$ as one can get for the analogous problem when the maximum likelihood estimator is approximated by a Gaussian random variable or by the average of i.i.d.…

平衡的同时置信集AI 解读

Suppose that T(θ) = {Tu (θ)} is a family of parametric functions indexed by the variable u. For each u, an approximate confidence set Cn,u for Tu (θ) may be obtained by referring a function of Tu (θ) and the sample to an estimated quantile dn,u of that function's sampling distribution. This approach is sometimes called the pivotal method for constructing a confidence set, even when it is not based on a true pivot. A simultaneous confidence set Cn for T(θ) is then obtained by…

二元反应模型的识别AI 解读

This article studies identification of the threshold-crossing model of binary response. Most research on binary response has considered specific estimators and tests. The study of identification exposes the foundations of binary response analysis by making explicit the assumptions needed to justify different methods. It also clarifies the connections between reduced-form and structural analyses of binary response data. Assume that the binary outcome z is determined by an…

用于 Bootstrap 置信区间的重要性抽样AI 解读

The use of importance-sampling methods to substantially reduce the amount of resampling necessary for the construction of nonparametric bootstrap confidence intervals is investigated. The classical importance-sampling method of Hammersley and Handscomb (1964) is modified to apply to the estimation of quantiles. Based on this method, a resampling procedure is introduced that recenters the bootstrap distribution of a robust estimator of location to produce estimated confidence…

检验统计量的预枢轴化：渐近改进的 Bootstrap 视角AI 解读

Approximate tests for a composite null hypothesis about a parameter θ may be obtained by referring a test statistic to an estimated critical value. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. The simple asymptotic test ϕA refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated. The bootstrap approach used here is based on the concept of prepivoting. Prepivoting is…