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贝叶斯 bootstrap 的大样本研究AI 解读

An asymptotic justification of the Bayesian bootstrap is given. Large-sample Bayesian bootstrap probability intervals for the mean, the variance and bands for the distribution, the smoothed density and smoothed rate function are also provided.

弯曲指数族的置信域：一般相对风险函数下的配对病例对照与生存研究AI 解读

Differential geometric methods are used to construct approximate confidence regions for curved exponential families. The $\alpha$-connection geometries discussed by Amari (1982), and another geometry introduced here, the $c$ geometry, are exploited to construct confidence regions. Survival and case-control studies with general relative risk functions are interpreted in the context of curved exponential families, and an example illustrates the construction of confidence…

比例风险模型部分似然方法向一般变换模型的扩展AI 解读

Estimates of the linear model parameters in a linear transformation model with unknown increasing transformation are obtained by maximizing a partial likelihood. A resampling scheme (likelihood sampler) is used to compute the maximum partial likelihood estimates. It is shown that for a certain "local" parameter set where the "signal to noise ratio" is small, it is asymptotically possible to estimate the linear model parameters using the partial likelihood as well as if the…

离散与连续混合观测的标准化对数似然比统计量AI 解读

When the $\log$-likelihood statistic is divided by its mean, or an approximation to its mean, the limiting chi-squared distribution is often correct to order $n^{-3/2}$. Similarly, when the signed version of the likelihood ratio statistic is standardized with respect to its mean and variance the normal approximation is correct to order $n^{-3/2}$. Proofs for these statements have been given in great generality in the literature for the case of continuous observations. In…

提前终止的单样本非参数位置偏移检验能节省多少样本AI 解读

Asymptotic moments of normed sample size savings are derived for one-sided curtailed nonparametric tests for symmetry for testing location shift hypotheses under the null hypothesis and contiguous alternatives. The contiguous alternative results are derived using Feller's (1943) central limit theorem result for large deviations and techniques used in Albers, Bickel and van Zwet (1976).

以可容许性为试金石AI 解读

Consider the problem of estimating simultaneously the means $\theta_i$ of independent normal random variables $x_i$ with unit variance. Under the weighted quadratic loss $L(\theta, a) = \sum_i\lambda_i(\theta_i - a_i)^2$ with positive weights it is well known that: (1) An estimator which is admissible under one set of weights is admissible under all weights. (2) Estimating individual coordinates by proper Bayes estimators results in an admissible estimator. (3) Estimating…

测量误差回归模型中最小二乘估计的极限分布AI 解读

It is well-known that the ordinary least squares (OLS) estimator $\hat{\beta}$ of the slope and intercept parameters $\beta$ in a linear regression model with errors of measurement for some of the independent variables (predictors) is inconsistent. However, Gallo (1982) has shown that certain linear combinations of $\beta$. In this paper, it is shown that under reasonable regularity conditions such linear combinations of $\hat{\beta}$ are (jointly) asymptotically normally…

回归中的k统计量与离散程度效应AI 解读

By the term $k$-statistic or polykay, we mean an unbiased estimate of a cumulant or product of cumulants [Fisher (1929) and Tukey (1950), 1956)]. In this paper, two sets of unbiased estimates are given for the case where the mean response, $E(Y)$, depends linearly on known covariates $x$. The $k$'s are symmetric functions of the least-squares residuals and have previously been discussed by Anscombe (1961; 1981, Appendix 2). The $l$'s are optimal in the sense of having…

回归曲线的可变带宽核估计AI 解读

In the model $Y_i = g(t_i) + \varepsilon_i,\quad i = 1,\cdots, n,$ where $Y_i$ are given observations, $\varepsilon_i$ i.i.d. noise variables and $t_i$ nonrandom design points, kernel estimators for the regression function $g(t)$ with variable bandwidth (smoothing parameter) depending on $t$ are proposed. It is shown that in terms of asymptotic integrated mean squared error, kernel estimators with such a local bandwidth choice are superior to the ordinary kernel estimators…

核密度估计带宽选择所固有的噪声AI 解读

In the setting of kernel density estimation, data-driven bandwidth, i.e., smoothing parameter, selectors are considered. It is seen that there is a well-defined, and surprisingly restrictive, bound on the rate of convergence of any automatic bandwidth selection method to the optimum. The method of least squares cross-validation achieves this bound.

密度估计中交叉验证方法的比较AI 解读

In the setting of nonparametric multivariate density estimation, theorems are established which allow a comparison of the Kullback-Leibler and the least-squares cross-validation methods of smoothing parameter selection. The family of delta sequence estimators (including kernel, orthogonal series, histogram and histospline estimators) is considered. These theorems also show that either type of cross validation can be used to compare different estimators (e.g., kernel versus…

高阶样本累积量谱泛函的极限行为AI 解读

This paper is concerned with establishing a broad class of estimators (and the limiting behavior, thereof) for parametrizations of higher than second-order structure. This includes parametrizations which reflect, for example, such properties as nonlinearity and/or non-Gaussianity and/or time irreversibility. Asymptotic distributions, almost-sure convergence, and probability-one bounds for such estimators are established. Several applications of such estimators are discussed.

平稳 ARMA 过程中的自适应估计AI 解读

We consider the estimation problem for the parameter $\vartheta_0$ of a stationary ARMA $(p, q)$ process, with independent and identically, but not necessary normally distributed errors. First we prove local asymptotic normality (LAN) for this model. Then we construct locally asymptotically minimax (LAM) estimators, which asymptotically achieve the smallest possible covariance matrix. Utilizing these, we finally obtain strongly adaptive estimators, by using usual kernel…

非平稳分类时间序列回归模型：渐近估计理论AI 解读

For the analysis of nonstationary categorical time series, a parsimonious and flexible class of models is proposed. These models are generalizations of regression models for stochastically independent categorical observations. Consistency, asymptotic normality and efficiency of the maximum likelihood estimator are shown under weak and easily verifiable requirements. Some models for binary time series are discussed in detail. To demonstrate asymptotic properties, a theorem is…

高斯过程均值函数的筛估计量AI 解读

A new sieve estimator for the mean function $m(t)$ of a general Gaussian process of known covariance is presented. The estimator $\hat{m}(t)$ is given explicitly from the data and has a simple distribution. It is shown that $\hat{m}(t)$ is asymptotically unbiased and consistent (weakly and in mean square) at each $t$, and that $\hat{m}$ is strongly consistent for $m$ in an appropriate norm. No assumptions are made about the "time" parameter or the covariance.

非平稳过程预测误差的渐近分布及相关拟合检验AI 解读

Limit distributions of prediction errors when the parameters are estimated are obtained for a general class of nonstationary processes that includes supercritical branching processes and explosive autoregressive processes as typical examples. The estimated prediction errors are used to construct new tests of fit whose limit distributions are also derived.

线性模型中的截尾均值AI 解读

For the general linear model with independent errors, we propose and examine the large sample properties of an estimator of the regression parameter. In the location model, the estimator has the same properties as the trimmed mean and the robustness and efficiency properties of the trimmed mean carry over to the general model. The estimator depends on a preliminary estimate of the regression parameter and the residuals based on it. The properties of the adaptive estimator…

重复显著性检验与序贯检验后估计中的 Anscombe 定理渐近展开AI 解读

Let $x_1, x_2, \cdots$ be independent and normally distributed with unknown mean $\theta$ and variance 1. Let $\tau = \inf \{n \geq 1: |s_n| \geq \sqrt{2a(n + c)}\}$. Then a repeated significance test for a normal mean rejects the hypothesis $\theta = 0$ if and only if $\tau \leq N_0$ for some positive integer $N_0$. The problem we consider is estimation of $\theta$ based on the data $x_1,\cdots, x_T, T = \min\{\tau, N_0\}$. We shall solve this problem by obtaining the…

多元散布的分布无关 M 估计量AI 解读

The existence and uniqueness of a limiting form of a Huber-type $M$-estimator of multivariate scatter is established under certain conditions on the observed sample. These conditions hold with probability one when sampling randomly from a continuous multivariate distribution. The existence of the estimator is proven by showing that it is the limiting point of a specific algorithm. Hence, the proof is constructive. For continuous populations, the estimator of multivariate…

稳健谱回归AI 解读

This paper addresses the problem of linear regression estimation when the disturbances follow a stationary process with its spectral density known only to be in a neighborhood of some specified spectral density, for instance, that of white noise. Rather than trying to adapt to a small unspecified autocorrelation, we follow here the robustness approach, and establish the extent of the regressors and disturbance spectra interaction which require serial correlation correction.…

二项分布参数 n 的可容许估计AI 解读

Suppose that $X$ has a binomial distribution $B(n, p)$, with known $p \in (0, 1)$ and unknown $n \in \{1, 2, \cdots\}$. A natural estimator for $n$ is given by $T(0) = 1, T(x) = x/p, x = 1, 2, \cdots$. This estimator is shown to be inadmissible under quadratic loss. It is shown that modifying $T(0)$ to $T(0) = -(1 - p)/(p \ln p)$ results in an admissible estimator. For $p \geq \frac{1}{2}$ it is further shown that this is the only admissible modification of $T(0)$. A partial…

未知协方差与一般二次损失下正态均值向量的极小极大估计AI 解读

The problem of finding classes of estimators which dominate the usual estimator $X$ of the mean vector $\mu$ of a $p$-variate normal distribution $(p \geq 3)$ under general quadratic loss is analytically difficult in cases where the covariance matrix is unknown. Estimators of $\mu$ in this case depend upon $X$ and an independent Wishart matrix $W$. In the present paper, integration-by-parts methods for both the multivariate normal and Wishart distributions are combined to…

椭圆对称分布中离散程度偏移的多元异常值检测AI 解读

In this paper Ferguson's univariate normal results for detection of outliers with variance slippage are extended to the multivariate elliptically symmetric case with dispersion slippage. The locally optimum test we derive possesses all three robustness properties, optimality, null and nonnull, and is based on Mardia's multivariate kurtosis statistic.

不满足合法密度约束的密度估计量之改进AI 解读

In order to improve the rate of decrease of the IMSE for nonparametric kernel density estimators with nonrandom bandwidth beyond $O(n^{-4/5})$ all current methods must relax the constraint that the density estimate be a bona fide function, that is, be nonnegative and integrate to one. In this paper we show how to achieve similar improvement without relaxing any of these constraints. The method can also be applied for orthogonal series, adaptive orthogonal series, spline,…