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交叉验证密度估计的一致性AI 解读

Application of nonparametric density estimators generally requires the specification of a "smoothing parameter." The kernel estimator, for example, is not fully defined until a window width, or scaling, for the kernels has been chosen. Many "data-driven" techniques have been suggested for the practical choice of smoothing parameter. Of these, the most widely studied is the method of cross-validation. Our own simulations, as well as those of many other investigators, indicate…

最近邻拟合优度检验的无限维近似AI 解读

Let $X_1, \cdots, X_n$ be i.i.d. $\mathbb{R}^m$-valued observations from a bounded density $g(x)$ continuous on its support. Let $W_i = \exp\{- ng(X_i)V(R_i)\}, i = 1, \cdots, n$, where $V(R_i)$ is the volume of the nearest neighbor sphere around $X_i$, and let $w(x)$ be any bounded continuous weight function on $\mathbb{R}^m$. An infinite-dimensional approximation to the asymptotic form of the weighted empirical distribution function of the $W_i$'s is presented. The…

基于最近邻统计量加权经验分布的多维拟合优度检验AI 解读

Let $X_1, \cdots, X_n$ be a random sample in $\mathbb{R}^m$ from an unknown density $f(x)$. Recently Bickel and Breiman have introduced a goodness of fit test for this situation based on the empirical distribution function of the variables $W_i = \exp\{-ng(X_i)V(R_i)\}, i = 1, \cdots, n$, where $g(x)$ is the hypothesized density and $V(R_i)$ represents the volume of the nearest neighbor sphere centered at $X_i$. Under the null hypothesis $H : f(x) = g(x)$, the empirical…

不变充分的等变统计量与平移族中的正态性刻画AI 解读

It is shown that an equivariant statistic $S$ is invariantly sufficient iff the generated $\sigma$-algebra and the $\sigma$-algebra of the invariant Borel sets are independent, and that if $S$ is invariantly sufficient and equivariant, then the Pitman estimator for location parameter $\gamma$ is given by $S - E_0(S)$. For independent $X_1, \cdots, X_n$, the existence of an invariantly sufficient equivariant linear statistic is characterized by the normality of $X_1, \cdots, X_n$…

随机变量的负关联及其应用AI 解读

Random variables, $X_1, \cdots, X_k$ are said to be negatively associated (NA) if for every pair of disjoint subsets $A_1, A_2$ of $\{1, 2, \cdots, k\}, \operatorname{Cov}\lbrack f(X_1, i \in A_1), g(X_j, j \in A_2) \rbrack \leq 0$, for all nondecreasing functions $f, g$. The basic properties of negative association are derived. Especially useful is the property that nondecreasing functions of mutually exclusive subsets of NA random variables are NA. This property is shown…

基于观测 Fisher 信息的序贯抽样与最大似然估计精度保障AI 解读

A sequential sampling scheme in which observations are taken until the observed Fisher information exceeds a target value is considered for a restricted class of parametric families. It is proved that the theorems for fixed sample size asymptotic likelihood inference conditional on ancillary statistics are directly applicable to the sequential sampling scheme for location families. The fixed sample-size approximate ancillary statistic for two-dimensional curved exponential…

乘积极限估计量在整条实线上的大样本性质AI 解读

Weak convergence results are proved for the product-limit estimator on the whole line. Applications are given to confidence band construction, estimation of mean lifetime, and to the theory of $q$-functions. The results are obtained using stochastic calculus and in probability linear bounds for empirical processes.

若干时间序列模型的最优与渐近最优设计札记AI 解读

The problem of regression design in the presence of correlated errors is considered. Under the assumption that derivative information is available on the response process, uniqueness results for optimal designs given in Eubank, Smith and Smith (1981) for the BLUE of the regression coefficient are extended to a wider class of processes. In the event that derivatives cannot be sampled, an asymptotic solution is developed and extended to the multiparameter setting.

具有最小期望测度的不变置信集AI 解读

A method is given for constructing a confidence set having smallest expected measure within the class of invariant level $1 - \alpha$ confidence sets. The main assumptions are (i) that the invariance group acts transitively on the parameter space and also acts on the parametric function of interest, and (ii) that the measure satisfies a certain equivariance property. When the invariance group satisfies the conditions of the Hunt-Stein Theorem, the optimal invariant…

Since Robbins (1956), a large literature has evolved treating the empirical Bayes formulation of a sequence of decision problems. In this paper we look at the formulation and the asymptotic optimality criterion as a classical iid problem and a consistency property. With this point of view and the finite state component problem, we evoke classical results on Bayes procedures to prove a complete class theorem and to establish the asymptotic optimality of Bayes empirical Bayes…

一致性度量的渐近理论：以平均 Kendall tau 为重点AI 解读

The problem of $n$ rankings is considered and the asymptotic distributions of measures of concordance based on rank correlations are derived under the null model of complete randomness. The Bahadur efficiencies of the measures are computed. A matrix analysis then reveals the asymptotic distribution and superior efficiency of average Kendall tau. Some interpretation of the results is also made.

狄利克雷分布混合与列联表估计AI 解读

Assuming a multinomial sampling model, prior distributions are developed which can accept prior information about symmetry and independence in a two-way contingency table. Bayesian estimates for the cell probabilities are obtained from the posterior distributions which are attractive alternatives to the usual classical estimates when vague prior information about symmetry or independence is available.

Likelihood ratio statistics for (i) testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped (cf. Shaked, Ann. Statist., 1979), and for (ii) testing the null hypothesis that this parameter vector is starshaped, are considered. For both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the Chi-bar-square distribution.…

线性模型中的异方差自适应估计AI 解读

In a heteroscedastic linear model, it is known that if the variances are a parametric function of the design, then one can construct an estimate of the regression parameter which is asymptotically equivalent to the weighted least squares estimate with known variances. We show that the same is true when the only thing known about the variances is that they are determined by an unknown but smooth function of the design or the mean response.

核估计中的带宽变化：平方根法则AI 解读

We consider kernel estimation of a smooth density $f$ at a point, but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density. Proportionally varying the bandwidths like $f^{-1/2}$ at the contributing readings lowers the bias to a vanishing fraction of the usual value, and makes for performance seen in well-known estimators that force moment conditions on the kernel (and so sacrifice positivity of the curve…

季节性时间序列模型中非平稳参数设定的检验AI 解读

Let $Y_t$ be an autoregressive process satisfying $Y_t = \alpha_1 Y_{t - 1} + \alpha_2 Y_{t - d} + \alpha_3 Y_{t - d - 1} + e_t$, where $\{e_t\}^\infty_{t = 0}$ is a sequence of $\operatorname{iid}(0, \sigma^2)$ random variables and $d \geq 2$. Such processes have been used as parametric models for seasonal time series. Typical values of $d$ are 2, 4, and 12 corresponding to time series observed semi-annually, quarterly, and monthly, respectively. If $\alpha_1 = 1, \alpha_2 = 1, \alpha_3 = - 1$…

NonGaussian linear processes are considered. It is shown that the phase of the transfer function can be estimated under broad conditions. This is not true of Gaussian linear processes and in this sense Gaussian linear processes are atypical. The asymptotic behavior of a phase estimate is determined. The phase estimates make use of bispectral estimates. These ideas are applied to a problem of deconvolution which is effective even when the transfer function is not minimum…

模型选择与反正弦定律AI 解读

Generalizations of the arc sine laws are shown to provide insight into the operating characteristics of certain techniques for selecting models to fit a given data set, when the available models are nested. As a corollary, one sees that a popular technique may be expected to include about one superfluous parameter, even if the sample size is large.

观测依赖对卡方拟合检验的影响AI 解读

Tests of fit that assume i.i.d. observations may be affected by dependence among the observations. This effect is studied for the Pearson chi squared test when the data form a stationary stochastic process. The general form of asymptotic distribution theory under the null hypothesis is outlined and examples are given. For testing fit to a specified normal law, it is shown that when observations come from a quite general class of Gaussian stationary processes, positive…

多重决策与假设检验中的自适应程序AI 解读

Necessary and sufficient conditions for the existence of adaptive procedures for identification of one of several probability distributions or for testing a simple hypothesis against a simple alternative are obtained. By definition, adaptive procedures are required to exhibit the same asymptotic behavior for several parametric families as do the optimal (minimax) estimators for each of these families. The proofs are based on a multivariate version of Chernoff's theorem,…

离散指数族可容许估计量的半尾部上界及其应用AI 解读

Admissibility problems involving simultaneous estimation in discrete exponential families are studied by solving difference inequalities. It is shown that if an estimator is admissible under the loss function $L_m(\mathbf{\theta, a)} = \sum^p_{i = 1} \theta^{m_i}_i (\theta_i - a_i)^2$, then in the tail (i.e., for large values of the observations), this estimator has to be less than certain bounds. Specific bounds, called Semi Tail Upper Bounds (STUB), are given here. These…

空间模式的非参数最大似然估计AI 解读

Let $\mathbf{X}$ be an absolutely continuous random variable in $\mathbb{R}^k$ with distribution function $F(\mathbf{x})$ and density $f(\mathbf{x})$. Let $\mathbf{X}_1, \cdots, \mathbf{X}_n$ be independent random variables distributed according to $F$. Mapping the spatial distribution of $\mathbf{X}$ normally entails drawing a map of the isopleths, or level curves, of $f$. In this paper, it is shown how to map the isopleths of $f$ nonparametrically according to the…

对 Andersen 与 Gill《计数过程中的 Cox 回归模型：大样本研究》的评论AI 解读

In this issue Andersen and Gill (hereafter AG) present a stimulating development of asymptotic distribution theory for the Cox regression model with time-dependent covariates. They use a counting process formulation for the failure time data and martingale covergence results. This approach involves such conditions as $\sigma$-algebra right continuity and predictable, locally bounded, covariate processes. In this commentary we consider the implications of such assumptions for…

信赖推断的函数模型基础AI 解读

The role of functional models and their associated fiducial analysis is explored in an attempt to uncover a general theory of fiducial inference.